New idea and algorithm are proposed to compute asymptotic expression of limit cycles bifurcated from the isochronous center. Compared with known inverse integrating factor method, new algorithm to analytically computing shape of limit cycle proposed in this paper is simple and easy to apply. The applications of new algorithm to some examples are also given. Rewrite f(θ) into the following Fourier series:θ from 0 to θ, we obtain (φ(θ) is a periodic function with period 2π.For  series:f(θ)=a02+F(θ) is a periodic function with period 2π, then we get g · (2π − 0) + φ(2π) − φ(0) = 0. From φ(2π) = φ(0), we conclude that g = 0.If The proof of the lemma is completed.ɛ, of the limit cycles bifurcated from linear isochronous center. As applications, in x′ = y, y′ = −x + ɛ(1 − x2)y up to order o(ε7). In The main goal of this paper is to develop a new approach for computing analytically the global shape of the bifurcated limit cycles from an isochronous center and the paper is organized as follows. In pk and qk are both analytic functions, pk = qk = 0, k = 1,2,…, and ɛ is a small real parameter. System , qk are both analytic functions, we rewrite system , R2,… are analytic functions about r, cos(θ), sin(θ).Firstly, we make a polar coordinates transformation o system . By elime obtaindrdθ=ɛ is limit cycle of system (r(θ), we get that rk(θ) is periodic function with period 2π, where k = 0,1, 2,….From f system , then frkth order terms of ɛ in the obtained system, then we can obtain a series of equations:Thirdly, substituting  into 1111 and coAs to the formula of fk is analytic function about rk−1, rk−2,…, r1, r0, cos(θ), sin(θ), k = 0,1, 2,….Functions ained in  have theAccording to  and 12)12), we gk = 0, it is easy to get that For k = 1, the term on the right hand side of . In detail, the constant in the term R1 determines r(θ) = r0 + ɛr1(θ) + ε2r2(θ)+⋯, we get that r0. In other words, if function rn, n ≥ 1, then rn appearing in the terms in the right hand side of  of system (rk(θ) in  in  in , k = 0,1k(θ) in  are recudr0/dθ = 0, we get r0(θ) ≡ r0(constant). To determine the constant r0 in (r1(θ).From t r0 in , the newdr1/dθ = f1, we obtainFrom f1 is a periodic function, according to φ1 is periodic function with period 2π.By noting that r = r(θ) is a limit cycle of system (r(θ) and r1(θ) in  in  are perir0.So from ɛ=0.In fact, the function d system  near clod system |ɛ=0.L : x2 + y2 = r02. From [In detail, 2. From , 9, we kπ-periodic differential equation. First order  averaging method to study the existence and number of periodic orbits of planar differential equation is proposed in [It should be pointed out that the function posed in , 11. Ther0 into . Thus we obtainSubstitute the value of r0 into ; we can c1, new algorithm proposed in this paper needs the expression of r2(θ). From dr2/dθ = f2, we getTo determine the value of From r2(θ) is a periodic function and From the fact that c1. Thus we have obtained r1(θ) by  by .r0, r1(θ),…, rk−1(θ), now we start to determine rk(θ).Assuming that we have obtained the explicit expressions of drk/dθ = fk, we getFrom rk(θ) is to determine the value of ck. According to the algorithm proposed in this paper, we resort to the expression of rk+1(θ). From drk+1/dθ = fk+1, we getTo determine the expression of From rk+1(θ) is a periodic function, from Because ck; thus we determine rk(θ) by  by .ɛ explicitly and recursively.Thus we can compute the shape of limit cycles of system  to any go(ε7).In this section we will apply the method just described in the above section to compute the analytic expansion of the unique limit cycle of the Van der Pol systemx = rcos(θ)y = rsin(θ) to system  = ∑k=0∞εkrk(θ) is the polar coordinates form of the limit cycles of ,  k = 3,4,…, 8 is omitted.Assume ycles of  and subsycles of . By compdr0/dt = 0, we get that r0 is arbitrary constant.From r0, we compute the following expression of r1(θ):g1 = 0, so we get r0 = 2.To determine the value of r0 = 2 into :To determine the value of g2 = 0, so we get c1 = 0.Because r1(θ) is given by (c1 = 0.Thus explicit expression of given by  with c1r1(θ) and r0 = 2 into  are similar to the ones given in Section 3 of [r7(θ) obtained in our method which was omitted in [ɛ = 0, (1/5), (1/2), (9/10) are plotted in x2 + y2 = 4 of system . To utilize new algorithm introduced in Consider the following perturbed system:3x′=−y+2xyx′=−y+2xyx′=−y+2xyx′=−y+2xyɛ.In this subsection we start to compute the analytic expansion of the limit cycle of the perturbed system  to the su = rcosθ, v = rsinθ; then system , r2(θ) are given in the following:By applying the algorithm described in f system ,(41)r is given in  is unstable and its parametric form is the following:ɛ = 1/20 is plotted by using formula 2 + y2 = 2 of unperturbed system (ɛ = 0 is drawn in sold line, and the limit cycle of the perturbed system (ɛ = 1/20 is drawn in dash line.In d system  for ɛ = d system  for ɛ =


M, N)-soft intersection set, which is a generalization of soft intersection sets. We introduce the concepts of -SI filters of BL-algebras and establish some characterizations. Especially, -soft congruences in BL-algebras are concerned.The purpose of this paper is to give a foundation for providing a new soft algebraic tool in considering many problems containing uncertainties. In order to provide these new soft algebraic structures, we discuss a new soft set-( In fact, the MV-algebras, Gödel algebras, and product algebras are the most known classes of BL-algebras. BL-algebras are further discussed by many researchers; see [It is well known that certain information processing, especially inferences based on certain information, is based on classical two-valued logic. In making inference levels, it is natural and necessary to attempt to establish some rational logic system as the logical foundation for uncertain information processing. BL-algebra has been introduced by Hájek as the algebraic structures for his Basic Logic . A well-ers; see –12.We note that the complexities of modeling uncertain data in economics, engineering, environmental science, sociology, information sciences, and many other fields cannot be successfully dealt with by classical methods. Based on this reason, Molodtsov  proposedo groups , near-rio groups , and BL-o groups , respectM, N)-SI filters of BL-algebras. In particular, some important properties of -soft congruences of BL-algebras are discussed in In this paper, we organize the recent paper as follows. In L =  is a BL-algebra [L, ⊙, 1) is a commutative monoid;;x → y)∨(y → x) = 1.:, 6:L, thx ≤ y⇔x → y = 1;x → (y → z) = (x⊙y) → z = y → (x → z);x⊙y ≤ x∧y;x → y ≤ (z → x)→(z → y), x → y ≤ (y → z)→(x → z);x → x′ = x′′ → x;x∨x′ = 1⇒x∧x′ = 0;x → y)⊙(y → z) ≤ x → z;A,1 ∈ (I2)x ∈ A, ∀y ∈ L, x → y ∈ A⇒y ∈ A.∀ A of L is a filter of L if and only if it satisfiesIt is easy to check that a nonempty subset (I3)x, y ∈ L, x⊙y ∈ A,∀(I4)x ∈ A, ∀y ∈ L, x ≤ y⇒y ∈ A (see [∀∈ A (see ).A nonempty subset L be a BL-algebra, U an initial universe, E a set of parameters, and P(U) the power set of U and A, B, C⊆E.From now on, we let fA over U is a set defined by fA : E → P(U) such that fA(x) = ∅ if x ∉ A. Here fA is also called an approximate function. A soft set over U can be represented by the set of ordered pairs fA = {) | x ∈ E, fA(x) ∈ P(U)}. It is clear to see that a soft set is a parameterized family of subsets of U. Note that the set of all soft sets over U will be denoted by S(U).A soft set fA, fB ∈ S(U).fA is said to be a soft subset of fB and denoted by fA(x)⊆fB(x), for all x ∈ E. fA and fB are said to be soft equal, denoted by fA = fB, if fA and fB, denoted by fA∪B(x) = fA(x) ∪ fB(x), for all x ∈ E.The union of fA and fB, denoted by fA∩B(x) = fA(x)∩fB(x), for all x ∈ E.The intersection of Let fL over U is called an SI- filter of L over U if it satisfies S1)  fL(x)⊆fL(1) for any x ∈ L,( S2)  fL(x → y)∩fL(x)⊆fL(y) for all x, y ∈ L.-SI filters in BL-algebras and investigate some characterizations. From now on, we let ∅⊆M ⊂ N⊆U.In this section, we introduce the concept of -soft intersection filter -SI filter) of L over U if it satisfies1)(SIfL(x)∩N⊆fL(1) ∪ M for all x ∈ L,2)(SIfL(x → y)∩fL(x)∩N⊆fL(y) ∪ M for all x, y ∈ L.A soft set fL is an -SI filter of L over U, then fL is an -SI filter of L over U. Hence every SI-filter of L is an -SI filter of L, but the converse need not be true in general. See the following example.If U = S3, the symmetric 3-group is the universal set, and let L = {0, a, b, 1}, where 0 < a < b < 1. We define x∧y : = min{x, y}, x∨y : = max{x, y} and ⊙ and → as follows:L, ∧, ∨, ⊙, →, 1) is a BL-algebra. Let M = {(13), (123)} and N = {(1), (12), (13), (123)}. Define a soft set fL over U by fL(1) = {(1), (12), (123)}, fL(b) = {(1), (12), (13), (123)} and fL(a) = fL(0) = {(1), (12)}. Then we can easily check that fL is an -SI filter of L over U, but it is not SI-filter of L over U since fL(b)⊈fL(1).Assume that The following proposition is obvious.fL over U is an -SI filter of L over U, thenIf a soft set S(U) as follows: for any fL, gL ∈ S(U), ∅⊆M ⊂ N⊆U, we define M,N)-soft intersection filter -SI filter) of L over U if it satisfies1′)-SI filter of L over U, then fL* = {x ∈ L | (fL(x)∩N) ∪ M = (fL(1)∩N) ∪ M} is a filter of L.If fL is an -SI filter of L over U. Then it is clear that 1 ∈ fL*. For any x, x → y ∈ fL*, (fL(x)∩N) ∪ M = (fL(x → y)∩N) ∪ M = (fL(1)∩N) ∪ M. By fL(y)∩N) ∪ M⊆(fL(1)∩N) ∪ M. Since fL is an -SI filter of L over U, we havefL(y)∩N) ∪ M = (fL(1)∩N) ∪ M, which implies y ∈ fL*. This shows that fL* is a filter of L.Assume that fL over U is an -SI filter of L, then for any x, y, z ∈ L,fL(x⊙y)=M,N)(fL(x)∩fL(y)=M,N)(fL(x∧y),fL(0)=M,N)(fL(x)∩fL(x′),If a soft set x, y ∈ L be such that x ≤ y. Then x → y = 1, and hence(1) Let x, y ∈ L be such that fL(x → y) = fL(1). Then,(2) Let a3), we have x⊙y ≤ x∧y for all x, y ∈ L. By (1), x ≤ y → x⊙y, we obtain 2) that fL(x⊙y)  =M,N)(  fL(x)∩fL(y).(3) By (y ≤ x → y and x⊙(x → y) ≤ x∧y, we have Since fL(x)∩fL(y)=M,N)(fL(x∧y). Thus fL(x⊙y)=M,N)(fL(x)∩fL(y)=M,N)(fL(x∧y).x⊙x′ = 0.(4) It is a consequence of (3), since a4).(5) By (a7).(6) By (a8).(7) By (a9).(8) By -SI filter of L over U if and only if it satisfiesA soft set fL over U is an -SI filter of L over U if and only if it satisfies  SI5)  ∀ x, y ∈ L, fL(x⊙y)=M,N)(fL(x)∩fL(y).(A soft set (⇒) By x, y ∈ L. Since x ≤ 1, by (SI3), we have 1′) holds. Since x⊙(x → y) ≤ y, by (SI3) and (SI4), we have 2′) holds. Therefore, fL is an -SI filter of L over U.(⇐) Let M, N)-soft congruences, -soft congruences classes, and quotient soft BL-algebras.In this section, we investigate (θ from fL × fL to P(U × U) is called an -congruence in L over U × U if it satisfies C1)  θ=M,N), ∀x ∈ L,( C2)  θ=M,N), ∀x ∈ L,-congruence in BL-algebra L over U × U and x ∈ L. Define θx in L as θx(y) = θ, ∀y ∈ L. The set θx is called an -congruence class of x by θ in L. The set L/θ = {θx | x ∈ L} is called a quotient soft set by θ.Let θ is an -congruence in L over U × U, then If C1) and (C3), we have By -congruence in L over U × U, then θ1 is an -SI filter of L over U.If x ∈ L, we have1′) holds.For any x, y ∈ L, by (C3) and (C5), we obtain2′) holds. Thus, θ1 is an -SI filter of L over U.For any fL be an -SI filter of L over U. Then θ = fL(x → y)∩fL(y → x) is an -soft congruence in L.Let x, y, z ∈ L, we have the following. C1) Consider(This proves that (C1) holds. C2) It is clear that (C2) holds.( C3) By (Thus (C3) holds.For any C4) Since x → y ≤ (x⊙z)→(y⊙z) and y → x ≤ (y⊙z)→(x⊙z), we haveC4) holds.(C5) Finally, we prove condition (C5):C5) holds. Therefore θf is an -soft congruence in L.-SI filter of L over U and x ∈ L. In the following, let fx denote the -congruence class of x by θf in L and let L/f be the quotient soft set by θf.Let fL is an -SI filter of L over U, then fx=M,N)(fy if and only if fL(x → y)=M,N)(fL(y → x)=M,N)(fL(1) for all x, y ∈ L.If fL is an -SI filter of L over U, then fμ(ν) = θfμ(ν) = θf = fL(μ → ν)∩fL(ν → μ); that is, fμ(ν) = fL(μ → ν)∩fL(ν → μ) for all x, y ∈ L. If fx=M,N)=M,N)(fy(x), and hence fL(x → x) = fL(1)=M,N)(fL(y → x)∩fL(x → y). Thus, fL(y → x)=M,N)(fL(x → y)=M,N)(fL(1).If fL(y → x)=M,N)(fL(x → y)=M,N)(fL(1), then fL(x → z)⊇M,N)(fL(y → z) and fL(y → z)⊇M,N)(fL(x → z). Thus fL(x → z)=M,N)(fL(y → z). Similarly, we can prove that fL(z → x)=M,N)(fL(z → y). This implies thatz ∈ L. Hence, fx=M,N) by ff(1) : = {x ∈ L | f(x)=M,N)(f(1)}.We denote f is an -SI filter of L over U, then fx=M,N)(fy if and only if x~ff(1)y, where x~ff(1)y if and only if x → y ∈ ff(1) and y → x ∈ ff(1).If f be an -SI filter of L over U. For any fx, fy ∈ L/f, we defineLet f is an -SI filter of L over U, then L/f =  is a BL-algebra.If L/f are well defined. In fact, if fx=M,N))(y and a~ff(1)b, and so x∨a~ff(1)y∨b. Thus fx∨a=M,N)(” on L/f as follows:fL(x → y)=M,N)(fL(1). If fx, fy, fz ∈ L/f, thenWe claim that the above operations on fL is an -SI filter of L over U, then L/f≅L/ff(1).If φ : L → L/f by φ(x) = fx for all x ∈ L. For any x, y ∈ L, we haveφ is an epic. Moreover, we haveL/f≅L/ff(1).Define M, N)-SI (implicative) filters of BL-algebras. We investigate their characterizations. In particular, we describe -soft congruences in BL-algebras.As a generalization of soft intersection filters of BL-algebras, we introduce the concept of -SI prime (semiprime) filters of BL-algebras. Maybe one can apply this idea to decision making, data analysis, and knowledge based systems.To extend this work, one can further investigate (


We first investigate sufficient and necessary conditions of stability of nonlinear distributed order fractional system and then we generalize the integer-order Chen system into the distributed order fractional domain. Based on the asymptotic stability theory of nonlinear distributed order fractional systems, the stability of distributed order fractional Chen system is discussed. In addition, we have foundthat chaos exists in the double fractional order Chen system. Numerical solutions are used to verify the analytical results. Since then, many researchers have completed further studies on the stability of linear fractional differential systems  and satisfy ∫01b(α)sαdα ≠ 0 for Re(s) > 0 and f ∈ L1 for every α ∈ ; then initial value problem (Let the function  problem  has a unX(t) = (x1(t), x2(t),…, xn(t))T ∈ ℝn and b(α) = (b1(α), b2(α),…, bn(α))T, 0 < α < 1. The n-dimension distributed order fractional system is described as follows:Furthermore, the above definition in one dimension can naturally be generalized to the case of multiple dimensions; that is, let follows:Dctb(α)X(X(t) = (x1(t), x2(t),…, xn(t))T ∈ ℝn, the matrix A ∈ ℝn×n, and b(α) = (b1(α), b2(α),…, bn(α))T, 0 < α < 1. Then Saberi Najafi et al. , and r = eiπ.In this section, we generalize the main stability properties for systems of distributed order fractional. The linear distributed order fractional systems are expressed asi et al.  have obtDctb(α)XI − A) = 0 have negative real parts.The distributed order fractional system of  is asympB(s)I − A) = 0 is the characteristic function of the matrix A with respect to the distributed function B(s), where B(s) = ∫01b(α)sαdα is the distributed function with respect to the density function b(α) ≥ 0.The value of det(A), υB(s)(A), and δB(s)(A) are, respectively, the number of roots of det(B(s)I − A) = 0 with positive, negative, and zero real parts, where B(s) = (B1(s), B2(s),…, Bn(s))T is the distributed function with respect to the density function b(α) = (b1(α), b2(α),…, bn(α))T.The inertia of the system  is the tπnB(s)(A) = δnB(s)(A) = 0,s of the characteristic function of A with respect to B(s) = (B1(s), B2(s),…, Bn(s))T satisfy |arg(s)|>π/2.all roots The linear distributed order fractional system  is asympX(0) = X0, whereX(t) = (x1(t), x2(t),…, xn(t))T ∈ ℝn, and b(α) = (b1(α), b2(α),…, bn(α))T, 0 < α < 1.Next, we will mainly discuss the stability of a nonlinear autonomous distributed order fractional system, which can be described bys of the characteristic function of J with respect to B(s) = (B1(s), B2(s),…, Bn(s))T satisfy |arg(s)|>π/2.Let f system ; that isζ(t) = (ζ1(t), ζ2(t),…, ζn(t)) is a small disturbance from a fixed point. ThereforeLet hereforeDctb(α)ζ = (B1(s), B2(s),…, Bn(s))T satisfy |arg(s)|>π/2, which implies that the equilibrium The analytical procedure of linearization is based on the fact that if the matrix r system . Hence, l system  is as asπnB(s)(J) = δnB(s)(J) = 0.The nonlinear distributed order fractional system  in the pR3 [x, y, and z are the state variables and a, b, and c are three system parameters. The above system has a chaotic attractor when a = 35, b = 3, and c = 28 as shown in bi(α) for i = 1,2, 3 denote the nonnegative density function of order α ∈  = (x(t), y(t), z(t))T, b(α) = (b1(α), b2(α), b3(α))T, 0 < α ≤ 1, and i = 1,2, 3. The Jacobian matrix of distributed order fractional Chen system  is given byThe Chen system is described by the following nonlinear differential equations on R3 , 23:(21shown in Dtb1(α)cxshown in Dtb1(α)cxbi(α) = δ(α − qi), where 0 < qi ≤ 1 for i = 1,2, 3 and δ(α) is the Dirac delta function, then, we have the following fractional incommensurate-order Chen system [If n system :(25)DtαInB(s)(J) in the case that the density function varies. The results are shown in a, b, and c.Based upon h is assumed to be very small [tk = kh   and ciα) are binomial coefficients, which can be computed as [ciαj)=quations∑i=1nγiDc6Dctαx(t)ry small , 43(28)Dctαx(t)=quations∑i=1nγiDcuted as [c0(α)=1, tion of (h−α∑i=0kcDctαx(t)=quations∑i=1nγiDcT = 10, h = 0.005 with the initial conditions .To verify the efficiency of the obtained results in a, b, c) =  and  =  is asymptotically stable in the equilibrium a, b, c) =  and  =  is asymptotically stable in the equilibrium a, b, c) =  and  =  is chaotic. a, b, c) =  and  =  has chaotic attractor. a, b, c) =  and  = . From a, b, c) =  and  = .In this paper, we introduced the nonlinear distributed order fractional differential equations with respect to a nonnegative density function; hence the asymptotical stability for such systems has been investigated. In addition, we presented the distributed order fractional Chen system and then in two special cases the stability for the distributed order fractional Chen is discussed. Numerical solutions were coincident with results of


Some fixed point results are given for a class of Meir-Keeler contractive maps acting on metric spaces endowed with locally transitive relations. Technical connections with the related statements due to Berzig et al. (2014) are also being discussed. X be a nonempty set. Call the subset Y of X, almost-singleton (in short: asingleton), provided y1, y2 ∈ Y implies y1 = y2 and singleton if, in addition, Y is nonempty; note that, in this case, Y = {y}, for some y ∈ X. Take a metric  d : X × X → R+∶ = .(A) Let (P0)ℛ is amorphous; that is, it has no specific properties at all;(P1)ℛ is an order; that is, it is reflexive , transitive [xℛy and yℛz imply xℛz], and antisymmetric [xℛy and yℛx imply x = y];(P2)ℛ is a quasiorder; that is, it is reflexive and transitive;(P3)ℛ is transitive (see above).Among the classes of relations to be used, the following ones  are important for us:N, ≤); here, N = {0,1,…} is the set of natural numbers and (≤) is defined as m ≤ n if and only if m + p = n, for some p ∈ N. For each n ∈ N, let N: = {0,…, n − 1} stand for the initial interval (in N) induced by n. Any set P with P ~ N (in the sense: there exists a bijection from P to N) will be referred to as effectively denumerable. In addition, given some natural number n ≥ 1, any set Q with Q ~ N will be said to be n-finite; when n is generic here, we say that Q is finite. Finally, the (nonempty) set Y is called (at most) denumerable if and only if it is either effectively denumerable or finite.A basic ordered structure is x, z) ∈ ℛ∘𝒮 if there exists y ∈ X with  ∈ ℛ,  ∈ 𝒮.ℛ0 = ℐ, ℛn+1 = ℛn∘ℛ, n ∈ N. ; x ∈ X} is the identical relation over X). The following properties will be useful in the sequel:Given the relations k ∈ N, let us say that ℛ is k-transitive, if ℛk⊆ℛ; clearly, transitive is identical with 2-transitive. We may now complete the increasing scale above as(P4)ℛ is finitely transitive; that is, ℛ is k-transitive for some k ≥ 2;(P5)ℛ is locally finitely transitive; that is, for each (effectively) denumerable subset Y of X, there exists k = k(Y) ≥ 2, such that the restriction to Y of ℛ is k-transitive;(P6)ℛ is trivial; that is, ℛ = X × X; hence, .Given zn; n ≥ 0) in X, ℛ-ascending, if ziℛzi+1 for all i ≥ 0.Concerning these concepts, the following property will be useful. Call the sequence  in X and the natural number k ≥ 2 be such that (b03)ℛ is k-transitive on Z : = {zn; n ≥ 0}. Then, necessarily,Let the r. First, by the choice of our sequence,  ∈ ℛ; whence, the case r = 0 holds. Moreover, by definition,  ∈ ℛk; and this, along with the k-transitive property, gives  ∈ ℛ; hence, the case of r = 1 holds too. Suppose that this property holds for some r ≥ 1; we claim that it holds as well for r + 1. In fact, let i ≥ 0 be arbitrary fixed. Again by the choice of our sequence, (zi+1+r(k−1), zi+1+(r+1)(k−1)) ∈ ℛk−1, so that, by the inductive hypothesis :k-transitive condition, gives (k−1)) ∈ ℛ. The proof is thereby complete.We will use the induction with respect to X, d) be a metric space. We introduce a d-convergence and d-Cauchy structure on X as follows. By a sequence in X, we mean any mapping x : N → X. For simplicity reasons, it will be useful to denote it as (x(n); n ≥ 0) or ; moreover, when no confusion can arise, we further simplify this notation as (x(n)) or (xn), respectively. Also, any sequence (yn : = xi(n); n ≥ 0) with i(n) → ∞ as n → ∞ will be referred to as a subsequence of . Given the sequence (xn) in X and the point x ∈ X, we say that (xn),  d-converges to x  → 0 as n → ∞; that is,x will be denoted limn(xn); note that it is an asingleton, because d is triangular symmetric; if limn(xn) is nonempty, then (xn) is called d-convergent. We stress that the introduced convergence concept xn),  d-Cauchy when d → 0 as m, n → ∞, m < n; that is,d is triangular symmetric, any d-convergent sequence is d-Cauchy too; but, the reciprocal is not in general true. Concerning this aspect, note that any d-Cauchy sequence  is d-semi-Cauchy; that is,(B) Let  ↦ d is d-Lipschitz, in the sensed-continuous; that is,The mapping ; we do not give details.The proof is immediate, by the usual properties of the ambient metric X, d) be a metric space; and let ℛ⊆X × X be a (nonempty) relation over X; the triple  will be referred to as a relational metric space. Further, take some T ∈ ℱ(X). Call the subset Y of X, ℛ-almost-singleton (in short: ℛ-asingleton) provided y1, y2 ∈ Y, y1ℛy2  ⇒  y1 = y2 and ℛ-singleton when, in addition, Y is nonempty. We have to determine circumstances under which Fix(T) is nonempty; and, if this holds, to establish whether T is fix-ℛ-asingleton  is ℛ-asingleton) or, equivalently, T is fix-ℛ-singleton (in the sense: Fix(T) is ℛ-singleton); to do this, we start from the working hypotheses:(b04)T is ℛ-semi-progressive: X: = {x ∈ X; xℛTx} ≠ ∅;(b05)T is ℛ-increasing: xℛy implies TxℛTy.(C) Let ((2a)T is a Picard operator ) if, for each x ∈ X,  is d-convergent.We say that (2b)T is a strong Picard operator ) when, for each x ∈ X,  is d-convergent and limn(Tnx) ∈ Fix(T).We say that (2c)T is a globally strong Picard operator ) when it is a strong Picard operator ) and T is fix-ℛ-asingleton .We say that The basic directions under which the investigations be conducted are described by the list below, comparable with the one in Turinici :(2a)We s ascending orbital concepts (in short: (a-o)-concepts). Remember that the sequence  in X is called ℛ-ascending, if ziℛzi+1 for all i ≥ 0; further, let us say that  is T-orbital, when it is a subsequence of , for some x ∈ X; the intersection of these notions is just the precise one.(2d)X, -complete, provided (for each (a-o)-sequence) d-Cauchy ⇒  d-convergent.Call (2e)T is -continuous, if ((zn)=(a-o)-sequence and We say that (2f)ℛ, -almost-self-closed, if: whenever the (a-o)-sequence  in X and the point z ∈ X fulfill wn : = zi(n); n ≥ 0) of  with wnℛz, for all n ≥ 0.Call The sufficient (regularity) conditions for such properties are being founded on ascending notions (in short: a-notions). On the other hand, when the ascending properties are ignored, the same conventions give us orbital notions (in short: o-notions). The list of these is obtainable from the previous one; so, further details are not needed. Finally, when ℛ = X × X, the list of such notions is comparable with the one in Rus  = X.When the orbital properties are ignored, these conventions give us in Rus  be a relational metric space; and let T be a self-map of X, supposed to be ℛ-semi-progressive and ℛ-increasing. The basic directions and sufficient regularity conditions under which the problem of determining the fixed points of  T  is to be solved were already listed. As a completion of them, we must formulate the specific metrical contractive conditions upon our data. These, essentially, consist in a “relational” variant of the Meir-Keeler condition ; so,  irreflexive [x ∈ X]. Denote for x, y ∈ X (c03)A1 = d, B1 = diam{x, Tx, y, Ty}, A2 = (1/2)[d + d], A3 = max{d, d}, A4 = (1/2)[d + d]. Then, let us introduce the functions  (c04)B2 = max{A1, A2}, B3 = max{A1, A3}, B4 = max{A1, A4}, C1 = max{A1, A2, A4}, C2 = max{A1, A3, A4}, 𝒢 = {A1, B2, B3, B4, C1, C2}, 𝒢1 = {A1, B2, B4, C1}, 𝒢2 = {B3, C2}.Note that, for each G ∈ 𝒢, we haveG is sufficient; note that, by the properties of d, we must haveG is diameter bounded.Let -contractive, if(c05)d < G, expressed as T is strictly -nonexpansive;(c06)ε > 0, ∃δ > 0: [ε < G < ε + δ] ⇒  d ≤ ε, expressed as T has the Meir-Keeler property ).for all  Note that, by the former of these, the Meir-Keeler property may be written as (c07)ε > 0, ∃δ > 0: [G < ε + δ] ⇒  d ≤ ε.for all Given In the following, two basic examples of such contractions will be given.ℱ(re)(R+) stand for the class of all φ ∈ ℱ(R+) with the (strong) regressive property: . We say that φ ∈ ℱ(re)(R+) is Meir-Keeler admissible, if(c08)γ > 0, ∃β∈]0, γ0, γ0, ε0, ∞(c15)ε > 0): limsupnφ(tn) > ψ(ε + 0) − ψ(ε), whenever tn → ε + +. Let us say that  in R and the point r ∈ R, we denoted rn → r+ , if rn → r and rn ≥ r , for all n ≥ 0 large enough.Here, given the sequence  of functions in ℱ(R+), let us say that T is )-contractive, provided(c16)ψ) ≤ ψ) − φ), for all x, y ∈ X, Given T is )-contractive, for a pair  of weak generalized altering functions in ℱ(R+). Then, T is Meir-Keeler -contractive (see above).Suppose that x, y ∈ X be such that G is sufficient), G > 0, so that (by the choice of our pair), φ) > 0; wherefrom ψ) < ψ). This via (ψ = increasing) yields d < G, so that T is strictly -nonexpansive.(i) Let T does not have the Meir-Keeler property ); that is, for some ε > 0,δn) in R+0, we get a couple of sequences  and  in X, so asψ = increasing), we getG → ε + +, so that passing to limsup as n → ∞,ψ, φ), these relations are contradictory. This ends the argument.(ii) Assume by contradiction that X, so as(∀n):xnℛ~X, d, ℛ) be a relational metric space. Further, let T be a self-map of X, supposed to be ℛ-semi-progressive and ℛ-increasing. The basic directions and regularity conditions under which the problem of determining the fixed points of T is to be solved, were already listed; and the contractive type framework was settled. It remains now to precise the regularity conditions upon ℛ. Denote, for each x ∈ X,x), but the possibility of spec(x) = {1} cannot be removed. This fact remains valid even if x ∈ X is orbital admissible, in the sense [i ≠ j implies Tix ≠ Tjx], when the associated orbit TNx : = {Tnx; n ≥ 0} is effectively denumerable. But for the developments below, it is necessary that these spectral subsets of N should have a finite Hausdorff-Pompeiu distance to N; hence, in particular, these must be infinite. Precisely, given k ≥ 1, let us say that ℛ is k-semirecurrent at the orbital admissible x ∈ X, if  n ∈ N, there exists q ∈ spec(x) such that q ≤ n < q + k.for each  A global version of this convention is the following: call ℛ, finitely semirecurrent if, for each orbital admissible x ∈ X, there exists k(x) ∈ N, such that ℛ is k(x)-semirecurrent at x.Let ((d01)ℛ is finitely semirecurrent and nonidentical.Assume in the following thatOur main result in this exposition is the following.T is Meir-Keeler -contractive, for some G ∈ 𝒢. In addition, let X be -complete; and one of the following conditions holds:T is -continuous;ℛ is -almost-self-closed and G ∈ 𝒢1;ℛ is -almost-self-closed and T is -contractive, for a certain Meir-Keeler admissible function φ ∈ ℱ(re)(R+);ℛ is -almost-self-closed and T is )-contractive, for a certain pair  of weak generalized altering functions in ℱ(R+). Then T is a globally strong Picard operator ).Assume that ℛ-asingleton property. Let z1, z2 ∈ Fix(T) be such that z1ℛz2; and assume by contradiction that z1 ≠ z2; whence G = d. This, via T being strictly -nonexpansive, yields an evaluation liked, ℛ)). The argument will be divided into several steps.First, we check the fix-Part 1. We firstly assert thatx ∈ X be such that T is strictly -nonexpansive, one has d < G. On the other hand,Part 2. Take some x0 ∈ X; and put . If xn = xn+1 for some n ≥ 0, we are done, so, without loss, one may assume that, for each n ≥ 0, (d02)xn ≠ xn+1; hence, ρn : = d > 0. From the preceding part, we deriveρn; n ≥ 0) is strictly descending. As a consequence, ρ : = limnρn exists as an element of R+. Assume by contradiction that ρ > 0; and let δ > 0 be the number given by the Meir-Keeler -contractive condition upon T. By definition, there exists a rank n(δ) such that n ≥ n(δ) implies ρ < ρn < ρ + δ; hence (by a previous representation) ρ < G = ρn < ρ + δ. This, by the Meir-Keeler contractive condition we just quoted, yields (for the same n), ρn+1 = d ≤ ρ; contradiction. Hence, ρ = 0, so thatxn; n ≥ 0) is d-semi-Cauchy.Part 3. Suppose that (d03)i, j ∈ N such that i < j, xi = xj.there exist  Denoting p = j − i, we thus have p > 0 and xi = xi+p, so thatρi = ρi+np, for all n ≥ 0. This, along with ρi+np → 0 as n → ∞, yields ρi = 0, in contradiction with the initial choice of . Hence, our working hypothesis cannot hold; wherefromPart 4. As a consequence of this, the map i ↦ xi : = Tix0 is injective; hence, x0 is orbital admissible. Let k : = k(x0) ≥ 1 be the semirecurrence constant of ℛ at x0 (assured by the choice of this relation). Further, let ε > 0 be arbitrary fixed; and δ > 0 be the number associated by the Meir-Keeler -contractive property; without loss, one may assume that δ < ε. By the d-semi-Cauchy property and triangular inequality, there exists a rank n(δ) ≥ 0, such thatxn; n ≥ 0) is d-Cauchy. To do this, an induction argument upon s ≥ 1 will be used. The case s ∈ {1,…, 2k} is evident, by the preceding evaluation. Assume that it holds for all s ∈ {1,…, p}, where p ≥ 2k; we must establish its validity for s = p + 1. As ℛ is k-semirecurrent at x0, there exists q ∈ spec(x0) such that q ≤ p < q + k; note that the former of these yields (from the ℛ-increasing property of T), B1 < ε + δ, so that (by the diameter boundedness property), (0<)  G < ε + δ. Taking the Meir-Keeler -contractive assumption imposed upon T into account givesuch that(∀n≥n(δ))uch that(∀n≥n(δ))Part 5. As X is -complete, z ∈ X. If there exists a sequence of ranks (i(n); n ≥ 0) with [i(n) → ∞ as n → ∞] such that xi(n) = z +1 = Tz) for all n, then, as (xi(n)+1; n ≥ 0) is a subsequence of , one gets z = Tz. So, in the following, we may assume that the opposite alternative is true: (d04)h ≥ 0: n ≥ h  ⇒  xn ≠ z.∃ There are several cases to discuss.Case  5a. Suppose that T is -continuous. Then n → ∞. On the other hand,  is a subsequence of (xn); whence d is sufficient), z = Tz.Case  5b. Suppose that ℛ is -almost-self-closed. Put, for simplicity reasons, b : = d. By definition, there exists a subsequence (un : = xi(n); n ≥ 0) of , such that unℛz, for all n. Note that, as limn i(n) = ∞, one may arrange for i(n) ≥ n, for all n, so that, from (d04),Tun = xi(n)+1; n ≥ 0) being as well a subsequence of , gives (via (ves (via  and Lemmves (via (37)A1 yields a contradiction. Two alternatives must be treated.We now show that the assumption Alter 1. Suppose that G ∈ 𝒢1. By the Meir-Keeler contractive condition,G → b. This, along with , , Alter 1Alter 2. Suppose that G ∈ 𝒢2. The above convergence properties of  tell us that, for a certain rank n(b) ≥ h, we must haved-Lipschitz property of d, givesb/2 < d < 3b/2,  ∀n ≥ n(b). Combining these yieldsAlter 2a. Suppose that T is -contractive, for a certain Meir-Keeler admissible function φ ∈ ℱ(re)(R+).  By (b ≤ φ(b); contradiction; hence, z = Tz.tep.) By  and thisives (by  above), Alter 2b. Suppose that T is )-contractive, for a certain pair  of weak generalized altering functions in ℱ(R+).  From this contractive condition,d < b, for all n ≥ n(b). Passing to limit as n → ∞ and taking (φ(b) ≤ ψ(b) − ψ(b − 0). This, however, contradicts the choice of , so that z = Tz. The proof is complete.ing with  above),d taking  into accℛ is transitive, this result is comparable with the one in Turinici . Then, if we take ℛ : = 𝒮 and G = A1, the alternative (i1) of Let (1-2)ℛ = X × X . Then, Suppose that The following particular cases of this result are to be noted.As another consequence of T is ℛ-semiprogressive, ℛ-increasing, and -contractive, for some G ∈ 𝒢 and a certain Meir-Keeler admissible function φ ∈ ℱ(re)(R+). In addition, let ℛ be finitely semirecurrent nonidentical,  X be -complete, and one of the conditions below holds:(j1)T is -continuous;(j2)ℛ is -almost-self-closed. Then T is a globally strong Picard operator ).Assume that (2-1)ℛ = X × X  and G = A1. Then, Suppose that (e03)T(Y)) ≤ φ(diam(Y)), for all Y ∈ CB(X).diam is the class of all (nonempty) closed bounded subsets of X.) Clearly, this condition is stronger than the one we already used in (2-2)ℛ = X × X; and φ ∈ ℱ(re)(R+) is BWM-admissible . Then, if G = A1, φ is Boyd-Wong admissible; and, respectively, the Matkowski's result [φ is Matkowski admissible. Moreover, when G = C2, Suppose that s result  when φ i(2-3)ℛ is an order on X. Then, Suppose that The following particular cases of this result are to be noted.As a final consequence of T is ℛ-semiprogressive, ℛ-increasing, and )-contractive, for a certain G ∈ 𝒢 and some pair  of generalized altering functions in ℱ(R+). In addition, let ℛ be finitely semirecurrent nonidentical,  X be -complete, and one of the conditions below holds:(k1)T is -continuous;(k2)ℛ is -almost-self-closed. Then T is a globally strong Picard operator ).Assume that the self-map (3-1)α(·),  β(·) be a couple of functions in ℱ; and 𝒜,  ℬ stand for the associated relations:Let Then, if we take ℛ : = 𝒜∩ℬ and G ∈ 𝒢, this result includes (cf. Lemma  1) the one in Berzig et al. [T is -contractive. In particular, when G = A1, this last result reduces to the one in Berzig and Karapınar [The following particular cases of this result are to be noted.g et al. , based oet al. [4ψ be a metric space; and T be a self-map of Y. Given p ≥ 2, let {A1,…, Ap} be a finite system of closed subsets of Y with(e04)T(Ai)⊆Ai+1, for all i ∈ {1,…, p} (where Ap+1 = A1). Define a relation ℛ over Y as (e05)ℛ = (A1 × A2)∪⋯∪(Ap × Ap+1);then, put X = A1 ∪ ⋯∪Ap. Clearly, T is a self-map of X; and the relation ℛ is p-semirecurrent at each orbital admissible point of X. The corresponding version of (3-2) Let  transitive type requirements. Further aspects  may be found in Berzig [Finally, we should remark that none of these particular theorems may be viewed as a genuine extension for the fixed point statement due to Samet and Turinici ; becausen Berzig .


For a smooth bivariate function defined on a general domain with arbitrary shape, it isdifficult to do Fourier approximation or wavelet approximation. In order to solve these problems, in this paper,we give an extension of the bivariate function on a general domain with arbitrary shape to a smooth, periodicfunction in the whole space or to a smooth, compactly supported function in the whole space. These smoothextensions have simple and clear representations which are determined by this bivariate function and somepolynomials. After that, we expand the smooth, periodic function into a Fourier series or a periodic waveletseries or we expand the smooth, compactly supported function into a wavelet series. Since our extensions aresmooth, the obtained Fourier coefficients or wavelet coefficients decay very fast. Since our extension tools arepolynomials, the moment theorem shows that a lot of wavelet coefficients vanish. From this, with the help ofwell-known approximation theorems, using our extension methods, the Fourier approximation and the waveletapproximation of the bivariate function on the general domain with small error are obtained. In the recent several decades, various approximation tools have been widely developed –14. For This paper is organized as follows. In T = 2 and the interior of T by To and always assume that Ω is a simply connected domain. We say that f ∈ Cq(Ω) if the derivatives (∂i+jf/∂xi∂yj) are continuous on Ω for 0 ≤ i + j ≤ q. We say that f ∈ C∞(Ω) if all derivatives (∂i+jf/∂xi∂yj) are continuous on Ω for i, j ∈ ℤ+. We say that a function h is a γ-periodic function if h = h   ∈ T; k, l ∈ ℤ), where γ is an integer. We appoint that 0! = 1 and the notation [α] is the integral part of the real number α.Throughout this paper, we denote In this section, we state the main results of smooth extensions and their applications in Fourier analysis and wavelet analysis.Our main theorems are stated as follows.f ∈ C∞(Ω), where Ω ⊂ To and the boundary ∂Ω is a piecewise infinitely many time smooth curve. Then for any r ∈ ℤ+ there is a function F ∈ Cr(T) such that (i)F = f   ∈ Ω);(ii)i+jF/∂xi∂yj) = 0 on the boundary ∂T for 0 ≤ i + j ≤ r;(∂(iii)T∖Ω, F can be expressed locally in the formson the complement where L is a positive integer and each coefficient cij is constant.Let f ∈ C∞(Ω), where Ω is stated as in r ∈ ℤ+, there exists a 1-periodic function Fp ∈ Cr(ℝ2) such that Fp = f   ∈ Ω).Let f ∈ C∞(Ω), where Ω is stated as in r ∈ ℤ+, there exists a function Fc ∈ Cr(ℝ2) with compact support T such that Fc = f   ∈ Ω).Let f ∈ C∞(Ω). If f ∈ Cq(Ω) (q is a nonnegative integer), by using the similar method of arguments of Theorems In Sections Here we show some applications of these theorems.F be the smooth extension of f from Ω to T which is stated as in F ∈ Cr(T) and F = f on Ω. By ΔN, denote the set of all bivariate polynomials in the form ∑n1,n2=−NNcn1,n2xn1yn2. ThenLp(D) is the norm of the space Lp(D). The right-hand side of formula  and Fp = f on Ω. By the well-known results 2 byFo is an odd function. By Fo ∈ Cr and (∂i+jFo/∂xi∂yj) = 0 on ∂ for 0 ≤ i + j ≤ r. Again, doing a 2-periodic extension, we obtain a 2-periodic odd function Fpo and Fpo ∈ Cr(ℝ2). By the well-known results 2 byFo={ssin sum  of the FFe on 2 as follows:Fe is an even function on 2. By Fe ∈ Cr and (∂i+jFe/∂xi  ∂yj) = 0 on ∂ for 0 ≤ i + j ≤ r. Again, doing a 2-periodic extension, we obtain a 2-periodic even function Fpe and Fpe ∈ Cr(ℝ2). By the well-known result , where α* = inf{g(x), x1 ≤ x ≤ x2}. Then we haveE1 = { : x1 ≤ x ≤ x2, 0 ≤ y ≤ g(x)} and g(x) ≥ α*(x1 ≤ x ≤ x2). Soψ chosen by us is L time smooth, then, by using the moment theorem and suppψm*,n2* ⊂ , we haveI2 = 0. Similarly, since F is bivariate polynomials on rectangles H1 and H3 ∈∂Ω   such that T∖Ω can be divided into the four rectanglesg ∈ C∞, h ∈ C∞, g* ∈ C∞, and h* ∈ C∞ andT can be expressed into a disjoint union as follows:Eν is a trapezoid with a curved side and each Hν is a rectangle (see Since ngle see .f to each Eν and continue to extend to each Hν such that the obtained extension F satisfies the conditions of In Sections E1 with a curved side y = g(x)(x1 ≤ x ≤ x2) is represented asak,1}0∞ and {bk,1}0∞ as follows:x1 ≤ x ≤ x2,By , the trafollows:a0,1E1, we define a sequence of functions {S1k)}0∞ by induction.On LetThen, by ,(30)S1, we, we(31)Sk ∈ ℤ+, one has S1k)( ∈ C∞(E1) andFor any f ∈ C∞(Ω) and g ∈ C∞, and g(x) > 0  (x1 ≤ x ≤ x2), by the above construction, we know that S1k)( ∈ C∞(E1) for any k = 0,1,….Since k = 0, sincek = l − 1; that is,al,1 and bl,1 are polynomials of y whose degrees are both 2l + 1. From this and )g(x)y, holds. W))g(x)y, holds fothis and , it follthis and  holds foduction,  holds folS1k)(/∂yl)  (0 ≤ l ≤ k) on the curved side Γ1 = {) : x1 ≤ x ≤ x2} and the bottom side Δ1 = { : x1 ≤ x ≤ x2} of E1.Below we compute derivatives (∂S1k) be stated as above. For any k ∈ ℤ+, one hasLet k = 0, ) = 0, ) = 0. So we getl = k, note that ) = 1 and ) = 0. By ).By , we havex2, by (∂lS1(k)∂yrmula of  holds fol = 0,…, k − 1, by the assumption of induction and (lS1k−1)(/∂yl) = 0 and  =  = 0. Sol = k, since  = 0,  = 1, by ∂yrmula of  holds. Brmula of  holds foS1k) on the curved side Γ1 and bottom side Δ1 of E1.Now we compute the mixed derivatives of 1 and Δ1 be the curved side and the bottom side of E1, respectively. Then, for k ∈ ℤ+, i+jS1k)(/∂xi∂yj) = (∂i+jf/∂xi∂yj)   ∈ Γ1), (∂i+jS1k)(/∂xi∂yj) = 0   ∈ Δ1),(∂where 0 ≤ i + j ≤ k.Let Γx1 ≤ x ≤ x2. Then we havef by S1k)) = f); that is, (i) holds for i = j = 0. So we get (i).Now we start from the equalityent from  to 50),,(51)ddx(jS1k)(/∂yj) = 0  (0 ≤ j ≤ k). From this and S1k)( ∈ C∞(E1), we haveBy From this, we get the following.r, denote lr = r(r + 1)(r + 2)(r + 3). LetF ∈ Clr(Ω⋃E1)   and   F = f  ∈ Ω); (ii) (∂i+jF/∂xi∂yj) = 0∈(E1⋂∂T), 0 ≤ i + j ≤ lr).For any positive integer f ∈ C∞(Ω), S1k)( ∈ C∞(E1), and 1 = Ω⋂E1, we get (i). By E1⋂∂T = Δ1, we get (ii). By the assumption ν = 2,3, 4, by using a similar method, we define Sνk) on the each trapezoid Eν with a curve side. The representations of Sνk) are stated in For ν = 1,2, 3,4, letlr = r(r + 1)(r + 2)(r + 3). Then, for ν = 1,2, 3,4, one has the following: (i)F ∈ Clr(Ω⋃Eν  );(ii)i+jF/∂xi∂yj) = 0, ∈(Eν⋂∂T) for 0 ≤ i + j ≤ lr;(∂(iii)F can be expressed in the form:For any ν = 2,3, 4, we haveBy The proof of (iii) is similar to the argument of f to each trapezoid Eν with a curved side. In this subsection we complete the smooth extension of the obtained function F to each rectangle Hν. First we consider the smooth extension of F to H1. We divide this procedure in two steps.We have completed the smooth extension of F = S4lr on E4. Now we construct the smooth extension of S4lr) from E4 to H1, where S4lr) is stated in lr = r(r + 1)(r + 2)(r + 3).In τr = r(r + 2).LetJl1,}14 be four sides of the rectangle H1:M1τr) = ∑i,j=0lr+12di,j(1)xiyj, where di,j(1) is a constant;i+jM1τr)(/∂xi∂yj) = (∂i+jS4lr)(/∂xi∂yj)   ∈ J1,1);(∂i+jM1τr)(/∂xi∂yj) = 0   ∈ J1,2);(∂i+jM1τr)(/∂xi∂yj) = 0   ∈ J1,3),(∂where 0 ≤ i + j ≤ τr.Let {kS4lr)(/∂yk) is a polynomial of degree 2lr + 1 with respect to x. Since ατr,11(y) and βτr,11(y) are both polynomials of degree 2τr + 1, (i) follows from (By ows from .Similar to the argument of y1)∈(E4⋂∂T), by M1(0) and (i + j ≤ lr − (1/2)k(k + 1), we have 0 ≤ i + j ≤ lr − (1/2)k(k − 1) and 0 ≤ i + k ≤ lr − (1/2)k(k − 1). Again, by the assumption of induction, we getk+iS4lr)(/∂xi∂yk) = 0. From this and (τr + 1) = (lr/2) ≥ τr, we get (iii). Since , by ∂i+jS4(lr−1)).By , we get⋂∂T), by ∂i+jS4(lr1)).By (∂i+jM1(k)F = S1lr on E1. We consider the difference S1lr) − M1τr). Obviously, it is infinitely many time differentiable on E1 since M1τr) is a polynomial. Now we construct its smooth extension from E1 to the rectangle H1 as follows. LetIn N1r) possesses the following properties: i+jN1r)(/∂xi∂yj) = (∂i+jS1lr)(/∂xi∂yj) − (∂i+jM1τr)(/∂xi∂yj) on J1,4;(∂i+jN1r)(/∂xi∂yj) = 0 on J1,2;(∂i+jN1r)(/∂xi∂yj) = 0 on J1,1;(∂i+jN1r)(/∂xi∂yj) = 0 on J1,3, where 0 ≤ i + j ≤ r and {Jν1,}14 are stated in  = ∑i,j=0lr+12τi,j(1)xiyj, where τi,j(1) is a constant.The arguments similar to lr ≥ τr, we get that, for 0 ≤ i + j ≤ τr,By τr − (1/2)r(r + 1) ≥ r, we get (iii). By Now we assume that−1)).By  and 73)(75)∂i+jNthis and , by usinS1lr)( − M1τr) is a polynomial of degree 2lr + 1 with respect to y. From this and , i+jF/∂xi∂yj) = 0,    ∈ ((E1⋃E4⋃H1)⋂∂T) for 0 ≤ i + j ≤ r; = ∑i,j=0lr+12cij(1)xiyj ∈ H1), where each cij(1) is constant.LetF ∈ Cr(Ω⋃E1⋃E4). Since S1lr)( ∈ Cr(E1),F ∈ Cr(E1⋃H1). Since S4lr)( ∈ Cr(E4),F ∈ Cr(H1⋃E4). So we get (i).By J1,4,by , we deduJ1,1,by , we deduH1⋂∂T = J1,2⋃J1,3, by  = M1τr) + N1r)  ∈ H1), we get (iii). From ν = 2,3, 4, by using a similar method, we define F = Mντr) + Nνr)  ∈ Hν), where representations of Mντr) and Nνr) see Section For F has been defined on the unit square T. The argument similar to i + j ≤ r,Ω⋂∂T = ∅, by (F ∈ Cr(T) and (∂i+jF/∂xi∂yj) = 0   ∈∂T, 0 ≤ i + j ≤ r). So we get (i) and (ii).Let(87)By , F has b = ∅, by , we havecijν) can be expressed locally in the formSimilar to the argument of F satisfying the conditions of The representation of F be the smooth extension of f from Ω to T which is stated as in Fp byFp is a 1-periodic function of ℝ2. By Fp ∈ Cr(T) andTn1,n2 = ×. Since Fp is 1-periodic function, we have Fp ∈ Cr and for any n1, n2 ∈ ℤ,2 = ⋃n1,n2∈ℤTn1,n2, we have Fp ∈ Cr(ℝ2). By  ∈ Cr(ℝ2). By ={n Fc byFc={f and Ω be stated as in Ω be divided as in F satisfying conditions of Hν}14 and the trapezoids {Eν}14 with a curved side are stated in (lr = r(r + 1)(r + 2)(r + 3) and τr = r(r + 2).Let tated in  and 23)f and Ω bSνk)}14, {Mνk)}14, and {Nνk)}14.Below we write out the representations of {S1k) by induction as follows:(i) DenoteS2k) by induction as follows:(ii) DenoteS3k) by induction as follows:(iii) DenoteS4k) by induction as follows:(iv) DenoteM1k) by induction as follows:(i) DenoteN1k) by induction as follows:DenoteM2τr) by induction as follows:(ii) DenoteN2r) by induction as follows:DenoteM3k) by induction as follows:(iii) DenoteN3k) by induction as follows:DenoteM4k) by induction as follows:(iv) DenoteN4k) by induction as follows:DenoteBy using the extension method given in Ω be a trapezoid with two curved sides:L1 < η(x) < ξ(x) < L2 (x1 ≤ x ≤ x2), η, ξ ∈ Cm. Denote the rectangle D = ×. Then D = G1⋃Ω⋃G2, where G1 and G2 are both trapezoids with a curved side:Let f ∈ Cq(Ω) (q is a nonnegative integer). We will smoothly extend f from Ω to the trapezoids G1 and G2 with a curved side, respectively, as in F is smooth on the rectangle D. Moreover, we will give a precise formula. It shows that the index of smoothness of F depends on not only smoothness of f but also smoothness of η, ξ.Suppose that a0,1 = (y − L1)/(η(x) − L1) andS1k)} on G1 as follows. Letk0 be the maximal integer satisfying 1 + 2 + ⋯+k0 ≤ q. For k = 1,2,…, k0, we defineS1k)( ∈ Cλk(G1), where λk = min{q − 1 − 2 − ⋯−k, m}.Denote a0,2 = (L2 − y)/(L2 − ξ(x)) andS2k)} on G2 as follows. Letk = 1,2,…, k0, defineS2k)( ∈ Cλk(G2), where λk is stated as above.Denote   k ≤ k0 and 0 ≤ i + j ≤ min{k, λk},k satisfying k ≤ λk, where [·] expresses the integral part. So τ ≤ λτ.An argument similar to Lemmas i + j ≤ τ,f ∈ Cq(Ω). Now we define a function on D byFq,m ∈ Cτ(D). This implies the following theorem.By , we get τ,∂i+jS1(τ(Ω and the rectangle D be stated as above. If f ∈ Cq(Ω), then the function Fq,m, defined in (D), where τ is stated in  = 0, and so Fm0, ∈ C(D); for q = 2 and m ≥ 1, we have τ = 1, and so F2,1 ∈ C1(D); for q = 5 and m ≥ 2, we have τ = 2, and so F5,2 ∈ C2(D).Especially, for f ∈ Cq and ⊂. In order to extend smoothly f from  to , we construct two polynomialsS0(0)(x) = f(x1)(x/x1) and for k = 1,…, q,S0q)((x) is a polynomial of degree ≤2q + 1.Let Similar to the proof of f from  to , we construct two polynomialsS1(0)(x) = f(x2)((1 − x)/(1 − x2)) and for k = 1,…, q,S1q)((x) is a polynomial of degree ≤2q + 1.Again extend smoothly Similar to the proof of F from  to  byS0q)((x) and S1q)((x) are polynomials of degree 2q + 1 defined as above, and F ∈ Cq and Fl)((0) = Fl)((1) = 0 . From this, we get the following.Therefore, we obtain the smooth extension f ∈ Cq and ⊂. Then there exists a function F ∈ Cq satisfying F(x) = f(x) (x1 ≤ x ≤ x2) and Fl)((0) = Fl)((1) = 0 .Let f ∈ Cq and ⊂, and let F be the smooth extension of f from  to  which is stated as in Fp be the 1-periodic extension satisfying Fp(x + n) = F(x) . Then Fp ∈ Cq(ℝ) and Fp(x) = f(x) . We expand F(x) into the Fourier series which converges fast. From this, we get trigonometric approximation of f ∈ Cq. We also may do odd extension or even extension of F from  to , and then doing periodic extension, we get the odd periodic extension Fpo ∈ Cq(ℝ) or the even periodic extension Fpe ∈ Cq(ℝ). We expand Fpo or Fpe into the sine series and the cosine series, respectively. From this, we get the sine polynomial approximation and the cosine polynomial approximation of f on . For F ∈ Cq(x) , we pad zero in the outside of  and then the obtained function Fc ∈ Cq(ℝ). We expand Fc into a wavelet series which converges fast. By the moment theorem, a lot of wavelet coefficients are equal to zero. From this, we get wavelet approximation of f ∈ Cq.Let


R denotes a ring with identity I. We study conditions under which I − Aλ and I − Bλ are left coprime or right coprime, where A, B ∈ R. As applications, we get sufficient conditions under which the Kneading determinant of a finite rank pair of operators on an infinite dimensional space is rational.In this section,  A is a k × k matrix with rational coefficients, one has the well-known identity between formal power series:An) denotes the trace of matrix An (matrix A raised to the nth power) and I denotes the k × k identity matrix. This identity plays a significant role in the discussion of an important problem in dynamical systems theory. For more details see ]:For any pair (Kneading determinant was first studied by Milnor and Thurston in .ℚ[[z]]q×q be the ring of the q × q matrices whose entries lie in ℚ[[z]]. If φ ∈ L(ℋ), Let φ, ψ) ∈ L(ℋ) × L(ℋ), I the q × q identity matrix. Then,ℚ[[z]].Let  with finite rank . So it The following is the main result of this paper.I − φλ and I − ψλ are left coprime or right coprime, and h = ψ − φ with finite rank, then Δφ,ψ), Y(λ) ∈ R[λ] such thatWe say I − Aλ and I − Bλ are left coprime if and only if there exist X0, X1,…, Xm ∈ R such thatH = A − B.I − Aλ and I − Bλ are left coprime, there exist X(λ) = X0 + X1λ + ⋯+Xmλm and Y(λ) = Y0 + Y1λ + ⋯+Ysλs such thatIf s equals m. And we haveWe can assume that So,H = A − B, then,If we write So,X0, X1,…, Xm ∈ R such thatConversely, if there exist Then,I − Aλ and I − Bλ are right coprime if there exist polynomials X(λ), Y(λ) ∈ R[λ] such thatNow we give the definition of right coprime. We say I − Aλ and I − Bλ are right coprime if and only if there exist X0, X1,…, Xs ∈ R such thatH = A − B.It is similar to the proof of φ, ψ ∈ L(ℋ), where ℋ is an infinite dimensional vector space over ℚ, and we denote by L(ℋ) the ring of linear transforms on ℋ.In this section, I − φλ and I − ψλ are left coprime or right coprime, h = ψ − φ is of finite rank, and W =Im(h). Then there exists a finite dimensional space W such that k = 1,2,….Suppose that I − φλ and I − ψλ are right coprime, then by X0, X1,…, Xs ∈ L(ℋ) such thatFirst we assume that if ℋ under h. It is not very hard to check that φ.Write We use induction to prove the conclusion.k = 1, then (φk − ψk)(ℋ) = W.If k ≤ l we have (φk − ψk)(ℋ) ⊂ W + φW + ⋯+φmW. Then for k = l + 1, we have Suppose that for I − φλ and I − ψλ are left coprime; then there exist s ∈ ℤ and x0, x1,…, xs ∈ L(ℋ) such thatl ≥ s + 1, φl(ℋ) ⊂ x0W + x1W + ⋯+xsW.Now we assume that We use induction to prove the conclusion.k = 1, then (φk − ψk)(ℋ) = W.If k ≤ l we have (φk − ψk)(ℋ) ⊂ W + φW + ⋯+φmW. Then for k = l + 1, we have Suppose that for The proof is complete.Now we will give the proof of I − φλ and I − ψλ are coprime, by If q = rank(φ − ψ); we denote by Iq the identity operator. Then i, j ∈ {1,2,…, q}, we have ∑n≥0αiφn(uj)zn = fij(z)/p(z) for some fij(z) ∈ ℚ[z]. For more details see [Suppose ails see . So by Lℋ is the ring of countable infinite matrix with finite nonzero entries in each column.Suppose α1, α2, α3,… are three-dimensional row vectors. We see that (φ − h)3 = 0. So I − ψλ and I − φλ are left coprime. It is easy to check thatLet


The unique feature of this equation is the coexistence of an equilibrium solution and the minimal period-two solution both of which are locally asymptotically stable.We investigate the basins of attraction of equilibrium points and minimal period-two solutions of the difference equation of the form  Equation  ≪ y =   if  x⪯y  with  x1 ≠ y1  and  x2 ≠ y2.Consider a partial ordering  ⪯  on  ℝT on a nonempty set ℛ ⊂ ℝ2 is a continuous function T : ℛ → ℛ. The map T is monotone if x⪯y implies that T(x)⪯T(y) for all x, y ∈ ℛ, and it is strongly monotone on ℛ if x≺y implies that T(x) ≪ T(y) for all x, y ∈ ℛ. The map is strictly monotone on ℛ if x≺y implies that T(x)≺T(y) for all x, y ∈ ℛ. Clearly, being related is invariant under iteration of a strongly monotone map.A map North-East ordering (NE) for which the positive cone is the first quadrant; that is, this partial ordering is defined by   ⪯ne  if  x1 ≤ x2  and  y1 ≤ y2  and the South-East (SE) ordering defined as   ⪯se  if  x1 ≤ x2  and  y1 ≥ y2.Throughout this paper, we will use the T on a nonempty set ℛ ⊂ ℝ2 which is monotone with respect to the North-East ordering is called cooperative and a map monotone with respect to the South-East ordering is called competitive.A map T is a differentiable map on a nonempty set ℛ, a sufficient condition for T to be strongly monotone with respect to the SE ordering is that the Jacobian matrix at all points x has the following sign configuration:ℛ is open and convex.If x ∈ ℝ2, define Qℓ(x) for ℓ = 1,…, 4 to be the usual four quadrants based at x and numbered in a counterclockwise direction; for example, Q1(x) = {y ∈ R2 : x1 ≤ y1, x2 ≤ y2}. Basin of attraction of a fixed point T, denoted as x0, y0) for which the sequence of iterates Tn) converges to p. The next five results, from [For ts, from , 14, arets, from , 16.T  be a competitive map on a rectangular region ℛ ⊂ ℝ2. Let T  such thatℛ), and  T  is strongly competitive on  Δ. Suppose that the following statements are true.T  has a  C1  extension to a neighborhood of  The map  T at λ, μ such that 0 < |λ | <μ, where |λ | <1, and the eigenspace Eλ associated with λ is not a coordinate axis.The Jacobian Then, there exists a curve 𝒞 ⊂ ℛ through 𝒞 is tangential to the eigenspace Eλ at 𝒞 is the graph of a strictly increasing continuous function of the first coordinate on an interval. Any endpoints of 𝒞 in the interior of ℛ are either fixed points or minimal period-two points. In the latter case, the set of endpoints of 𝒞 is a minimal period-two orbit of T.Let  𝒞 are boundary points of ℛ is of interest. The following result gives a sufficient condition for this case.We will see in 𝒞 of ℛ, it is sufficient that at least one of the following conditions is satisfied.T  has no fixed points nor periodic points of minimal period-two in  Δ.The map  T  has no fixed points in  Δ,x ∈ Δ.The map  T  has no points of minimal period-two in  Δ,x ∈ Δ.The map  For the curve λ | <1. This follows from a change of variables [For maps that are strongly competitive near the fixed point, hypothesis (b) of ariables  that allThe next result is useful for determining basins of attraction of fixed points of competitive maps.𝒞 be the curve whose existence is guaranteed by 𝒞 belong to ∂ℛ, then 𝒞 separates ℛ into two connected components, namely,𝒲−  is invariant, andn → ∞  for every  x ∈ 𝒲−.  𝒲+  is invariant, andn → ∞  for every  x ∈ 𝒲+.  (B) If, in addition to the hypotheses of part (A),ℛ and  T  is  C2  and strongly competitive in a neighborhood ofT  has no periodic points in the boundary of(iii)x ∈ 𝒲−, there exists  n0 ∈ ℕsuch thatn ≥ n0.For every  (iv)x ∈ 𝒲+, there exists  n0 ∈ ℕsuch thatn ≥ n0.For every  (A) Assume the hypotheses of T  is a map on a set ℛ and ifT, the stable setIf  T  is noninvertible, the setℛ, the setsWhen  μ > 1 and that the eigenspace Eμ associated with μ is not a coordinate axis. If the curve 𝒞 of ℛ, then 𝒞 is the stable set ℛ that is tangential to Eμ at ℛ are fixed points of T.In addition to the hypotheses of part (B) of f  is strongly decreasing in the first argument and strongly increasing in the second argument if it is differentiable and has first partial derivative  D1f  negative and first partial derivative  D2f  positive in a considered set. The connection between the theory of monotone maps and the asymptotic behavior of  = ). The second iterate  T2  is given byI × I  (see [Set  n  in  to obtai I  (see ).λ, μ  which satisfy  λ < 0 < μ  and | λ | <μ, whenever  f  is strictly decreasing in first and increasing in second variable. Thus, the applicability of Theorems The characteristic equation of  at an eqThere are several global attractivity results for . Some off : I × I → I  is a continuous function and  f  is decreasing in the first argument and increasing in the second argument. Assume thatφ, ψ)  and    are minimal period-two solutions which are saddle points such that𝒲s) and 𝒲s). More precisely,ℬ) = 𝒲s) and ℬ) = 𝒲s) are exactly the global stable sets of  and .Consider  where  fx−1, x0) ∈ 𝒲+)  or   ∈ 𝒲−), then  Tn)  converges to the other equilibrium point or to the other minimal period-two solutions or to the boundary of the region  I × I.If    is decreasing in the first variable and increasing in the second variable. By xn}n=−1∞  of =It is clear that  has a una + b − c < 0, then equilibrium solutionIf  3a + b − c > 0, then equilibrium solutionIf  3a + b − c = 0, then equilibrium solutionλ1 = −1  and  λ2 = 1/2).If  3Equation  has a unD = (2a + b)(6a + 5b + 4c)  . It is clear that  λ− < 0  and  λ+ > 0. Now, we prove that  λ+ ∈   and(i)(ii)also,(iii)λ− > −1⇔3a + b − c < 0  and  λ− = −1⇔3a + b − c = 0.  Also, if  3a + b − c = 0, then  c = 3a + b, and we have  λ+ = 1/2.By , a lineaIn this section, we present results for the existence of minimal period-two solutions of .a, b, and  c.(a) Equation  has the a + b − c < 0, then ((b) If 3 0, then  has the φ, ψ, φ, ψ,…}  of  follows from  = u2/(av2 + buv + cu2).  The second iteration of the map  T  isT2  is competitive by T2  isThe map  e system  is of thc),  ,  , and    of  The minimal period-two pointsa < c and 3a + b − c < 0, then the minimal period-two pointsφ  and  ψ  satisfy ((ii) If  satisfy , are sadF = 0, for periodic point  , we haveλ1,2 = 0, which implies that    is locally asymptotically stable.(i) Since  F = 1/c, for periodic solution  , we haveλ1,2 = 0, which implies that    is locally asymptotically stable.Similarly, since  (ii) By , we haveF=φ,ψ)=φ,∂F∂u(φψ)T2  at the point    is of the formNow, we obtain that Jacobian matrix of the map  The corresponding characteristic equation isNotice that(i)m2 + n2 ≥ 2mn.   Consider that(ii)Notice thatWe need to show thatThis implies thatIn this section, we present global dynamics results for .T = ,  u > 0, and  T = ,  v > 0.Notice that  a + b − c < 0, then , , which is a saddle point and has the minimal period-two solution ,  which is locally asymptotically stable. The basin of attraction 𝒲s) and 𝒲s). The basins of attraction ℬ) = 𝒲s) and ℬ) = 𝒲s) are exactly the global stable sets of  and . Furthermore, the basin of attraction of the minimal period-two solution ,  is the union of the regions above 𝒲s) and below 𝒲s) in SE ordering; that is,x−1, x0) ∈ 𝒲−), then  limn→∞xn2 = 1/c  and  limn→∞xn+12 = 0;if   ∈ 𝒲+, then  limn→∞xn2 = 0  and  limn→∞xn+12 = 1/c.if   and (T2  as well. Equation (Using assumption  3< 0) and , it is end x−1, x0) ∈ 𝒲−), thenif  ((ii)ifThe conclusion follows from (i)Tn2) →   and  Tn+12) → , that is,if(ii)x−1, x0) ∈ 𝒲+, then  Tn+12) →   and  Tn2) → , that is,if  ,  which is locally asymptotically stable. There exists a set 𝒞 which is an invariant subset of the basin of attraction of 𝒞 is a graph of a strictly increasing continuous function of the first variable on an interval and separates ℛ∖, where ℛ = [0, ∞) × [0, ∞), into two connected and invariant components n→∞xn2 = 1/c  and  limn→∞xn+12 = 0;ifn→∞xn2 = 0  and  limn→∞xn+12 = 1/c.ifIf 3 0, then  has a unT2  as well. The existence of the set 𝒞 with stated properties follows from (i)if(ii)ifConsequently,(i)Tn2) →   and  Tn+12) → ; that is,if(ii)Tn+12) →   and  Tn2) → ; that is,if,  which are locally asymptotically stable points. There exists a set 𝒞 which is an invariant subset of the basin of attraction of 𝒞 is a graph of a strictly increasing continuous function of the first variable on an interval and separates ℛ∖, where ℛ = [0, ∞) × [0, ∞), into two connected and invariant components n→∞xn2 = 1/c  and  limn→∞xn+12 = 0;ifn→∞xn2 = 0  and  limn→∞xn+12 = 1/c.ifIf 3 0, then  has a unT  at the equilibrium pointλ1 = −1  and  λ2 = 1/2, which means that  μ1 = λ12 = 1  and  μ2 = λ22 = 1/4  are the eigenvalues of the map  T2.  Using (Q = (2a + b)/(a + b + c).  A straightforward calculation yields that the eigenvector corresponding to the eigenvalue  μ2 = 1/4  is of the formIn view of .  Using , 52),  × . As a consequence of this and using  →   and   → ;ifun2, vn2) →   and   → .ifWe see that eigenvector  nd using , we have(i)Tn2) →   and  Tn+12) → ; that is,if(ii)Tn+12) →   and  Tn2) → ; that is,if.As one may notice from the figures all stable manifolds of either saddle point equilibrium points or saddle period-two solutions are asymptotic to the origin, which is the point where  is not d


L2-error bound of finite element accuracy and of second order in time. Numerical examples are included to confirm our theoretical analysis.We analyze a fully discrete leapfrog/Galerkin finiteelement method for the numerical solution of the space fractional order  diffusion equation. The generalized fractional derivative spaces aredefined in a bounded interval. And some related properties are further discussed for thefollowing finite element analysis. Then the fractional diffusion equationis discretized in space by the finite element method and in time by the explicitleapfrog scheme. For the resulting fully discrete, conditionally stable scheme,we prove an  Fractional calculus and fractional partial differential equations (FPDEs) have many applications in various aspects such as in viscoelastic mechanics, power-law phenomenon in fluid and complex network, allometric scaling laws in biology and ecology, colored noise, electrode-electrolyte polarization, dielectric polarization, boundary layer effects in ducts, electromagnetic waves, quantitative finance, quantum evolution of complex systems, and fractional kinetics . And a lGenerally speaking, the finite difference method and the finite element method are the two main means to solve FPDEs. Recently, some typical fractional difference methods have been utilized to solve FPDEs numerically –4. On thIn this paper, we use the explicit leapfrog difference/Galerkin finite element mixed method to numerically solve the space fractional diffusion equation in order to get a higher convergence order.RLDa,xα2 is commonly referred to the left (right) sided Lévy stable distribution, where the underlying stochastic process is Lévy α-stable flights; see , time T > 0. Here the spatial fractional differential operator Δα is denoted by κ1 · RLDa,xα2 + κ2 · RLDx,bα2, where 0 ≤ κ1, κ2 ≤ 1, and κ1 + κ2 = 1. When α = 1, the problem models a Brownian diffusion process. And f is a source term, λ is a positive constant.Here, we mainly focus on constructing and analyzing a kind of efficient numerical schemes for approximately solving space fractional diffusion equation. The considered problem reads as follows: for 1/2 < The rest of this paper is constructed as follows. In L2 sense to the Lp sense.In this section, we first give the definition of fractional derivatives. There are several definitions for the fractional derivatives, but Riemann-Liouville derivative is one of the most often used fractional derivatives, which is a reasonable generalization of the classical derivative , 19–22. αth order left and right Riemann-Liouville integrals of function u(x) are defined in a finite interval  as follows:α > 0.The αth order left and right Riemann-Liouville derivatives of function u(x) defined in a finite interval  are given asn − 1 < α < n ∈ Z+. Obviously, they are the integer derivatives of the left and right fractional integrals, respectively.The Now, we give some lemmas and corollaries which are necessary to define the generalized fractional derivative spaces.Ω =  be bounded and α > 0. Then u ∈ Lp(Ω) satisfiesLet p ≠ 1, q ≠ 1 in the case 1/p + 1/q = 1 + α.The relationu(x) ∈ RLDa,xα−(Lp(Ω)), v(x) ∈  RLDx,bα−(Lq(Ω)), 1/p + 1/q ≤ 1 + α, where the function space RLDa,xα−(Lp(Ω)) = {f(x) | f(x) = RLDa,xα−ϕ(x), ϕ(x) ∈ Lp(Ω)}, RLDx,bα−(Lq(Ω)) = {f(x) | f(x) = RLDx,bα−ψ(x), ψ(x) ∈ Lq(Ω)}.The formulau(x) ∈ RLDa,xα−2(Lp(Ω)), v(x) ∈ RLDx,bα−(Lq(Ω)), 1/p + 1/q ≤ 1 + α.One can further give the following corollary:u(x) ∈ RLDa,xα−2(Lp(Ω)) implies u(x) ∈ RLDa,xα−(Lp(Ω)) one can prove that by using Note that the above assumption u(x) ∈ RLDx,bα−2(Lp(Ω)), v(x) ∈ RLDa,xα−(Lq(Ω)), 1/p + 1/q ≤ 1 + α.ConsiderRLDa,xα−(Lp(Ω)), we can get that if u(x) ∈ RLDa,xα−(Lp(Ω)), then u(x) = RLDa,xα−ϕ(x), and RLDa,xαu(x) = ϕ(x), where ϕ(x) ∈ Lp(Ω), such that u ∈ Lp(Ω), which is obtained by RLDa,xαu(x) ∈ Lp(Ω) naturally holds. So, by the above idea, we define the following fractional derivative spaces from the L2 sense to the Lp sense, which will be proved to be equivalent with the fractional Sobolev spaces under some certain conditions.Note that, from the definition of the function space WLα,p) fractional derivative space and the right  fractional derivative space in a bounded interval Ω =  as follows correspondingly, where 1 < p < +∞:Define the following norms of the left (with symbol HSα) in a bounded interval Ω =  in the L2 senseDefine the symmetric fractional derivative space , WR,0α,p(Ω), and HS,0α(Ω) as the closures of C0∞(Ω) under their respective norms.Define the spaces L2 sense.From , we can WL,0α,2(Ω), WR,0α,2(Ω), HS,0α(Ω), and H0α(Ω) are equal to equivalent seminorms and norms, where Hα(Ω) is the fractional Sobolev space in terms of the Fourier transform.The spaces H0α when p = 2, to denote the fractional derivative space equipped with the norm ||·||α which can be any one of (Hα−(Ω) is denoted as the dual space of H0α(Ω), with norm ||·||α−.Therefore, in this paper we always use y one of , 15), a, aH0α y one of , and H−Lp sense.Moreover, we can present some new properties about norms for the above left and right fractional derivative spaces in the α > 0 and Ω =  ⊂ R be bounded. Then the following mapping properties hold:RLDa,xα−u(x) : Lp(Ω) → Lp(Ω) is a bounded linear operator;  RLDx,bα−u(x) : Lp(Ω) → Lp(Ω) is a bounded linear operator;  RLDa,xαu(x) : WLα,p(Ω) → Lp(Ω) is a bounded linear operator;  RLDx,bαu(x) : WRα,p(Ω) → Lp(Ω) is a bounded linear operator;  RLDa,xα−u(x) : Lp(Ω) → WLα,p(Ω) is a bounded linear operator;  RLDx,bα−u(x) : Lp(Ω) → WRα,p(Ω) is a bounded linear operator.  Let Properties (1) and (2) follow directly from WLα,p(Ω) and WRα,p(Ω) asProperty (3) follows directly from the definition of WLα,p(Ω) and the semigroup property of fractional operator,ck, k = 0,…, [α] such thatProperty (5) follows from the definition of p < ∞. And if 1 ≤ p ≤ q < ∞, one hasConsiderLp(Ω).It is obviously true by using the norms of fractional derivative spaces and imbedding theorems for Ω =  ⊂ R be bounded. Then for u ∈ WL,0α,p(Ω), one hass < α, one hasLet u ∈ WL,0α,p(Ω) by using Lemmas If erefore,  is true.erefore,  holds.Sh denote a uniform partition on Ω, with grid parameter h. For k ∈ N, let Pk(Ω) denote the space of polynomials on Ω with degree not greater than k. Then we define Xh as the finite element space on Sh with the basis of the piecewise polynomials of order k ∈ Z+; that is,D is the unit of Sh.In this section, we firstly give a fully discrete scheme, where we use the leapfrog difference method in the temporal direction and the finite element method in the spatial direction and then analyze the error estimate. Let u ∈ Hk+1(Ω), 0 ≤ μ ≤ k + 1, there exists v ∈ Xh such thatThe following property of finite element spaces is necessary for our subsequent analysis : for u ∈t, H and an, bn, cn, γn (for integer n ≥ 0) be nonnegative numbers such thatN ≥ 0. Suppose that Δtγn < 1 for all n, and set σn = (1 − Δtγn)−1; thenN ≥ 0.Let Δt denote the step size for t so that tn = nΔt, n = 1,2,…, N − 1. For notational convenience, we denote un : = u andIn the following, we give the fully discrete scheme of . Let Δt uhn of (t = tn of the following fully discrete scheme:L2 inner product and (Δα(λ · uhn), v) = κ1 · , RLDx,bαv) + κ2 · , RLDa,xαv). For brevity, we always use (Δα(λ · uhn), v) instead of the right hand side of this equation.Let uhn of  be the ft > 0, there exists a unique solution uhn ∈ Xh satisfying /2Δt − (Δα(λ · uhn), uhn) is positive, which is one of the sufficient conditions for the existence and uniqueness of uhn.Firstly, we prove that /2Δt − (Δα(λ · uhn), uhn). Hence, by using the Lax-Milgram theorem, we have that ), ut ∈ L2), and uttt ∈ L2), with u0 ∈ Hk+1(Ω). uhn is the solution of  as Δt, h → 0. And the approximation solution uh satisfies the following error estimates:Assume that  has a soution of , and uhximation  is convet = tn,  n = 1,2,…, N − 1. Let un = u represent the solution of 42), we fun also satisfiesϵn+1 = Λn+1 + En+1, v = En+1 + En−1 into . For the second term of the right hand side, one hast, by using the Cauchy-Schwarz inequality, we obtainn = 1 to N − 1, one hasWe now estimate each term in An+1 is positive and comparable to ||En||2 + ||En+1||2. To this end, we use the inverse inequality ||v||α ≤ C12hα−||v||, v ∈ Xh, and this yieldst · hα−2 is sufficiently small such that C12Δt · hα−2 ≤ C13 ≤ 1, we getWe now show that, under our stability assumption , An+1 ian = ||EN||2, bn = ||En||α2, H = C154||uttt||0,02).Therefore, we obtainG = hk+22||ut||k+10,2 + hk+2−2α2||u||k+10,2 + (Δt)4||uttt||0,02 and using the condition  Letα = 0.6, κ1 = κ2 = 1/2, λ = 1, Ω = , T = 1, and  f is numerically obtained.(ii) LetThe experiential error results and convergence rates are displayed in t · hα−2 be sufficiently small, the error analysis for the fully discrete scheme is discussed, which is an L2-error bound of finite element accuracy and of second order in time. Numerical examples are given to demonstrate the efficiency of the theoretical results.In this paper, we study the finite element method for fractional diffusion equation. We use the simple, second order accurate explicit scheme, leapfrog difference method in time, and the finite element method in space. Under the suitably accurate initial conditions and the stability requirement that Δ


We obtain some equivalent conditions of (strictly) pseudoconvex and quasiconvex fuzzy mappings. These results will be useful to present some characterizations of solutions for fuzzy mathematical programming. The occurrence of randomness and fuzziness in the real world is inevitable owing to some unexpected situations. In , Zadeh iNanda and Kar  proposedMotivated by the earlier works of Panigrahi et al. , KaramarIn this section, we quote some preliminary notations and definitions.μ : ℝ →  with the following properties:μ  is normal; that is, [μ]1 = {x ∈ ℝ : μ(x) = 1} ≠ ∅,μ is upper semicontinuous,μ is convex; that is, μ(λx + (1 − λ)y) ≥ min{μ(x), μ(y)} for all x, y ∈ ℝ, λ ∈ ,μ, supp(μ) = {x ∈ ℝ : μ(x) > 0} and its closure cl(suppμ) is compact.the support of Let ℝ be the set of all real numbers. A fuzzy number is a mapping ℱ be the set of all fuzzy numbers on ℝ. The α-level set of a fuzzy number μ ∈ ℱ, 0 ≤ α ≤ 1, denoted by [μ]α, is defined asLet α-level set of a fuzzy number is a closed and bounded interval . μ∗(α) denotes the left-hand end point of [μ]α and μ*(α) denotes the right-hand end point of [μ]α. Also any m ∈ ℝ can be regarded as a fuzzy number t = 0 and otherwise μ can be identified by parameterized triples {(μ∗(α), μ*(α), α) : α ∈ }.It is clear that the μ and υ parameterized byk, we define the addition kμ as follows:r,μ to be the fuzzy number −μ satisfying (−μ)(x) = μ(−x). In other words, if μ is represented by the parametric form {(μ∗(α), μ*(α), α) : 0 ≤ α ≤ 1}, then −μ is represented by the corresponding parametric form {(−μ*(α), −μ∗(α), α) : 0 ≤ α ≤ 1}. We represent a fuzzy number μ as .For fuzzy numbers μ =  is said to be a triangular fuzzy number if μ∗(1) = μ*(1). Moreover, if μ∗(α) and μ*(α) are linear, then we say that  μ  is a linear triangular fuzzy number. We denote 〈μ∗(0); μ*(1); μ*(0)〉.A fuzzy number u, v ∈ ℱ, we say that u≼v if for each α ∈ , u∗(α) ≤ v∗(α), u*(α) ≤ v*(α). If u≼v, v≼u, then u = v. We say that u≺v, if u≼v and there exists α0 ∈  such that u∗(α0) < v∗(α0) or u*(α0) < v*(α0). For u, v ∈ ℱ, if either u≼v or v≼u, then we say that u and v are comparable, otherwise noncomparable.For F : K⊆ℝn → ℱ is said to be a fuzzy mapping. For any α ∈  and for any x ∈ K, we denote F(x) = .A mapping F : K⊆ℝn → ℱ be a fuzzy mapping. Then, F is said to be comparable if for each pair x ≠ y ∈ K, F(x) and F(y) are comparable. Otherwise, F is said to be noncomparable.Let K⊆ℝn be an open set and assume that F : K → ℱ is a fuzzy mapping. Let x =  ∈ K and let Dxi, i = 1,2,…, n stand for the partial differentiation with respect to the ith variable xi. Assume that for all α ∈ , F∗ and F* have continuous partial derivatives. Definei = 1,2,…, n, DxiF(x)[α] defines the α-cut of a fuzzy number, then we say that F is differentiable at x, and we denoteF at x. A fuzzy mapping F is said to be differentiable at x if F∗ and F* for each α ∈  are differentiable at x.Let K⊆ℝn be a nonempty open convex set and let F : K → ℱ be a differentiable fuzzy mapping. F is said to be pseudoconvex if for each x, y ∈ K, F(x)≼F(y).Let K⊆ℝn be a nonempty convex set and let F : K → ℱ be a fuzzy mapping. F is said to be quasiconvex if for each x, y ∈ K and for each λ ∈ , the following implications hold:F(x) and F(y) are comparable.Let K⊆ℝn be a nonempty open convex set and let F : K → ℱ be a differentiable fuzzy mapping. F is said to be strictly pseudoconvex if for each x, y ∈ K, F(x)≺F(y).Let K⊆ℝn be a nonempty open convex set, F : K⊆ℝn → ℱ be a differentiable fuzzy mapping, and F be comparable.In the following sections, we always assume that In this section, we establish the equivalent conditions of pseudoconvex and strictly pseudoconvex fuzzy mappings. We first give some lemmas which will be used in the sequel.F is a pseudoconvex fuzzy mapping. Then F is a quasiconvex fuzzy mapping.Assume that F is a quasiconvex fuzzy mapping if and only if for each x, y ∈ K, F(x)≼F(y) implies that F is a pseudoconvex fuzzy mapping if and only if for each x, y ∈ K, F is a pseudoconvex fuzzy mapping. Let x, y ∈ K be such thatF and (Suppose that uch that0~≼∇~F(x)f F and (F(x)≼F(y)ary that∇~F(y)T≼F(y). Assume the contrary, that is, F(y)≺F(x). Hence, for each α ∈ ,α0 ∈  such thatλ ∈ . From < we haveF∗−uch that0~≼∇~F(x)). From ary that∇~F(y)T(y we haveF(x)≺F(y)x, y ∈ K, x ≠ y such thatF(x)≺F(y). Assume to the contrary thatα ∈ ,λ ∈ .Conversely, let α ∈ ,From  and 31)31), we hnd, from , it foll0≤∇F∗≼F(x) is not possible. Otherwise, it will imply thatSuppose that uch that0~≺∇~F(x)rding to F(x)≺F(y)F is not a quasiconvex fuzzy mapping. Then, there exists x, y ∈ K, λ ∈  such that F(x)≼F(y) and α ∈  such thatz1, z2 such thatx, y ∈ K,Conversely, assume that ume thatF∗≤Fuch thatF∗−),wherez1=x+λ1,0<∇F∗(z2,


Moreover, we obtain a strong convergence theorem for the sequence generated by this process in the framework Banach spaces. The results presented in this paper improve and generalize some well-known results in the literature.In this paper, we construct the hybrid block iterative algorithm for finding a common element of the set of common fixed points of an infinite family of closed and uniformly quasi -  In the theory of variational inequalities, variational inclusions, and equilibrium problems, the development of an efficient and implementable iterative algorithm is interesting and important. Equilibrium theory represents an important area of mathematical sciences such as optimization, operations research, game theory, financial mathematics and mechanics. Equilibrium problems include variational inequalities, optimization problems, Nash equilibria problems, saddle point problems, fixed point problems, and complementarity problems as special cases.C be a nonempty closed convex subset of a real Banach space E with ∥·∥ and E∗ the dual space of E and A : C→E∗ be an operator. The classical variational inequality problem for an operator A is to find x∗ ∈ C such that Let VI. Recall that let A : C → E∗ be a mapping. Then A is calledThe set of solution of (1.1) is denoted by monotone if (i) α-inverse-strongly monotone if there exists a constant α > 0 such that (ii) x∗ ∈ E satisfying Ax∗ = 0.Such a problem is connected with the convex minimization problem, the complementary problem, the problem of finding a point fi}i∈Γ : φi}i∈Γ : Γ is an arbitrary index set. The system of equilibrium problems, is to find x ∈ C such that Let {SEP. If Γ is a singleton, then problem (1.2) reduces to the equilibrium problem, is to find x ∈ C such that The set of solution of (1.2) is denoted by EP(f). The above formulation (1.3) was shown in (Blum and Oettli EP(f). In other words, the EP(f) is an unifying model for several problems arising in physics, engineering, science, optimization, economics, etc. In the last two decades, many papers have appeared in the literature on the existence of solutions of EP(f); see, for example (Blum and Oettli EP(f); see, for example, (Blum and Oettli The set of solution of (1.3) is denoted by p > 1, the generalized duality mappingJp : For each x ∈ E. In particular, J = J2 is called the normalized duality mapping. If E is a Hilbert space, then J = I, where I is the identity mapping. Consider the functional defined by  for all C is a nonempty closed convex subset of a Hilbert space H and PC : H → C is the metric projection of H onto C, then PC is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. It is obvious from the definition of function ϕ that As well know that if E is a Hilbert space, then ϕ = ∥x - y∥2, for all x, y ∈ E. On the author hand, the generalized projection , that is, If on Alber  πC : E →πC follows from the properties of the functional ϕ = 0 if and only if x = y. It is sufficient to show that if ϕ = 0 then x = y. From (1.4), we have ∥x∥ = ∥y∥. This implies that 〈x, Jy〉 = ∥x∥2 = ∥Jy∥2. From the definition of J, one has Jx = Jy. Therefore, we have x = y; see  the set of fixed points of T; that is, F(T) = {x ∈ C : Tx = x}. Recall that a point p in C is said to be an asymptotic fixed point of T ≤ ϕ for all x ∈ C and p ∈ F(T). The asymptotic behavior of a relatively nonexpansive mapping was studied in  ≤ ϕ for x, y ∈ C. T is said to be relatively quasi-nonexpansive if F(T) ≠ ∅ and ϕ ≤ ϕ for all x ∈ C and p ∈ F(T). T is said to be quasi-ϕ-asymptotically nonexpansive if F(T) ≠ ∅ and there exists a real sequence {kn} ⊂ ; see E is a uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E.If (ii)E is a reflexive and strictly convex Banach space, then J-1 is norm-weak ∗-continuous.If (iii)E is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping If (iv)E is uniformly smooth if and only if E∗ is uniformly convex.A Banach space (v)E has the Kadec-Klee property, that is, for any sequence {xn} ⊂ E, if xn∥ → ∥x∥, then xn → x.Each uniformly convex Banach space The following basic properties can be found in Cioranescu .(i)If EWe also need the following lemmas for the proof of our main results.If E be a 2-uniformly convex Banach space. Then for all x, y ∈ E, we have. If where J is the normalized duality mapping of E and 0 < c ≤ 1.p-uniformly convex constant of E.The best constant IfEbe a p-uniformly convex Banach space and letpbe a given real number withp ≥ 2. Then for allx, y ∈ E, jx ∈ Jp(x) andjy ∈ Jp(y)  → 0 and either {xn} or {yn} is bounded, then ∥xn - yn∥ → 0.(Kamimura and Takahashi ). Let E LetCbe a nonempty closed convex subset of a smooth Banach spaceEand x ∈ E. Then x0 = πCxif and only if(Alber ). LetCbeLetEbe a reflexive, strictly convex and smooth Banach space, letCbe a nonempty closed convex subset ofEand letx ∈ E. Then = ϕ).for all LetEbe a reflexive, strictly convex smooth Banach space and letVbe as in (2.1). Then(Alber ). LetEbefor allx ∈ Eandx∗, y∗ ∈ E∗.A be an inverse-strongly monotone mapping of C into E∗ which is said to be hemicontinuous if for all x, y ∈ C, the mapping F of  into E∗, defined by F(t) = A(tx+(1-t)y), is continuous with respect to the weak ∗ topology of E∗. We define by NC(v)the normal cone for C at a point v ∈ C, that is, Let LetCbe a nonempty, closed convex subset of a Banach spaceEandAis a monotone, hemicontinuous operator ofCintoE∗. LetB ⊂ E × E∗be an operator defined as follows:(Rockafellar ). LetCbeThenBis maximal monotone andB-10 = VI.LetEbe a uniformly convex Banach space,r > 0 be a positive number andBr(0) be a closed ball ofE. Then, for any given sequenceand for any given sequenceof positive number withthere exists a continuous, strictly increasing, and convex functiong:, {rj,n} ⊂ for somea,bwith 0 < a < b < c2α/2, whereis the 2-uniformly convexity constant ofE. Iffor alln ≥ 0, lim infn→∞βn(1 - βn) > 0 and lim infn→∞αn,0αn,i > 0 for all i ≥ 1, then {xn} converges strongly top ∈ F, wherep = πFx0.Cn+1 is closed and convex for each n ≥ 0. Clearly C1 = C is closed and convex. Suppose that Cn is closed and convex for each z ∈ Cn, we known that We first show that  is equivalent to Cn+1 is closed and convex. Hence, F ⊂ Cn for all n ≥ 0. Since by the convexity of ∥ · ∥2, property of ϕ, Lemma 2.9 and by uniformly quasi- ϕ-asymptotically nonexpansive of Sn for each q ∈ F ⊂ Cn, we have Next, we show that and It follows from Lemma 2.7, that q ∈ VI and A is an α-inverse-strongly monotone mapping, we have Since Axn∥ ≤ ∥Axn - Aq∥, ∀q ∈ VI, we also have By Lemma 2.2 and ∥Substituting (3.5) and (3.6) into (3.4), we have Substituting (3.7) into (3.3), we also have and substituting (3.8) into (3.2), we obtain q ∈ Cn+1 implies that F ⊂ Cn+1 and hence, F ⊂ Cn for all n ≥ 0. This implies that the sequence {xn} is well defined. From definition of Cn+1 that Thus, this show that Form Lemma 2.6, it follows that ϕ} are nondecreasing and bounded. So, we obtain that xn∥ - ∥x0∥)2} is bounded. This implies {xn} is also bounded. We denote By (3.10) and (3.11), then {θn and (3.12), it follows that Moreover, by the definition of xn} is a Cauchy sequence in C. Since m > n, by Lemma 2.6, we have Next, we show that {n→∞ϕ exists and we taking m, n → ∞ then, we get ϕ → 0. From Lemma 2.4, we have limn→∞∥xm - xn∥ = 0. Thus {xn} is a Cauchy sequence and by the completeness of E and there exist a point p ∈ C such that  Since limJun - Jxn∥ → 0, as n → ∞. By definition of Now, we claim that ∥ Since Again form Lemma 2.4, that J is uniformly norm-to-norm continuous on bounded subsets of E, we obtain From Cn+1, we have Since  By (3.13) and (3.15), that Applying Lemma 2.4, we have Since  It follows from (3.23) and (3.19), that J is uniformly norm-to-norm continuous on bounded subsets of E, we also have Since Next, we will show that i) We show that ( By (3.13) and (3.15), that Form Lemma 2.4, that J is uniformly norm-to-norm continuous, we obtain Since From (3.45), we note that  and hence From (3.17), (3.24) and J-1 is uniformly norm-to-norm continuous on bounded sets, we have Since Using the triangle inequality, that  From (3.23) and (3.27), we have On the other hand, we observe that θn → 0, ∥xn - un∥ → 0 and ∥Jxn - Jun∥ → 0, that  It follows from From (3.2), (3.3) and (3.7), we compute  and hence λn} ⊂ for some a, b with 0 < a < b < c2α / 2, liminfn→∞(1 - βn) > 0 and liminfn→∞αn,0αn,i > 0, for i ≥ 0 and kn → 1 as n → ∞, we obtain that From (3.29), {From Lemma 2.6, Lemma 2.7 and (3.6), we compute  Applying Lemma 2.4 and (3.31) that and we also obtain i ≥ 1 From Again by the triangle inequality, we get  From (3.28) and (3.34), we have By using the triangle inequality, we have  That is i ≥ 1, Si is uniformly Li-Lipschitz continuous, hence we have. By the assumption that ∀Si, we have Sip = p, for all i ≥ 1. This imply that By (3.23) and (3.36), it follows that ii) We show that j = 1, 2, 3, ..., m, Ωn0 = I, for q ∈ F, we observe that , (3.21), By using triangle inequality, we have  From (3.20) and (3.39), we have Again by using triangle inequality, we have  From (3.40),we also have Since J is uniformly norm-to-norm continuous, we obtain rj,n > 0 we have n → ∞, ∀j = 1, 2, 3, ..., m, and  From  By (A2), that f ≤ 0 for all y ∈ C. For 0 < t < 1, define yt = ty + (1 - t)p. Then yt ∈ C which imply that fj ≤ 0. From (A1), we obtain that  and fj ≥ 0. From (A3), we have fj ≥ 0 for all y ∈ C and j = 1, 2, 3, ..., m. Hence p ∈ EP(fj), ∀j = 1, 2, 3, ..., m. This imply that  Thus iii) We show that xn → p ∈ VI. Indeed, define B ⊂ E × E∗ by . Let  ∈ G(B). Since w ∈ Bv = Av + NC(v), we get w - Av∈NC(v). From vn ∈ C, we have By Lemma 2.8, vn = πCJ-1(Jxn - λnAxn). Then by Lemma 2.5, we have On the other hand, since  and thus A is monotone and It follows from (3.43), (3.44) and H = supn≥1∥v - vn∥. Take the limit as n → ∞, (3.32) and (3.33), we obtain 〈v - p, w〉 ≥ 0. By the maximality of B we have p ∈ B-10, that is p ∈ VI. where p = πFx0. From Jx0 - Jxn, xn - z〉 ≥ 0, ∀z ∈ Cn. Since F ⊂ Cn, we also have Finally, we show that n → ∞, we obtain  Taking limit p = πFx0 and xn → p as n → ∞. This completes the proof. □ By Lemma 2.5, we can conclude that Si = S for each If LetCbe a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach spaceE. For eachj = 1, 2, ..., mletfjbe a bifunction fromC × Ctowhich satisfies conditions (A1)-(A4). LetAbe anα-inverse-strongly monotone mapping ofCintoE∗satisfying ∥Ay∥ ≤ ∥Ay - Au∥, ∀y∈Candu ∈ VI ≠ ∅. LetS : C → Cbe a closedL-Lipschitz continuous and quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn} ⊂ , {rj,n} ⊂ for somea, bwith 0 < a < b < c2α / 2, where is the 2-uniformly convexity constant ofE. If lim infn→∞(1 - βn) > 0 and lim infn→∞αn(1 - αn) > 0, then {xn} converges strongly top ∈ F, wherep = πFx0.i = 1, 2, we can obtain the following results on a pair of quasi- ϕ-asymptotically nonexpansive mappings immediately from Theorem 3.1.For a special case that C be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. For each j = 1, 2, ..., m let fj be a bifunction from C × C to A be an α-inverse-strongly monotone mapping of C into E∗ satisfying ∥Ay∥ ≤ ∥Ay - Au∥, ∀y ∈ C and u ∈ VI ≠ ∅. Let S, T : C → C be two closed quasi- ϕ-asymptotically nonexpansive mappings and LS, LT-Lipschitz continuous, respectively with a sequence {kn} ⊂ , {rj,n} ⊂ for some a, b with 0 < a < b < c2α / 2, where E. If αn + βn + γn = 1 for all n ≥ 0 and lim infn→∞αnβn > 0, lim infn→∞αnγn > 0, lim infn→∞βnγn > 0 and lim infn→∞δn(1 - δn) > 0, then {xn} converges strongly to p ∈ F, where p = πFx0.where C be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. For each j = 1, 2, ..., m let fj be a bifunction from C × C to A be an α-inverse-strongly monotone mapping of C into E∗ satisfying ∥Ay∥ ≤ ∥Ay - Au∥, ∀y ∈ C and u ∈ VI ≠ ∅. Let ϕ- nonexpansive mappings such that x0∈E with C1 = C, we define the sequence {xn} as follows: Let J is the duality mapping on E, {αn,i} and {βn} are sequences in , {rj,n} ⊂ for some a, b with 0 < a < b < c2α / 2, where E. If n ≥ 0, lim infn→∞(1 - βn) > 0 and lim infn→∞αn,0αn,i > 0 for all i ≥ 1, then {xn} converges strongly to p ∈ F, where p = πFx0.where ϕ-nonexpansive mappings, it is an infinite family of closed and uniformly quasi- ϕ-asymptotically nonexpansive mappings with sequence kn = 1. Hence the conditions appearing in Theorem 3.1F is a bounded subset in C and for each i ≥ 1,Si is uniformly Li-Lipschitz continuous are of no use here. By virtue of the closeness of mapping Si for each i ≥ 1, it yields that p ∈ F(Si) for each i ≥ 1, that is, Since Let Cbe a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach spaceE. Letfbe a bifunction fromC × Ctosatisfying (A1)- (A4). LetA be an α-inverse-strongly monotone mapping ofCintoE∗satisfying ∥Ay∥ ≤ ∥Ay - Au∥, ∀y ∈ Candu ∈ VI ≠ ∅. Letbe a finite family of closed quasi-ϕ-nonexpansive mappings such thatFor an initial pointx0 ∈ EwithandC1 = C, we define the sequence {xn} as follows: for somed > 0 and {λn} ⊂ for somea, bwith 0 < a < b < c2α / 2, whereis the 2-uniformly convexity constant ofE. Ifαi ∈  such thatthen {xn} converges strongly top ∈ F, wherep = πFx0.LetCbe a nonempty closed and convex subset of a uniformly convex and uniformly smooth Banach spaceE. Letfbe a bifunction fromC × Ctosatisfying (A1)- (A4). Letbe an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn} ⊂, {rn} ⊂, {rj,n} ⊂ for somea, bwith 0 < a < b < α/2. Iffor alln ≥ 0 and lim infn→∞αn,0αn,i > 0 for alli ≥ 1, then {xn} converges strongly top ∈ F, wherep = πFx0.ϕ-asymptotically nonexpansive mappings.Theorem 4.1 improve and extend the Corollary 3.7 in Zegeye  in the aE into E∗. Assume that A satisfies the conditions:Next, we consider the problem of finding a zero point of an inverse-strongly monotone operator of A is α-inverse-strongly monotone,(C1)A-10 = {u ∈ E : Au = 0} ≠ ∅.(C2) LetCbe a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach spaceE. For eachj = 1, 2, ..., mletfjbe a bifunction fromC × Ctowhich satisfies conditions (A1)-(A4). LetAbe an operator ofEintoE∗satisfying (C1) and (C2). Letbe an infinite family of closed uniformlyLi-Lipschitz continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn} ⊂, {rj,n}⊂ for somea, bwith 0 < a < b < c2α / 2, whereis the 2-uniformly convexity constant ofE. Iffor alln ≥ 0, lim infn → ∞(1 - βn) > 0 and lim infn→∞αn,0αn,i > 0 for alli ≥ 1, then {xn} converges strongly top∈F, wherep = πFx0.C = E in Corollary 3.4, we also get πE = I. We also have VI = VI = {x ∈ E : Ax = 0} ≠ ∅ and then the condition ∥Ay∥ ≤ ∥Ay - Au∥ holds for all y ∈ E and u ∈ A-10. So, we obtain the result. □Setting K be a nonempty, closed convex cone in E. We define the polarK∗ of K as follows: Let A : K → E∗ is an operator, then an element u ∈ K is called a solution of the complementarity problem .The set of solutions of the complementarity problem is denoted by LetKbe a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach spaceE. For eachj = 1, 2, ..., mletfjbe a bifunction fromC × Ctowhich satisfies conditions (A1)-(A4). LetAbe anα-inverse-strongly monotone mapping ofKintoE∗satisfying ∥Ay∥ ≤ ∥Ay - Au∥, ∀y ∈ Kandu ∈ CP ≠ ∅. Letbe an infinite family of closed uniformlyLi-Lipschitz continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn} ⊂, {rj,n} ⊂ for somea,bwith 0 < a < b < c2α / 2, whereis the 2-uniformly convexity constant ofE. Iffor alln ≥ 0, lim infn → ∞(1 - βn) > 0 and lim infn→∞αn,0αn,i > 0 for all i ≥ 1, then {xn} converges strongly top ∈ F, wherep = πFx0.VI = CP. So, we obtain the result. □As in the proof of Takahashi in (Takahashi


ℳ = (Mk). We also examine some topological properties of the resulting sequence spaces.In the present paper, we introduce some sequence spaces using ideal convergence and Musielak-Orlicz function  I-convergence of generalized sequences with respect to Musielak-Orlicz function.The notion of ideal convergence was first introduced by Kostyrko et al.  as a genℐ ⊂ 2X of subsets of a nonempty set X is said to be an ideal in X ifϕ ∈ ℐ,A, B ∈ ℐ imply A ∪ B ∈ ℐ,A ∈ ℐ, B ⊂ A imply B ∈ ℐ,while an admissible ideal ℐ of X further satisfies {x} ∈ ℐ for each x ∈ X; see [xn)n∈ℕ in X is said to be I-convergent to x ∈ X. If for each ϵ > 0, the set A(ϵ) = {n ∈ ℕ : ||xn − x||≥ϵ} belongs to ℐ; see [A family ∈ X; see . A sequeo ℐ; see . For moro ℐ; see –10 and rλ-convergent and λ-bounded sequences as follows.Mursaleen and Noman  introducλ = (λk)k=1∞ be a strictly increasing sequence of positive real numbers tending to infinity; that is,x = (xk) ∈ w is λ-convergent to the number L, called the λ-limit of x, if Λm(x) → L, as m → ∞, whereLet x = (xk) ∈ w is λ-bounded if supm | Λm(x)|<∞. It is well known [m xm = a in the ordinary sense of convergence, thenThe sequence ll known  that if mΛm(x) = a and hence x = (xk) ∈ w is λ-convergent to a.This implies thatX be a linear metric space. A function p : X → ℝ is called paranorm ifp(x) ≥ 0, for all x ∈ X,p(−x) = p(x), for all x ∈ X,p(x + y) ≤ p(x) + p(y), for all x, y ∈ X,λn) is a sequence of scalars with λn → λ as n → ∞ and (xn) is a sequence of vectors with p(xn − x) → 0 as n → ∞, then p(λnxn − λx) → 0 as n → ∞.if (Let p for which p(x) = 0 implies that x = 0 is called total paranorm and the pair  is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm  = 0, M(x) > 0 for x > 0 and M(x) → ∞ as x → ∞.An Orlicz function w be the space of all real or complex sequences x = (xk). Then,ℓM is a Banach space with the normLindenstrauss and Tzafriri  used theℓM contains a subspace isomorphic to ℓp (p ≥ 1). The Δ2-condition is equivalent to M(Lx) ≤ kLM(x) for all values of x ≥ 0 and for L > 1.It is shown in  that eveℳ = (Mk) of Orlicz function is called a Musielak-Orlicz function see; [𝒩 = (Nk) defined byℳ. For a given Musielak-Orlicz function ℳ, the Musielak-Orlicz sequence space tℳ and its subspace hℳ are defined as follows:Iℳ is a convex modular defined byA sequence ion see; , 18. A stℳ equipped with the Luxemburg normWe consider ℳ = (Mk) be a Musielak-Orlicz function and let p = (pk) be a bounded sequence of positive real numbers. We define the following sequence spaces:Let We can writep = (pk) = 1, for all k ∈ ℕ, we haveIf we take pk ≤ suppk = H, D = max, thenk, and ak, bk ∈ ℂ. Also |a|pk ≤ max for all a ∈ ℂ.The following inequality will be used throughout the paper. If 0 ≤ ℳ = (Mk). We also make an effort to study some topological properties and prove some inclusion relations between these spaces.The main aim of this paper is to study some ideal convergent sequence spaces defined by a Musielak-Orlicz function ℳ = (Mk) be a Musielak-Orlicz function and let p = (pk) be a bounded sequence of positive real numbers. Then, the spaces cI, c0I, mI, and m0I are linear.Let x, y ∈ cI and let α, β be scalars. Then, there exist positive numbers ρ1 and ρ2 such thatLet ϵ > 0, we haveFor a given ρ3 = max{2 | α | ρ1, 2 | β | ρ2}. Since ℳ = (Mk) is nondecreasing convex function, so by using inequality . Hence cI is a linear space. Similarly, we can prove that c0I, mI, and m0I are linear spaces.Therefore, ℳ = (Mk) be a Musielak-Orlicz function. Then,Let x ∈ cI. Then, there exist L ∈ ℂ and ρ > 0 such thatLet We havek on both sides, we get x ∈ l∞. The inclusion c0I ⊂ cI is obvious. Thus,Taking supremum over This completes the proof of the theorem.ℳ = (Mk) be a Musielak-Orlicz function and let p = (pk) be a bounded sequence of positive real numbers. Then, l∞ is a paranormed space with paranorm defined byLet g(x) = g(−x). Since Mk(0) = 0, we get g(0) = 0. Let us take x, y ∈ l∞ . LetIt is clear that ρ1 ∈ B(x) and ρ2 ∈ B(y). If ρ = ρ1 + ρ2, then we haveLet kMk(|Λ(x + y)|/(ρ1 + ρ2))pk ≤ 1 andThus, supσs → σ, where σ, σs ∈ ℂ and let g(xs − x) → 0 as s → ∞. We have to show that g(σsxs − σx) → 0 as s → ∞. LetLet ρs ∈ B(xs) and ρs′ ∈ B(xs − x), then we observe thatIf From the above inequality, it follows thatThis completes the proof.ℳ′ = (Mk′) and ℳ′′ = (Mk′′) be Musielak-Orlicz functions that satisfy the Δ2-condition. Then,Let Z⊆Z,(i)  Z∩Z⊆Z for Z = cI, c0I, mI, m0I.(ii)  x ∈ c0I. Then, there exists ρ > 0 such that(i) Let ϵ > 0 and choose δ with 0 < δ < 1 such that Mk′(t) < ϵ for 0 ≤ t ≤ δ. Write yk = Mk′′(|Λk(x)|/ρ)pk and considerLet ℳ = (Mk) satisfies Δ2-condition, we haveSince yk > δ, we haveFor ℳ′ = (Mk′) is nondecreasing and convex, it follows thatSince ℳ′ = (Mk′) satisfies Δ2-condition, we haveSince x = (xk) ∈ c0I. Thus, c0I⊆c0I. Similarly, we can prove the other cases.Hence,k).From , 34), a, a38)lili(38)limx ∈ c0I∩c0I. Then, there exists ρ > 0 such that(ii) Let The rest of the proof follows from the following equality:ℳ = (Mk) be a Musielak-Orlicz function which satisfies Δ2-condition. Then, Z⊆Z holds for Z = cI, c0I, mI, and m0I.Let Mk′′(x) = x and Mk′(x) = Mk(x)  ∀x ∈ [0, ∞).The proof follows from c0I and m0I are solid.The spaces c0I. Let x ∈ c0I. Then, there exists ρ > 0 such thatWe will prove for the space αk) be a sequence of scalars with |αk | ≤1  ∀k ∈ ℕ. Then, the result follows from the following inequality:m0I.Let  and m0I are monotone.The spaces It is easy to prove, so we omit the details.cI and c0I are sequence algebra.The spaces x, y ∈ c0I. Then,Let ρ = ρ1 + ρ2. Then, we can show thatLet x · y) ∈ c0I. Hence, c0I is a sequence algebra. Similarly, we can prove that cI is a sequence algebra.Thus, (