We show that if p = 1 (or p ≥ 2 and k is odd), then every well-defined solution of this equation is eventually periodic with period k, which generalizes the results of (Elsayed and Stevip ≥ 2 and k being even which has a well-defined solution that is not eventually periodic.We study the following max-type difference equation  The max operator arises naturally in certain models in automatic control theory (see ). In recAn}n=1∞+ is a periodic sequence with period p and k, r ∈ {1,2,…} with gcd = 1 and k ≠ r, and the initial conditions xd1−, xd2−,…, x0 are real numbers with In this paper, we consider the following max-type equation:k = 4, r = 1, and p = 1. Elsayed and Stevik = 3, r = 1, and p = 1. In [k = 2, r = 1, and p = 1, then every well-defined solution of  = 1 and k ≠ r.p ≥ 2 and k is odd, then every well-defined solution of kr < 0 eventually for all 1 ≤ i ≤ k − 1 and every {xnk−ir}n=r∞+ (1 ≤ i ≤ k − 1) is not constant sequence eventually.We claim that {p ≥ 2 and k is odd, then we have xnkxnk−kr∏i=1k−1xnk−irxnk−ir < 0 eventually. This is a contradiction.If p = 1, then we write An = A for all n ≥ 1 and choose m0 ≥ m1 ≥ ⋯≥mk2 such that xmjk−jr < xmj+1)k−jr( and xmj+1)k−(j+1)r( = xmj+1+1)k−(j+1)rk+rk+rk+r. By induction, we can show that xnk+jr is a constant sequence eventually for every 1 ≤ j ≤ k. Note {jr : 1 ≤ j ≤ k}kmod = {0,1, 2,…, k − 1} since gcd = 1. Then xnk+i is a constant sequence eventually for every i ∈ {0,1,…, k − 1}, which implies that {xn}n=1−d∞+ is eventually periodic with period k.By the above claim we may choose an From the proof of k, r ∈ {1,2,…} with k ≠ r. If {An}n=1∞+ is a periodic sequence, then every positive (or negative) solution of  = 1 and k ≠ r. If As ≥ 0 for some s ∈ {1,2,…, p}, then every well-defined solution of k = 1. Then xnk+i is a constant sequence eventually for every i ∈ {0,1,…, k − 1}, which implies that {xn}n=1−d∞+ is eventually periodic with period k.Using arguments similar to the ones developed in the proof of p ≥ 2 and k being even which has a well-defined solution that is not eventually periodic.Now we construct an example with k, r ∈ {1,2,…} and k is even with gcd = 1 and k ≠ r and An is a periodic sequence with Ai2 = A < Ai−12 = B < 0 for all i ≥ 1. Choose the initial conditions xi− = B for odd i ∈ {0,1,…, d} and xi− = 1 for even i ∈ {0,1,…, d} with d = max{r, k}; we can obtain a solution {xn}n=−1∞ of  If  It is easy to verify that limn→∞xn2 = ∞ and limn→∞xn−12 = 0. r > k, then(2) If It is easy to verify that limn→∞xn2 = ∞ and limi→∞xn−12 = 0.Consider the max-type equationxn=max{AAn}n=1∞+ is a periodic sequence with period ps and s, k, r ∈ {1,2,…} with gcd = 1 and k ≠ r, and p = 1 (or p ≥ 2), and the initial conditions xd1−, xd2−,…, x0 are real numbers with yni = xns+i for every 1 ≤ i ≤ s and n = 0,1, 2,…. Then ((1)p = 1 (or p ≥ 2 and k is odd), then it follows from i ≤ s, every well-defined solution of equation k. Thus every well-defined solution of (sk.If ution of  is event(2)p ≥ 2 and, for every 1 ≤ i ≤ s, there exists some ji such that Ajis+i ≥ 0, then it follows from i ≤ s, every well-defined solution of equation k. Thus every well-defined solution of (sk.If ution of  is event(3)p ≥ 2 and k is even, then it follows from i ≤ s, we can construct an equation If such that it has a well-defined solution which is not eventually periodic.Consider the max-type equation,…. Then  reduces


Molodtsov introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainty. In this paper, we initiate the study of soft congruence relations by using the soft set theory. The notions of soft quotient rings, generalized soft ideals and generalized soft quotient rings, are introduced, and several related properties are investigated. Also, we obtain a one-to-one correspondence between soft congruence relations and idealistic soft rings and a one-to-one correspondence between soft congruence relations and soft ideals. In particular, the first, second, and third soft isomorphism theorems are established, respectively. To solve complicated problems in economics, engineering, environmental science, medical science, and social science, methods in classical mathematics are not always successfully used because various uncertainties are typical for these problems. Therefore, there has been a great deal of alternative research and applications in the literature concerning some special tools such as probability theory, fuzzy set theory , 2, rougCurrently, works on soft set theory are progressing rapidly. Maji et al.  discusseThe rest of the paper is organized as follows. In In this section, we recall some notions and definitions see , 22, 23), 2322, 2U be an initial universe and let E be a set of parameters. Let ℘(U) denote the power set of U and let A and B be nonempty subsets of E.From now on, let F, A) is called a soft set over U, where F is a mapping given by F : A → ℘(U).A pair  and  over a common universe U, we say that  is a soft subset of  if it satisfiesA⊆B;F(α)⊆G(α) for all α ∈ A.For two soft sets  is said to be a soft super set of .In this case,  and  over a common universe U is the soft set  where C = A × B and for all  ∈ C, H = F(α)∩G(β). In this case, we write The AND-operation of two soft sets  and  over a common universe U is the soft set  where C = A × B and for all  ∈ C, H = F(α) ∪ G(β). In this case, we write G, B) = .The OR-operation of two soft sets  and  over a common universe U is the soft set  where C = A ∪ B and for all α ∈ C,The extended intersection of two soft sets ∩ɛ = .In this case, we write  and  over a common universe U is the soft set  where C = A∩B ≠ ∅ and for all α ∈ C, H(α) = F(α)∩G(α). In this case, we write ∩R = .The restricted intersection of two soft sets ∩R = ∅∅, where ∅∅ is the unique soft set over U with an empty parameter set.If F, A) and  over a common universe U is the soft set  where C = A ∪ B and for all α ∈ C,The extended union of two soft sets ∪ɛ = .In this case, we write  and  over a common universe U is the soft set  where C = A∩B ≠ ∅ and for all α ∈ C, H(α) = F(α) ∪ G(α). In this case, we write ∪R = .The restricted union of two soft sets ∪R = ∅∅.If Fi, Ai) | i ∈ I} of soft sets over U as follows in . Then RE2 and ∅∅2 are the greatest element and the least element of SC(R)E, respectively.Let (R)A the set of all those soft congruence relations defined over R with a fixed parameter set A. Then RA2 and ∅A2 are the greatest element and the least element of SC(R)A, respectively.In a similar way, denote by SC = {(f(x), f(y)) ∈ R′ × R′ | xηy}; then f(η) is a congruence relation on R′.Let x′, y′ ∈ R, and  ∈ f(η), since f is a ring epimorphism, there exists x, y ∈ R, such that x′ = f(x), y′ = f(y), and xηy. It is easy to verify that f(η) is a congruence relation on R′.For all f : R → R′ be a mapping of rings and let  be a soft set over R. Then we can define a soft set (f(F), A) over R′ where f(F) : A → ℘(R′) is defined as f(F)(α) = f(F(α)) for all α ∈ A. Here, by definition, we see that Supp(f(F), A) = Supp.Let f : R → R′ be a ring epimorphism. If  is a soft congruence relation over R, then (f(ρ), A) is a soft congruence relation over R′, where f(ρ)(α) = {(f(x), f(y)) ∈ R′ × R′ |  ∈ ρ(α)} for all α ∈ A.Let f(ρ), A) is a nonnull soft set since  is a soft congruence relation over R, which is a nonnull soft set by α ∈ Supp(f(ρ), A), we have f(ρ)(α) = f(ρ(α)) ≠ ∅. Then the nonempty set ρ(α) is a congruence relation on R, and so we deduce that its onto homomorphic image f(ρ(α)) is a congruence on R′. Hence f(ρ)(α) is a congruence on R′ for all ∈Supp(f(ρ), A). That is, (f(ρ), A) is a soft congruence relation over R′.Note first that  be a soft congruence relation over R, where f(ρ)(α) = {(f(x), f(y)) ∈ R′ × R′ |  ∈ ρ(α)} for all α ∈ A.ρ, A) is trivial, then (f(ρ), A) is the trivial soft congruence relation over R′.If  is whole, then (f(ρ), A) is the whole soft congruence relation over R′.If  = { | x ∈ R}. Since f is a ring epimorphism, we have that f(ρ)(α) = f(ρ(α)) = { | x′ = f(x), x ∈ R} for all α ∈ A. So, (f(ρ), A) is the trivial soft congruence relation over R′ by (1) By α ∈ A, ρ(α) = { | x, y ∈ R}. Since f is a ring epimorphism, we have that f(ρ)(α) = { | x′ = f(x), y′ = f(y),  ∈ ρ(α), x, y ∈ R} for all α ∈ A. It follows from f(ρ), A) is the whole soft congruence relation over R′.(2) By This completes the proof.Let us define now some definitions about soft quotient rings that we will use in the following paragraphs.ρ, A) be a soft congruence relation over R; we have that  = 0 is an idealistic soft ring over R by α ∈ A, xρ(α) = x + F(α). Let R/ = {x + F(α) | x ∈ R, α ∈ A} be an initial universe set and consider the set-valued function F* : A → ℘) given by F*(α) = {x + F(α) | x ∈ R} for all α ∈ A. We say that  is a soft quotient ring of R with respect to the idealistic soft ring , where 〈F*(α), +*, ·*〉 is a quotient ring of 〈R, +, ·〉 with respect to an ideal F(α).Let  be an idealistic soft ring over 〈Z, +, ·〉 and A = N+. For all α ∈ A, we have F(α) = {αx | x ∈ Z}⊲Z. Let Z/ = {x + F(α) | x ∈ Z, α ∈ A} be an initial universe set and consider the set-valued function F* : A → ℘) given by F*(α) = Z/F(α) = {0 + F(α), 1 + F(α),…, (α − 1) + F(α)} = Z/〈α〉 for all α ∈ A; it is clear that F*(α) is a quotient ring of Z with respect to F(α). Then  is a soft quotient ring of Z with respect to .Let  be a soft quotient ring of R with respect to an idealistic soft ring  and let  be a soft quotient ring of R′ with respect to an idealistic soft ring , respectively. We say that  is soft homomorphic to , denoted by ~Q, if, for all α ∈ A, there exists β ∈ B such that a mapping f : F*(α) → G*(β) is a homomorphism; that is, F*(α) ~ G*(β).Let  is soft isomorphic to , which is denoted by ≃Q.If S is a subring of R, we write S ≤ R; if H is an ideal of R, we write H⊲R.Finally, we establish the first soft isomorphism theorem. In order to do this, we need to introduce some relative notions and results. Note that, if F, A) and  be soft sets over a common universe U.  is said to be contained by , denoted by α ∈ A, there exists β ∈ B such that F(α)⊆G(β).Let  and  over a common universe U are said to be soft set equal, denoted by  = , if it satisfies the following:α ∈ A, ∃β ∈ B such that F(α) = G(β);for all b ∈ B, ∃a ∈ A such that G(b) = F(a).for all Two soft sets  and G : B → ℘(R) be set-valued functions defined as follows: F(α) = {x, z}, F(α′) = {u, v}, G(β) = {u, v}, and G(β′) = {x, z}. Therefore,  and  are soft set equal.Let f : R → R′ be a ring epimorphism.ρ : A→℘(R × R) be a set-valued function given by ρ(α) = { | x, y ∈ R, f(x) = f(y)} for all α ∈ A. Then  is a soft congruence relation over R.Let F : A → ℘(R) be a set-valued function given by F(α) = Kerf = {x ∈ R | f(x) = 0} for all α ∈ A. Then  is an idealistic soft ring over R.Let Let f : R → R′ be a ring epimorphism. Let S = {H | H ≤ R, H⊇K = Kerf} and T = {H′ | H′ ≤ R′}. Then, for H ∈ S, we make H correspond to Hf = {f(x) | x ∈ H}, which is a bijective mapping between S and T. And H⊲R⇔Hf⊲R′; here, R/H≅R′/Hf.Let f : R → R′ be a ring epimorphism and consider an idealistic soft ring  over R defined as G, B) ∈ SR(R), let f(G) : B → ℘(R′) be a set-valued function given by f(G)(α) = f(G(α)) = {f(x) | x ∈ G(α)} for all α ∈ B. We make  correspond to (f(G), B), which is a bijective mapping between SR(R) and SR(R′).For  is an idealistic soft ring over R if and only if (f(G), B) is an idealistic soft ring over R′. And ≃Q(f(G)*, B), where  is a soft quotient ring of R with respect to , and (f(G)*, B) is a soft quotient ring of R′ with respect to (f(G), B).,  ∈ SR(R) if (f(G1), B1) = (f(G2), B2); that is, for all α ∈ B1, ∃β ∈ B2 such that f(G1)(α) = f(G2)(β) and for all b ∈ B2, ∃a ∈ B1 such that f(G2)(b) = f(G1)(a), but by G1(α) = G2(β) and G2(b) = G1(a), which means that  = . Thus it is injection. Let  ∈ SR(R); then we have, for all α ∈ B, G(α) ≤ R and its homomorphism image f(G)(α) = f(G(α)) ≤ R′, so f(G), B) ∈ SR(R′). Let (f(G), B) ∈ SR(R′); then we have for all α ∈ B, f(G)(α) ≤ R′. Define G(α) = {x ∈ R | f(x) ∈ f(G)(α)} for all α ∈ B; then G(α) ≤ R. And for all a ∈ A, ∃α ∈ B, Kerf = F(a)⊆G(α). So G, B) ∈ SR(R). Thus it is surjection. Therefore, we make  correspond to (f(G), B), which is a bijective mapping between SR(R) and SR(R′).(1) Let  is an idealistic soft ring over R⇔G(α)⊲R ⇔ (by f(G(α))⊲R′  ⇔(f(G), B) is an idealistic soft ring over R′. From G*(α)≅f(G)*(α) for all α ∈ B, which means that ≃Q(f(G)*, B) by (2) (∈ B)⇔ by  f(G(α))⊲f : Z → Z/〈6〉 be a ring epimorphism, A = Z and B = C = {α | α  is  divisor  of  6}. Consider the set-valued function F : A → ℘(Z) given by F(α) = 6Z⊲Z. Then  is an idealistic soft ring over Z. Let G, B) ∈ SR(Z), we have G(β) = βZ. And let H, C) ∈ SR(Z/〈6〉), we have H(c) = cZ/〈6〉.B = C = {1,2, 3,6} and let f(G) : B → ℘(R′) be a set-valued function given by f(G)(β) = βZ/〈6〉 for all β ∈ B. Then, (f(G), B) ∈ SR(Z/〈6〉). It is clear that we make  correspond to (f(G), B), which is a bijective mapping between SR(Z) and SR(Z/〈6〉).Let B = {2,3} and G*(β) = {x + G(β) | x ∈ Z} = Z/〈β〉. Then  is an idealistic soft ring over R iff G(β) = βZ⊲Z , f(G(β)) = βZ/〈6〉⊲Z/〈6〉 , and (f(G), B) is an idealistic soft ring over Z/〈6〉. It is clear that  is a soft quotient ring of Z with respect to , and (f(G)*, B) is a soft quotient ring of Z/〈6〉 with respect to (f(G), B). Let β ∈ B; we have G*(β) = {x + G(β) | x ∈ Z} = Z/〈β〉 and f(G)*(β) = {x + f(G)(β) | x ∈ Z/〈6〉} = (Z/〈6〉)/(βZ/〈6〉). Note that Z/〈2〉≅(Z/〈6〉)/(2Z/〈6〉) and Z/〈3〉≅(Z/〈6〉)/(3Z/〈6〉); we have ≃Q(f(G)*, B) by Let Let In this section, we study internal connections between soft congruence relations and soft ideals of soft rings and obtain the second and third soft isomorphism theorems. In order to do this, we recall the following notions.F, A) and  be soft rings over R. Then  is called a soft subring of , denoted by B⊆A;G(α) is a subring of F(α) for all α∈Supp.Let  be a soft ring over R. A nonnull soft set  over R is called a soft ideal of , denoted by B⊆A;G(α) is an ideal of F(α) for all α∈Supp.Let  be a soft ring over R. A nonnull soft set  over R × R is called a soft binary relation of  if it satisfies the following:B⊆A;ρ(α) is a binary relation of F(α) for all α ∈ B.Let  be a soft ring over R. A soft binary relation  of  is called a soft congruence relation of  if ρ(α) is a congruence relation on F(α) for all ∈Supp.Let  be a soft ring over R. Then we have the following.ρ, B) be a soft congruence relation of . If  = 0, then  is a soft ideal of , and we have ρ(α) = { | x, y ∈ F(α), x − y ∈ G(α)} for all α ∈ B.Let  be a soft ideal of  and consider a soft binary relation  of  defined by ρ(α) = { | x, y ∈ F(α), x − y ∈ G(α)} for all α ∈ B. Then  is a soft congruence relation of , and 0 = .Let , ρ(α) is a congruence relation on a ring F(α). Since  = 0, we have G(α) = 0ρ(α)⊲F(α), and ρ(α) = { | x, y ∈ F(α), x − y ∈ G(α)} for all α ∈ B by G, B) is a non-null soft set, we have (1) By G, B), G(α)⊲F(α). By the hypothesis, we know that ρ(α) is a congruence relation on F(α) and 0ρ(α) = G(α) by ρ, B) is a soft congruence relation of  and 0 = .(2) By ρ, B) of a soft ring  can be represented by the soft ideal generated by . Also, we observe that any soft ideal  of  is the soft congruence class of 0 with respect to the soft congruence relation  generated by .Here, we obtain that any soft congruence relation )E the set of all soft congruence relations and by SI)E the set of all soft ideals of a soft ring . We can establish the following two mappings:ψ : SC)E → SI)E, ψ) = , where  = 0;φ : SI)E → SC)E, φ) = , where, for all α ∈ B, we define  ∈ ρ(α) in F(α), which is equivalent to x − y ∈ G(α).Denote by SC)E and SI)E of .The above two mappings F, A) and  be soft rings over R. We say that  is a generalized soft subring of , denoted by β ∈ B, there exists α ∈ A such that G(β) is a subring of F(α).Let  and  be soft rings over R. We say that  is a generalized soft ideal of , denoted by β ∈ B, there exists α ∈ A such that G(β) is an ideal of F(α).Let  be a generalized soft ideal of a soft ring  which is over 〈R, +, ·〉 and let / = {x + F(α) |  ∈ A×∗B, x ∈ G(β)} be an initial universe set, where A×∗B = { ∈ A × B | F(α)⊲G(β)}. Let us consider the set-valued function F∗ : A×∗B → ℘/) given by F∗ = {x + F(α) | x ∈ G(β)} for all  ∈ A×∗B. We say that  is a generalized soft quotient ring of soft ring  with respect to the generalized soft ideal , where 〈F∗, +∗, ·∗〉 is a quotient ring of ring 〈G(β), +, ·〉 with respect to F(α).Let  given by G(β) = {x ∈ R | x · β = {0,2}}. Then G(0) = R, G(1) = {0,2}, G(2) = Z/〈4〉, and G(3) = {0,2}. As we see, all these sets are subrings of R. Hence,  is a soft ring over R. On the other hand, consider the function F : A → ℘(R) given by F(α) = {x ∈ R | x · α = 0}. As we see,Let F, A) is a generalized soft ideal of . For all  ∈ A × B, where F(α)⊲G(β), F∗ = {x + F(α) | x ∈ G(β)}; we haveHence,  are quotient rings of ring 〈G(β), +, ·〉 with respect to F(α). Thus  is a generalized soft quotient ring of soft ring  with respect to the generalized soft ideal .As we see, F, A) and  be generalized soft ideals of soft rings  and , respectively. Suppose  is a generalized soft quotient ring of  with respect to , and  is a generalized soft quotient ring of  with respect to . We say that  is soft homomorphic to , denoted by ∼G, if, for all  ∈ A×∗A′, there exists  ∈ B×∗B′ such that a mapping f : F∗ → G∗ is a homomorphism; that is, F∗ ~ G∗.Let  is soft isomorphic to , which is denoted by ≃G.If F, A) and  be soft rings over the rings R and R′, respectively. Let f and g be two mappings. The pair  is called a soft ring homomorphism if the following conditions are satisfied:f is a ring epimorphism;g is surjective;f(F(α)) = G(g(α)) for all α ∈ A.Let  and ,  is said to be soft homomorphic to , denoted by ~. In addition, if f is a ring isomorphism and g is a bijective mapping, then  is called a soft ring isomorphism. In this case, we say that  is soft isomorphic to , denoted by ≃.If we have a soft ring homomorphism between  and  over R is the soft set  where C = A × B and for all  ∈ C, H = F(α) + G(β). In this case, we write + = .The basic sum of two soft rings  and  be soft rings over rings R and R′, respectively. Let  be a soft ring homomorphism between  and , and A′⊆A.ρ, A′) of  defined by ρ(α) = { | x, y ∈ F(α), f(x) = f(y)} for all α ∈ A′. Then  is a soft congruence relation of .Let us consider a soft binary relation (F′ : A′ → ℘(R) be a set-valued function given by F′(α) = Kerf(F(α)) = {x ∈ F(α) | f(x) = 0} for all α ∈ A′. Then  is a soft ideal of .Let Let  and  be soft rings over rings R and R′, respectively. Let  be a soft ring epimorphism between  and , and consider a soft ideal  of  defined as F1, A1) ∈ SS) and consider the set-valued function f(F1) : A1 → ℘(R′) given by f(F1)(α) = f(F1(α)) = {f(x) | x ∈ F1(α)} for all α ∈ A1. Then we make  correspond to (f(F1), A1), which is a bijective mapping between SS) and SS).Let  is a soft ideal of  if and only if (f(F1), A1) is a generalized soft ideal of . And ≃G(f(F1)∗, A1×∗B), where  is a generalized soft quotient ring of  with respect to , and (f(F1)∗, A1×∗B) is a generalized soft quotient ring of  with respect to (f(F1), A1).,  ∈ SS). If (f(F1), A1) = (f(F2), A2), that is, for all α ∈ A1, ∃β ∈ A2 such that f(F1)(α) = f(F2)(β) and for all b ∈ A2, ∃a ∈ A1 such that f(F2)(b) = f(F1)(a), but by F1(α) = F2(β) and F2(b) = F1(a), which means that  = . Thus it is injection. Let  ∈ SS); we have that, for all α ∈ A1, F1(α) ≤ F(α); then its homomorphism image f(F1(α)) ≤ f(F(α)) = G(g(α)), and so f(F1), A1) ∈ SS). Let (f(F1), A1) ∈ SS), we have that, for all α ∈ A1, ∃g(α) ∈ B such that f(F1(α)) ≤ G(g(α)). So we can define F1(α) = {x ∈ R | f(x) ∈ f(F1(α))}; then for all α ∈ A1, F1(α) ≤ F(α) and for all β ∈ A′, ∃α ∈ A1, Kerf(F(β)) = F′(β)⊆F1(α). So F1, A1) ∈ SS). Thus it is surjection. Therefore, we make  correspond to (f(F1), A1), which is a bijective mapping between SS) and SS).(1) Let (α ∈ A1)⇔ (by f(F1(α))⊲f(F(α)) = G(g(α))  ∈ B) α, α) ∈ A1×∗A, ∃) ∈ A1×∗B such that F1∗≅f(F1)∗), which means that ≃G(f(F1)∗, A1×∗B).(2) A1)⇔ by  f(F1(α)S be a subring and let H be an ideal of R, respectively. Then S + H = {x + y | x ∈ S, y ∈ H} is a subring of R, S∩H is an ideal of S, and a mapping f : S/(S∩H)→(S + H)/H, for x + (S∩H) ∈ S/(S∩H), f(x + (S∩H)) = x + H, is an isomorphism. That is, S/(S∩H)≅(S + H)/H.Let F, A) be a soft ring and  be an idealistic soft ring over R.H, C) = + is a soft ring over R, and  is a generalized soft ideal of ; = ∩R is a generalized soft ideal of ; ∈ B×∗C, there exists  ∈ D×∗A such that a mapping f : G∗ → K∗, with respect to x + K(d) ∈ K∗ and f(x + K(d)) = x + G(b), is an isomorphism, where x ∈ F(a). That is, ≃G, where  is a generalized soft quotient ring of  with respect to , and  is a generalized soft quotient ring of  with respect to .For all  ≤ R and for all β ∈ B, G(β)⊲R. By α, β) ∈ C, F(α) + G(β) = H ≤ R; then β ∈ B, ∃ ∈ C such that G(β)⊲H. Hence, (1) By given conditions, we observe that, for all α ∈ D = A∩B ≠ ∅, K(α) = F(α)∩G(α). Since F(α) ≤ R, G(α)⊲R, then by α ∈ D, F(α)∩G(α) = K(α)⊲F(α). Therefore  is a generalized soft ideal of .(2) By β ∈ B, ∃ ∈ C such that G∗ = {x + G(β) | x ∈ H}, and we observe that, for all α ∈ D = A∩B, ∃α ∈ A such that K∗ = {y + F(α) | y ∈ K(α)}, where G(β)⊲H and K(α)⊲F(α). By α, β) ∈ B×∗C, ∃ ∈ D×∗A, such that a mapping f, with respect to x + K(α) ∈ K∗ and f(x + K(α)) = x + G(α), is an isomorphism, where x ∈ F(α). That is, G∗≅K∗, which means that ≃G by (3) By In this paper, concepts of soft congruence relations are presented. Two types of soft congruence relations that are soft congruence relation over a ring and soft congruence relation of a soft ring are defined in a natural way. Then we study relations among soft congruence relations and homomorphisms, soft ring homomorphisms. Also, we obtain a one-to-one correspondence between soft congruence relations and idealistic soft rings and a one-to-one correspondence between soft congruence relations and soft ideals. In particular, we established the first, second, and third soft isomorphism theorems, respectively. In the light of these results, our future work on this topic will be focused on applying soft congruence relations to fuzzy information over rings.


By these properties, the following important results are proved: (1) a fuzzy filter  of a pseudo-BCI algebra is a fuzzy associative filter if and only if it is a fuzzy a-filter; (2) a filter  of a pseudo-BCI algebra is associative if and only if it is an a-filter ; (3) a fuzzy filter of a pseudo-BCI algebra is fuzzy a-filter if and only if it is both a fuzzy p-filter and a fuzzy q-filter.Some new properties of fuzzy associative filters , fuzzy  BCI algebra, which originated from BCI-logic; it is a kind of nonclassical logic and inspired by the calculus of combinators  is a fuzzy filter of X if and only if the level set μt = {x ∈ X | μ(x) ≥ t} is filter of X for all t ∈ Im(μ).Let X be a pseudo-BCI algebra. Then a fuzzy set μ : X →  is a fuzzy filter of X if and only if it satisfiesx, y, z ∈ X, x ≤ y → z⇒μ(z) ≥ min{μ(x), μ(y)};for all x, y, z ∈ X, x ≤ y⇝z⇒μ(z) ≥ min{μ(x), μ(y)}.for all Let μ : X →  is called a fuzzy p-filter of a pseudo-BCI algebra X if it satisfies (FF1) and(FPF1)x, y, z ∈ X, μ(z) ≥ min{μ((x → y)⇝(x → z)), μ(y)};for all (FPF2)x, y, z ∈ X, μ(z) ≥ min{μ((x⇝y)→(x⇝z)), μ(y)}.for all A fuzzy set μ : X →  is called a fuzzy a-filter of a pseudo-BCI algebra X if it satisfies (FF1) and(FaF1)x, y, z ∈ X, μ(z → x) ≥ min{μ((x⇝1)→(y⇝z)), μ(y)};for all (FaF2)x, y, z ∈ X, μ(z⇝x) ≥ min{μ((x → 1)⇝(y → z)), μ(y)}.for all A fuzzy set μ : X →  is called a fuzzy associative filter of a pseudo-BCI algebra X if it satisfies (FF1) and(FAF1)x, y, z ∈ X, μ(z) ≥ min{μ(x → (y⇝z)), μ(x → y)};for all (FAF2)x, y, z ∈ X, μ(z) ≥ min{μ(x⇝(y → z)), μ(x⇝y)}.for all A fuzzy set μ : X →  is called a fuzzy q-filter of a pseudo-BCI algebra X if it satisfies (FF1) and(FqF1)x, y, z ∈ X, μ(x → z) ≥ min{μ((x⇝y) → z), μ(y)};for all (FqF2)x, y, z ∈ X, μ(x⇝z) ≥ min{μ((x → y)⇝z), μ(y)}.for all A fuzzy set μ be a fuzzy a-filter of a pseudo-BCI algebra X. Then μ satisfiesLet μ be a fuzzy a-filter of a pseudo-BCI algebra X. Then μ satisfiesLet μ be a fuzzy a-filter of a pseudo-BCI algebra X. Then the following statements hold for all x, y, z ∈ X:x, y ∈ X, μ(x) ≥ min{μ(y), μ(x → y)}, μ(x) ≥ min{μ(y), μ(x⇝y)};for all x, y ∈ X, μ(x → y) = μ(x⇝y);for all x ∈ X, μ((x → 1) → x) = μ((x⇝1)⇝x) = μ(1);for all x, y, z ∈ X, μ(x → z) ≥ min{μ((x⇝y) → z), μ(y)};for all x, y, z ∈ X, μ(x⇝z) ≥ min{μ((x → y)⇝z), μ(y)};for all x ∈ X, μ(x → (x⇝1)) = μ(x⇝(x → 1)) = μ(1);for all x, y, z ∈ X, μ((x → y) → y) ≥ μ(x), μ((x⇝y)⇝y) ≥ μ(x).for all Let x, y ∈ X, by y → 1 ≤ (x → y)→(x → 1). From this, applying μ(x → 1) = μ(x) and μ(y → 1) = μ(y). Therefore, μ(x) ≥ min{μ(y), μ(x → y)}.(1) For any μ(x) ≥ min{μ(y), μ(x⇝y)}.Similarly, we have x, y ∈ X, by x → y) → 1 = (x → 1)⇝(y → 1). By μ(x → y) ≤ μ(x⇝y). Similarly, we can get μ(x⇝y) ≤ μ(x → y). By μ(x → y) = μ(x⇝y).(2) For any x ∈ X, applying μ((x → 1) → x) = μ(1).(3) For any μ((x⇝1)⇝x) = μ(1).Similarly, we have x, y ∈ X, by μ((x⇝y)⇝x) ≥ μ(y). From this and (2), we get μ((x⇝y) → x) ≥ μ(y).(4) For any x⇝y) → x ≤ (x → z)⇝((x⇝y) → z). Using On the other hand, by (5) The proof is similar to (4).x ∈ X, by (4), we haveμ(x → (x⇝1)) = μ(1). Similarly, μ(x⇝(x → 1)) = μ(1).(6) For any x ∈ X, by x ≤ (x → y)⇝y. From this, (2), and (7) For any μ((x⇝y)⇝y) ≥ μ(x).Similarly, we have μ be a fuzzy associative filter of a pseudo-BCI algebra X. Then μ satisfiesLet It is easily proved by μ be a fuzzy associative filter of a pseudo-BCI algebra X. Then the following statements hold:x ∈ X, μ(x → (x⇝1)) = μ(x⇝(x → 1)) = μ(1);for all x, y ∈ X, μ(y) ≥ μ((x → 1)⇝(x → y));for all x, y ∈ X, μ(y) ≥ μ((x⇝1)→(x⇝y));for all x, y ∈ X, μ(x → y) = μ(x⇝y);for all x, y ∈ X, μ(x → (x → y)) ≥ μ(y), μ(x⇝(x⇝y)) ≥ μ(y);for all x ∈ X, μ(x) ≥ μ((x → 1) → 1)), μ(x) ≥ μ((x⇝1)⇝1));for all x, y ∈ X, μ(x → (y → 1)) ≥ μ(x → y), μ(x⇝(y⇝1)) ≥ μ(x⇝y);for all x, y ∈ X, μ(y → x)) ≥ μ(x → y), μ(y⇝x)) ≥ μ(x⇝y);for all x ∈ X, μ((x⇝1) → x) = μ((x → 1)⇝x) = μ(1);for all x, y ∈ X, μ((x → y) → y) ≥ μ(x), μ((x⇝y)⇝y) ≥ μ(x);for all x ∈ X, μ(x) = μ(x → 1) = μ(x⇝1);for all x, y ∈ X, μ(x) ≥ min{μ(y), μ(x → y)}, μ(x) ≥ min{μ(y), μ(x⇝y)}.for all Let x ∈ X (by μ(x → (x⇝1)) = μ(1). Similarly, μ(x⇝(x → 1)) = μ(1).(1) For any x ∈ X by ,(16)) = μ(1), μ(y) ≥ μ(x⇝(x → y)). Therefore,(2) For any (3) It is similar to (2).x, y ∈ X, by (2), we have μ(x⇝y) ≥ μ((y → 1)⇝(y → (x⇝y))). On the other hand, applying μ((y → 1)⇝(y → (x⇝y))) ≥ μ(x → y). Thus, μ(x⇝y) ≥ μ(x → y). Similarly, we can get μ(x → y) ≥ μ(x⇝y). Therefore, μ(x → y) = μ(x⇝y).(4) For any x, y ∈ X, since (by μ(y⇝(x → (x → y))) = μ(x → (x⇝1)). By (1), μ(x → (x⇝1)) = μ(1); hence, μ(y⇝(x → (x → y))) = μ(1). Moreover, by (5) For any since by (22)y⇝(xμ(x⇝(x⇝y)) ≥ μ(y).Similarly, (6) By (2) and (3), we can get (6).x, y ∈ X, by y → (y → 1))≤(x → y)→(x → (y → 1)).(7) For any Applying (1) and μ(x⇝(y⇝1)) ≥ μ(x⇝y).Similarly, x, y ∈ X, by μ(y → x) ≥ μ(x⇝(x → (y → x)))  . And, using μ(y → x) ≥ μ(x → (y → 1)). From this and (7), we get(8) For any μ(y⇝x) ≥ μ(x⇝y).Similarly, (9) By (1) and (8), we can get (8).(10) It is similar to the proof of x ∈ X, by μ(x) ≥ μ(x → (x⇝x)) = μ(x → 1). On the other hand, using (8), μ(x → 1) ≥ μ(1 → x) = μ(x). Hence, μ(x) = μ(x → 1) = μ(x⇝1).(11) For any (12) It is similar to the proof of Checking the proof of μ be a fuzzy filter of a pseudo-BCI algebra X. If μ satisfies(C1)x, y ∈ X, μ(y) ≥ μ(x → (x⇝y));for all (C2)x, y ∈ X, μ(y) ≥ μ(x⇝(x → y)),for all  then the following statements hold:(C3)x ∈ X, μ(x → (x⇝1)) = μ(x⇝(x → 1)) = μ(1);for all (C4)x, y ∈ X, μ(x → y) = μ(x⇝y);for all (C5)x ∈ X, μ(x) = μ(x → 1) = μ(x⇝1);for all (C6)x, y ∈ X, μ(x) ≥ min{μ(y), μ(x → y)}, μ(x) ≥ min{μ(y), μ(x⇝y)}.for all Let It is similar to μ be a fuzzy filter of a pseudo-BCI algebra X. Then μ is a fuzzy associative filter of X if and only if it satisfies(C1)x, y ∈ X, μ(y) ≥ μ(x → (x⇝y));for all (C2)x, y ∈ X, μ(y) ≥ μ(x⇝(x → y)).for all Let μ is a fuzzy associative filter of X; by Assume that μ satisfies conditions (C1) and (C2). For any x, y, z ∈ X, by y → (x → z)≤(x → y)→(x → (x → z)). Using μ(y → (x → z)) = μ(y⇝(x → z)) = μ(x → (y⇝z)).Conversely, assume that Thus,μ(x → (x → z)) ≥ min{μ(x → (y⇝z)), μ(x → y)}.(P1) On the other hand, applying μ(z) ≥ min{μ(1), μ(x → (x → z))} = μ(x → (x → z)).(P2) μ(z) ≥ min{μ(x → (y⇝z)), μ(x → y)}. That is, (FAF1) holds. Similarly, condition (FAF2) holds. Therefore, by μ is a fuzzy associative filter of X.Combining (P1) and (P2), we get that μ be a fuzzy filter of a pseudo-BCI algebra X. Then μ is a fuzzy a-filter of X if and only if it satisfies(a1)x, y ∈ X, μ(x → y) ≥ μ((y⇝1) → x);for all (a2)x, y ∈ X, μ(x⇝y) ≥ μ((y → 1)⇝x).for all Let μ be a fuzzy filter of a pseudo-BCI algebra X. Then the following conditions are equivalent:μ is a fuzzy a-filter of X;μ is a fuzzy associative filter of X.Let μ is a fuzzy a-filter of X. For any x, y ∈ X, by μ((x⇝1)⇝(x⇝y)) = μ((x⇝1)→(x⇝y)). Thus,y ≤ 1 → y ≤ (x⇝1)⇝(x⇝y). From this and y ≤ (x⇝1)⇝(x⇝y), we have y → ((x⇝1)⇝(x⇝y)) = 1; it follows thatμ(y) ≥ μ(x⇝(x → y)). By μ is a fuzzy associative filter of X.(i) ⇒ (ii). Suppose that μ is a fuzzy associative filter of X. For any x, y ∈ X, by y⇝1) → x ≤ (x → y)⇝(y⇝1). Applying μ((y⇝1) → y) = μ(1). It follows that(ii) ⇒ (i). Suppose that μ(x⇝y) ≥ μ((y → 1)⇝x). By μ is a fuzzy a-filter of X.Similarly, we can get a-filters of pseudo-BCI algebras. At first, we give the following results (the proofs are omitted).Now, we discuss the relationship between associative filters and F of pseudo-BCI algebra X is a filter  of X if and only if the characteristic function χF of F is a fuzzy filter  of X.A nonempty subset X be a pseudo-BCI algebra. Then a fuzzy set μ : X →  is a fuzzy associative filter (fuzzy a-filter) of X if and only if the level set μt = {x ∈ X | μ(x) ≥ t} is associative filter (a-filter) of X for all t ∈ Im(μ).Let  Transfer Principle for Fuzzy Sets in [In fact, the above proposition is a consequence of the so-called Sets in .Combining Propositions F be a filter of a pseudo-BCI algebra X. Then the following conditions are equivalent:F is an a-filter of X;F is an associative filter of X.Let In , the auta-filters, and fuzzy q-filters in pseudo-BCI algebras.Finally, we discuss the relationship among fuzzy associative filters, fuzzy μ be a fuzzy p-filter of a pseudo-BCI algebra X. Then μ satisfiesx ∈ X, μ(x) = μ((x → 1)⇝1);for all x ∈ X, μ(((x → 1)⇝1)⇝x) = μ(1).for all Let x ∈ X, by μ((x → 1)⇝1) ≥ μ(x). It follows that μ(x) = μ((x → 1)⇝1).(1) For any x ∈ X, by (2) For any μ be a fuzzy filter of a pseudo-BCI algebra X. Then the following conditions are equivalent:μ is a fuzzy a-filter of X;μ is both a fuzzy p-filter and a fuzzy q-filter of X.Let μ is a fuzzy a-filter of X. It is easy to prove that μ is both a fuzzy p-filter and a fuzzy q-filter of X.Assume that μ be both a fuzzy p-filter and fuzzy q-filter of X. For any x, y ∈ X, by μ is a fuzzy a-filter of X.Conversely, let


Particle size versus ground time clearly shows the existence of a size-induced regime transition . Thermoelectric properties of β-MnO2 powders as a function of electrical resistance in the range of RP = 10 - 80Ω were measured. Based on the data presented, we propose a model for the β-MnO2 system in which nanometer-scale MnO2 crystallites bond together through weak van der Waals forces to form larger conglomerates that span in size from nanometer to micrometer scale.Particle sizes of manganese oxide (β-MnO One of the most known and commercially available TE material is bismuth telluride (Bi2Te3), which exhibits one of the highest ZT values at room temperature . Thermoelectric devices (TED) made out of Bi2Te3 are already commercially available and are used for small-scale energy harvesting , [101], [200], [111], [210], [211], [220], [002], and [310], respectively. The position and orientation of these peaks confirmed that our particles have β-MnO2 crystal structure   with different properties, and from a TE standpoint, β-MnO2 is preferred since the electrical conductivity of α-phase MnO2 is approximately six orders of magnitude lower [Shown in Fig. te) Ref. . This crde lower . Althoug2 system, the group of Wallia et al. [2 nanopowder by using ball-milling method with crystals ranging from 400 to 700 nm in size; however, their SEM micrograph shows particle conglomerates that are ranging in size from a few hundreds of nm up to 2 μm. They present no XRD results that would support their crystalline size claim. The groups of Xia et al. [2 powders, but no supporting SEM/TEM or XRD measurements are presented. Finally, the group of Song et al. [Extracting information about the crystalline size based on the broadening of the XRD peaks is not a trivial task. It involves understanding diffraction peak broadening and also choosing the right method of analysis . For exaa et al.  claimed a et al.  and Preia et al.  obtainedg et al.  reports D = (k ⋅ λ)/(β ⋅ cos(θ)), where D is the particle size in nanometers, λ is the wavelength of X-ray radiation , k is a constant equal to 0.94, and β is the measured FWHM of the peak, and θ is the peak position shown in Fig. D4 = 306.30 nm smaller than the particle size for sample S6  D6 = 546.26 nm. Therefore, it is important that the data should be obtained from more than one peak in order to get consistent and reliable results.Just to solidify our findings, we also used Scherrer formula to estimate the particle size. The Scherrer formula is RP), for samples S4 through S8, varies exponentially with tube length (L). We believe that this response induces further exponential behavior observed in σ and power factor (σ ⋅ S2) versus RP  versus electrical resistance. The S measured varies linearly with RP, and it has the tendency to have larger values at smaller resistances  at values of − 316μV/K (−288μV/K) and at resistances of 9.8 Ω (69Ω)  (Table S observed by Song and co-workers [Rp = 10 - 80 Ω) as compared to Song’s resistance values . Also, our method of particle modification was quite different. For example, we used a simple mortar and pestle  while Song et al. used a ball-milling method. Furthermore, our data also suggests that the S increases with decreasing RP, which is opposite to the work of Song et al.  and   9Ω) Fig. . Our absof Refs. , 13, 14 -workers . This isRP for samples S4 through S8. We find that the largest  σ data were measured for samples S4 (S8) at values of (0.058 S/cm) and at resistances of Rp = 9.8 Ω (Rp = 79 Ω).Figure σ ⋅ S2) versus RP. The largest  power factors we had obtained came for samples S4 (S8) 5.8 × 10− 7 W/(m ⋅ K2) 5.7 × 10− 8 W/(m ⋅ K2) at resistances of Rp = 9.8 Ω ⋅ (Rp = 79Ω).In Figure , we present the literature data compared to our data for power factor versus electrical conductivity. Our power factor values are lower than other works by Walia et al. ) (Ref. [− 5 W/(m ⋅ K2)) (Ref. [− 6 W/(m ⋅ K2)) (Ref. [− 6 W/(m ⋅ K2)) (Ref. [− 6 W/(m ⋅ K2) and 5.09 × 10− 6 W/(m ⋅ K2)) ) (Ref. ), Islam )) (Ref. ), Xia et)) (Ref. ), and So)) (Ref. ). It is l. (Ref. ).Fig. 6LZ (1/K) and the unitless figure of merit ZT that were reported in the literature. It appears that the highest Z and ZT values were obtained by the work of Walia et al. . If we were to assume a thermal conductivity value of k = 0.2096  for the other works that did not report thermal conductivity  [Z = 3.66 × 10− 51/K and ZT = 1.1 × 10− 2) [Z = 3.42 × 10− 51/K and ZT = 1.02 × 10− 2) [Z = (7.29 - 9.84) × 10− 31/K and ZT = (2.43 - 3.28) × 10− 5) [Z = (0.254 - 1.59) × 10− 131/K and ZT = (3.0 - 8.3) × 10− 11) [S values of |S| = 20,000 - 40,000 μV/V, their reported electrical conductivity was σ = (3.18 - 12.7) × 10− 5 S/cm, which has a strong contribution toward lowering their ZT values.In Table ty Table . Still ta et al.  followed× 10− 2) , Song et× 10− 5) , and Bha 10− 11) , 9, 10). 10− 11)  reported2 system behaves differently as compared to other works. For example, the work of Song et al. [2 crystallites at nanometer scale bond together through weak van der waals forces to form larger conglomerates that span anywhere from mesoscopic to microscopic scale.All our data lead us to believe that our β−MnOg et al.  might ha2 powders as a function of electrical resistance in the range of R = 10 - 80Ω. We found two distinct particle size regimes , which were further confirmed by our thermoelectric measurements. According to SEM and TEM data, most of the MnO2 show a wide range of crystallite sizes that span from nanometer all the way to micrometer sizes. The data presented suggest that the thermoelectric properties of β-MnO2 depend heavily on particle size distribution and particle morphology. The details of particle agglomeration is presented as a model in which smaller MnO2 crystallites  bond together through weak van der Waals forces to form larger conglomerates with sizes ranging from mesoscopic to microscopic scale. Future research consists in a systematic study of thermoelectric properties as a function of high-energy ball-milling process parameters such as ground time, and angular speed.We have investigated thermoelectric properties of β-MnO


Using certain results of exponential Diophantine equations, we prove that (i) if p ≡ ±3(mod  8), then the equation 8x + py = z2 has no positive integer solutions ; (ii) if p ≡ 7(mod  8), then the equation has only the solutions  = (q + 2), 2, 2q + 1), where q is an odd prime with q ≡ 1(mod  3); (iii) if p ≡ 1(mod  8) and p ≠ 17, then the equation has at most two positive integer solutions .Let  Z, N be the sets of all integers and positive integers, respectively. Let p be a fixed odd prime. Recently, the solutions  of the equationp = 19, then ((Sroysang ) if p = p = 13, then ((Sroysang ) if p = p = 17, then  = , , and .(Rabago ) if p = Let In this paper, using certain results of exponential Diophantine equations, we prove a general result as follows.p ≡ ±3(mod  8), then . If p ≡ 7(mod  8), then (q is an odd prime with q ≡ 1(mod  3).If 8), then  has no s8), then  has onlyp ≡ 1(mod  8) and p ≠ 17, then .If Obviously, the above theorem contains the results of , 2. Finap ≠ 17, then .If n − 1 is a prime, where n is a positive integer, then n must be a prime.If 2See Theorem 1.10.1 of .p is an odd prime with p ≡ 1(mod  4), then the equationu, v).If See Section 8.1 of .X, Y, m, n) = .The equationSee Theorem 8.4 of .D be a fixed odd positive integer. If the equationu, v), then the equationX, n), except the following cases:D = 2r2 − 3 · 2r+1 + 1,  = , , , and , where r is a positive integer with r ≥ 3;D = ((1/3)(2r+12 − 17))2 − 32,  = ((1/3)(2r+12 − 17), 5), (1/3), and ((1/3)(17 · 2r+12 − 1), 4r + 7), where r is a positive integer with r ≥ 3;D = 2r12 + 2r22 − 2r1+r2+1 − 2r1+1 − 2r2+1 + 1,  = , , and , where r1, r2 are positive integers with r2 > r1 + 1 > 2.Let See .D is an odd prime and D belongs to the exceptional case (i) of D = 17.If D is an odd prime with D = 2r2 − 3 · 2r+1 + 1. Then we haver, since r ≥ 3, then r ≥ 4, and by (r − 1) − 2r/2+1 ≡ −1(mod  8), a contradiction.We now assume that  we have2r−12−2r+ and by /2 of D = 17.If Using the same method as in the proof of D belongs to the exceptional case (ii), then  with 3∣n.If i), then  has at mr, there exists at most one number of 5, 2r + 3, and 4r + 7 which is a multiple of 3. Thus, by Notice that, for any positive integer X, Y, m, n) = .The equationSee .x, y, z) is a solution of  = 1.We now assume that  = 2, then from .If 2∣hen from  we get, a, a15), ap satisfies (p ≡ 7(mod  8). Otherwise, since 2∤y, we see from (p ≡ py ≡ z2 − 8x ≡ 1(mod  8). It implies that if p ≡ ±3(mod  8), then . If p ≡ 7(mod  8), then . By the above analysis, we have 2∤y. If y > 1, then y ≥ 3 and  =  with 3∣m. But, by X, n) =  with 3∣n. Since p ≡ 1(mod  8), by u, v). Therefore, by Lemmas x, y, z). Thus, the theorem is proved.Here and below, we consider the remaining cases that  ≥ 3 and  has the  But, by y=1.Subs


BE-algebras.We study several degrees in defining a fuzzy positive implicative filter, which is a generalization of a fuzzy filter in  BE-algebra. Ahn and So  byμ is a fuzzy implicative filter of X with degree  and a fuzzy filter of X with degree , but it is neither a fuzzy filter of X nor a fuzzy positive implicative filter of X with degree  sinceConsider a self-distributive X : = {1, a, b, c} be a BE-algebra . But it is neither a fuzzy filter of X nor a fuzzy positive implicative filter of X with degree  sinceLet algebra  with thX with degree  sinceAlso, it is not a fuzzy implicative filter of μ is a fuzzy positive implicative filter of a BE-algebra X with degree , then μ is a fuzzy filter of X with degree .If z : = y in (e4), we havex, y ∈ X. Thus, μ is a fuzzy filter of X with degree .By putting The converse of λ, κ) is a fuzzy filter if and only if  = .Note that a fuzzy filter with degree . Then, the following holds:Let μ is a fuzzy positive implicative filter of a BE-algebra X with degree  and let x, y ∈ X. Using (e4) and (e1), we haveAssume that μ be a fuzzy filter of a BE-algebra X with degree  satisfyingμ is a positive implicative filter of X with degree .Let x, y, z ∈ X. Using (e2), we haveLet μ is a positive implicative filter of a BE-algebra X with degree .Thus, μ be a fuzzy filter of X. Then, μ is a fuzzy positive implicative filter of X, if and only ifLet It follows from Propositions BE-algebra X with degree  satisfies the following assertions:x, y ∈ X)(μ(x∗y) ≥ λκμ(y));(x ≤ y⇒μ(y) ≥ λκμ(x)).. If λ = κ, thenx, y ∈ X)(μ(x∗y) ≥ λ2μ(y)),(x ≤ y⇒μ(y) ≥ λ2μ(x)).. Then,Let x, y ∈ X. Let μ be a fuzzy positive implicative filter of X with degree . By μ is a fuzzy filter of X with degree . Since x ≤ (y∗x)∗x, using y∗x)∗x)∗y ≤ x∗y. Hence,Let X be a BE-algebra. X is said to be commutative if the following identity holds:(C)x∗y)∗y = (y∗x)∗x; that is, x∨y = y∨x, where x∨y = (y∗x)∗x, for all x, y ∈ X. is a fuzzy implicative filter of X with degree .Let μ be a fuzzy positive implicative filter of X with degree . By μ is a fuzzy filter of X with degree . Using (BE4) and x∗(y∗z))∗((x∗y)∗(x∗(x∗z))) = 1, for any x, y, z ∈ X. Hence, by μ(x∗(x∗z)) ≥ min{κμ(x∗(y∗z)), λκ2μ(x∗y)}. On the other hand, using (BE4) and (C), we obtainLet FPI(X) the set of all positive implicative filters of a BE-algebra X. Note that a fuzzy subset μ of a BE-algebra X is a fuzzy positive implicative filter of X, if and only ifμ of a BE-algebra X, there exist λ, κ ∈  and t ∈  such thatμ is a fuzzy positive implicative filter of X with degree ,U ∉ FPI(X)∪{∅}. Denote by BE-algebra X = {1, a, b, c, d} which is given in μ : X →  byt ∈  = {1, b} is not a positive implicative filter of X, since b∗((a∗d)∗a) = 1, b ∈ {1, b}, and a ∉ {1, b}. But μ is a fuzzy positive implicative filter of X with degree .Consider a μ be a fuzzy subset of a BE-algebra X. For any t ∈  with t ≤ max{λ, κ}, if U is an enlarged positive implicative filter of X related to U, then μ is a fuzzy positive implicative filter of X with degree .Let μ(1) < t ≤ λμ(x), for some x ∈ X and t ∈  ≥ t/λ ≥ t/max{λ, κ} and so x ∈ U; that is, U ≠ ∅. Since U is an enlarged filter of X related to U, we have 1 ∈ U; that is, μ(1) ≥ t. This is a contradiction, and thus μ(1) ≥ λμ(x), for all x ∈ X.Assume that a, b, c ∈ X such that μ(b) < κmin{μ(a∗((b∗c)∗b)), μ(a)}. If we take t : = κmin{μ(a∗((b∗c)∗b)), μ(a)}, then t ∈ ∗b) ∈ U⊆U and a ∈ U⊆U. It follows from b ∈ U so that μ(b) ≥ t, which is impossible. Therefore,x, y, z ∈ X. Thus, μ is a fuzzy positive implicative filter of X with degree .Now suppose that there exist μ be a fuzzy subset of a BE-algebra X. For any t ∈  with t ≤ k/n, if U is an enlarged positive implicative filter of X related to Ut), then μ is a fuzzy positive implicative filter of X with degree .Let t ∈  be such that U  (≠∅) is not necessarily a positive implicative filter of a BE-algebra X. If μ is a fuzzy positive implicative filter of X with degree , then U is an enlarged positive implicative filter of X related to U.Let tmin{λ, κ} ≤ t, we have U⊆U. Since U ≠ ∅, there exists x ∈ U and so μ(x) ≥ t. By (e1), we obtain μ(1) ≥ λμ(x) ≥ λt ≥ tmin{λ, κ}. Therefore, 1 ∈ U.Since x, y, z ∈ X be such that x∗((y∗z)∗y) ∈ U and x ∈ U. Then μ(x∗((y∗z)∗y)) ≥ t and μ(x) ≥ t. It follows from (e4) thaty ∈ U. Thus, U is an enlarged positive implicative filter of X related to U.Let


Some numerical examples are given to verify our theoretical results.We study the boundedness and persistence, existence, and uniqueness of positive equilibrium, local and global behavior of positive equilibrium point, and rate of convergence of positive solutions of the following system of rational difference equations:  Systems of nonlinear difference equations of higher order are of paramount importance in applications. Such equations also appear naturally as discrete analogues and as numerical solutions of systems differential and delay differential equations which model diverse phenomena in biology, ecology, physiology, physics, engineering, and economics. For applications and basic theory of rational difference equations, we refer to –3. In 44–10, appαi, βi, ai, and bi for i ∈ {1, 2} and initial conditions x0, x−1, y0, and y−1 are positive real numbers.Gibbons et al.  investigMore precisely, we investigate the boundedness character, persistence, existence, and uniqueness of positive steady state, local asymptotic stability, and global behavior of unique positive equilibrium point and rate of convergence of positive solutions of system  which coThe following theorem shows the boundedness and persistence of every positive solution of system .β1 < a1 and β2 < a2; then every positive solution {} of system } of system } be a positive solution of system }n=−1∞ of system  ∈ I × J for i ∈ {−1,0}. Along with system . An equilibrium point of ε > 0 there exists δ > 0 such that, for every initial condition , i ∈ {−1,0} if n > 0, where ||·|| is usual Euclidian norm in R2.An equilibrium point (ii)An equilibrium point (iii)η > 0 such thatAn equilibrium point (iv)n → ∞.An equilibrium point (v)An equilibrium point Let e system .(i)An eqF = , where f and g are continuously differentiable functions at FJ is Jacobian matrix of system , n = 0,1,…, is a system of difference equations such that F. If all eigenvalues of the Jacobian matrix JF about λ| < 1, then Assume that The following theorem shows the existence and uniqueness of positive equilibrium point of system .β1 < a1 and β2 < a2; then there exists unique positive equilibrium point of system  ∈  × ; then it follows from (f(x) = α2/(a2 − β2 + b2x) and x ∈ . Then, we obtain thatF(x) = 0 has at least one positive solution in .Consider the following system of equations:x=α1+β1xF(x) = 0 has a unique positive solution in . The proof is therefore completed.Furthermore, assume that condition  is satisb1b2U1U2 + β1β2 + β1(a2 + b2L1) + β2(a1 + b1L2) < (a1 + b1L2)(a2 + b2L1).The unique positive equilibrium point f system  is localλ) = λ4 and b1b2U1U2 + β1β2 + β1(a2 + b2L1) + β2(a1 + b1L2) < (a1 + b1L2)(a2 + b2L1) and |λ| = 1; then one hasλ) and Φ(λ) − Ψ(λ) have the same number of zeroes in an open unit disk |λ| < 1. Hence, all the roots of  =λ4iven by4P(λ) =λArguing as in , we havef : ×→ and g : ×→ are continuous functions and a, b, c, and d are positive real numbers with a < b, c < d. Moreover, suppose that f : ×→ and g : ×→ such that following conditions are satisfied: f is increasing in x and decreasing in y, and g is decreasing in x and increasing in y;m1,  M1,  m2, and M2 be real numbers such that m1 = f, M1 = f, m2 = g, and M2 = g; then m1 = M1 and m2 = M2.let Assume that xn+1 = f, yn+1 = g has a unique positive equilibrium point Then, the system of difference equations a1 − β1 + b1L2)2(a2 − β2 + b1L1)2 > α1α2b1b2.The unique positive equilibrium point of system  is globaf = (α1 + β1x)/(a1 + b1y) and g = (α2 + β2y)/(a2 + b2x). Then, it is easy to see that f is increasing in x and decreasing in y. Moreover, g is decreasing in x and increasing in y. Let  be a solution of the systemK = (a1 − β1 + b1L2)2(a2 − β2 + b1L1)2. Finally, from } be an arbitrary solution of the system } is a positive solution of the system } of } of  the folla1 < β1; then it follows from xn ≤ α1/(a1 − β1) = U1, n = 1,2,…. Furthermore, from system  > 1; then we obtain that {wn} is divergent. Hence, by comparison, we have yn → ∞ as n → ∞.(i) Suppose that uation:wn+1=c2+a2 < β2; then from yn ≤ α2/(a2 − β2) = U2,  n = 1,2,…. Moreover, from system  > 1; then one has {zn} that is divergent. Hence, by comparison we have xn → ∞ as n → ∞.(ii) Assume that uation:zn+1=c1+a1 > β1 and a2 > β2; then system (β1 − a1) + b2α1 − b1α2 and ν = (a2 − β2)(β1 − a1) − b2α1 + b1α2. From ,p,q2,)2,q1+q2=ν+. From ((p1+p2)2, from ((q1+q2)2α1 = 0.5, β1 = 12, a1 = 13, b1 = 0.2, α2 = 0.1, β2 = 17, a2 = 17.5, and b2 = 0.3. Then, system (x0 = 0.46, x−1 = 0.5, y−1 = 0.11, and y0 = 0.14.Let xn is shown in yn is shown in In this case, the unique positive equilibrium point of the system  is givenα1 = 10,  β1 = 1.5,  a1 = 1.6,  b1 = 0.003,  α2 = 12,  β2 = 23,  a2 = 23.1, and  b2 = 0.02.  Then, system (x−1 = 82,  x0 = 89,  y−1 = 5.9, and  y0 = 6.Let  xn is shown in yn is shown in In this case, the unique positive equilibrium point of the system  is givenα1 = 3.2,  β1 = 8,  a1 = 8.1,  b1 = 5.5,  α2 = 4.2,  β2 = 16,  a2 = 16.1,  and b2 = 8.5. Then, system (x−1 = 3.9,  x0 = 3.5,  y−1 = 0.1, and  y0 = 0.12.Let xn is shown in yn is shown in In this case, the unique positive equilibrium point of the system  is givenIn literature, several articles are related to qualitative behavior of competitive system of planar rational difference equations . It is v


SS,Σ* and £S,Σ of biunivalent functions denoted by subordination. The results presented in this paper improve the recent work of Crisan (2013).We obtain the Fekete-Szegö inequalities for the classes  A denote the class of analytic functions in the unit disk S we will denote the class of all functions in A which are univalent in U.Let U under every function f from S contains a disk of radius 1/4. Thus every such univalent function has an inverse f−1 which satisfies The Koebe one-quarter theorem  states tU if both U. Let Σ denote the class of biunivalent functions defined in the unit disk U.A function f and g are analytic in U, then f is said to be subordinate to g, written as w(z), analytic in U, with If the functions a2|. Subsequently, Brannan and Clunie [f ∈ Σ. Netanyahu [a2| = 4/3 if f(z) ∈ Σ. Brannan and Taha [an| for n ∈ ℕ∖{1,2}; ℕ = {1,2, 3,…} is presumably still an open problem.Lewin  studied d Clunie  conjectuetanyahu  showed tand Taha  introducand Taha –12. The ϕ be an analytic and univalent function with positive real part in U with ϕ(0) = 1, ϕ′(0) > 0, and ϕ maps the unit disk U onto a region starlike with respect to 1 and symmetric with respect to the real axis. Taylor's series expansion of such function is of the form B1 > 0.Let S*(ϕ) and C(ϕ) we denote the following classes of functions: By S*(ϕ) and C(ϕ) are the extensions of classical sets of starlike and convex functions and in such a form were defined and studied by Ma and Minda [S*(ϕ) and C(ϕ) as well as Fekete-Szegö inequalities for S*(ϕ) and C(ϕ). Their proof of Fekete-Szegö inequalities requires the univalence of ϕ. Ali et al. [ϕ. So in this paper, we assume that ϕ has series expansion ϕ(z) = 1 + B1z + B2z2 + ⋯, B1, B2 are real, and B1 > 0. A function f is bistarlike of Ma-Minda type or biconvex of Ma-Minda type if both f and f−1 are, respectively, Ma-Minda starlike or convex. These classes are denoted, respectively, by SΣ*(ϕ) and CΣ(ϕ)  (see ).SS* of starlike functions with respect to symmetric points in U, consisting of functions f ∈ A that satisfy the condition Re(zf′(z)/(f(z) − f(−z))) > 0, z ∈ U. Similarly, in [CS of convex functions with respect to symmetric points in U, consisting of functions f ∈ A that satisfy the condition Re((zf′(z))′/(f′(z) + f′(−z))) > 0, z ∈ U. In the style of Ma and Minda, Ravichandran (see [SS*(ϕ) and CS(ϕ).In , Sakagucarly, in , Wang etran (see ) definedf ∈ A is in the class SS*(ϕ) if CS(ϕ) if A function SS,Σ* and £S,Σ. These inequalities will result in bounds of the third coefficient which are, in some cases, better than these obtained in [In this paper, motivated by the earlier work of Zaprawa , we obtaained in .In order to derive our main results, we require the following lemma.p(z) = 1 + p1z + p2z2 + p3z3 + ⋯ is an analytic function in U with positive real part, then If f ∈ Σ is said to be in the class SS,Σ* if the following subordination holds: g(w) = f−1(w).A function α = 0, the class SS,Σ* reduces to the class SS*(ϕ) introduced by Ravichandran [We note that, for chandran .f given by  and μ ∈ ℝ. Then Let given by  be in thf ∈ SS,Σ* and g be the analytic extension of f−1 to U. Then there exist two functions u and v, analytic in U with u(0) = v(0) = 0, |u(z)| < 1, |v(w)| < 1, z, w ∈ U, such that p and q by Rep(z) > 0 and Req(w) > 0. From , a, af ∈ S0. From  then If f ∈ SS,Σ* then If If ualities  and 31)(32)ϕz=1+If ualities  and 31)(34)ϕz=1+a3| obtained by Crisan [Corollaries y Crisan .f ∈ Σ is said to be £S,Σ if the following subordination holds: g(w) = f−1(w).A function α = 0, the class £S,Σ reduces to the class CS(ϕ) introduced by Ravichandran [We note that, for chandran .f given by  and μ ∈ ℝ. Then Let given by  be in thf ∈ £S,Σ and g be the analytic extension of f−1 to U. Then there exist two functions u and v, analytic in U with u(0) = v(0) = 0, |u(z)| < 1, |v(w)| < 1, z, w ∈ U, such that μ = 1 or μ = 0 we get the following.Let uch that 2zf′z′f′zw. From  then If f ∈ £S,Σ then If If ualities  and 51)(52)ϕz=1+If ualities  and 51)(54)ϕz=1+a3| obtained by Crisan [Corollaries y Crisan .


Using three critical points theorems, we prove the existence of at least three solutions for a quasilinear biharmonic equation. Ω ⊂ ℝN  (4 > N ≥ 1) is a nonempty bounded open set with a sufficient smooth boundary ∂Ω, λ > 0, f : Ω × ℝ → ℝ is an L1-Carathéodory function, and g : ℝ → ℝ is a Lipschitz continuous function with Lipschitz constant L > 0; that is, t1, t2 ∈ ℝ and g(0) = 0.In this paper, we show the existence of at least three weak solutions for the Navier boundary value problem N = 1, in  × ℝ → ℝ is a L2-Carathéodory function. Later, some authors generalized this type of equation  is a nonempty bounded open set with a sufficient smooth boundary ∂Ω, p > max{1, N/2}, λ > 0, and f : Ω × ℝ → ℝ is an L1-Carathéodory function. After that some authors used different critical point theorems to get one nontrivial, at least three, and infinitely many solutions , 19N = 1u and Su  also useu and Su  to estabons see , 16, 18), 18N = 1ons ∩W01,2(Ω).The goal of the present paper is to establish some new criteria for  to have or which  admits aA special case of our main results is the following theorem.g : ℝ → ℝ be a Lipschitz continuous function with the Lipschitz constant L > 0 and g(0) = 0 such that L < 1/K2m(Ω), where K is defined by (f : ℝ → ℝ be a continuous function and put F(t) = ∫0tf(ξ)dξ for each t ∈ ℝ. Assume that F(d) > 0 for some d > 0 and F(ξ) ≥ 0 in  and λ* > 0 such that for each λ > λ* the problem Let fined by . Let f :First we here recall for the reader's convenience our main tools to prove the results. The first result has been obtained in  and the X be a separable and reflexive real Banach space, Φ : X → ℝ a nonnegative continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X*, and Ψ : X → ℝ a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Assume that there exists x0 ∈ X such that Φ(x0) = Ψ(x0) = 0 and that r > 0, x1 ∈ X, such that r < Φ(x1) and −1 in the weak topology. Then, for each X and, moreover, for each h > 1, there exists an open interval σ such that, for each λ ∈ Λ2, (X whose norms are less than σ.Let equation Φ′(u)−λΨ′X be a reflexive real Banach space; let Φ : X → ℝ be a sequentially weakly lower semicontinuous, coercive, and continuously Gâteaux differentiable whose Gâteaux derivative admits a continuous inverse on X*, and let Ψ : X → ℝ be a sequentially weakly upper semicontinuous and continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Assume that there exist r ∈ ℝ and u1 ∈ X with 0 < r < Φ(u1), such that (A1)u∈Φ−1Ψ(u) < r(Ψ(u1)/Φ(u1));sup(A2)λ ∈ Λr∶ = ]Φ(u1)/Ψ(u1), r/supu∈Φ−1Ψ(u);(A5)Ωsupt∈Fdx < (1 − K2Lm(Ω))c2Fdx/((1 − K2Lm(Ω))c2+(1 + K2Lm(Ω))(θd)2)).∫Then, for each λ in X and, moreover, for each h > 1, there exist an open interval σ such that, for each λ ∈ Λ2′, problem (X whose norms are less than σ.Assume that there exist a positive function w given by r∶ = ((1 − K2Lm(Ω))/2)(c/K)2. It is easy to verify that w ∈ W2,2(Ω)∩W01,2(Ω), and, in particular, one has w(x) ≤ d for each x ∈ Ω, the condition (A4) ensures that 1′⊆Λ1 and Λ2⊆Λ2′, we have the desired conclusion directly from We claim that all the assumptions of tly from  we see tc and d with c < θd such that the assumption (A4) in (B4)Ωsupt∈Fdx/(1 − K2Lm(Ω))c2 < ∫BFdx/(1 + K2Lm(Ω))(θd)2;∫(B5)t|→+∞|F/t2 < ∫Ωsupt∈Fdx/c2.limsupThen, for each Assume that there exist two positive constants w as given in (r∶ = ((1 − K2Lm(Ω))/2)(c/K)2 and bearing in mind that All the assumptions of given in  and r∶ =λ > λ*∶ = m(Ω)(1 + K2Lm(Ω))(θd)2/m)F(d) for some d > 0. Since cm}m∈ℕ⊆  ]0, +∞[ such that limm→+∞cm = 0 and F(ξcm) = supξ|≤cm|F(ξ). Hence, there is Fix u ∈ X, we let the functionals Φ, Ψ : X → ℝ be defined by u ∈ X is the functional Φ′(u) ∈ X*, given by v ∈ X. Furthermore, the differential Φ′ : X → X* is a Lipschitzian operator. Indeed, for any u, v ∈ X, there holds g is Lipschitz continuous and the embedding X↪L2(Ω) is compact, the claim is true. In particular, we derive that Φ is continuously differentiable. The inequality , it turns out that Φ′ is a strongly monotone operator. So, by applying Minty-Browder theorem [X → X* admits a Lipschitz continuous inverse. On the other hand, the fact that X is compactly embedded into C0(Ω) implies that the functional Ψ is well defined, continuously Gâteaux differentiable, and with compact derivative, whose Gateaux derivative at the point u ∈ X is given by v ∈ X. Note that the weak solutions of  = 0, we have from  > r. From the definition of Φ and by using )dx ≥ 0  = 0 so x0 = 0 and x1 = w from λ ∈ Λ1, the problem  for any fixed [, using , taking [, using  into accby using  we have (A2) and , we haveX → ℝ as given in the proof of w) > r. From the definition of Φ we have η, ϑ ∈ ℝ with x ∈ Ω and all t ∈ ℝ. Fix u ∈ X. Then x ∈ Ω. Now, to prove the coercivity of the functional Φ − λΨ, first we assume that η > 0. So, for any fixed η ≤ 0, clearly we obtain limu||→+∞||(Φ(u) − λΨ(u)) = +∞. Both cases lead to the coercivity of functional Φ − λΨ.To apply  X. Then F)X.So, assumptions A1) and (A2) in  and (A2)


A stochastic SIS-type epidemic model with general nonlinear incidence and disease-induced mortality is investigated. It is proved that the dynamical behaviors of the model are determined by a certain threshold value  Our real life is full of randomness and stochasticity. Therefore, using stochastic dynamical models can gain more real benefits. Particularly, stochastic dynamical models can provide us with some additional degrees of realism in comparison to their deterministic counterparts. There are different possible approaches which result in different effects on the epidemic dynamical systems to include random perturbations in the models. In particular, the following three approaches are seen most often. The first one is parameters perturbation; the second one is the environmental noise that is proportional to the variables; and the last one is the robustness of the positive equilibrium of the deterministic models.In recent years, various types of stochastic epidemic dynamical models are established and investigated widely. The main research subjects include the existence and uniqueness of positive solution with any positive initial value in probability mean, the persistence and extinction of the disease in probability mean, the asymptotical behaviors around the disease-free equilibrium and the endemic equilibrium of the deterministic models, and the existence of the stationary distribution as well as ergodicity. Many important results have been established in many literatures, for example, –16 and tThe stochastic epidemic models with general nonlinear incidence are not investigated. Up to now, only some special cases of nonlinear incidence, for example, saturated incidence rate, are considered. But, we all know that the nonlinear incidence rate in the theory of mathematical epidemiology is very important.For the stochastic epidemic models with the standard incidence, up to now, we do not find any interesting researches.The conditions obtained on the existence of unique stationary distribution are very rigorous. Whether there is a unique stationary distribution only when the model is permanent in the mean with probability one is still an open problem.However, from articles –16 and tS and I denote the susceptible and infectious individuals, Λ denotes the recruitment rate of the susceptible, μ is the natural death rate of S and I, α is the disease-related death rate, the transmission of the infection is governed by a nonlinear incidence rate βf, where β denotes the transmission coefficient between compartments S and I, f is a continuously differentiable function of S and I, and γ denotes the per capita disease contact rate.Motivated by the above work, in this paper, we consider the following deterministic SIS epidemic model with nonlinear incidence rate and disease-induced mortality:dStdt=Λ−ββ of disease in deterministic model (B(t) is a one-dimensional standard Brownian motion defined on some probability space. Thus, model , for example, f = SI/N (standard incidence) and f = h(S)g(I), many idiographic criteria on the extinction, permanence, and stationary distribution are established. Lastly, some affirmative answers for the open problems which are proposed in this paper also are given by the numerical examples .The organization of this paper is as follows. In R+2 = { : x1 > 0, x2 > 0}, R+0 = [0, ∞), and R+ = . Throughout this paper, we assume that model  with a filtration {Ft}t≥0 satisfying the usual conditions; that is, {Ft}t≥0 is right continuous and F0 contains all P-null sets.Denote S and I denote the susceptible and infected fractions of the population, respectively, and N = S + I is the total size of the population among whom the disease is spreading; the parameters Λ, μ, β, and γ are given as in model (βSg(I); B(t) denotes one-dimensional standard Brownian motion defined on the above probability space; and σ represents the intensity of the Brownian motion B(t). Throughout this paper, we always assume the following. is two-order continuously differentiable for any S ≥ 0, I ≥ 0, and S + I > 0. For each fixed I ≥ 0, f is increasing for S > 0 and for each fixed S ≥ 0, f/I is decreasing for I > 0. f = f = 0 for any S > 0 and I > 0, and ∂f/∂I > 0, where S0 = Λ/μ.In model , S and Iin model ; the traf = h(S)g(I), then assumption (H) becomes in the following form: (h(S) and g(I) are continuously differentiable for S ≥ 0 and I ≥ 0, h(S) is increasing for S ≥ 0, and g(I)/I is decreasing for I > 0.Particularly, when H), by simple calculating, we can obtain that for any S > 0 and I > 0, 0 ≤ f ≤ /∂I)I, and for any S2 > S1 > 0, ∂f/∂I ≥ ∂f/∂I.From  = SI/N (standard incidence), where N = S + I, f = SI/(1 + ω1I + ω2S) (Beddington-DeAngelis incidence) with constants ω1 ≥ 0 and ω2 ≥ 0, and f = SI/(1 + ωI2) with constant ω ≥ 0, then (H) is satisfied.When f.Now, we give the following result for function p > q > 0, let D = { : S > 0, I > 0, q ≤ S + I ≤ p}. Then,For any constants H), we have H), we haveH and G are continuous for  ∈ D. Therefore, conclusion , I(0)) ∈ R+2, model (S(t), I(t)) defined on t ∈ R+0 satisfying (S(t), I(t)) ∈ R+2 for all t ≥ 0 with probability one. Furthermore, when α > 0 then S0 ≤ lim inft→∞N(t) ≤ lim supt→∞N(t) ≤ S0, and when α = 0 then limt→∞N(t) = S0, where N(t) = S(t) + I(t) and S0 = Λ/(μ + α).For any initial value (R0 is the basic reproduction number of deterministic model (Define the constantsic model . On the (a)σ2 ≤ β//∂I) and (b)σ2 > β2/2(μ + γ + α).Then disease I in model (S(0), I(0)) ∈ R+2, solution (S(t), I(t)) of model (t→∞I(t) = 0 a.s.Assume that one of the following conditions holds:S(t), I(t)) ∈ R+2 a.s. for all t ≥ 0 and lim supt→∞(S(t) + I(t)) ≤ S0. For any η > 0 there is T0 > 0 such that S(t) + I(t) < S0 + η for all t ≥ T0. Hence, for any t ≥ T0,ε > 0, σ = 0, g(u) is monotone increasing for u ∈ R+, and when σ > 0, g(u) is monotone increasing for u ∈ [0, (β + ε)/σ2) and monotone decreasing for u ∈ [(β + ε)/σ2, ∞).By ula see , 18), we, weS(t),σ = 0, from /∂I ≤ β/σ2, we can choose η > 0 such that η ≤ ε and ∂f/∂I < (β + ε)/σ2. From  holds, then when  0, from , we dire2. From  we also 0, from , we obtσ > 0, g(u) has maximum value (β + ε)2/2σ2 − (μ + γ + α) at u = (β + ε)/σ2, and for any t ≥ 0, we have ε, we obtaint→∞I(t) = 0 a.s. This completes the proof.If condition (b) holds, then, since a.s.From  and 19)σ > 0, g and (b) of ic model .R0 ≤ 1, then, for any σ > 0, σ > 0, the conclusions of R0 ≤ 2. From σ1 ≤ σ2, we easily prove that when R0 > 2, we have σ1 > σ2 and σ > σ1 the conclusions of When (a)R0 ≤ 1 and σ > 0;(b)R0 ≤ 2 and 1 < (c)R0 > 2 and σ > σ1.Then disease I in model  = SI/N (standard incidence). Assume that one of the following conditions holds:(a)σ2 ≤ β and (b)σ2 > β2/2(μ + γ + α).Then disease I in model  = h(S)g(I). Assume that (H∗) holds and one of the following conditions holds:(a)σ2 ≤ β/h(S0)g′(0) and (b)σ2 > β2/2(μ + γ + α).Then disease I in model , I(0)) ∈ R+2, solution (S(t), I(t)) of model (If in model  is permaε > 0 such thatS(0), I(0)) ∈ R+2, solution (S(t), I(t)) of model (t→∞(1/t)∫0tI(s)ds ≤ S0 and for above ε > 0 there is T0 > 0 such that S0 − ε ≤ S(t) + I(t) ≤ S0 + ε a.s. for all t ≥ T0. Denote the set Dε = { : S0 − ε ≤ S + I ≤ S0 + ε}. Since dN(t) = (Λ − μN(t) − αI(t))dt, we obtain for any t > T0t ≥ T0,f/I for S > 0 and I > 0 is continuously differentiable, limI→0/I) = ∂f/∂I exists for any S > 0, and set Dε is convex and connected, by the Lagrange mean value theorem when t ≥ T0 we have ξ(t), ϕ(t)) ∈ Dε. Let constantsM1, M2 < ∞. For any t ≥ T0, we havet→∞(H(t)/t) = 0 a.s. Therefore, from Lemma  5.2 given in [t→∞(1/t)∫0tI(s)ds ≥ θ/θ0 a.s. This completes the proof.From such that3β∂fS0,0∂Ntμ.From , for any, I(0)) ∈ R+2, solution (S(t), I(t)) of model , I(0)) ∈ R+2, solution (S(t), I(t)) of model (t→∞(1/t)∫0tS(s)ds ≤ S0 and for any small enough constant ε > 0 there is T0 > 0 such that S0 − ε ≤ S(t) + I(t) ≤ S0 + ε for all t ≥ T0. Hence, by t ≥ T0 we have f(S(t), I(t)) ≤ MSS(t), where MS = maxDε{f/S} < ∞. Integrating the first equation of model  = SI/N (standard incidence). If Let  R~0=β-1/σ2/ = h(S)g(I). Assume that (H∗) holds and Let We further have the result on the weak permanence of model  in probaξ > 0 such that, for any initial value (S(0), I(0)) ∈ R+2, solution (S(t), I(t)) of model , I(0)) ∈ R+2, solution (S(t), I(t)) of model (ξ > 0 satisfies the equationLet in model . If R~0>t→∞(S(t) + I(t)) = S0. Without loss of generality, we assume that S(t) + I(t) ≡ S0 for all t ≥ 0. From  = f/I. Then, for any t ≥ 0,g(u) = βu − (σ2/2)u2 − (μ + γ). With condition g(0) = −(μ + γ) < 0 andg(u) = 0 has a positive root η in /∂I) which isu(I) is monotone decreasing for I ∈ , u(S0) = 0, andξ ∈  such that u(ξ) = f/ξ = η and g(u(ξ)) = g(η) = 0.From  0. From , for anyσ > 0 and β/σ2 < ∂f/∂I, since function g(u) has maximum value g(β/σ2) at u = β/σ2 and g(β/σ2) > g/∂I), there is a unique η ∈ /∂I) and g(η) = 0 we have η < β/σ2. Hence, When g(u(I)) > 0 is strictly increasing on g(u(I)) > 0 is strictly decreasing on g(u(I)) < 0 is strictly decreasing on I ∈ .From the above discussion we obtain that σ2 ≤ β//∂I), similarly to the above discussion, we can obtain that g(u(I)) > 0 is strictly decreasing on I ∈  and g(u(I)) < 0 is strictly decreasing on I ∈ .When ε0 ∈  such that P(Ω1) > ε0, where Ω1 = {lim supt→∞I(t) < ξ}. Hence, for every ω ∈ Ω1, there is a constant T1 = T1(ω) ≥ T0 such thatg(u(I(t))) ≥ g(u(ξ − ε0)) > 0 for all t ≥ T1. From (t ≥ T1t→∞(logI(t)/t) ≤ g(u(ξ − ε0)) > 0, which implies I(t) → ∞ as t → ∞. This leads to a contradiction with  such that P(Ω2) > ε1, where Ω2 = {lim inft→∞I(t) > ξ}. Hence, for every ω ∈ Ω2, there is T2 = T2(ω) ≥ T0 such thatg(u(I(t))) ≤ g(u(ξ + ε1)) < 0 for all t ≥ T2. Together with (t ≥ T2t→∞(logI(t)/t) ≤ g(u(ξ + ε1)) < 0, which implies I(t) → 0 as t → ∞. This leads to a contradiction with , I(t)) of model (S(0), I(0)) ∈ R+2 oscillates about a positive number ξ. Therefore, an interesting open problem is whether there is a more less positive m than number ξ such that any solution (S(t), I(t)) of model (S(0), I(0)) ∈ R+2, lim inft→∞I(t) ≥ m a.s. In of model  with iniξ will arise from the change when the noise intensity σ changes. Therefore, it is very interesting and important to discuss how number ξ changes along with the change of σ. We have the following result.From α = 0 in model /∂I)/(μ + γ). Then one has the following.(a)ξ as the function of σ is defined for (b)ξ is monotone decreasing for (c)σ→0ξ = I∗, where  is the endemic equilibrium of deterministic model (limic model .(d)R0 ≤ 2, then R0 > 2, then ξ2 satisfies If 1 ≤ Assume that in model  and R~0>ξ as the function of η is defined for η ∈ /∂I). Fromη ∈ /∂I) when ξ as a function of σ is defined for Sinceη with respect to σ, we havedη/dσ > 0. From the definition of ξ, we easily see that ξ is monotone decreasing for η. From (H), we obtain that dξ/dη exists and is continuous for η. Since (∂/∂ξ)/ξ) < 0, we have dξ/dη < 0. Therefore, dξ/dσ = (dξ/dη)(dη/dσ) < 0. It follows that ξ is monotone decreasing as σ increases. Thus, both limσ→0ξ and σ→0ξ = ξ1 and σ→0/ξ) = limσ→0η = (μ + γ)/β. This shows that f/ξ1 = (μ + γ)/β. Let  be the endemic equilibrium of deterministic model /I∗ = (μ + γ)/β. Hence, ξ1 = I∗. This shows that limσ→0ξ = I∗.Computing the derivative of  η. From  and (H), we have5dηdσ−2βσ3R0 ≤ 2, then from . Therefore, 0 < ξ < I∗ when σ > 0. If 1 ≤ R0 ≤ 2, then when ξ has a minimum value 0 and if R0 > 2 then when ξ has a minimum value ξ2 > 0 by Conclusion (d).Conclusion (b) of in model , number α = 0 then R0 > 2 then when σ > σ1, where σ1 is given in (t→∞I(t) = 0 a.s. for any solution (S(t), I(t)) of model (S(0), I(0)) ∈ R+2, which implies that ξ = 0. Therefore, when R0 > 2, we can propose an interesting open problem: whether there is a critical value σ ∈  we have the fact that ξ is monotonically decreasing and ξ > 0 and when σ > σ∗ we have ξ = 0.It is clear that when in model  α = 0 th=σ¯ from . On the given in , we haveR0 > 2, then from  is the endemic equilibrium of deterministic model  = SI, we easily validate that Theorems When α > 0 in model , I(t)) of model (S(0), I(0)) ∈ R+2, one has limt→∞I(t) ≥ m, a.s. In For the case in model , an inteof model  with iniFrom Theorems >1 model  is perma>1 model  also hasIf en model  is positS, I) : S ≥ 0, I ≥ 0, S0 ≤ S + I ≤ S0}. Let (S(t), I(t)) be any solution of model (S(0), I(0)) ∈ Γ a.s. for all t ≥ 0. Let a > 0 be a large enough constant, and letS, I) ∈ Γ∖D, then either 0 < S < 1/a or 0 < I < 1/a. The diffusion matrix for model  ≥ σ2/(aS0 − 1))2.Here, the method given in the proof of Theorem  5.1 in  is improor model  is(63)ASv < 1 is a constant. Computing LΨ1, by ξ, ϕ) ∈ Γ andM1; M2 < ∞. We hence haveLΨ2, by LΨ3, we haveM0 = maxΓ{f/S} < ∞. From the above calculations, we obtain that for any  ∈ Γ∖Dv > 0 is small enough, it follows thata > 0 is large enoughξ such thatChoose a Lyapunov function as follows:given in , we knowComparing Since en model  is positf, we have the following idiographic results on the stationary distribution as the consequences of Particularly, for some special cases of nonlinear incidence f = SI/N (standard incidence). If Let f = h(S)g(I). Assume that (H∗) holds and Let σ of stochastic perturbation and basic reproduction number R0 of deterministic model (Combining ic model .R0 ≤ 1. Then for any σ > 0 the disease in model ((a) Let in model  is extinR0 ≤ 2. Then for any (b) Let 1 < R0 > 2. Then for any σ > σ1, where σ1 is given in ((c) Let given in , the disf = SI/(1 + ωI), where ω > 0 is a constant. The corresponding discretization system of model  are the Gaussian random variables which follow standard normal distribution N.In this section we analyze the stochastic behavior of model  by meansof model  is givenβ = 0.60, μ = 11, γ = 13, σ = 0.075, and α = 2.In model  we choosR0 = 4.195 > 2, β/S0 − σ2 = −0.0023 < 0, and σ2 − β2/2(μ + γ) = −0.0019 < 0 which is the case of By computing we have β = 0.9, μ = 30, γ = 12, and σ = 0.09.In model , choose By computing we have β = 0.5, μ = 30, γ = 20, σ = 0.02, and α = 2.In model  choose ΛR0 = 1.2500, and ξ = 0.1037. The numerical simulations are found in I(t) of model g(I). A series of criteria in the probability mean on the extinction of the disease, the persistence and permanence in the mean of the disease, and the existence of the stationary distribution are established. Furthermore, the numerical examples are carried out to illustrate the proposed open problems in this paper.In this paper we investigated a class of stochastic SIS epidemic models with nonlinear incidence rate, which include the standard incidence, Beddington-DeAngelis incidence, and nonlinear incidence It is easily seen that the research given in  for the The researches given in this paper show that stochastic model  has moreσ > 0 of the stochastic perturbation, then R0 > 1 we still can have In addition, we easily see that when intensity ic model  the dise


There are 2 errors in the first paragraph of the Results subsection titled “HIV tat and gp 120 and Cell-free HIV Virions Induce Activation of MAPK Signaling in Polarized Oral Epithelial Cells.” In the first sentence, αϖ should be αv. In the third sentence, ϖ should be αv.In the Discussion section, there are 2 errors in the fourth sentence of the third paragraph. In the fourth sentence, “ωηιχη” should be “which” and “λεαδing” should be “leading.”


T be a singular integral operator with its kernel satisfying |K(x − y) − ∑k=1ℓBk(x)ϕk(y)| ≤ C | y|γ/|x − y|n+γ, |x | > 2 | y | > 0, where Bk and ϕk   are appropriate functions and γ and C are positive constants. For bj ∈ BMO(ℝn), the multilinear commutator T and Lp-boundedness and the weighted weak type LlogL estimate for the multilinear commutator Let  In the classical Calderón-Zygmund theory, the Hörmander's condition∫|x|>2|y|Lp-boundedness of certain singular integral operators, Grubb and Moore |2 ∈ RH∞(Rnl), where yi ∈ Rn and i, k = 1,…, l;there exist functions K4) , we defined the convolution operator associated to the kernel K byFor (1)p < ∞ and ω ∈ Ap. Then there exists a constant C > 0 such thatLet 1 < (2)ω ∈ A1. Then there exists a constant C > 0 such that for all λ > 0Let Let l = 1, B1(x) = K(x), and ϕ1(y) ≡ 1, then condition (K(x) = sinx/x satisfies conditions (K1)–(K4), but does not satisfy the Hörmander's condition   . The generalized commutator, the so-called the multilinear commutator, For In 2002, Pérez and Trujillo-González  studied In 1993, Alvarez et al.  establisK be a linear operator and 1 < p < ∞. Suppose that for all ω ∈ Ap(Rn), the linear operator K satisfies the following weighted estimateC depends only on n, p, and the Ap constant of ω. Then for b ∈ BMO(Rn) and any weight function ν ∈ Ap, the commutator  is bounded from Lp(ν) to Lp(ν) with bound depending on n, p, and the Ap constant of ω.Let T defined by (K1)–(K4).The goal of this paper is to study the weighted norm inequalities for multilinear commutator of the convolution operator fined by  with itsm-times, we can easily get the following weighted Lp inequalities for the multilinear commutator By T be the singular integral operator defined by (K1)–(K4). If 1 < p < ∞, ω ∈ Ap, and bj ∈ BMO(Rn)  , then there exists a positive constant C such thatLet fined by  with itsH1(Rn) into L1(Rn) when b ∈ BMO(Rn); see ).).3]). Byϕ1,…, ϕl} is a finite family of bounded functions on Rn such that |det[ϕk(yi)]|2 ∈ RH∞(Rnl). Then, for any cube Q centered at the origin and any f ∈ L1(Q), there exists the projection PQf of f onto span{ϕ1,…, ϕl} ⊂ L1(Q) and satisfiesC depends only on n, l, and the RH∞ constant of |det[ϕk(yi)]|2.Suppose {∞)→[0, ∞) is said to be a Young function, if Φ is continuous, convex, and increasing with Φ(0) = 0 and limt→∞Φ(t) = ∞. We use A function Φ : [0, f over a cube Q ⊂ Rn is defined byThe Φ-average of a locally integrable function ies  = t(1 + log+t)α  (α > 0) with its complementary Young function The Young function that we are going to use is Φα = 1, we simply write Φ(t) = t(1 + log+t) and f||L(logL), Q = ||f||QΦ,  and When The following generalized Hölder's inequality holds (see (2.5) in ):(24)1|ω ∈ A∞ and a cube Q ⊂ Rn, denoteWe also need the following notations . For 0 ≤ t < ∞, define φj−1(t) = inf{s : φj(s) > t}  . If for all 0 ≤ t < ∞t1, t2,…, tk < ∞, there existk(t) = t(1 + log+t)k   and Ψ(t) = et − 1, we have Φk−1(t) ≈ t/(logt)k and Ψ−1(t) ≈ logt A(t) = Φj(C−1t) since A−1(t) = CΦj−1(t), then it follows from s, t1, t2,…, tm−j < ∞, we haveNoting that f and a cube Q, denoteFor a locally integrable function ω ∈ A∞ and b ∈ BMO(Rn). Then, for any cube Q ⊂ Rn,C0 and C are positive constants independent of b and Q, and ||b||∗ is the BMO-norm of b.Let p < ∞, ωp ∈ A1, bj ∈ BMO(Rn)  , and Q be a cube. Then for any positive integer m and k = 0,1,…,Let 1 ≤ m = 1, we write b = b1 and ω ∈ A1 and b ∈ BMO(Rn), there exists constant C > 0 such that, for all λ > 0,When λ > 0, we consider the Calderón-Zygmund decomposition of f at height λ and get a sequence of nonoverlapping cubes {Qi}, where Qi = Q is a cube centered at yi with radius ri, such thatFor any fixed f|Qi the restriction of f to Qi. Let gi(x) be the projection of f|Qi onto Yi = span{ϕ1(·−yi),  ϕ2(·−yi),…, ϕl(·−yi)}. We decompose f into two parts, f = g + h, whereh(x) = f(x) − g(x) = ∑ihi(x) with hi(x) = f(x) − gi(x) for x ∈ Qi.Denote by hi is supported on Qi and it follows from ),g(x)|≤λ, for a.e. x ∈ Rn∖∪iQi. On the other hand, for any x ∈ ∪iQi there exists an i so that x ∈ Qi, and noting that gi(x) is the projection of f|Qi onto Yi, then it follows from Furthermore, we havedeed, by  and 38)40)|g(x)|g(x)(40)ω ∈ A1, then by , a, aω ∈ A then by , we haveQi, by Qi, by  first. Applying (K4), and Let us consider Applying , conditiows from  that(50J(2). By the weak type  estimate of T , .of T see , and LemNote that  impliesThen by  we haveJ(1) and J(2), we haveCombining the estimates for ong with  and 46)J(1) andong with , which iIn this section, we will use an induction argument to prove m and j  (1 ≤ j ≤ m), we denote by Cjm the family of all finite subsets σ = {σ(1), σ(2),…, σ(j)} of {1,2,…, m} of j different elements. For any σ ∈ Cjm, we write σ′ = {1,2,…, m}∖σ.As in , given pbj ∈ BMO(Rn) and σ = {σ(1), σ(2),…, σ(j)} ∈ Cjm  (1 ≤ j ≤ m), we denote by Q is a cube in Rn and For m = 1 in j < m; namely, for all 1 ≤ j < m and any σ ∈ Cjm, we haveWe have proved that λ > 0, we consider the Calderón-Zygmund decomposition of f at height λ as in Qi}, Qi*, g, h, hi, and Ω as there.For any fixed For the same reason as in , we haveSimilar to , we haveReasoning as the proof of Lemma 3.1 in  , and J(1) in Applying , conditiJ2, by the weak type  estimate for T (see J(2) in For or T see , 27), 43) we haJ3 by applying the induction hypothesis.Now, let us consider hi(x) = (f(x) − gi(x))χQi(x)  , we can split J3 into two partsNoting that σ ∈ Cjm, we denote by σ′ = {σ′(1), σ′(2),…σ′(m − j)}, so thatFor Cs,0 and Cs such that for s = 1 ⋯ m − jFrom γs = ||∗)−1  ; then it follows from the induction hypothesis and (Set esis and  that(72By  and 43)43), we hm(ab) ≤ CΦm(a)Φm(b) for a, b > 0, we haveNoting that ΦJ3(2). By Jensen's inequality,Finally, we consider J3(1), we haveBy the induction hypothesis, , and 7575, similApplying , we haveong with  and 74)(77)∫Qi{1J1, J2, and J3, we obtainBy , 65), a, a65), aThe proof of the general case of


R0 has been worked out. The local and global asymptoticalstability analysis of the equilibria are performed, respectively. Furthermore, if we take the treatedrate τ as the bifurcation parameter, periodic orbits will bifurcate from endemic equilibrium whenτ passes through a critical value. Finally, some numerical simulations are given to support ouranalytic results.An SEIV epidemic model for childhood disease with partial permanent immunity isstudied. The basic reproduction number  S(t), I(t), and V(t), respectively, represent the number of susceptible individuals at time t, infective individuals at time t, and vaccinated individuals at time t. At the earliest, most researches on these types of models assume that the disease incubation is negligible, so that each susceptible individual, once infected, instantaneously turns into infectious and later recovers obtaining a permanent immunity. Soon afterwards, the models become more general. Researchers assume that a susceptible individual first goes through a latent period after infection before becoming infectious .It is primarily important for health administrators to protect children from disease that can be prevented by vaccination. Although preventive vaccines have reduced the incidence of infectious diseases among children, childhood disease is an important public health problem. We often use mathematical models to realize the transmission dynamics of childhood diseases and to estimate control programs –4. RecenA represents the number of additional populations of childhood; ω represents the rate at which vaccine wanes; μ represents the natural death rate; β represents the rate at which susceptible individuals become infected by those who are infectious; p represents the fraction of recruited individuals who are vaccinated; τ represents the rate at which infected individuals are treated; and σ represents the rate at which exposed individuals become infectious.In , the autβSI is called incidence rate which plays an important role in the transmission dynamics. In addition, incidence rate can determine the tendency of epidemics. At the earliest, in the classical epidemic disease model, scholars made much focus on the bilinear incidence /μ(μ + ω), pA/(μ + ω), 0,0); that is, E = I = 0. And {∣S > 0, E ≥ 0, I ≥ 0, V > 0} is a positively invariant set of system (t→∞(S + E + I + V) ≤ A/μ. Therefore, the set Ω. Set x = T; then system (FV−1 develops a meaningful definition of R0 and is the expected number of new infections for system (ρ(FV−1) = σβA[μ(1 − p) + ω]/μ(μ + ω)(μ + σ)(μ + τ + ϵ) is the spectral radius of matrix FV−1. Thus by /μ(μ + ω), pA/(μ + ω), 0,0) when R0 = 1. In other words, we will discuss under what conditions system  is the linearization matrix of system  right eigenvector V and a left eigenvector W with respect to the zero eigenvalue.Matrix Suppose the following.fk as the kth component of f, and a and b totally decide the local dynamic of system . And the disease-free equilibrium changes its local stability at the point  /μ(μ + ω), pA/(μ + ω), 0,0) and notice that the condition R0 = 1 can be seen as β = β* = μ(μ + τ + ϵ)(μ + σ)(μ + ω)/σA[(1 − p)μ + ω] in terms of the parameter β.We will show that system  may exhiλ1 = −μ, λ2 = −μ − ω, λ3 = −2μ − σ − τ, and λ4 = 0.Calculate the eigenvalues of the following matrix:J has a simple eigenvalue of 0; and all others have negative real parts. Thus, we can make use of the center manifold theory. The disease-free equilibrium P0 is a nonhyperbolic equilibrium when β = β* . This completes the verification with respect to (A1) of The matrix V = T as a right eigenvector associated with the zero eigenvalue λ4 = 0. It is calculated by W =  satisfying W · V = 1 is obtained by W turns out to be Now we set lated by −μω0τ−β∗Apanding  and (∂2fk/∂xi∂β) which are nonzero, the following are deduced: So, we evaluate m system , and the0,β∗).By  and 39)a and b ab is always positive. Therefore the sign of the coefficient a determines the local dynamics around the disease-free equilibrium for β = β* by Obviously α0 = (M − στ(μ + ω))/σA[μ(1 − p) + ω]. The coefficient a is positive if and only if α > α0. Under this circumstance, the direction of the bifurcation for system τ = τ*, τ = τ*, we have τ, the roots are all in the following general forms: λj(τ) = α1(τ) + iα2(τ) into (For =c3.From  and 46)τ = τ*, c3.From ((λ2+c2) < 1. If R1* > 1 the disease-free equilibrium P0 is globally asymptotically stable in Ω; if R1* < 1 the disease-free equilibrium P0 is globally asymptotically stable in Ω when R0 < R0*.Let R0(1 + αA/μ) < 1, R0 < 1. If R1* > 1, P0 is the only equilibrium of  (Ω. From the first equation of  (dS/dt ≤ (1 − p)A + (A/μ)(ω + τ)−(μ + ω)S. A solution of the equation dy/dt = (1 − p)A + (A/μ)(ω + τ)−(μ + ω)y is a upper solution of S(t). Due to that y → A[μ(1 − p)+(ω + τ)]/μ(μ + ω) when t → ∞, we can easily get that, for a small enough ε > 0 which is sufficiently small, there exists a t0 such that S(t) ≤ y(t) ≤ A[μ(1 − p)+(ω + τ)]/μ(μ + ω) + ε as t > t0.When rium of   which istion of  , we obtaL = σE + (μ + σ)I, thus Considering the Lyapunov function R0(1 + αA/μ) < 1, we can choose ε small enough such that (1 + αA/μ)R0 − 1 + (1 + αA/μ)εσβ/(μ + σ)(μ + τ + ϵ) < 0. Thus,L′ = 0 if and only if I = 0. The singleton P0 is the maximum positive invariant set in { ∈ Ω, L′ = 0}. The global stability of P0 for every solution follows from LaSalle's Invariance Principle.For R1* < 1, system  so that the disease can be eliminated. Thus, when R1* < 1 the disease-free equilibrium P0 is globally asymptotically stable in Ω when R0 < R0*.If , system  has two , system  shows a P* for R0 > 1. Due to S + V + E + I → A/μ when t → ∞, we can determine V(t) by S(t), E(t), and I(t). So system  → g(x) when t → ∞ locally uniformly for x ∈ Rn, then system ,dydt=g(y)P is a locally asymptotically stable equilibrium of (x is the ω-limit set of a forward bounded solution x(t) of (x includes a point y0 such that the solution of (y(0) = y0 converges to P when t → ∞, then ω = P; that is, x(t) → P when t → ∞.Set brium of  and x is x(t) of . If x inution of  with y(0P of the limit system (x(t) of system (x(t) → P when t → ∞.If solutions of system  are bount system  is globaf system  satisfieP* is globally asymptotically stable for R0 > 1 by the geometrical approach :Next we will look for conditions which satisfied (H3). Consider the Jacobian matrix  and get p(x) = P = diag{E/I, E/I, E/I}. Then PfP−1 = diag{E′/E − I′/I, E′/E − I′/I, E′/E − I′/I}. Thus, the matrix B = PfP−1 + PJ[2]P−1 can be written in block form as Let R3. We choose a standard in R3 as μ be the LozinskiSet nique in , the folc ≤ S, I ≤ A/μ, where c is the constant of uniform persistence; it is obvious that d > 0.From the second and third equations of system , we can S(t), E(t), I(t)) ∈ Ω, we obtain For each  = 0.4800 and R0 = 0.00031987 < 1 which satisfies R0(1 + αA/μ) = 0.0011 < 1 which satisfies P0 and it is globally asymptotically stable  = 63.8400, R0 = 48.1293, R1* = 267.6519, and R0* = 0.0032. Therefore R0* < 1 < R0 < R1*, due to P0 and an endemic equilibrium P*. And we can calculate d = 45.53 > 0 which guarantees τ = 40 > τ*, then the endemic equilibrium P* becomes unstable and a periodic orbits bifurcates from P*, which is demonstrated by We take parameters R0 and found that when R0 = 1 and R1* < 1 system  < 1, respectively. Also we have studied the local and global asymptotic stability of the endemic equilibrium. Moreover, taking the disease-caused death rate τ as bifurcation parameter, we discussed the Hopf bifurcation of system  < 1, if R1* > 1 holds or R1* < 1, R0 < R0* hold; system  < 0, which means that vaccinating more susceptible populations decreases the likelihood of the occurrence of backward bifurcation [From the sense of epidemiology, when ; system  has one equality  or 78)  R0(1 + equality  has a un, system  has a unurcation .


On PDF page 5, first column, the GenBank accession number “1910160” should be updated to “KX174310.” The correct URL is http://www.ncbi.nlm.nih.gov/nuccore/KX174310.Volume 7, no 3, doi:mon, +700/− less than 5,062 bp but greater than 700 bp)” should be updated to “RD1mon, +309 bp/−513 bp).”On PDF page 11, second column, “RD1