𝒫0(X) and 𝒦0(X) of a tvs-cone metric space , where 𝒫0(X) and 𝒦0(X) are the space consisting of nonempty subsets of X and the space consisting of nonempty compact subsets of X, respectively. The purpose of this paper is to establish some relationships between the lower topology and the lower tvs-cone hemimetric topology  on 𝒫0(X) and 𝒦0(X), which makes it possible to generalize some results of superspaces from metric spaces to tvs-cone metric spaces.This paper investigates superspaces  K-metric and K-normed spaces were introduced in the mid-20th century . . K-metricentury  and 𝒦0(X) be the space consisting of nonempty subsets of X and the space consisting of nonempty compact subsets of X, respectively. And then the following two theorems are well known .X, d) be a metric space and let 𝒞 be a subset of 𝒫0(X). Then the following hold.𝒞 is open in the lower topology on 𝒫0(X), then 𝒞 is open in the lower hemimetric topology on 𝒫0(X).If 𝒞 is open in the upper hemimetric topology on 𝒫0(X), then 𝒞 is open in the upper topology on 𝒫0(X).If Let  be a metric space. Then the following hold.𝒦0(X).The lower topology and the lower hemimetric topology coincide on 𝒦0(X).The upper topology and the upper hemimetric topology coincide on 𝒦0(X).The Vietoris topology and the Hausdorff hemimetric topology coincide on Let  and 𝒦0(X) of a tvs-cone metric space . The purpose of this paper is to establish some relationships between the lower topology and the lower tvs-cone hemimetric topology  on 𝒫0(X) and 𝒦0(X), respectively. These results answer This paper investigates superspaces ℕ, ℝ+, and ℝ* denote the set of all natural numbers, the set of all positive real numbers, and the set of all nonnegative real numbers, respectively.Throughout this paper, E be a topological vector space with its zero vector θ. A subset P of E is called a tvs-cone in E if the following are satisfied.P is a closed in E with a nonempty interior P°.α, β ∈ P and a, b ∈ ℝ*⇒aα + bβ ∈ P.α, −α ∈ P  ⇒  α = θ.Let E be a topological vector space with a tvs-cone P. It is clear that θ ∈ P from θ ∉ P°. In fact, pick α ∈ E − {θ}. Then {(1/n)α} → θ and {−(1/n)α} → θ when n → ∞. If θ ∈ P°, then there is n ∈ ℕ such that {(1/n)α, −(1/n)α}⊆P°⊆P. By n)α = θ. This contradicts that α ≠ θ. So θ ∉ P°.Let E be a topological vector space with a tvs-cone P. Some partial orderings ≤, <, and ≪ on E with respect to P are defined as follows, respectively. Let α, β ∈ E.α ≤ β if β − α ∈ P.α < β if α ≤ β and α ≠ β.α ≪ β if β − α ∈ P°.Let E with respect to P. The meanings of these notations are clear and the following hold:α ≥ β⇔α − β ≥ θ⇔α − β ∈ P,α > β⇔α − β > θ⇔α − β ∈ P − {θ},α ≫ β⇔α − β ≫ θ⇔α − β ∈ P°,α ≫ β⇒α > β⇒α ≥ β.For the sake of convenience, we also use notations “≥”, “>,” and “≫” on P in a topological vector space E is called strongly minihedral if each subset of E bounded above has a supremum, equivalently, if each subset of E bounded below has an infimum.A tvs-cone P in a topological vector space E is strongly minihedral.In this paper, we always suppose that a tvs-cone E be a topological vector space with a tvs-cone P. Then the following hold.α ≫ θ, then rα ≫ θ for each r ∈ ℝ+.If α1 ≫ β1 and α2 ≥ β2, then α1 + α2 ≫ β1 + β2.If α ≫ θ and β ≫ θ, then there is γ ≫ θ such that γ ≪ α and γ ≪ β.If Let α ≫ θ; that is, α ∈ P°. Then there is an open neighborhood B of α in E such that B⊆P. If r ∈ ℝ+, then rB⊆P from rB = {rβ : β ∈ B}. Note that rα ∈ rB and rB is an open subset of E. So rα ∈ P°; that is rα ≫ θ.(1) Let α1 ≫ β1 and α2 ≥ β2. Then α1 − β1 ≫ θ and α2 − β2 ≥ θ; that is, α1 − β1 ∈ P° and α2 − β2 ∈ P. So there is an open neighborhood B of α1 − β1 in E such that B⊆P. Write (α2 − β2) + B = {(α2 − β2) + β : β ∈ B}. Note that (α2 − β2) + B is an open subset of E, and (α2 − β2)+(α1 − β1)∈(α2 − β2) + B⊆P from α2 − β2)+(α1 − β1) ∈ P°; that is, (α2 − β2)+(α1 − β1) ≫ θ; hence, (α1 + α2)−(β1 + β2) ≫ θ. It follows that α1 + α2 ≫ β1 + β2.(2) Let α ≫ θ and β ≫ θ; that is, α, β ∈ P°. Then there is n1, n2 ∈ ℕ such that α − ((α + β)/n) ∈ P° for all n ≥ n1 and β − ((α + β)/n) ∈ P° for all n ≥ n2. Put γ = (α + β)/n0, where n0 = max{n1, n2}. Then γ ≫ θ from the above (1) and (2). It is clear that α − γ ∈ P° and β − γ ∈ P°; that is, α − γ ≫ θ and β − γ ≫ θ. So γ ≪ α and γ ≪ β.(3) Let We give the definition of tvs-cone metric, which is very similar to the well-known definition of metric.X be a nonempty set and let E be a topological vector space with a tvs-cone P. A mapping d : X × X → E is called a tvs-cone metric on X, and  is called a tvs-cone metric space if the following are satisfied.d ≥ θ for all x, y ∈ X and d = θ if and only if x = y.d = d for all x, y ∈ X.d ≤ d + d for all x, y, z ∈ X.Let ∞ as a possible value of the mapping d in the following definition, where ∞ ∉ E and the following hold.∞ + α = ∞ + ∞ = ∞ for each α ∈ E.α ≪ ∞ for each α ∈ E.Note that hemimetric takes values in the extended nonnegative real numbers . We letX be a nonempty set and let E be a topological vector space with a tvs-cone P. A mapping d : X × X → E⋃{∞} is called a tvs-cone hemimetric on X, and  is called a tvs-cone hemimetric space if the following (1) and (2) are satisfied.d ≤ d + d for all x, y, z ∈ X.d = θ for all x ∈ X.Let X, d) be a tvs-cone hemimetric space. Put B = {y ∈ X : d ≪ ɛ} for x ∈ X and ɛ ≫ θ, and put ℬ = {B : x ∈ X  and  ɛ ≫ θ}. Then ℬ is a base for some topology on X.Let , B ∈ ℬ and z ∈ B⋂B. Since z ∈ B, d ≪ α. Put γ1 = α − d; then γ1 ≫ θ. We claim that B⊆B. In fact, if u ∈ B, then d ≪ γ1; hence, d ≤ d + d ≪ d + γ1 = α, and so u ∈ B. Using the same way, we can obtain that there is γ2 ≫ θ such that B⊆B. By γ ≫ θ such that γ ≪ γ1 and γ ≪ γ2. Let v ∈ B; then d ≪ γ ≪ γ1 and d ≪ γ ≪ γ2, so v ∈ B⊆B and v ∈ B⊆B, and hence v ∈ B⋂B. This has proved that B⊆B⋂B. Note that z ∈ B ∈ ℬ. Consequently, ℬ is a base for some topology on X. In fact, put τ = {U⊆X : there  is  ℬ′⊆ℬ such that U = ⋃ℬ′}; then τ is a topology on X and ℬ is a base for τ.It is clear that X, d) be a tvs-cone metric space and let τ denote the topology on X induced by the tvs-cone metric d described in G of X,  and IG denote the subfamilies {P ∈ 𝒫0(X) : P⊆G} and {P ∈ 𝒫0(X) : P⋂G ≠ ∅} of 𝒫0(X), respectively.𝒯ℒ is called the lower topology on 𝒫0(X), where 𝒯ℒ is generated by the subbase ℒ = {IG : G ∈ τ}.𝒯𝒰 is called the upper topology on 𝒫0(X), where 𝒯𝒰 is generated by the base 𝒰 = { : G ∈ τ}.𝒯𝒱 is called the Vietoris topology on 𝒫0(X), where 𝒯𝒱 is generated by ℒ and 𝒰 together.Let  be a tvs-cone metric space. For P, Q ∈ 𝒫0(X), put δl = inf{ɛ ≫ θ : P⊆S}, δu = inf{ɛ ≫ θ : Q⊆S}, and δ = inf{ɛ ≫ θ : ɛ ≫ δl and ɛ ≫ δu}, where inf∅ = ∞. Then δl, δu, and δ are tvs-cone hemimetrics on 𝒫0(X). Let 𝒯l, 𝒯u, and 𝒯 denote the topologies on 𝒫0(X) induced by δl, δu, and δ described in 𝒯l is called the lower tvs-cone hemimetric topology.The topology 𝒯u is called the upper tvs-cone hemimetric topology.The topology 𝒯 is called the Hausdorff tvs-cone hemimetric topology.The topology Let , and the relative topologies on 𝒦0(X) will still be denoted by 𝒯ℒ, and so forth. Also, if δu  is restricted to 𝒦0(X), then it does not take ∞.It is often preferable to restrict the six topologies In this section, we need to use the following notation.Notation. Let  be a tvs-cone metric space, x ∈ X, P ∈ 𝒫0(X), K ∈ 𝒦0(X), and ɛ ≫ θ. Consider the following:B = {y ∈ X : d ≪ ɛ},S = ⋃{B : x ∈ P},Bl = {P′ ∈ 𝒫0(X) : δl ≪ ɛ},Bu = {P′ ∈ 𝒫0(X) : δu ≪ ɛ},Cl = {K′ ∈ 𝒦0(X) : δl ≪ ɛ},Cu = {K′ ∈ 𝒦0(X) : δu ≪ ɛ},C = {K′ ∈ 𝒦0(X) : δ ≪ ɛ}.X, d) be a tvs-cone metric space and let 𝒞 be a subset of 𝒫0(X). Then the following hold.𝒞 is open in the lower topology 𝒯ℒ on 𝒫0(X), then 𝒞 is open in the lower tvs-cone hemimetric topology 𝒯l on 𝒫0(X).If 𝒞 is open in the upper tvs-cone hemimetric topology 𝒯u on 𝒫0(X), then 𝒞 is open in the upper topology 𝒯𝒰 on 𝒫0(X).If Let (𝒞 be open in the lower topology 𝒯ℒ on 𝒫0(X). Without loss of generality, we can assume that 𝒞 is an element in the subbase ℒ for the lower topology 𝒯ℒ; that is, 𝒞 = IG for some G ∈ 𝒯. Let P ∈ 𝒞, then P⋂G ≠ ∅. Pick x ∈ P⋂G; then there is ɛ ≫ θ such that B⊆G since G is open in X. Let Q ∈ Bl; then δl ≪ ɛ; that is, P⊆S. Since x ∈ P, x ∈ S, hence x ∈ B for some y ∈ Q. Thus, y ∈ B⊆G, which means that Q⋂G ≠ ∅. It follows that Q ∈ 𝒞. This proves that Bl⊆𝒞. So P is an interior point of 𝒞 in the lower tvs-cone hemimetric topology 𝒯l and the proof is completed.(1) Let 𝒞 be open in the upper tvs-cone hemimetric topology 𝒯u on 𝒫0(X). Let P ∈ 𝒞; then there is ɛ ≫ θ such that Bu⊆𝒞. Note that S is open in X. So  is open in the upper topology on 𝒫0(X). Clearly, P ∈ . On the other hand, if Q ∈ , that is, Q⊆S, then δu ≪ ɛ, and hence Q ∈ Bu⊆𝒞. This has proved that ⊆𝒞. Consequently, 𝒞 is an open neighborhood of P for the upper topology 𝒯𝒰 on 𝒫0(X) and the proof is completed.(2) Let X, d) is a metric space). Moreover, there is no simple relationship between the Vietoris topology 𝒯𝒱 and the Hausdorff tvs-cone hemimetric topology 𝒯 on 𝒫0(X), which is similar to (1) or (2) in (1) The converses of both (1) and (2) in 𝒫0(X)” in 𝒦0(X). Furthermore, we have the better results for the topologies on superspaces 𝒦0(X) (see the following).(2) It is clear that “X, d) be a tvs-cone metric space. If K is a compact subset of X, then, for any ɛ ≫ θ, there is a finite subset F of K such that K⊆S.Let  : x ∈ K} is an open cover of K; there is a finite subset F of K such that {B : x ∈ F} covers K. It follows that K⊆S.Let X, d) be a tvs-cone metric space. If K⊆U with K compact in X and U open in X, then there is ɛ ≫ θ such that S⊆U.Let ⊆U. Put εx = (1/2)ηx; then εx ≫ θ from B : x ∈ K} is an open cover of K and K is compact, there is a finite subset F of K such that {B : x ∈ F} covers K. By ɛ ≫ θ such that ɛ ≪ εx for each x ∈ F. We claim that S⊆U. In fact, let u ∈ S; then there is y ∈ K such that u ∈ B, that is, d ≪ ɛ. Furthermore, there is z ∈ F such that y ∈ B; that is, d ≪ εz. By d ≤ d + d ≪ ɛ + εz ≪ 2εz = ηz. It follows that u ∈ B⊆U. This has proved that S⊆U.Let X, d) be a tvs-cone metric space. Then the following hold.𝒯ℒ and the lower tvs-cone hemimetric topology 𝒯l coincide on 𝒦0(X).The lower topology 𝒯𝒰 and the upper tvs-cone hemimetric topology 𝒯u coincide on 𝒦0(X).The upper topology 𝒯𝒱 and the Hausdorff tvs-cone hemimetric topology 𝒯 coincide on 𝒦0(X).The Vietoris topology Let (𝒞 is open in the lower hemimetric topology 𝒯l on 𝒦0(X). Let K ∈ 𝒞; then there is ɛ ≫ θ such that Cl⊆𝒞. Since K is compact, by F of K such that K⊆S. We write Gx = B for each x ∈ F and put 𝒲 = ⋂{IGx : x ∈ F}. Note that B ∈ 𝒯 for each x ∈ F. It is clear that K ∈ 𝒲 and 𝒲 is an element of the base for the lower topology 𝒯ℒ on 𝒦0(X). Let K′ ∈ 𝒲. For each x ∈ F, we claim that Gx⊆S. In fact, let y ∈ Gx; then d ≪ ɛ/2. Since K′ ∈ IGx, K′⋂Gx ≠ ∅. Pick z ∈ K′⋂Gx; then d ≪ ɛ/2; hence, d ≤ d + d ≪ ɛ/2 + ɛ/2 = ɛ; that is, y ∈ B⊆S. This proves that Gx⊆S. Furthermore, K⊆S = ⋃{Gx : x ∈ F}⊆S. Thus, δl ≪ ɛ; that is, K′ ∈ Cl⊆𝒞. This proves that 𝒲 ⊂ 𝒞. It follows that K is an interior point of 𝒞 for the lower topology 𝒯ℒ on 𝒦0(X). Consequently, 𝒞 is open in the lower topology 𝒯ℒ on 𝒦0(X). Combining (1) Assume that 𝒞 be open in the upper topology 𝒯𝒰 on 𝒦0(X). Without loss of generality, we can assume that 𝒞 is an element of the base for the upper topology 𝒯𝒰 on 𝒦0(X); that is, 𝒞 = ⋂𝒦0(X) for some G ∈ 𝒯. Let K ∈ 𝒞; then K⊆G and K is compact. By ɛ ≫ θ such that S⊆G. If K′ ∈ Cu; then δu ≪ ɛ; hence, K′⊆S⊆G. It follows that K′ ∈ ⋂𝒦0(X) = 𝒞. Consequently, Cu⊆𝒞. This has proved that 𝒞 is an open neighborhood of K for the upper tvs-cone hemimetric topology 𝒯u on 𝒦0(X). Combining (2) Let 𝒯𝒱⊆𝒯 and 𝒯⊆𝒯𝒱. Let 𝒞 be open in the Vietoris topology 𝒯𝒱 on 𝒦0(X). Without loss of generality, we only need to assume that 𝒞 is an element of the subbase ℒ for the lower topology 𝒯ℒ or an element of the base 𝒰 for the upper topology 𝒯𝒰. If 𝒞 ∈ ℒ, then 𝒞 ∈ 𝒯l by (1). So, for each K ∈ 𝒞, there is ɛ ≫ θ such that Cl⊆𝒞. It is clear that C⊆Cl. In fact, if K′ ∈ C, then δ ≪ ɛ. Note that δl ≤ δ ≪ ɛ. So K′ ∈ Cl. It follows that K ∈ C⊆Cl⊆𝒞. So K is an interior point of 𝒞 for the Hausdorff tvs-cone hemimetric topology 𝒯. Consequently, 𝒞 is open in the Hausdorff tvs-cone hemimetric topology 𝒯 on 𝒦0(X). By a similar way, if 𝒞 ∈ 𝒰, then 𝒞 is open in the Hausdorff tvs-cone hemimetric topology 𝒯 on 𝒦0(X). This has proved that 𝒯𝒱⊆𝒯. Conversely, let 𝒞 be open in the Hausdorff tvs-cone hemimetric topology 𝒯 on 𝒦0(X). Then, for each K ∈ 𝒞, there is ɛ ≫ θ such that C⊆𝒞. We claim that Cl⋂Cu⊆C. In fact, if K′ ∈ Cl⋂Cu, then δl ≪ ɛ/2 and δu ≪ ɛ/2. So δ ≤ δl + δu ≪ ɛ/2 + ɛ/2 = ɛ; hence, K′ ∈ C. This proves that Cl⋂Cu⊆C. By (1) and (2), Cl and Cu are open in 𝒯ℒ and 𝒯𝒰, respectively. It follows that Cl⋂Cu is open in 𝒯𝒱. So K is an interior point of 𝒞 for the Vietoris topology 𝒯𝒱 on 𝒦0(X). Consequently, 𝒞 is open in the Vietoris topology 𝒯𝒱 on 𝒦0(X). This has proved that 𝒯⊆𝒯𝒱.We need to prove that


In this paper, a Meir-Keeler contraction is introduced to propose a viscosity-projection approximation method for finding a common element of the set of solutions of a family of general equilibrium problems and the set of fixed points of asymptotically strict pseudocontractions in the intermediate sense. Strong convergence of the viscosity iterative sequences is obtained under some suitable conditions. Results presented in this paper extend and unify the previously known results announced by many other authors. H be a real Hilbert space with inner product 〈·, ·〉 and norm ||·||, respectively. Let C be a nonempty closed convex subset of H. Let A : C → H be a nonlinear mapping and F : C × C → ℝ be a bifunction, where ℝ denotes the set of real numbers. We consider the following generalized equilibrium problem: Find x ∈ C such thatEP to denote the set of solution of problem  to denote the set of solution of problem  to denote the set of solution of problem +〈Auch thatF+〈Auch thatF≥0,uch thatF+〈Auch that〈Ax,y−x〉≥uch thatF+〈Auch thatF+〈AT : C → C is said to be nonexpansive ifT is said to be uniformly L-Lipschitz continuous if there exists a constant L > 0 such thatT is said to be asymptotically nonexpansive if there exists a sequence kn ∈ N + i(n), i = i(n) = 1,2,…, N. They also obtain weak and strong convergence theorems based on the cyclic scheme above.In 2009, Qin et al.  introducRecently, Sahu et al.  considerC be a nonempty closed and convex subset of a real Hilbert space H and T : C → C be a uniformly continuous asymptotically λ-strictly pseudocontractive mapping in the intermediate sense with a sequence {kn} such that Fix(T) is nonempty and bounded. Let {αn} be a sequence in  such that 0 < δ ≤ αn ≤ 1 − λ for all n ∈ ℕ. Let {xn} ⊂ C be a sequence generated by the following (CQ) algorithm:θn = (kn − 1)ρn2 + en and ρn = sup{||xn − p||:p ∈ Fix(T)} < ∞. Then, {xn} converges strongly to PT)Fix((u), where PT)Fix( is metric projection from H onto Fix(T).Let θn = (kh(n) − 1)ρn2 + eh(n) → 0 as n → ∞ and ρn = sup{||xn − p||:p ∈ Ω} < ∞. Moreover, they obtained convergence theorems under some suitable conditions.In 2011, Hu and Cai  modifiedf is a contractive mapping. He proved that the viscosity iterative sequence {xn} convergence strongly to a fixed point of T, which is the unique solution of the variational inequality:On the other hand, Moudafi  introducakahashi  and Inchakahashi  modifiedIn 2012, Kimura and Nakajo  introducC be a nonempty closed convex subset of H, and let {Tn} be a sequence of mappings of C into itself with Ω = ⋂n=1∞Fix(Tn) ≠ Ø which satisfies the following condition: there exists {an} ⊂ ℝ with liminfn→∞an > −1 such that ||Tnx − z||2 ≤ ||x − z||2 − an||x − Tnx||2 for every n ∈ ℕ, x ∈ C, and z ∈ Ω. Let f be a Meir-Keeler contraction of C into itself, and let {xn} be a sequence generated byn ∈ ℕ. For every sequence {zn} ⊂ C and zn → z ∈ C and Tnzn → z imply that z ∈ Ω. Then, {xn} converges strongly to q ∈ Ω, which satisfies q = PΩf(q).Let θn = (kh(n) − 1)ρn2 + eh(n) → 0 as n → ∞ and ρn = sup{||xn − p||:p ∈ Ω} < ∞.In this paper, inspired and motivated by research going on in this area, we introduce a new viscosity-projection method for a family of general equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense, which is defined in the following way:Our purpose is not only to extend the viscosity-projection method with a Meir-Keeler contraction to the case of a family of general equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense, but also to obtain a strong convergence theorem by using the proposed schemes under some appropriate conditions. Results presented in this paper extend and unify the corresponding ones of –13, 16.C be a nonempty closed convex subset of a real Hilbert space H with inner product 〈·, ·〉 and norm ||·||, respectively. We use notation ⇀ for weak convergence and → for strong convergence of a sequence. For every point x ∈ H, there exists a unique nearest point in C, denoted by PCx, such thatPC is called the metric projection of H onto C defined by PC(x) = argminy∈C||x − y||. It is well known that PC is nonexpansive mapping, and u = PCx is equivalent to  the folA : C → H is said to be monotone ifA is said to be r-strongly monotone if there exists a constant r > 0 such thatA is said to be α-inverse strongly monotone if there exists a constant α > 0 such thatA is an α-inverse strongly monotone mapping from C into H, then A is 1/α-Lipschitz continuous.Recall that a mapping F : C × C → ℝ satisfies the following conditions:  (A1)F = 0  for all x ∈ C;  (A2)F is monotone, that is, F + F ≤ 0  for all x, y ∈ C;  (A3)x, y, z ∈ C, limt→0F(tz + (1 − t)x, y) ≤ F;  for each  (A4)x ∈ C, y ↦ F is convex and lower semi-continuous. for each To study the generalized equilibrium problem , we may F : C × C → ℝ be a bifunction satisfying (A1)–(A4). Then, for any r > 0 and x ∈ H, there exists z ∈ C such thatFrx = {z ∈ C : F + (1/r)〈y − z, z − x〉≥0, ∀y ∈ C}, then the following hold: Fr is single-valued; Fr is firmly nonexpansive, that is, ||Frx−Fry||2 ≤ 〈Frx − Fry, x − y〉 for all x, y ∈ H; Fr) = EP(F); Fix(EP(F) is closed and convex. Let H, there hold the following identities: x+y||2 ≤ ||x||2 + 2〈y, (x + y)〉, for all x, y ∈ H; ||tx+(1−t)y||2 = t||x||2 + (1 − t)||y||2 − t(1 − t)||x−y||2, for all t ∈ , for all x, y ∈ H.||In a Hilbert space C be a nonempty closed convex subset of a real Hilbert space H. For any x, y, z ∈ H and given also a real number a ∈ ℝ, the setLet C be a nonempty closed convex subset of a real Hilbert space H and T : C → C be a uniformly L-Lipschitz continuous and asymptotically λ-strict pseudocontraction in the intermediate sense. Then Fix(T) is closed and convex. Let C be a nonempty closed convex subset of a real Hilbert space H and T : C → C be a uniformly L-Lipschitz continuous and asymptotically λ-strict pseudocontraction in the intermediate sense. Then I − T is demiclosed at zero, that is, if the sequence {xn} ⊂ C such that xn⇀x and xn − Txn → 0 as n → ∞, then x ∈ Fix(T). Let C be a nonempty closed convex subset of a real Hilbert space H and T : C → C be an asymptotically λ-strict pseudocontraction in the intermediate sense with γn = kn − 1. ThenLet f of a complete metric space  into itself is called a contraction with coefficient r ∈  if ||f(x) − f(y)||≤r||x − y||, for all x, y ∈ X. It is known that f has a unique fixed point  < ϵ + δ implies thatRecall also that a mapping e, e.g., ). On thee, e.g.,  defined roved in . A Meir-Keeler contraction defined on a complete metric space has a unique fixed point.f be a Meir-Keeler contraction on a convex subset C of a Banach space E. Then, for every ϵ > 0, there exists r ∈  such that ||x − y||≥ϵ implies that Let Cn} be a sequence of nonempty closed convex subsets of H. We define a subset s-LinCn of H as follows: x ∈ s-LinCn if and only if there exists {xn} ⊂ H such that xn → x and xn ∈ Cn for all n ∈ ℕ. Similarly, a subset w-LsnCn of H is defined by y ∈ w-LsnCn if and only if there exists a subsequence {Cni} of {Cn} and a sequence {yi} ⊂ H such that yi⇀y and yi ∈ Cni for all i ∈ ℕ. If C0 ⊂ H satisfiesCn} converges to C0 in the sense of Mosco  such that 0 < a ≤ αn ≤ 1, 0 < b ≤ βn ≤ 1 − λ and {rm,n}⊂ such that rm,n ∈ ⊂, for each m = 1,2,…, M and n ∈ ℕ. Then the sequence {xn} generated by (q = PΩf(q).Let rated by  convergeWe split the proof into six steps.Step  1. We prove that PΩf exists a unique fixed point. To do this, we first show that  is nonexpansive for each m = 1,2,…, M. Indeed,I − rm,nAm) is nonexpansive. By m=1MEP is closed and convex. We also know from i=1NFix(Ti) is closed, and convex. Hence, Ω = (⋂i=1NFix(Ti))∩) is a nonempty, closed and convex subset of C. Consequently, PΩ is well-defined. Since PΩ is nonexpansive, the composed mapping PΩf of C into itself is a Meir-Keeler contraction on C; see N + i(n), where i = i(n) = 1,2,…, N. By  = h(n − N) + 1 and i(n) = i(n − N), we observe thatL-Lipschitzian of Ti, we havei = 1,2,…, N, we havexn → q as n → ∞. It follows from (q ∈ ⋂i=1NFix(Ti).||.From  and 68)Step  6. …, N. By  and the |).From  and (68)ows thatlimn→∞||).From (limn→∞||).From . From nmxn = Frm,nΘnm−1xn, m = 1,2,…, M, we havezt = ty + (1 − t)q for all t ∈  and the monotonicity of Am, we obtaint → 0, from (A3) and , m = 1,2,…, M. Therefore, q ∈ ⋂m=1MEP. Consequently, we obtain that q ∈ Ω. This completes the proof. Next, we show that n).From , we havee obtain〈Amzt,zt−nd henceFm+We also obtain the following results by using the viscosity-hybrid projection methods, which extend and improve the hybrid method (CQ) proposed by Sahu et al.  and Hu aC be a nonempty closed convex subset of Hilbert space H. Let Fm : C × C → ℝ be a bifunction satisfying (A1)–(A4), and let Am : C → H be an αm-inverse strongly monotone mapping, for each m = 1,2,…, M. Let Ti : C → C be a uniformly Li-Lipschitz continuous and asymptotically λi-strict pseudocontractive mapping in the intermediate sense with the sequences {kn,i} and {en,i} for each i = 1,2,…, N. If f is a Meir-Keeler contraction of C into itself and Ω = (⋂i=1NFix(Ti))∩) is nonempty and bounded. Let {xn} be a sequence defined byθn = (kh(n) − 1)ρn2 + eh(n) → 0 as n → ∞ and ρn = sup{||xn − p||:p ∈ Ω} < ∞. Assume that {αn} and {βn} are sequences in  such that 0 < a ≤ αn ≤ 1, 0 < b ≤ βn ≤ 1 − λ and {rm,n}⊂ such that rm,n ∈ ⊂, for each m = 1,2,…, M. Then the sequence {xn} generated by (q = PΩf(q).Let fined byx1∈C, Q1=Cn and Qn are closed convex subsets of H and Ω ⊂ Cn for every n ∈ ℕ. We only prove that Ω ⊂ Qn for every n ∈ ℕ and that a sequence {xn} is well-defined. We have x1 ∈ C and Ω ⊂ Q1 = C. Assume that xk ∈ C and Ω ⊂ Qk for some k ∈ ℕ. Since Ω ⊂ Ck∩Qk, there exists a unique element xk+1 = PCk∩Qkf(xk), and henceΩ ⊂ Qk+1. Therefore, we prove that Ω ⊂ Qn.We have that Pn=1∞Qn⋂f is a Meir-Keeler contraction on C, there exists a unique element q = Pn=1∞Qn⋂f(q) ∈ ⋂n=1∞Qn by zn = PQnf(q) for each n ∈ ℕ. Since Ω ⊂ Qn+1 ⊂ Qn, it follows from zn → q = Pn=1∞Qn⋂f(q). We also have xn = PQnf(xn−1) by the definition of Qn. Therefore, as in the proof of xn → q, and the desired conclusion follows immediately from On the other hand, M = N = 1, we obtain the following corollary for a general equilibrium problem and asymptotically strict pseudocontraction in the intermediate sense as a special cases.If C be a nonempty closed convex subset of Hilbert space H. Let F : C × C → ℝ be a bifunction satisfying (A1)–(A4) and A : C → H be an α-inverse strongly monotone mapping. Let T : C → C be a uniformly L-Lipschitz continuous and asymptotically λ-strict pseudocontractive mapping in the intermediate sense with the sequences {kn} and {en}. If f is a Meir-Keeler contraction of C into itself and Ω = Fix(T)∩EP is nonempty and bounded. Let {xn} be a sequence defined byθn = (kn − 1)ρn2 + en → 0 as n → ∞ and ρn = sup{||xn − p|| : p ∈ Ω} < ∞. Assume that {αn} and {βn} are sequences in  such that 0 < a ≤ αn ≤ 1, 0 < b ≤ βn ≤ 1 − λ and {rn}⊂ such that rn ∈ ⊂. Then the sequence {xn} generated by (q = PΩf(q).Let fined byx1∈C, C1=M = N = 1 and F = 0, the general equilibrium problem ∩VI is nonempty and bounded. Let {xn} be a sequence defined byθn = (kn − 1)ρn2 + en → 0 as n → ∞ and ρn = sup{||xn − p|| : p ∈ Ω} < ∞. Assume that {αn} is a sequence in  such that 0 < b ≤ αn ≤ 1 − λ and {rn}⊂ such that rn ∈ ⊂. Then the sequence {xn} generated by (q = PΩf(q).Let fined byx1∈C, C1=F = 0, the general equilibrium problem (un = PC(xn − rnAxn). The desired conclusion follows immediately from αn = 1, βn = αn). This completes the proof. If  problem  reduces  problem , and(94


As an application we prove certain fixed point results in the setup of such spaces for different types of contractive mappings. Finally, some periodic point results in b-metric-like spaces are obtained. Two examples are presented in order to verify the effectiveness and applicability of our main results.We discuss topological structure of  The number s is called the coefficient of .A partial X is a mapping σ : X × X → ℝ+ such that for all x, y, z ∈ Xσ1) = 0 implies x = y,σ2) = σ,σ3) ≤ σ + σ.A metric-like on a nonempty set X, σ) is called a metric-like space.The pair  = x + y − xy is a metric-like on X.Let X = ℝ; then the mappings σi : X × X → ℝ+ , defined byX, where a ≥ 0 and b ∈ ℝ.Let X be a nonempty set and s ≥ 1 a given real number. A function σb : X × X → ℝ+ is b-metric-like if, for all x, y, z ∈ X, the following conditions are satisfied:σb1) = 0 implies x = y,σb2) = σb,σb3) ≤ s[σb + σb].Let b-metric-like space is a pair  such that X is a nonempty set and σb is b-metric-like on X. The number s is called the coefficient of .A b-metric-like space  if x, y ∈ X and σb = 0, then x = y, but the converse may not be true and σb may be positive for x ∈ X. It is clear that every partial b-metric space is a b-metric-like space with the same coefficient s and every b-metric space is also a b-metric-like space with the same coefficient s. However, the converses of these facts need not hold.In a X = ℝ+, p > 1 a constant, and σb : X × X → ℝ+ be defined byX, σb) is a b-metric-like space with coefficient s = 2p−1, but it is not a partial b-metric space. Indeed, for any 0 < y < x we have σb = (x+x)p > (x+y)p = σb, so (pb2) of Let b-metric-like spaces.The following propositions help us to construct some more examples of X, σ) be a metric-like space and σb = [σ]p, where p > 1 is a real number. Then σb is b-metric-like with coefficient s = 2p−1.Let (a+b)p ≤ 2p−1(ap + bp), where a, b ∈ ℝ+.The proof follows from the fact that  = (x+y−xy)p, where p > 1 is a real number, is b-metric-like on X with coefficient s = 2p−1.Let X = ℝ. Then the mappings σbi : X × X → ℝ+ , defined byb-metric-like on X, where p > 1, a ≥ 0, and b ∈ ℝ.Let X be a nonempty set such that d and pb are b-metric and partial b-metric, respectively, s > 1, and σ is a metric-like on X. Then the mappings σbi : X × X → ℝ+ , defined byx, y ∈ X are b-metric-like on X.Let X, pb) be a partial b-metric space and  a b-metric space with s > 1. Then conditions (σb1), (σb2), and (σb3) are obvious for the function σb5. For instance, if x, y, z ∈ X are arbitrary then, as pb is partial b-metric and d is b-metric on X, we haveσb3) is satisfied and so  is a b-metric-like space. Similarly, one can show that  and  are b-metric-like spaces.Let  = x + y −  xy + |x−y|p, where p > 1 is a real number, is b-metric-like on X with coefficient s = 2p−1.Let X = ℝ. Then the mappings σbi : X × X → ℝ+ , defined byb-metric-like on X with coefficient s = 2p−1, where p > 1, a ≥ 0, and b ∈ ℝ.Let b-metric-like σb on X generates a topology τσb on X whose base is the family of all open σb-balls {Bσb : x ∈ X, ɛ > 0}, where Bσb = {y ∈ X : |σb − σb| < ɛ} for all x ∈ X and ɛ > 0.Each b-metric-like space.Now, we define the concepts of Cauchy sequence and convergent sequence in a X, σb) be a b-metric-like space with coefficient s, and let {xn} be any sequence in X and x ∈ X. ThenLet ((i)xn} is said to be convergent to x with respect to τσb, if limn→∞σb = σb;the sequence {(ii)xn} is said to be a Cauchy sequence in  if limn,m→∞σb exists and is finite;the sequence {(iii)X, σb) is said to be a complete b-metric-like space if for every Cauchy sequence {xn} in X there exists x ∈ X such that.It is clear that the limit of a sequence in a We start our work by proving the following crucial lemma.X, σb) be a b-metric-like space with coefficient s > 1, and suppose that {xn} and {yn} are convergent to x and y, respectively. Then one hasσb = 0, then one has limn→∞σb = 0.Let  = 0, thenMoreover, for each b-metric-like space it is easy to see thatn → ∞ in the first inequality and the upper limit as n → ∞ in the second inequality we obtain the first desired result. If σb = 0, then by the triangle inequality we get σb = 0 and σb = 0. Therefore, we have limn→∞σb = 0. Similarly, using again the triangle inequality the other assertions follow.Using the triangle inequality in a f on a metric space  is said to be a Banach contraction mapping, if there exists a number k ∈ [0,1) such thatx, y ∈ X. A mapping f : X → X is called a quasicontraction if for some constant α ∈ [0,1) and for every x, y ∈ Xf has a unique fixed point.It is well known that a self-map  in 1974 . A resulThe existence of fixed points in partially ordered metric spaces was first investigated in 2004 by Ran and Reurings  and thenb-metric-like space. We investigate also the so-called P-property for mappings in such spaces.In this paper, we establish some fixed point theorems for quasicontractive type mappings in a partially ordered complete X, ⪯) be a partially ordered set, and let  be a b-metric-like space  is a partially ordered b-metric-like space). Further, let F(f) = {x ∈ X : fx = x} be the fixed point set of f, (LF)f = {x ∈ X : x⪯fx} the lower fixed point set of f, andb-metric-like space.Throughout this paper, let  is said to have the sequential limit comparison property if for every nondecreasing sequence (nonincreasing sequence) {xn}n∈ℕ in X, xn → x implies that xn⪯x  (x⪯xn), for all n ∈ ℕ.An ordered X, ⪯, σb) be a complete partially ordered b-metric-like space. If f : X → X is a nondecreasing map such that, for all elements x, y ∈ X with x⪯y,α ∈ [0, 1/2s2), then F(f) ≠ ∅ provided that there exists an x0 ∈ (LF)f and one of the following two conditions is satisfied:f is a continuous self-map on X,X, ⪯, σb) has the sequential limit comparison property.f and f is nondecreasing, therefore fnx0⪯fn+1x0 for each n ∈ ℕ. Define a sequence {xn} in X with xn = fnx0 and so xn+1 = fxn for all n ∈ ℕ. If there exists a positive integer n such that xn = xn+1, then fnx0 = fn+1x0 = ffnx0 which implies that fnx0 is a fixed point of f. Assume that xn ≠ xn+1 for every positive integer n. Since xn−1⪯xn, therefore by replacing x by xn−1 and y by xn in  > σb > 0, then according to the above inequality σb ≤ 2sασb < σb which is a contradiction.Since y xn in , we haven, σb ≤ σb and thereforeh = 2αs. Obviously, 0 ≤ h < 1. Repeating the above process, we getn ≥ 1, and so, for m > n, we haveh < 1/s, it follows that limm,n→∞σb = 0. Since X is complete, there exists an element u ∈ X such thatf is a continuous self-map on X, then fu = u. Indeed, by the triangle inequality we haven → ∞ in the above inequality, the desired result is obtained.Hence, for all xn = fnx0⪯u for all n ∈ ℕ, and it follows thatn → ∞,  = 0) it follows thatσb ≤ αs2σb, or equivalently, fu = u.If condition (b) is fulfilled then f are comparable. Let w be another fixed point of f such that w ≠ u. Assume, for example, that u⪯w. Using  is a complete b-metric-like space with coefficient s = 2 (see Let  = 2 see .f : X → X be defined by fx = ln(1 + x/4). It is easy to see that f is a nondecreasing and continuous self-map on X, and 0 ∈ (LF)f. Using the Mean Value Theorem for any x, y ∈ X with x ≤ y and that fx ≤ x/4, we haveα = 1/16 ∈ [0, 1/8). Thus all conditions of f.Let y).Thus  is satisX, ⪯, σb) be a partially ordered complete b-metric-like space with coefficient s > 1. If a nondecreasing map f : X → X satisfiesx, y ∈ X with x⪯y, whereφ : [0, ∞) → [0, ∞) is a continuous and nondecreasing function such that φ(t) > 0 for all t ∈  and φ(0) = 0, then F(f) ≠ ∅ provided that there exists an x0 ∈ (LF)f and one of the following two conditions is satisfied:f is a continuous self-map on X.X, ⪯, σb) has the sequential limit comparison property. ≤ σb and {σb} is nonincreasing and bounded from below. Hence, there exists r ≥ 0 such that limn→∞σb = r.Define the sequence {y xn in , we obtan → ∞, we gets > 1, possible only if r = 0. So we haveFrom the above argument we havexn} is a Cauchy sequence. If not, then there exists ɛ > 0 for which we can find subsequences {xmk} and {xnk} of the sequence {xn} where nk is the smallest index for which nk > mk > k withk → ∞, from .Using , we get∞, from (ɛ≤liminf).Using , we get.Using .From  is satisfied then xn = fnx0⪯u for all n ∈ ℕ; it follows thatn → ∞ in  = 0, implying that u = fu.If ows thatsσb = 0, which yields that u = w. The converse is trivial.Now, suppose that the fixed points of w. Using , we obtaX = [0, ∞) be endowed with the usual order ≤. Define σb : X × X → ℝ+ byx, y ∈ X. Then  is a complete ordered b-metric-like space with coefficient s = 2 f. Using the Mean Value Theorem for the function f2x = (1/16)ln(x2/3 + 1) for any x, y ∈ X with x ≤ y, we haveφ : [0, ∞) → [0, ∞) by φ(t) = t/8, and, for any x, y ∈ X with x ≤ y, we havef2x = (1/16)ln(x2/3 + 1) is an increasing function and f2x ≤ x2/48.Let f.Thus, all conditions of f and g be two self-maps on partially ordered set X. A pair  is said to befx⪯gfx and gx⪯fgx, for all x ∈ X [weakly increasing if ll x ∈ X , 20,fx⪯gfx, for all x ∈ X [partially weakly increasing if ll x ∈ X .Let X be a nonempty set and f : X → X a given mapping. For every x ∈ X, let f−1(x) = {u ∈ X : fu = x}.Let X, ⪯) be a partially ordered set and f, g, h : X → X mappings such that fX⊆hX and gX⊆hX. The ordered pair  is said to beh if and only if, for all x ∈ X, fx⪯gy for all y ∈ h−1(fx), and gx⪯fy for all y ∈ h−1(gx) [weakly increasing with respect to h−1(gx) ,h if fx⪯gy, for all y ∈ h−1(fx) [partially weakly increasing with respect to h−1(fx) .Let  with respect to h and (ii) if h = I (the identity mapping on X), then the above definition reduces to the weakly increasing  mapping (see [In the above definition (i) if ing see , 24)..f = g, wψ, φ)-weakly contractive mappings in partially ordered b-metric-like spaces.The study of unique common fixed points of mappings satisfying weakly contractive conditions has been at the center of vigorous research activity. Motivated by the work in , 23–30, ψ : [0, ∞) → [0, ∞) which satisfies thatψ1)(ψ(t) is increasing and continuous,ψ2)(ψ(t) = 0 if and only if t = 0.Recall  that an X, ⪯, σb) be an ordered b-metric-like space and f, g, R, S : X → X four self-mappings. Throughout this subsection, unless otherwise stated, for all x, y ∈ X, letf(X)⊆R(X) and g(X)⊆S(X); let x0 be an arbitrary point of X. Choose x1 ∈ X such that fx0 = Rx1 and x2 ∈ X such that gx1 = Sx2. Continuing in this way, construct a sequence {zn} defined by zn+12 = Rxn+12 = fxn2 and zn+22 = Sxn+22 = gxn+12, for all n ≥ 0. The sequence {zn} in X is said to be a Jungck-type iterative sequence with initial guess x0.Let  be a partially ordered b-metric-like space with sequential limit comparison property, f, g, R, S : X → X four mappings such that f(X)⊆R(X) and g(X)⊆S(X), and RX and SXσb-complete subsets of X. Suppose that, for comparable elements Sx, Ry ∈ X, one hasψ, φ : [0, ∞) → [0, ∞) are altering distance functions. Then, the pairs  and  have a coincidence point z* in X provided that the pairs  and  are weakly compatible, and the pairs  and  are partially weakly increasing with respect to R and S, respectively.Let  and x2 ∈ S−1(gx1), and the pairs  and  are partially weakly increasing with respect to R and S, so we haveRxn+12⪯Sxn+22, for n ≥ 0. We will complete the proof in three steps.As Step  I. We will prove that limk→∞σb = 0.σbk = σb. Suppose σbk0 = 0, for some k0. Then, zk0 = zk0+1. If k0 = 2n, then zn2 = zn+12 gives zn+12 = zn+22. Indeed,φ) = 0; that is, zn+12 = zn+22. Similarly, if k0 = 2n + 1, then zn+12 = zn+22 gives zn+22 = zn+32. Consequently, the sequence {zk} becomes constant for k ≥ k0 and hence limk→∞σb = 0.Define k. We now claim that the following inequality holds:k = 1,2,….Suppose thatk = 2n, and for an n ≥ 0, σb > σb > 0. Then, as Sxn2⪯Rxn+12, using  > σb > 0,  = 0; that is, σb = 0, a contradiction to  ≤ σb andk = 2n.Let 1, using  we obtai, using } is a nonincreasing sequence of nonnegative real numbers. Therefore, there is an r ≥ 0 such thatk → ∞ in  ≤ ψ(r) − φ(r). Therefore φ(r) = 0. Hence,φ.Similarly, it can be shown thatk → ∞ in , we obtan → ∞ in , using 2} and {zn(k)2} of {zn2} such that n(k) > m(k) ≥ k andn(k) is the smallest number such that the above statement holds; that is,k → ∞ in (k → ∞ in (k → ∞ in (k → ∞ and using (Sxm(k)2⪯Rxn(k)−12, so from (k → ∞ in (φ(liminfk→∞M(xm(k)2, xn(k)−12)) = 0. Hence,zn} is a σb-Cauchy sequence.We assume on the contrary that there exists  we haveσb(z2m(k) ≥ k andσb(z2m(k)k → ∞ in  we obtai we haveσb(z2m(k)that is,σb(z2m(k)k → ∞ in  we haveɛ.Also,σb(z2m(k)k → ∞ in  and 78)ɛ > 0 fornd using  and 82)ɛ > 0 for so from , we havend using , (82), ( → ∞ in (limsupk→d using (limsupk→so from (ψ(σb(z2m( → ∞ in (ɛs≤limsupd using (limsupk→ ≥ k andσb(z2m(k)Step  III. We will show that f, g, R, and S have a coincidence point.zn} is a σb-Cauchy sequence and R(X) and S(X) are σb-complete σb-metric spaces, there exists z* ∈ R(X)∩S(X) such thatu ∈ X such that z* = Ru andv ∈ X such that z* = Sv andz* is a coincidence point of g and R. For this purpose, we show that Ru = gu. Since Sxn+22 → z* = Ru as n → ∞, so Sxn+22⪯Ru.Since {n → ∞ in  = 0, and using Ru = z* = gu.Therefore, from , we haveψSimilarly it can be shown that X, ⪯, σb) be a partially ordered σb-complete σb-metric-like space. Let f, g : X → X be two mappings. Suppose that for every two comparable elements x, y ∈ X one hasψ, φ : [0, ∞) → [0, ∞) are altering distance functions andf and g be continuous and let the pair  be weakly increasing. Then, f and g have a common fixed point z* in X.Let ) = 0. Thus, we have fz* = gz* = z*.Let .  From , we haveerefore,max{σb be a complete ordered b-metric-like space, S, T : X → X dominated maps, and x0 an arbitrary point in X. Suppose that for k ∈ [0, 1/s) and for S ≠ T one hasxn} in xn} → u imply that u⪯xn. Then there exists σb = 0 and x* = Sx* = Tx*. Also, if for any two points x, y in z⪯x and z⪯y, that is, every pair of elements has a lower bound, then x* is a unique common fixed point in Let /2 so, using inequality /2) we obtainn ∈ N. This implies thatxn} is a Cauchy sequence in n→∞xn = x*. Also,n → ∞ and using the fact that x*⪯xn when xn → x*, we havex* = Sx*. Similarly, by usingx* = Tx*. Hence S and T have a common fixed point in Let equality , we obtaquality  ⊂ F(fn). However, the converse is false. For example, the mapping f : ℝ → ℝ, defined by fx = 1/2 − x, has a unique fixed point 1/4 but every x ∈ ℝ is a fixed point of f2. If F(f) = F(fn) for every n ∈ ℕ, then f is said to have property P. For more details, we refer to [Clearly, a fixed point of refer to , 34 and f : X → X with the constant α ∈ [0, 1/2), where X is a cone metric space, has the property P [X, d) is a cone metric space and T-Hardy-Rogers contraction f : X → X satisfies some appropriate conditions, then f has property P [Recently, the study of periodic points for contraction mappings has been considered by many authors; for instance, every quasicontraction X, ⪯) be a partially ordered set. A mapping f is called dominating on X if x⪯fx for each x in X.Let  be endowed with the usual ordering. Let f : X → X be defined by x ∈ [0,1) and fx = xn for x ∈ [1, ∞), for any n ∈ ℕ. Then, for all x ∈ X, x ≤ fx; that is, f is a dominating map.Let We have the following results.X, ⪯, σb) be a partially ordered complete b-metric-like space. Let f : X → X be a nondecreasing mapping such that for all x ∈ X with x⪯fx one hasλ ∈ [0,1). Then f has the property P provided that F(f) is nonempty and f is dominating on F(fn).Let (u ∈ F(fn) for some n > 1. We will show that u = fu. Since f is dominating on F(fn), therefore u⪯fu which implies that fn−1u⪯fnu as f is nondecreasing. Using  = 0, implying that u = fu.Let g. Using , we obtaX and f be as in f is dominating on X, then f satisfies property P.Let F(f) ≠ ∅. We will prove that  = 0, then it is easy to see that  > 0. Since f is dominating, we have x⪯fx. Also, fx⪯f2x as f is nondecreasing. Using  ≤ σb. On the contrary, if σb < σb, then from the above inequality we havesα < 2s · (1/2s2) < 1, a contradiction.If see that  is satisg. Using , we haveλ = 2sα. Obviously, λ ∈ [0,1). By f has property P.So we have


G =  is said to be r-regular if each vertex of G is of degree r. The vertex covering transversal domination number γvct(G) is the minimum cardinality among all vertex covering transversal dominating sets of G. In this paper, we analyse this parameter on different kinds of regular graphs especially for Qn and Hn3,. Also we provide an upper bound for γvct of a connected cubic graph of order n ≥ 8. Then we try to provide a more stronger relationship between γ and γvct.A simple graph  Kn, Km,n, Pn, Cn, and Wn and trees is dealt with in paper  and E = E1 ∪ E2 ∪ E3 ∪ E4 with   U = {u1, u2,…, up+12}, Vi = {vi1, vi2,…, vi(q + 2p)}, Wi = {wi1, wi2,…, wi(q + 2p − 1)},  E1 = {ujuk; 1 ≤ j < k ≤ 2p + 1},  E2 = ⋃i=1p+12{uivik; 1 ≤ k ≤ q},  E3 = ⋃i=1p+12{vi(q+2k−1)vi(q+2k); 1 ≤ k ≤ p},  E4 = ⋃i=1p+12{vikwil; 1 ≤ k ≤ q + 2p, 1 ≤ l ≤ q + 2p − 1}.Then(i)V(G)| = 2(2p + 1)(q + 2p),|(ii)G∗ is connected and (q + 2p)-regular,(iii)γvct) = 2(2p + 1).Given positive integers p = 1 and q = 2, the graph G∗ is as shown in (i) and (ii) are obvious. If Ij = [⋃i=1p+12(Wi)] ∪ {uj}, j = 1 to 2p + 1, is a maximum independent set in G∗. Therefore its complement Jj = V − Ij = [⋃i=1p+12(Vi)]∪[U − {uj}], j = 1 to 2p + 1, is an α0 set in G∗. Further J1, J2,…, Jp2 and Jp+12 are the only α0-sets of G∗. Now each Sjk = ⋃i=1p+12{vik, wij}, j = 1 to q + 2p − 1, k = 1 to q is a dominating set intersecting J1, J2,…, Jp2 and Jp+12 and also of minimum cardinality 2(2p + 1).It is clear that each γvct) = 2(2p + 1).Hence r ≥ 4, there exists a connected r-regular graph G of order n such that γvct(G) = n/r.For every G = G∗ be defined as in G is a connected r-regular graph with r = q + 2p. Also γvct(G)/n = 1/(q + 2p).Let γvct(G) = n/r.Thus γvct is replaced by γ.Theorems Hn3, defined in [In this section, we provide the vertex covering transversal domination number of some regular cubic graphs especially Harary graph fined in . We alsoY3 shown in Consider the triangular prism graph Y3 has 6 vertices and 9 edges. Assume that the graph Y3 is labelled as shown in the diagram. It is clear that {uimod3, vi + 1)mod3mod3mod3mod3mod3(} and Si = {ui + 1)mod3mod3mod3 = 2.Consider Peterson graph which is cubic regular shown in G is labelled as shown in Ii = {vimod5, vi + 3)mod5mod5mod5mod5mod5mod5mod5mod5mod5mod5(} are γ-sets intersecting each Ci. Hence γvct(G) = 3.Assuming that the graph Si = {vimod5, vi + 3)mod5mod5 = ⌊(n + 1)/3⌋.If Hn3, is a 3-regular graph and so n is even. By the definition of Hn3,, every vertex vi ∈ Hn3, is adjacent to the vertices vi+1, vi−1, and vi+k where n = 2k.V = {v0, v1, v2,…, vn−1}. The graphs H3,10 and H3,12 are shown in Let Case  1. Suppose n = 2k where k is odd.C1 = {v0, v2, v4,…, vn−2} and C2 = {v1, v3, v5,…, vn−1} are the only α0-sets of Hn3,.Then Subcase  1. Let n ≡ 0 (mod3).S = {v0, v3, v6,…, vn−3} is a γ-set which intersects C1 and C2 and |S| = n/3.Then Subcase  2. Suppose n ≡ 1 (mod3).S = {v0, v3, v6,…, vn−4} is a γ-set which intersects C1 and C2 and |S| = (n − 1)/3.Then Subcase  3. Suppose n ≡ 2 (mod3).S = {v0, v3, v6,…, vn−2} is a γ-set which intersects C1 and C2 with |S| = (n + 1)/3.Then γvct = ⌊(n + 1)/3⌋.Thus in all the subcases of Case  1, Case  2. Suppose n = 2k where k is even.Ii = {vimodn, vi + 2)modnmodnmodnmodnmodnmodnmodnmodnmodnmodnmodnmodnmodnmodn.Si = {vimodn, vi + 3)modnmodnmodn.Si = {vimodn, vi + 3)modnmodnmodn.Si = {vimodn, vi + 3)modnmodnmodn/3⌋.The γvct = ⌊(n + 1)/3⌋.Thus γvct = γ.In most of the graphs considered by us, it is observed that G is a connected cubic graph of order n with n ≥ 8, then γvct(G)≤⌈2n/5⌉.If I be an independent dominating set of cardinality i(G). Then I is a maximal independent set of minimum cardinality. Since I is independent, no two vertices of I are adjacent in G. Let J = V − I. Then the vertices in I are adjacent only to the vertices in J.Let Case  1. Suppose I itself is a β0-set. Then J is an α0-set. Let S = I ∪ {v} where v ∈ J. Then S is a vertex covering transversal dominating set of G. Therefore γvct(G) ≤ i(G) + 1. Hence γvct(G) ≤ 2n/5 + 1  ≤ 2n/5.Hence γvct(G)≤⌈2n/5⌉.Thus Cases  1 and 2 imply that γ and γvct than that proved in [γ = γvct and γ < γvct.In this section, we prove a more stronger relationship between roved in . In viewG is a simple connected graph, then γvct(G) ≤ γ(G) + 1.If D be a minimum dominating set. If D = V(G), then obviously γvct(G) = γ(G). If not, then D ⊂ V(G) and V(G) − D ≠ ϕ. Let u ∈ V(G) − D. Then u is dominated by some vertex v in D. Let S = D ∪ {u}. Since uv is an edge in G, either u or v is included in every minimum vertex covering set of G. This implies that S intersects every minimum vertex covering set in G. Hence γvct(G) ≤ γ(G) + 1.Let G be a simple connected graph. If there exists a γ-set which is not independent, then γvct(G) = γ(G).Let D be a minimum dominating set which is not an independent set of G. Then at least two vertices, say, u, v in D, are adjacent to each other. Therefore uv is an edge in G and hence either u or v lies in every minimum vertex covering set of G. So D intersects every α0-set of G. Therefore D itself is a γvct-set. Hence γvct(G) = γ(G).Let γvct(G) = γ(G), then there may exist a γ-set which is independent also. For example, consider C6, the cycle on 6 vertices as shown in The converse is not true. If v1, v3, v5} and {v2, v4, v6} are the only α0-sets of C6. Also {v1, v4} is a γ-set which is independent. Further, it is a γvct-set as it intersects both the α0-sets of C6. Thus there exists a γ-set which is independent in C6 even though γvct(C6) = γ(C6).Obviously {γvct(G) = γ(G), is every γ-set of G a γvct-set?” The answer is “not always.” The γ-sets and γvct-sets in the graphs Q2 and Y3 discussed in the previous sections are the best examples for it. So it is noted that this happens if there exists a γ-set which is also a βo-set. It obviously produces the result that “If γvct(G) = γ(G) = βo(G), then there exists at least one γ-set in G which is not a γvct-set.” The next general question is that “What happens if all the γ-sets of G are β0-sets?”. The following theorem provides the answer to it.Now, the obvious question is “If G be a simple connected graph. If every γ-set of G is a β0-set, then γvct(G) = γ(G) + 1.Let γ-set D of G is a β0-set, choose a vertex v in its complement. This is possible since D ≠ V(G) as D is a β0-set of a connected graph G. Obviously D is not a γvct-set as it does not intersect the α0-set V(G) − D. Let S = D ∪ {v}. We claim that S intersects every α0-set of G. Suppose that S∩C = Φ for some α0-set C in G. Then S⊆I where I = V(G) − C is a β0-set. This implies that β0(G) + 1 ≤ β0(G) which is a contradiction. Hence S intersects every α0-set of G. Also S is a γvct-set of G as it contains exactly one vertex more than that of the γ-set D. Thus γvct(G) = γ(G) + 1.Since every γvct = γ, there are graphs in which γ-sets do not become γvct-sets. This implies that the collection of γvct-sets in such graphs is contained in the collection of γ-sets. So this may lead to consider γvct-sets in the graphs for which γvct = γ when we are in a situation to select a minimum number of γ-sets in such graphs. This approach may affect a new variation in domination theory.It is easy to conclude that even though


Lmx1b Causes Glaucoma and Is Semi-lethal via LDB1-Mediated Dimerisation”In the title “LBD1” should appear as “LDB1”. The correct title should read “A Dominant-Negative Mutation of Mouse


Rapid elevation of sodium transport through insulin is mediated by AKT in alveolar cellsCharlott Mattes, Mandy Laube & Ulrich H. ThomePhysiol Rep, 2 (3), 2014, e00269, doi: 10.1002/phy2.269μmol/L” has been corrected to “20 nmol/L insulin”, and “200 μmol/L insulin” has been corrected to “200 nmol/L insulin”.In figure 1, the label “20 Please refer to the revised figure 1 below.


In this paper, not only strongly lacunary ward continuity, but also some other kindsof continuities are investigated in 2-normed spaces.A function  Menger  introducNθ-convergence was introduced in [Nθ-ward continuity and further studied in [Using the main idea in the definition of sequential continuity, many kinds of continuities were introduced and investigated, and not all but some of them are in –13. The duced in  and furtduced in . Stronglduced in  as namedudied in .The aim of this paper is to investigate strongly lacunary ward continuity in 2-normed spaces and prove interesting theorems.N and R will denote the set of all positive integers and the set of all real numbers, respectively. Now we recall the definition of a two-normed space. Let X be a real linear space with dimX > 1 and ||·, ·|| : X × X → R a function. Then  is called a linear 2-normed space if (i) ||x, y|| = 0⇔x and y are linearly dependent, (ii) ||x, y|| = ||y, x||, (iii) ||αx, y|| = |α|||x, y||, and (iv) ||x, y + z|| ≤ ||x, y|| + ||x, z|| for α ∈ R and x, y, z ∈ X. The function ||·, ·|| is called a 2-norm on X. Observe that in any 2-normed space  we have that ||·, ·|| is nonnegative, ||x − z, x − y|| = ||x − z, y − z||, and ||x, y + αx|| = ||x, y|| for all x, y ∈ X, α ∈ R. Throughout this paper by X we will mean a 2-normed space with a two-norm ||·, ·||. As an example of a 2-normed space we may take X = R2 being equipped with the 2-normx1 = , x2 = .Throughout this paper, xn) of points in X is said to converge to a point L of X in the 2-normed space X if limn→∞||xn − L, z|| = 0 for every z ∈ X. This is denoted by limn→∞||xn, z|| = ||L, z||. A sequence (xn) of points in X is said to be a Cauchy sequence with respect to the 2-norm if limn,m→∞||xn − xm, z|| = 0 for every z ∈ X. A sequence of functions (fn) is said to be uniformly convergent to a function f on a subset E of X if for each ε > 0, an integer N can be found such that ||fn(x) − f(x), z|| < ε for n ≥ N and for all x, z ∈ X [θ = (kr) is an increasing sequence of positive integers such that k0 = 0 and hr = kr − kr−1 → ∞ as r → ∞. The intervals determined by θ will be denoted by Ir =  of points in X is called strongly lacunary convergent, or Nθ-convergent to an element L of X ifz ∈ X and it is denoted by Nθ − limk→∞||xk, z|| = ||L, z|| for every z ∈ X [Nθ” instead of “strongly lacunary” and assume that liminfr qr > 1.A sequence  = ||x, z||, ∀x, for each z ∈ X, forms a locally convex topological vector space, and the topology formed by this family of seminorms gives the required topology on X. Since dimX ≥ 2, for each x ∈ X there exists a y ∈ X such that x and y are linearly independent, and hence by (i), py(x) = ||x, y|| ≠ 0. Thus the locally convex topological vector space induced by the set {pz : z ∈ X} of seminorms is Hausdorff so X is a Hausdorff space [The set of seminorms {ff space . In thisE of X is called Nθ-sequentially compact if any sequence of points in E has an Nθ-convergent sequence with an Nθ-limit in E.A subset Nθ-sequentially compact subsets of X is Nθ-sequentially compact, intersection of any Nθ-sequentially compact subsets is Nθ-sequentially compact, any compact subset of X is Nθ-sequentially compact, and any finite subset of X is Nθ-sequentially compact. Sum of Nθ-sequentially compact subsets of X is Nθ-sequentially compact where sum of two subsets A and B is defined as A + B = {a + b : a ∈ A, b ∈ B}.We note that union of two f defined on a subset E of X is said to be strongly lacunary sequentially continuous or Nθ-sequentially continuous at a point x0 of E if (f(xk)) is an Nθ-convergent sequence to f(x0) whenever (xk) is an Nθ-convergent to x0 sequence of points in E. If f is strongly lacunary sequentially continuous at every point of E, then it is said to be strongly lacunary sequentially continuous on E.A function f defined on a subset E of X is lacunary statistically sequentially continuous at a point x0, then (f(xk)) is an Nθ-convergent sequence with Nθ − lim||f(xk), z|| = ||f(x0), z|| for every z ∈ X whenever (xk) is an Nθ-convergent sequence with Nθ − lim||xk, z|| = ||x0, z|| for every z ∈ X. We see that a function f defined on a subset E of X is strongly lacunary sequentially continuous if and only if it preserves strongly lacunary convergent sequences without stating limit of the sequence. We note that sum of two Nθ-sequentially continuous functions at a point x0 of X is Nθ-sequentially continuous at x0, and composite of two Nθ-sequentially continuous functions at a point x0 of X is Nθ-sequentially continuous at x0. In the classical case, that is in the single normed case, it is known that uniform limit of sequentially continuous function is sequentially continuous; now we see that it is also true that not only uniform limit of sequentially continuous function is sequentially continuous, but also uniform limit of Nθ-sequentially continuous function is Nθ-sequentially continuous in 2-normed spaces. Now we give the latter in the following.If a function Nθ-sequentially continuous functions is Nθ-sequentially continuous.Uniform limit of fk) be a uniformly convergent sequence of each term defined on a subset E of X with uniform limit f and let (xk) be any Nθ-convergent sequence of points in E with Nθ − lim||xk, z|| = ||x, z|| for every z ∈ X. Take any ε > 0. By uniform convergence of (fk), there exists an n1 ∈ N such that ||f(x) − fk(x), z|| < ε/3 for k ≥ n1 and every x ∈ E and z ∈ X. Hence,r ≥ n1 and every x ∈ E and z ∈ X. As fn1 is Nθ-sequentially continuous on E, there exists an n2 ∈ N such that, for r ≥ n2,z ∈ X. Now write n0 = max{n1, n2}. Thus for r ≥ n0 we havevk(x) = f(x) − fn1(x) for every k ∈ N. HenceLet ) be any sequence of points in f(A) where xn ∈ A for each positive integer n. Nθ-sequentially compactness of A implies that there is a subsequence (γk) = (xnk) of (xn) with Nθ − limk→∞||γk, z|| = ||l, z|| for every z ∈ E. Write (tk) = (f(γk)). As f is Nθ-sequentially continuous, (f(γk)) is Nθ-sequentially convergent which is a subsequence of the sequence (f(xn)) with Nθ − limk→∞||tk, z|| = ||l, z|| for ∀z ∈ E. This completes the proof of the theorem.Assume that Nθ-quasi-Cauchy sequence.The concept of a quasi-Cauchy sequence in a 2-normed space was studied in . Now we xn) of points in a subset E of X is called Nθ-quasi-Cauchy if (Δxk) is Nθ-convergent to 0, that is,xk = xk+1 − xk.A sequence ) is an Nθ-quasi-Cauchy sequence whenever (xk) is.A function defined on a subset Nθ-ward continuous functions is Nθ-ward continuous, and sum of two Nθ-ward continuous functions is Nθ-ward continuous.We note that a composite of two Nθ-ward continuous image of any Nθ-ward compact subset of X is Nθ-ward compact.f is an Nθ-ward continuous function on a subset E of X and E is a Nθ-ward compact subset of A. Let (f(xn)) be any sequence of points in f(E) where xn ∈ E for each positive integer n. Nθ-ward compactness of E implies that there is a subsequence (γk) = (xnk) of (xn) with Nθ − limk→∞||Δγk, z|| = 0 for every z ∈ E. Write (tk) = (f(γk)). As f is Nθ-ward continuous, (f(γk)) is Nθ-quasi-Cauchy which is a subsequence of the sequence (f(xn)) with Nθ − limk→∞||Δtk, z|| = 0 for ∀z ∈ E. This completes the proof of the theorem.Assume that Nθ-ward continuous image of any compact subset of X is Nθ-ward compact.The proof follows from the preceding theorem.f defined on a subset E of X is sequentially continuous at x0, if for any sequence (xn) of points in E converging to x0, we have (f(xn)) converges to f(x0). f is sequentially continuous on E if it is sequentially continuous at every point of E  ∈ ΔNθ⇒(f(xk)) ∈ ΔNθ,(xk) ∈ ΔNθ⇒(f(xk)) ∈ c,(xk) ∈ c⇒(f(xk)) ∈ c,(xk) ∈ c⇒(f(xk)) ∈ ΔNθ,(xk) ∈ Nθ⇒(f(xk)) ∈ Nθ,(xk) ∈ ΔNθ⇒(f(xn))  is quasi-Cauchy,(xk) ∈ ΔNθ⇒(f(xn)) ∈ Nθ. is Nθ-sequentially continuity of f, and (3) is the ordinary sequential continuity of f. It is easy to see that (2) implies (1), and (1) does not imply (2); and (1) implies (4), and (4) does not imply (1); (2) implies (3), and (3) does not imply (2). (6) implies (1), but (1) does not imply (6) and (7) implies (1), but (1) does not imply (7). Now we give that the implication (1) implies (5), that is, any Nθ-ward continuous function is Nθ-sequentially continuous.We see that (1) is f is Nθ-ward continuous function on a subset E of X, then it is Nθ-sequentially continuous on E.If f is an Nθ-ward continuous function on a subset E of X. Let (xn) be any Nθ-convergent sequence in A with Nθ − limk→∞||xk, z|| = ||x0, z|| for all z ∈ X. Then the sequence x = (xn) defined byNθ-convergent to x0. Hence it is Nθ-quasi-Cauchy sequence. As f is Nθ-ward continuous on E, the transformed sequence (yn) obtained byNθ-quasi-Cauchy. Thus,z ∈ X. It follows from this that the sequence (f(xk)) is Nθ convergent to (f(x0)).Assume that f =  on the 2-normed space R2 with the usual 2-norm.The converse of this theorem is not valid in general, a counterexample can be easily constructed via the function fn) is a sequence of Nθ-ward continuous functions on a subset E of X and (fn) is uniformly convergent to a function f, then f is Nθ-ward continuous on E.If (xk) be any Nθ-quasi-Cauchy sequence of points in E, and let ε be any positive real number. By uniform convergence of (fk), there exists an n1 ∈ N such that ||f(x) − fk(x), z|| < ε/3 for k ≥ n1 and every x ∈ E and z ∈ X. Hence,r ≥ n1 and every x ∈ E and z ∈ X. As fn1 is Nθ-ward continuous on E, there exists an n2 ∈ N such that, for r ≥ n2,z ∈ X. Now write n0 = max{n1, n2}. Thus, for r ≥ n0, we havef preserves Nθ-quasi-Cauchy sequences. This completes the proof of the theorem.Let (n-normed spaces as another further study.In this paper, we investigate strongly lacunary continuity and some other kinds of continuities defined via a lacunary sequence and we prove interesting theorems related to these kinds of continuities. The results in this paper are extensively deeper than existing related results in the literature. We note that the notion of a strongly lacunary quasi-Cauchy sequence coincides with the notion of a strongly lacunary convergent sequence in a complete non-Archimedean 2-normed space, and so the set of strongly lacunary ward continuous functions coincides with the set of strongly lacunary sequentially continuous functions in a complete non-Archimedean 2-normed space (see  for the


ℳσ and ℳσ of meromorphic bi-univalent functions defined on Δ = {z : z ∈ ℂ, 1 < |z | <∞}. For functions belonging to these classes, estimates on the initial coefficients are obtained.We introduce and investigate two new subclasses  𝒜 denote the class of all functions of the form 𝒮 the class of all functions in the normalized analytic function class 𝒜 which are univalent in 𝕌.Let 𝕌. In fact, the Koebe one-quarter theorem [𝕌 under every univalent function f ∈ 𝒮 contains a disk of radius 1/4. Thus, every function f ∈ 𝒜 has an inverse f−1, which is defined by f−1 is given by Since univalent functions are one-to-one, they are invertible and the inverse functions need not be defined on the entire unit disk  theorem  ensures f ∈ 𝒜 is said to be bi-univalent in 𝕌 if both f and f−1 are univalent in 𝕌. Let Σ denote the class of bi-univalent functions in 𝕌 given by =z+∑nThe coefficient problem was investigated for various interesting subclasses of the meromorphic univalent functions .b0 and b1 of functions in the newly introduced subclasses are obtained.In the present investigation, certain subclasses of meromorphic bi-univalent functions are introduced and estimates for the coefficients g given by  if the following conditions are satisfied: h = g−1 is given by  if the following conditions are satisfied: h = g−1 is given by  and ℳσ reduce to the classes We note that, for m et al. .b0| and |b1| for functions in the above-defined classes ℳσ and ℳσ of the function class σℳ.The object of the present paper is to extend the concept of bi-univalent to the class of meromorphic functions defined on Δ and find estimates on the coefficients |Firstly, in order to derive our main results, we need the following lemma.p ∈ 𝒫, then |ck| ≤ 2 for each k, where 𝒫 is the family of all functions p analytic in 𝕌 for which z ∈ 𝕌.If b0| and |b1| for functions in the class ℳσ.We begin this section by finding the estimates on the coefficients |g(z) given by the series expansion (Let the function xpansion  be in thp(z) and q(w) are functions with positive real part in Δ and have the forms ℜ(p(z)) > 0 and ℜ(q(z)) > 0 in Δ, the functions p(1/z), q(1/z) ∈ 𝒫 and hence the coefficients pk and qk for each k satisfy the inequality in b0| as asserted in (It follows from  that (17ents in (−(1−λ)b0=−1)2p12,(1−λ)b0=α)b0=αp1,(1−λ)[(1−)b0=αq1,(1−λ)[(1−)p1=−q1,2(1−λ)2b0o, from (2(1−λ)2b0erted in .b1|, we subtract 2[. Hence |b1|≤α1−λ12. From |b1|≤5α21λ = 0, we have the following corollary of For g(z) given by the series expansion (Let the function xpansion  be in thg(z) given by the series expansion .Next we estimate the coefficients |g(z) given by the series expansion (Let the function xpansion  be in thp(z) and q(w) are functions with positive real part in Δ and have the forms (It follows from  that (40he forms  and 19)(40)zg′(ze forms (−(1−λ)b0=(1−β)p2,(1−λ)b0=((1−β)p1,(1−λ)[(1−(1−β)q1,(1−λ)[(1−ℜ(p(z)) > 0 and ℜ(q(z)) > 0 in Δ, the functions p(1/z), q(1/z) ∈ 𝒫 and hence the coefficients pk and qk for each k satisfy the inequality in b0| as asserted in b1|, we valently 4b12=(1−λ). Hence |b1|≤1−β12 from  given by the series expansion (Let the function xpansion  be in thg(z) given by the series expansion (Let the function xpansion  be in th


Ce-I functions and fuzzy completely Ce-I functions via fuzzy e-open sets. Some properties and several characterization of these types of functions are investigated. We introduced the notions of fuzzy  Ce-I functions, fuzzy Ce-continuous, fuzzy completely Ce-I functions, and fuzzy e-kernel via fuzzy e-open sets and studied their properties and several characterizations of these types of functions are investigated. In this paper, we denote fuzzy e-open, fuzzy e-closed, and fuzzy regular closed as, eo, ec, and rc, respectively.With the introduction of fuzzy sets by Zadeh  and fuzzX, τ) and  (or simply X and Y) represent nonempty fuzzy topological spaces on which no separation axioms are assumed, unless otherwise mentioned.Throughout this paper, , int(μ), clδ(μ), and intδ(μ), respectively. A fuzzy set μ of X is called fuzzy regular open [μ = int(cl(μ)) )).Let lar open  . The complement of fuzzy δ-open set is called fuzzy δ-closed ). A fuzzy set μ of X is called fuzzy δ-preopen [δ-semi open [μ ≤ int(clδ(μ)) )). The complement of a fuzzy δ-preopen set  is called fuzzy δ-preclosed .The fuzzy y δ-open  if μ = i-preopen   if μ ≤ μ of a fuzzy topological space X is called fuzzy e-open [μ ≤ cl(intδμ)∨int(clδμ). Fuzzy e-closed if μ ≥ cl(intδμ)∧int(clδμ).A fuzzy set y e-open  if μ ≤ cμ is called fuzzy e-closure of μ and is denoted by fe-cl(μ) and the union of all fuzzy e-open sets contained in μ is called fuzzy e-interior of μ and is denoted by fe-int(μ).The intersection of all fuzzy e-closed sets containing f : X → Y is said to be fuzzy e∗-open [X is fuzzy e-open set in Y.A mapping  e∗-open  if the if : X → Y is called fuzzy e-irresolute [f−1(λ) is fuzzy e-open in X for every fuzzy e-open set λ of Y.A function resolute . f−1(λ)μ is quasicoincident [λ denoted by μqλ iff there exist x ∈ X such that μ(x) + λ(x) > 1. If μ and λ are not quasicoincident, then we write A fuzzy set incident  with a fxp is quasicoincident [λ denoted by xpqλ iff there exist x ∈ X such that p + λ(x) > 1.A fuzzy point incident  with a fX, τ) is said to be fuzzy e-T1 [x and y of X there exist fuzzy e-open sets μ1and μ2 such that x ∈ μ1 and y ∈ μ2 and x ∉ μ2 and y ∉ μ1.A fuzzy topological space  is said to be fuzzy e-T2 [x and y of X there exists disjoint fuzzy e-open sets η and ρ such that x ∈ η and y ∈ ρ.A fuzzy topological space ∧cl(λ) = 0.A fuzzy topological space  Urysohn  if for eX, τ) is called fuzzy S-closed [X has a finite subcover.A space  if everf : X → Y is called fuzzy completely continuous [f−1(λ) is fuzzy regular open in X for every fuzzy open set λ of Y.A function ntinuous  if f−1 functions is introduced and some characteristics and properties are studied.In this section, the notion of fuzzy φ : → is called fuzzy Ce-I if the inverse image of every fuzzy e-open set of Y is fuzzy e-closed in X.A mapping Ce-I and fuzzy e-irresolute are independent notions as illustrated in the following example.The concepts of fuzzy X = {a, b, c} and Y = {x, y, z} and the fuzzy sets μ1, μ2, υ be defined as follows: τ = {0,1, μ1, μ2, μ1∨μ2, μ1∧μ2} and σ = {0, υ, 1}. Then, the mapping φ : → is defined by φ(a) = x, φ(b) = y, φ(c) = z. Then, φ is fuzzy Ce-I but not fuzzy e-irresolute.Let X = {a, b, c} and Y = {x, y, z} and the fuzzy sets η1, η2, ρ are defined as follows:Let τ = {0,1, η1, η2} and σ = {0, ρ, 1}. Then, the mapping φ : → is defined by φ(a) = x, φ(b) = y, φ(c) = z. Then, φ is fuzzy e-irresolute but not fuzzy Ce-I.Let φ : → is called fuzzy Ce-continuous if the inverse image of every fuzzy open set of Y is fuzzy e-closed in X.A mapping Ce-I function is fuzzy Ce-continuous, but not conversely from the following example.Every fuzzy X = {a, b, c} and Y = {x, y, z} and the fuzzy sets μ1, μ2, ν, υ are defined as follows: τ = {0,1, μ1, μ2, μ1∨μ2, μ1∧μ2} and σ = {0, ν, 1}. Then, the mapping φ : → is defined by φ(a) = x, φ(b) = y, and φ(c) = z. Then, φ is fuzzy Ce-continuous but not fuzzy Ce-I as the fuzzy set υ is fuzzy e-open in Y but φ−1(υ) is not fuzzy e-closed set in X.Let φ : X → Y, if φ(xε)qμ, the inverse image of for every fuzzy e-closed set of Y is fuzzy e-open in X iff for any xε ∈ X; if φ(xε)qμ, then xεqfe-int(φ−1(μ)).For a fuzzy function μ ≤ Y be a fuzzy e-closed set and φ(xε)qμ. Then, xεqφ−1(μ) and, by hypothesis, φ−1(μ) = fe-int(φ−1(μ)). We obtain, xεqfe-int(φ−1(μ)). Converse can be shown easily.Let φ : X → Y, if φ(xε)qμ, for any fuzzy e-closed set μ ≤ Y and for any xε ∈ X, xεqfe-int(φ−1(μ)) iff there exists a fuzzy e-open set δ such that xεqδ and φ(δ) ≤ μ.For a fuzzy function μ ≤ Y be any fuzzy e-closed set and let φ(xε)qμ. Then, xεqfe-int(φ−1(μ)). Take δ = fe-int(φ−1(μ)) then φ(δ) = φ(fe-int(φ−1(μ))) ≤ φ(φ−1(μ)) ≤ μ, and δ is fuzzy e-open in X and xεqδ.Let μ ≤ Y be any fuzzy e-closed set and letφ(xε)qμ. By hypothesis, there exists fuzzy e-open set δ such that xεqδ and φ(δ) ≤ μ. This implies, δ ≤ φ−1(μ) and then xεqfe-int(φ−1(μ)).Conversely, let φ : X → Y, the following statements are equivalent:(1)f  is fuzzy Ce-I.(2)μ in Y, φ−1(μ) is fuzzy e-open in X.For every fuzzy e- closed set (3)μ, φ−1(fe-int(μ)) is fuzzy e-closed.For every fuzzy open set (4)η, φ−1(fe-cl(η)) is fuzzy e-open.For every fuzzy closed set (5)xε ∈ X and each fuzzy e-closed set μ in Y containing φ(xε), there exists a fuzzy e-open set ρ in X containing xε such that φ(ρ) ≤ μ.For each (6)xε ∈ X and each fuzzy e-open set μ in Y noncontaining φ(xε), there exists a fuzzy e-closed set ν in X noncontaining xε such that φ−1(μ) ≤ ν.For each For a fuzzy function ρ be a fuzzy e-open set in Y. Then, 1Y − ρ is fuzzy e-closed. By (2), φ−1(1Y − ρ) = 1X − φ−1(ρ) is fuzzy e-open. Thus, φ−1(ρ) is fuzzy e-closed. Converse can be shown easily.(1) ⇔ (2): let μ be a fuzzy open set. Since fe-int(μ) is fuzzy e-open, then by (1) it follows that φ−1(fe-int(μ)) is fuzzy e-closed. The converse is easy to prove.(1) ⇔ (3): let η be a fuzzy closed set. Since fe-cl(η) is fuzzy e-closed set, then by (2) it follows that φ−1(fe-cl(η)) is fuzzy e-open. The converse is easy to prove.(2) ⇔ (4): let μ be any fuzzy e-closed set in Y containing φ(xε). By (2), φ−1(μ) is fuzzy e-open set in X and xε ∈ φ−1(μ). Take ρ = φ−1(μ). Then, φ(ρ) ≤ μ. The converse can be shown easily.(2) ⇔ (5): let μ be any fuzzy e-open set in Y noncontaining φ(xε). Then, 1 − μ is a fuzzy e-closed set containing φ(xε). By (5), there exists a fuzzy e-open set ρ in X containing xε such that φ(ρ) ≤ 1 − μ. Hence, ρ ≤ φ−1(1 − μ) = 1 − φ−1(μ) and φ−1(μ) ≤ 1 − ρ. Take ν = 1 − ρ. We obtain that ν is a fuzzy e-closed set in X noncontaining xε. The converse can be shown easily.(5) ⇔ (6): let ϕ : X → Y be a function and let φ : X → X × Y be the fuzzy graph function of ϕ, defined by φ(xε) = ) for every xε ∈ X. If φ is fuzzy Ce-I, then ϕ is fuzzy Ce-I.Let μ be a fuzzy e-closed set in Y; then, 1X × μ is a fuzzy e-closed set in X × Y. Since φ is fuzzy Ce-I, then ϕ−1(μ) = φ−1(1X × μ) is fuzzy e-open in X. Thus, ϕ is fuzzy Ce-I.Let Yλ : λ ∈ Λ} be a family of product spaces. If a function φ : X → ∏Yλ is fuzzy Ce-I, then Pλ∘φ : X → Yλ is fuzzy Ce-I for each λ ∈ Λ where Pλ is the projection of ∏Yλ onto Yλ.Let {δ be any fuzzy e-open set in Yλ. Since Pλ is a fuzzy continuous and fuzzy open set, it is a fuzzy e-open set. Now Pλ : ∏Yλ → Yλ, Pλ−1(δ) is a fuzzy e-open in ∏Yλ. Therefore, Pλ is a fuzzy e-irresolute function. Now (Pλ∘φ)−1(δ) = φ−1(Pλ−1(δ)), since φ is fuzzy Ce-I. Hence φ−1(Pλ−1(δ)) is a fuzzy e-closed set, since Pλ−1(δ) is a fuzzy e-open set. Hence, Pλ∘φ is fuzzy Ce-I.Let φ : ∏Xλ → ∏Yλ is fuzzy Ce-I, then φλ : Xλ → Yλ is fuzzy Ce-I for each λ ∈ Λ.If the function λ0 ∈ Λ be an arbitrary fixed index and let υλ0 be any fuzzy e-open set of Yλ0; then, ∏Yμ × υλ0 is fuzzy e-open in ∏Yλ, where λ0 ≠ μ ∈ Λ. Since φ is fuzzy Ce-I function, then φ−1(∏Yμ × υλ0) = ∏Xμ × φλ0−1(υλ0) is fuzzy e-closed in ∏Xλ and hence φλ0−1(Vλ0) is fuzzy e-closed in Xλ0. This implies φλ0 is fuzzy Ce-I.Let φ : X → Y is fuzzy Ce-I and δ is fuzzy closed set of X, then φ|δ : δ → Y is fuzzy Ce-I.If λ be a fuzzy e-open set of Y; then, (φ|δ)−1(λ) = φ−1(λ)∧δ. Since φ−1(λ) and δ are fuzzy closed, hence (φ|δ)−1(λ) is fuzzy e-closed in the relative topology of δ.Let η of a fuzzy topological space  containing μ is called the fuzzy e-kernel of μ , fe-Kμ = ⋀{η : μ ≤ η, η  is  fuzzy  e-open  set  of  X}.The intersection of all fuzzy e-open set μ, λ of X:(1)x ∈ fe-Kμ iff μ∧γ ≠ 0 for any fuzzy e-closed set γ containing x.(2)μ ≤ fe-Kμ and μ = fe-Kμ if μ is fuzzy e-open in X.(3)μ ≤ λ; then, fe-Kμ ≤ fe-Kλ.The following properties hold for fuzzy sets φ : X → Y, the following statements are equivalent:(1)φ is fuzzy Ce-I.(2)φ(fe-cl(μ)) ≤ fe-Kφ(μ) for every fuzzy set μ of X.(3)fe-cl(φ−1(η)) ≤ φ−1(fe-Kη) for every fuzzy set η of Y.For a fuzzy function μ ≤ X and y ∉ fe-Kφ(μ). There exists a fuzzy e-closed set γ in Y, such that y ∈ γ and φ(μ)∧γ = 0. Therefore, φ−1(φ(μ)∧γ) = 0. This implies that μ∧φ−1(γ) = 0 and fe-cl(μ)∧φ−1(γ) = 0. Thus, φ(fe-clμ)∧γ = 0 and y ∉ φ(fe-clμ). Hence, φ(fe-clφ(μ)) ≤ fe-Kφ(μ).(1)⇒(2): let η ≤ Y; then, φ−1(η) ≤ X. By hypothesis, φ(fe-clφ−1(η)) ≤ fe-Kφ(φ−1(η)) ≤ fe-Kη. Hence, fe-cl(φ−1(η)) ≤ φ−1(fe-Kη).(2)⇒(3): let η be any fuzzy e-open set of Y; we have fe-cl(φ−1(η)) ≤ φ−1(fe-Kη) = φ−1(η), since η is fuzzy e-open and fe-cl(φ−1(η)) = φ−1(η). This implies that φ−1(η) is fuzzy e-closed in X.(3)⇒(1): let γ of a fuzzy topological space X is given by fe-Fr(γ) = fe-cl(γ)∧fe-cl(1X − γ).The fuzzy e-Frontier of a fuzzy set xε ∈ X such that φ : X → Y is not fuzzy Ce-I is exactly the union of fuzzy e-Frontier if the inverse image of the fuzzy e-closed set in Y contains φ(xε).The fuzzy point φ is not fuzzy Ce-I at the point xε ∈ X; then there exists a fuzzy e-closed set γ such that φ(xε) ∈ γ and φ(μ)∧(1Y − γ) ≠ 0 for all fuzzy e-open set μ such that xε ∈ μ. It follows that μ∧φ−1(1Y − γ) ≠ 0 and hence xε ∈ fe-clφ−1(1Y − γ) = fe-cl(1X − φ−1(γ)). Thus, xε ∈ φ−1(γ) ≤ fe-cl(φ−1(γ)) and hence xε ∈ fe-Fr(φ−1(γ)).Suppose that xε ∈ fe-Fr(φ−1(γ)), γ is fuzzy e-closed set of Y containing φ(xε), and φ is fuzzy Ce-I at xε ∈ X. There exists fuzzy e-open set μ such that xε ∈ μ and μ ≤ φ−1(γ). Thus, xε ∈ fe-intφ−1(γ) and hence xε ∉ fe-Fr(φ−1(γ)) for each fuzzy e-closed set γ of Y containing φ(xε), a contradiction. Therefore, φ is not fuzzy Ce-I.Conversely, suppose that ϕ : X → Y and φ : Y → Z:(a)ϕ : X → Y is fuzzy Ce-I and φ : Y → Z is fuzzy Ce-continuous then φ∘ϕ : X → Z is fuzzy Ce-continuous.If (b)ϕ : X → Y is fuzzy Ce-I and φ : Y → Z is fuzzy e-irresolute then φ∘ϕ : X → Z is fuzzy Ce-I.If The following hold for functions ϕ : X → Y is a fuzzy e-irresolute surjective function and φ : Y → Z is a fuzzy function such that φ∘ϕ : X → Z is fuzzy Ce-I, then φ is fuzzy Ce-I.If η be any fuzzy e-closed set in Z. Since φ∘ϕ is fuzzy Ce-I, (φ∘ϕ)−1(η) is fuzzy e-open in X. Therefore, ϕ−1(φ−1(η)) = (φ∘ϕ)−1(η) is fuzzy e-open in X. Since ϕ is fuzzy e-irresolute, surjection implies ϕ(ϕ−1(φ−1(η))) = φ−1(η) is fuzzy e-open in Y. Thus, φ is fuzzy Ce-I.Let ϕ : X → Y is a fuzzy e∗ − open surjective function and φ : Y → Z is a fuzzy function such that φ∘ϕ : X → Z is fuzzy Ce-continuous, then φ is fuzzy Ce-continuous.If η be any fuzzy closed set in Z. Since φ∘ϕ is fuzzy Ce-continuous, (φ∘ϕ)−1(η) is fuzzy e-open in X. Therefore, ϕ−1(φ−1(η)) = (φ∘ϕ)−1(η) is fuzzy e-open in X. Since ϕ is fuzzy e∗-open, surjection implies ϕ(ϕ−1(φ−1(η))) = φ−1(η) is fuzzy e-open in Y. Thus, φ is fuzzy Ce-continuous.Let Ce-I functions is introduced and the relation between other functions is studied and further some structure preservation properties are investigated.In this section, the notion of fuzzy completely φ : → is called fuzzy completely Ce-I if inverse image of every fuzzy e-open set in Y is fuzzy regular closed in X.A mapping X = {x, y, z} and the fuzzy sets μ1, μ2 are defined as follows: τ = {0,1, μ1, μ2, μ1∨μ2, μ1∧μ2} and σ = {0, μ1, μ1∧μ2, 1}. Then, the mapping φ : → is defined by φ(x) = 1 − x. Then, φ is fuzzy completely Ce-I.Let Ce-I function is fuzzy Ce-I and fuzzy Ce-continuous, but the converse is not true, which can be seen in the following example.Every fuzzy completely X = {a, b, c} and Y = {x, y, z} and the fuzzy sets μ1, μ2, and ν are defined as follows: τ = {0,1, μ1, μ2, μ1∨μ2, μ1∧μ2} and σ = {0, ν, 1}. Then, the mapping φ : → is defined by φ(a) = x, φ(b) = y, φ(c) = z. Then, φ is fuzzy Ce-continuous and also fuzzy Ce-I but not fuzzy completely Ce-I as the fuzzy set υ is fuzzy e-open in Y but φ−1(υ) is not fuzzy regular closed set in X.Let From the above examples, we have the following implications.None of the these implications is reversible.φ : X → Y, if φ(xε)qμ, the inverse image of every fuzzy e-closed set of Y is fuzzy δ-open in X iff for any xε ∈ X if φ(xε)qμ, then xεqintδ(φ−1(μ)).For a fuzzy function μ ≤ Y be a fuzzy e-closed set and φ(xε)qμ. Then, xεqφ−1(μ) and, by hypothesis, φ−1(μ) = intδ(φ−1(μ)). From here, xεqintδ(φ−1(μ)). The converse can be shown easily.Let φ : X → Y, if φ(xε)qμ, for any fuzzy e-closed set μ ≤ Y and for any xε ∈ X, xεqintδ(φ−1(μ)) iff there exists a fuzzy δ-open set θ such that xεqθ and φ(θ) ≤ μ.For a fuzzy function μ ≤ Y be any fuzzy e-closed set and let φ(xε)qμ. Then, xεqintδ(φ−1(μ)). Take θ = intδ(φ−1(μ)); then, φ(θ) = φ(intδ(φ−1(μ))) ≤ φ(φ−1(μ)) ≤ μ; θ is fuzzy δ-open in X and xεqθ.Let μ ≤ Y be any fuzzy e-closed set and let φ(xε)qμ. By hypothesis, there exists fuzzy δ-open set θ such that xεqθ and φ(θ) ≤ μ. This implies θ ≤ φ−1(μ) and then xεqintδ(φ−1(μ)).Conversely, let φ : X → Y, the following statements are equivalent:(1)f is fuzzy completely Ce-I.(2)μ in Y, φ−1(μ) is fuzzy regular open in X.For every fuzzy e-closed set (3)μ, φ−1(fe-int(μ)) is fuzzy regular closed.For every fuzzy open set (4)η, φ−1(fe-cl(η)) is fuzzy regular open.For every fuzzy closed set (5)xε ∈ X and each fuzzy e-closed set μ in Y containing φ(xε), there exists a fuzzy regular open set ρ in X containing xε such that φ(ρ) ≤ μ.For each (6)xε ∈ X and each fuzzy e-open set μ in Y non containing φ(xε), there exists a fuzzy regular closed set ν in X noncontaining xε such that φ−1(μ) ≤ ν.For each For a fuzzy function ρ be a fuzzy e-open set in Y. Then, 1Y − ρ is fuzzy e-closed. By (2), φ−1(1Y − ρ) = 1X − φ−1(ρ) is fuzzy regular open. Thus, φ−1(ρ) is fuzzy regular closed. Thus, φ is fuzzy completely Ce-I.The converse can be shown easily.(1) ⇔ (2): let μ be a fuzzy open set. Since fe-int(μ) is fuzzy e-open, then by (1) it follows that φ−1(fe-int(μ)) is fuzzy regular closed. The converse is easy to prove.(1) ⇔ (3): let η be a fuzzy closed set. Since fe-cl(η) is fuzzy e-closed set, then by (2) it follows that φ−1(fe-cl(η)) is fuzzy regular open. The converse is easy to prove.(2) ⇔ (4): let μ be any fuzzy e-closed set in Y containing φ(xε). By (2), φ−1(μ) is fuzzy regular open set in X and xε ∈ φ−1(μ). Take ρ = φ−1(μ). Then, φ(ρ) ≤ μ. The converse can be shown easily.(2) ⇔ (5): let μ be any fuzzy e-open set in Y noncontaining φ(xε). Then, 1 − μ is a fuzzy e-closed set containing φ(xε). By (5), there exists a fuzzy regular open set ρ in X containing xε such that φ(ρ) ≤ 1 − μ. Hence, ρ ≤ φ−1(1 − μ) = 1 − φ−1(μ) and φ−1(μ) ≤ 1 − ρ. Take ν = 1 − ρ. We obtain that ν is a fuzzy regular closed set in X noncontaining xε. The converse can be shown easily.(5) ⇔ (6): let φ1 : X → Y be a function and let φ2 : X → X × Y be the fuzzy graph function of φ1, defined by φ2(xε) = ) for every xε ∈ X. If φ2 is fuzzy completely Ce-I, then φ1 is fuzzy completely Ce-I.Let μ be a fuzzy e-closed set in Y; then, 1X × μ is a fuzzy e-closed set in X × Y. Since φ2 is fuzzy completely Ce-I, then φ1−1(μ) = φ2−1(1X × μ) is fuzzy regular open in X. Thus, φ1 is fuzzy completely Ce-I.Let φ1 : X → Y and φ2 : Y → Z: (a)φ1 : X → Y is fuzzy Ce-I and φ2 : Y → Z is fuzzy completely Ce-I, then φ2∘φ1 : X → Z is fuzzy e-irresolute.If (b)φ1 : X → Y is fuzzy completely Ce-I and φ2 : Y → Z is fuzzy Ce-continuous, then φ2∘φ1 : X → Z is fuzzy completely continuous.If The following holds for functions ξ is said to be fuzzy e-convergent to a fuzzy point xε in X if for any fuzzy e-open set η in X containing xε there exists a fuzzy set ρ ∈ ξ such that ρ ≤ η.A fuzzy filter base φ : X → Y is fuzzy completely Ce-I for each fuzzy point xε ∈ X and each fuzzy filter base ξ in X is fuzzy rc-convergent to xε, then the fuzzy filter base φ(ξ) is fuzzy e-convergent to φ(xε).If a fuzzy function xε ∈ X and let ξ be any fuzzy filter base in X which is fuzzy rc-converging to xε. Since φ is fuzzy completely Ce-I, then for any fuzzy e-open set η in Y containing φ(xε), there exists a fuzzy regular closed set ρ in X containing xε such that φ(ρ) ≤ η. Since ξ is fuzzy rc-converging to xε, there exists a δ ∈ ξ such that δ ≤ ρ. This means that φ(δ) ≤ ρ and therefore the fuzzy filter base φ(ξ) is fuzzy e-convergent to φ(xε).Let φ : X → Y is a fuzzy completely Ce-I surjection and X is fuzzy S-closed, then Y is fuzzy e-compact.If φ : X → Y is a fuzzy completely Ce-I surjection and X is fuzzy S-closed. Let {νi}i∈I be a fuzzy e-open cover of Y. Since φ is a fuzzy completely Ce-I, then {φ−1(νi)}i∈I is fuzzy regular closed cover of X and hence there exists finite set I0 of I such that X = ⋁{φ−1(νi); i ∈ I0}. Therefore, we have Y = ⋁{νi; i ∈ I0} and Y is fuzzy e-compact.Suppose that φ : X → Y is a fuzzy completely Ce-I injection and Y is fuzzy e-T1, then X is fuzzy weakly Hausdorff.If Y is fuzzy e-T1. For any distinct fuzzy points xε and yυ in X, there exist fuzzy e-open sets η and ρ in Y. Since φ is injective, φ(xε) ∈ η, φ(yυ) ∉ η, φ(xε) ∉ ρ, and φ(yυ) ∈ ρ. Since φ is fuzzy completely Ce-I, φ−1(η) and φ−1(ρ) are fuzzy regular closed sets of X such that xε ∈ φ−1(η), yυ ∉ φ−1(η), xε ∉ φ−1(ρ), and yυ ∈ φ−1(ρ). This shows that X is fuzzy weakly Hausdorff.Suppose φ : X → Y is a fuzzy completely Ce-I injection and Y is fuzzy e-normal, then X is fuzzy strongly normal.If η and ρ be disjoint nonempty fuzzy closed sets of X. Since φ is injective, φ(η) and φ(ρ) are disjoint fuzzy closed sets. Since Y is fuzzy e-normal, there exist fuzzy e-open sets μ and λ such that φ(η) ≤ μ and φ(ρ) ≤ λ and μ∧λ = 0. This implies that fe-cl(μ) and fe-cl(λ) are fuzzy e-closed sets in Y. Then, since φ is fuzzy completely Ce-I, φ−1(fe-cl(μ)) and φ−1(fe-cl(λ)) are fuzzy regular open sets. Then, η ≤ φ−1(fe-cl(μ)) and ρ ≤ φ−1(fe-cl(λ)) and φ−1(fe-cl(μ)) and φ−1(fe-cl(λ)) are disjoint; by definition X is fuzzy strongly normal.Let X, τ) is said to be fuzzy e-T0 or fuzzy e-closed (fuzzy regular closed) sets of Y.A fuzzy topological space (0(r-T0 ) if for φ : X → Y is a fuzzy completely Ce-I injection and Y is fuzzy e-T0, then X is fuzzy r-T0.If η be a any fuzzy set of X. Since Y is fuzzy e-T0, φ(η) is fuzzy e-open set of Y. Then, φ(η) = ⋁i∈I⋀j∈Jλij, where λij are fuzzy e-open set or fuzzy e-closed sets of Y. Since φ is completely Ce-I injection we have η = φ−1(φ(η)) = φ−1(⋁i∈I⋀j∈Jλij) = ⋁i∈I⋀j∈Jφ−1(λij), where φ−1(λij) are fuzzy regular open sets or fuzzy regular closed sets of X. Thus, X is fuzzy r − T0.Let φ : X → Y is a fuzzy completely Ce-I injection and Y is fuzzy e-T2, then X is fuzzy Urysohn.If xε and yυ be any two distinct fuzzy points in X. Since φ is injective, φ(xε) ≠ φ(yυ) in Y. Since Y is fuzzy e-T2, there exist fuzzy e-open sets η and ρ in Y such that φ(xε) ∈ η and φ(yυ) ∈ ρ and η∧ρ = 0. This implies that fe-cl(η) and fe-cl(ρ) are fuzzy e-closed sets in Y. Then, since φ is fuzzy completely Ce-I, there exists fuzzy regular open sets δ and γ in X containing xε and yυ, respectively, such that φ(δ) ≤ fe-cl(η) and φ(γ) ≤ fe-cl(ρ). This implies that δ ≤ φ−1(fe-cl(η)) and γ ≤ φ−1(fe-cl(ρ)); we have that φ−1(fe-cl(η)) and φ−1(fe-cl(ρ)) are disjoint and hence cl(δ)∧cl(γ) = 0; by definition, X is fuzzy Urysohn.Let


We also study soft β-continuous functions and discuss their relations with soft continuous and other weaker forms of soft continuous functions.We introduce the concepts soft  The concept of soft sets was first introduced by Molodtsov  in 1999 In recent years, an increasing number of papers have been written about soft sets theory and its applications in various fields , 4. Shabβ-open sets was introduced in   β-open sX be an initial universe and let E be a set of parameters. Let P(X) denote the power set of X and let A be a nonempty subset of E. A pair  is called a soft set over X, where F is a mapping given by F : A → P(X). In other words, a soft set over X is a parameterized family of subsets of the universe X. For ε ∈ A, F(ε) may be considered as the set of ε-approximate elements of the soft set .Let F, A) over X is called a null soft set, denoted by Φ, if e ∈ A, F(e) = ∅.A soft set  over X is called an absolute soft set, denoted by e ∈ A, F(e) = X.A soft set  over X for which Y(e) = Y, for all e ∈ E.Let F, A) and  over the common universe X is the soft set , where C = A ∪ B and for all e ∈ C,The union of two soft sets of  of two soft sets  and  over a common universe X, denoted by C = A∩B, and H(e) = F(e)∩G(e) for all e ∈ C.The intersection  and  be two soft sets over a common universe X. A ⊂ B, and H(e) = F(e) ⊂ G(e) for all e ∈ A.Let  is called a soft topological space over X. Let  be a soft topological space over X; then the members of τ are said to be soft open sets in X. The relative complement of a soft set  is denoted by c and is defined by c =  where Fc : A → P(X) is a mapping given by Fc(e) = X − F(e) for all e ∈ A. Let  be a soft topological space over X. A soft set  over X is said to be a soft closed set in X, if its relative complement c belongs to τ. If  is a soft topological space with τ is called the soft indiscrete topology on X and  is said to be a soft indiscrete topological space. If  is a soft topological space with τ being the collection of all soft sets which can be defined over X, then τ is called the soft discrete topology on X and  is said to be a soft discrete topological space.The triplet  be a soft topological space over X and let  be a soft set over X.F, A) is the soft set Reference : the sofF, A) is the soft set Reference : the sofLet ) is the smallest soft closed set over X which contains  and int) is the largest soft open set over X which is contained in .Clearly cl of a soft topological space  is said to besoft open  if its cα-open [soft α-open  if soft preopen  if soft semiopen  if β-open [soft β-open  if A soft set (α-closed. (b) Every soft α-open set is soft preopen. (c) Every soft α-open set is soft semiopen. (d) Every soft semiopen set is soft β-open. (e) Every soft preclosed set is soft β-open.(a) Every soft open set is soft The proof is obvious from We have following implications; however, the converses of these implications are not true, in general, as shown in X = {x1, x2, x3}, E = {e1, e2, e3}, and F1, E), , , ... are soft sets over X, defined as follows: F1, E) = {, , }, = {, , }, = {, }, = {, , }, = {}, = {, }, = {, , }, = {, }, = {, , }, = {, , }. is a soft topological space over X. Clearly the soft closed sets are Then F, E) = {, , }; then F, E) is soft α-open set but not soft open set  is not soft open set).Then, let us take  = {, , }; then int) = {, , }, G, E) is soft semiopen set but not soft α-open set.Now, let us take  = {, }; then cl) = c, int)) = {, , }, and so L, E) is soft preopen set but not soft α-open set.Now, let us take  is soft β-open set, but it is neither soft semiopen set nor soft preopen set.Then F, A) of a soft topological space  is said to be soft β-open [β-open set is called soft β-closed. Soft β-closure and soft β-interior of a soft set are defined as follows.Recall that a soft set X, τ, E) be a soft topological space and let  be a soft set over X.β-interior of a soft set  in X is denoted by Soft β-closure of a soft set  in X is denoted by Soft Let ) is the smallest soft β-closed set over X which contains  and sβint) is the largest soft β-open set over X which is contained in .Clearly β-open sets  of a soft topological space  by SβOS ).We will denote the family of all soft β-open sets is a soft β-open set. (2) Arbitrary intersection of soft β-closed sets is a soft β-closed set.(1) Arbitrary union of soft X, τ, E) be a soft topological space and let  be a soft set over X; thenF, A) ∈ SβCS⇔ = sβcl); ∈ SβOS⇔ = sβint).(Let ((1) Let F, A) is soft β-closed set.Hence  be soft β-closed set.Conversely, let  is a soft β-closed set, Since F, E)'s.Further, (2) Similar to (1).X, τ, E), the following hold for soft β-closure:sβcl(Φ) = Φ.sβcl) is soft β-closed set in  for each soft subset  of X., if In a soft space  be a soft topological space and let  and  be two soft sets over X; thensβcl))c = sβintc);))c = sβclc);(sβcl(Φ) = Φ and sβint(Φ) = Φ and sβcl)) = sβcl);sβint)) = sβint).Let  and  be two soft sets over X.F, A) ∈ SβCS})cF, A) ∈ SβCS}F, A)c ∈ SβOS} = sβintc).We have Similar to (1).Follows from definition.β-closed sets so sβcl(Φ) = Φ and Since Φ and β-open sets so sβint(Φ) = Φ and Since Φ and We have Similar to (6).sβcl) ∈ SβCS, so by sβcl)) = sβcl).Since sβint) ∈ SβOS, so by sβint)) = sβint).Since Let  the following are valid.F, A) is a soft set in X and  is a soft preopen set in X such that F, A) is a soft β-open set.If  is a soft preopen set we have (a) The proof is obvious. (b) Since  is a soft β-open set.Then X, τ, E) be soft topological space and let Y be an ordinary subset of X. Then τY = /Y :  ∈ τ) is a soft topology on Y and is called the induced or relative soft topology. The pair  is called a soft subspace of :  is called a soft open/soft closed/soft β-open soft subspace if the characteristic function of Y, namely, XY, is soft open/soft closed/soft β-open, respectively.Let  be a soft topological space. Suppose Y, τY) is a soft β-open soft subspace of . Then Z is soft β-open soft subspace in X if and only if Z is soft β-open soft subspace in Y.Let  and  be soft classes. Let u : X → Y and p : E → K be mappings. Then a mapping f : → is defined as follows: for a soft set  in , , B),  B = p(A)⊆K is a soft set in  given by f(β) = u(∪F(α)α∈p−1(β)∩A) for β ∈ K. , B) is called a soft image of a soft set . If B = K, then we will write , K) as f.Let → be a mapping from a soft class  to another soft class , and let  be a soft set in soft class , where C⊆K. Let u : X → Y and p : E → K be mappings. Then , D), D = p−1(C), is a soft set in the soft classes , defined as follows: f−1(α) = u−1(G(p(α))) for α ∈ D⊆E.  , D) is called a soft inverse image of . Hereafter we will write , E) as f−1.Let f : →, u : X → Y, and p : E → K be mappings. Then for soft sets ,  and a family of soft sets  in the soft class , we have the following:f(Φ) = Φ,if f−1(Φ) = Φ,if Let X and Y stand for soft topological spaces with  and ) assumed unless otherwise stated. Moreover, throughout this paper, a soft mapping f : X → Y stands for a mapping, where f : →, u : X → Y, and p : E → K are assumed mappings unless otherwise stated.Throughout the paper, the spaces f : X → Y is called soft β-continuous [α-continuous [Y is soft β-open  set in X.A soft mapping ntinuous   if the We have the following implications; however, the converses of these implications are not true, in general, as shown in X = {x1, x2, x3}, Y = {y1, y2, y3}, E = {e1, e2, e3}, and K = {k1, k2, k3} and let  and  be soft topological spaces.Let u : X → Y and p : E → K as  u(x1) = {y1}, u(x2) = {y3}, u(x3) = {y2}, p(e1) = {k2}, p(e2) = {k1}, p(e3) = {k3}.Define τ on X given in   F, K) = {, , } and mapping;→ is a soft mapping. Then  is a soft open set in Y; f−1) = {, , } is a soft α-open set but not soft open set in X.Let us consider the soft topology f is a soft α-continuous function but not soft continuous function.Therefore, X = {x1, x2, x3}, Y = {y1, y2, y3}, E = {e1, e2, e3}, and K = {k1, k2, k3} and let  and  be soft topological spaces.Let u : X → Y and p : E → K as mapping given in τ on X given in  G, K) = {, } and mapping; f : → is a soft mapping. Then  is a soft open set in Y; f−1) = {, } is a soft preopen set but not soft α-open set in X. Thus, f is a soft precontinuous function but not soft α-continuous function.Let us consider the X = {x1, x2, x3}, Y = {y1, y2, y3}, E = {e1, e2, e3}, and K = {k1, k2, k3} and let  and  be soft topological spaces.Let u : X → Y and p : E → K as mapping given in τ on X given in  L, K) = {, } and mapping; f : → is a soft mapping. Then  is a soft open set in Y; f−1) = {, , } is a soft semiopen set but not soft α-open set in X. Hence, f is a soft semicontinuous function but not soft α-continuous function.Let us consider the X = {x1, x2, x3}, Y = {y1, y2, y3}, E = {e1, e2, e3}, and K = {k1, k2, k3} and let  and  be soft topological spaces.Let u : X → Y and p : E → K as mapping given in τ on X given in   f : → is a soft mapping. Then  is a soft open set in Y; β-open set, but it is neither soft semiopen set nor soft preopen set in X.Let us consider the f is a soft β-continuous function, but it is neither soft semicontinuous function nor soft precontinuous function.Therefore, f : X → Y be a function. f is called soft β-irresolute if the inverse image of soft β-open set in Y is soft β-open in X.Let f : X → Y be a function. f is called soft β-open if the image of each soft β-open set in X is soft β-open in Y.Let f : → be a soft continuous and soft open set. Then f is soft β-open set.Let F, A) be any soft β-open set. Then   f) is soft β-open.⇒Let  is soft closed and  is soft β-open then β-open.If → be soft continuous and soft open. Then f is soft β-irresolute.Let F, A) be any soft β-open set in Y. Then f is soft continuous and soft open it follows that  f−1))))  = cl)))). This shows that f−1) is soft β-open. = cl of Y, f−1) is soft β-closed.A function X, τ, E) the following are valid:F, A) is soft β-open ⇔sβint) = . is soft β-closed ⇔sβcl) = .→ is soft β-irresolute if and only if for every soft set  of X, f is soft β-irresolute. Now f)) is soft β-closed set. By hypothesis f−1))) is soft β-closed set.Suppose that β-closure, And That is F, A) is soft β-closed set in Y. Now by hypothesis f−1) = sβcl)).Conversely, suppose that ) is soft β-closed set and so f is soft β-irresolute.That is f : X → Y is soft β-irresolute if and only if for all soft sets  of Y, f is soft β-irresolute. Now sβcl) is soft β-closed set so that f−1)) is soft β-closed set. Since β-closure that Suppose F, K) is soft β-closed set in Y. Then sβcl) = .Conversely suppose that ))  = f−1).Now by hypothesis sβcl)) = f−1)) = f−1).Therefore, f−1) is soft β-closed set and so f is soft β-irresolute.Thus, The following results are easy to establish.f : X → Y and g : Y → Z are both soft β-irresolute. Then gof : X → Z is soft β-irresolute.Suppose f : X → Y be soft continuous and soft open. Thenf is soft β-irresolute;f−1)) = sβcl))), with  being a soft set in Y.Let X and Y be soft topological spaces. X and Y are said to be M-soft β-homeomorphic if and only if there exists f : X → Y such that f is 1-1, onto, M soft β-continuous and soft β-open. Such an f is called soft β-homeomorphism.Let f : X → Y is soft β-homeomorphism, then f−1)) = sβcl)), where  is a soft set in Y.If f : X → Y is a soft β-homeomorphism, thensβcl)) = f)),f)) = sβint)),f−1)) = sβint)).If β-interior and soft β-closure of a soft set in topological spaces and study some of their properties. We also introduce the concept of soft β-open sets and soft β-continuous functions in topological spaces and some of their properties have been established. We hope that the findings in this paper are just the beginning of a new structure and not only will form the theoretical basis for further applications of topology on soft sets but also will lead to the development of information system and various fields in engineering.In this paper, we introduce the concept of soft