Set-Valued Mappings

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Fixed set of set valued mappings with set valued domain in terms of start set on a metric space with a graph

Fixed set of set valued mappings with set valued domain in terms of start set on a metric space with a graph

Some influential work on the same subject, which we must take into account, is [–]. In the present paper, our goal is to consider the mapping T : CB(X) → CB(X) instead of T : X → X or T : X → CB(X) to study fixed point results of set valued mappings. The cur- rent corresponding author [] generalized some results of Jachymski [] by considering maps of the type T : CB(X) → CB(X), i.e., set valued contraction maps with set valued do- main. In a similar setting, recently, more fixed point results were investigated by Abbas et al. [, ]. In the current paper, we introduce the new concept of fixed set which is anal- ogous to the concept of fixed point for a multivalued map. We establish some existence results for fixed sets by defining another concept called the start set of a graph whose ver- tices are closed and bounded subsets of a metric space. Our results generalize and extend some existing results in the literature, especially those of [].

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Bilevel minimax theorems for non continuous set valued mappings

Bilevel minimax theorems for non continuous set valued mappings

ping and in [] for vector-valued mapping, respectively. Scalar minimax theorems and set minimax theorems for non-continuous set-valued mappings were first proposed by Lin et al. []. These results can be compared with the recent existing results [, ]. In this paper, we establish bilevel minimax results with a couple of non-continuous set-valued mappings (Theorem . in Section , Theorems .-. in Section ). These results might not hold for each individual non-continuous set-valued mapping since it always lack some conditions so that the existing minimax theorems are not applicable, such as Theorems .-. [], Theorem . [] or Proposition . [].

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Generalized hierarchical minimax theorems for set valued mappings

Generalized hierarchical minimax theorems for set valued mappings

Recently, Balaj [] proposed some minimax theorems for four real-valued functions by using some new alternative principles. Inspired by [–] we shall study some gener- alized hierarchical minimax theorems for set-valued mappings. The imposed conditions involve four set-valued mappings. In the second section, we introduce some notions and preliminary results. In the third section, we prove the hierarchical minimax theorem for scalar set-valued mappings. In the fourth section, we show some hierarchical minimax theorems for set-valued mappings in Hausdorff topological vector spaces by using the re- sults obtained in the previous section.

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Set valued mappings in partially ordered fuzzy metric spaces

Set valued mappings in partially ordered fuzzy metric spaces

Hence x ¯ ∈ F x ¯ = F x, consequently ¯ x ¯ is a fixed point of F . The theorem is proved. Corollary . Let (X, M, ∗) be a complete fuzzy metric space with Lukasiewicz t-norm and ‘ ’ be a partial order defined on X. Let Y ∈ C(X) and F : Y → C(Y) be a set-valued mapping with the property that there is α ∈ (, ) such that

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Generalized vector quasi equilibrium problems with set valued mappings

Generalized vector quasi equilibrium problems with set valued mappings

Throughout this paper, let Z , E , and F be topological vector spaces, let X ⊆ E and Y ⊆ F be nonempty, closed, and convex subsets. Let D : X → 2 X , T : X → 2 Y and Ψ : X × Y × X → 2 Z be set-valued mappings, and let C : X → 2 Z be a set-valued mapping such that C ( x ) is a closed pointed and convex cone with int C ( x ) = ∅ for each x ∈ X , where int C ( x ) denotes the interior of the set C(x). Then the generalized vector quasi-equilibrium prob- lem with set-valued mappings (GVQEP) is to find (x, y) in X × Y such that

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Stationary points for set valued mappings on two metric spaces

Stationary points for set valued mappings on two metric spaces

Example 2.5 . Let X = { 0 } ∪ { 1/n : n ≥ 1 } = Y with the usual metric. Define map- pings S, T by T 0 = { 1 } , T (1/n) = { 1/2n } for n ≥ 1 and S = T . It is easy to prove that all the conditions of Theorems 2.1 and 2.2 are satisfied except that the mappings S and T are continuous. But ST and T S have no stationary points.

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FIXED POINTS OF CONTRACTIVE SET VALUED MAPPINGS WITH SET VALUED DOMAINS ON A METRIC SPACE WITH GRAPH

FIXED POINTS OF CONTRACTIVE SET VALUED MAPPINGS WITH SET VALUED DOMAINS ON A METRIC SPACE WITH GRAPH

Let (X, d) be a metric space and ∆ = { (x, x) : x ∈ X } denote the diagonal of the Cartesian product X × X. Consider a directed graph G such that the set of its vertices coincides with X (i.e., V (G) = X) and the set of its edges E(G) is such that ∆ ⊆ E(G). We assume G has no parallel edges and thus we identify G with the pair (V (G), E(G)).

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Common fixed points of set valued mappings

Common fixed points of set valued mappings

Mappings satisfying the inequality (1.1) with a = 1 and b = c = 0 is called nonexpan- sive and it was considered by Kirk [6], whereas the mapping with a = 0, b = c = 1/2 by Wong [13]. Recently, Fisher et al. [3], Diviccaroet al. [2], Mukherjee et al. [9], and Murthy et al. [10] generalized Theorem 1.1 in many ways. In this context, we prove a common fixed point theorem for set-valued mappings using Greguš type condition. Before presenting our main theorem we need the following definitions and lemma for our main theorem.

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On locally contractive fuzzy set valued mappings

On locally contractive fuzzy set valued mappings

The Banach contraction theorem and its subsequent generalizations play a fundamental role in the field of fixed point theory. In particular, Heilpern introduced in [] the no- tion of a fuzzy mapping in a metric linear space and proved a Banach type contraction theorem in this framework. Subsequently several other authors [–] have studied and established the existence of fixed points of fuzzy mappings. The aim of this paper is to prove a common fixed-point theorem for a sequence of fuzzy mappings in the context of metric spaces without the assumption of linearity. Our results generalize and unify several typical theorems of the literature.

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The existence and stability for weakly Ky Fan’s points of set valued mappings

The existence and stability for weakly Ky Fan’s points of set valued mappings

In the same way, Corollary . and Corollary . can be promoted respectively as follows. Corollary . Let X be a nonempty convex subset of a Hausdorff topological vector space E, C is a closed, convex, pointed cone with int C = ∅. If a vector-valued function ϕ : X × X → H satisfies the following conditions:

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Related fixed points for set valued mappings on two metric spaces

Related fixed points for set valued mappings on two metric spaces

We let (X,d) be a complete metric space and let B(X) be the set of all nonempty sub- sets of X. As in [1, 2], we define the function δ(A,B) with A and B in B(X) by δ(A,B) = sup{d(a,b) : a ∈ A,b ∈ B}. If A consists of a single point a we write δ(A,B) = δ(a,B). If B also consists of single point b, we write δ(A,B) = δ(a,B) = δ(a,b) = d(a,b). It follows immediately that δ(A,B) = δ(B,A) ≥ 0, and δ(A,B) ≤ δ(A,C)+ δ(C,B) for all A, B in B(X).

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Common fixed point of multivalued graph contraction in metric spaces

Common fixed point of multivalued graph contraction in metric spaces

may depend on x) and is a fixed point of T. Let (X, d) be a metric space and G be a directed graph with set V (G) of its vertices coincides with X, and the set of its edges E(G) is such that (x, x) 6∈ E(G). Assume G has no parallel edges, we can identify G with the pair (V (G), E(G)), and treat it as a weighted graph by assigning to each edge the distance between its vertices. By G −1 we denote the conversion of a graph G, i.e., the graph obtained from G by reversing the direction of the edges. Thus we can write

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Gregus type fixed point results for tangential mappings satisfying contractive condition in fuzzy metric spaces

Gregus type fixed point results for tangential mappings satisfying contractive condition in fuzzy metric spaces

Bhaskar and Lakshmikantham [11] introduced the concepts of coupled fixed points and mixed monotone property and illustrated these results by proving the existence and uniqueness of the solution for a periodic boundary value problem. Later on these results were extended and generalized by Sedghi et al. [10] , Fang [4] etc. Recently, Abbas et.al. [1] proved coupled common fixed point theorem for a hybrid pair of mappings satisfying w-compatible in complete metric space.

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Characterizations of weak-ANODD set-valued mappings with applications to an approximate solution of GNMOQV inclusions involving ⊕ operator in ordered Banach spaces

Characterizations of weak-ANODD set-valued mappings with applications to an approximate solution of GNMOQV inclusions involving ⊕ operator in ordered Banach spaces

Let X be a real ordered Banach space with a norm · , a zero θ , a normal cone P, a normal constant N and a partial ordered relation ≤ defined by the cone P. For arbitrary x, y ∈ X, lub { x, y } and glb { x, y } express the least upper bound of the set { x, y } and the greatest lower bound of the set {x, y} on the partial ordered relation ≤, respectively. Suppose that lub{x, y} and glb{x, y} exist. Let us recall some concepts and results.

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Fixed point result and applications on a b-metric space endowed with an arbitrary binary relation

Fixed point result and applications on a b-metric space endowed with an arbitrary binary relation

On the other hand, Samet et al. [] introduced the concept of α-admissible mapping and using this concept proved a fixed point theorem for a single-valued mapping. They also showed that these results can be utilized to derive fixed point theorems in partially ordered spaces and coupled fixed point theorems. Moreover, they applied the main re- sults to ordinary differential equations. Recently, Mohammadi et al. [] introduced the

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Convergence of Inexact Iterative Schemes for Nonexpansive Set-Valued Mappings

Convergence of Inexact Iterative Schemes for Nonexpansive Set-Valued Mappings

In this section we show that if for any initial point, there exists a trajectory of the dynamical system induced by a nonexpansive set-valued mapping T, which converges to an invariant set F, then a convergent trajectory also exists for a nonstationary dynamical system induced by approximations of T .

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Ky Fan minimax inequalities for set-valued mappings

Ky Fan minimax inequalities for set-valued mappings

Shi and Ling [13] proved, respectively, a minimax theorem and a cone saddle point theorem for a class of vector-valued functions, which include the separated functions as its proper subset. Ferro [14,15] studied minimax theorems for general vector-valued functions. Gong [16] obtained a strong minimax theorem and established an equivalent relationship between the strong minimax inequality and a strong cone saddle point theorem for vector-valued functions. Li et al. [17] investigated a minimax theorem and a saddle point theorem for vector-valued functions in the sense of lexicographic order, respectively.

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On stationary points of nonexpansive set-valued mappings

On stationary points of nonexpansive set-valued mappings

have appeared in a natural way. See, for instance, [, ] and the notion of a strong ap- proximate fixed point sequence (which we call an approximate stationary point sequence here) in []. In the present work we show that some of the very well-known properties implying the existence of fixed points for nonexpansive single-valued mappings also imply the existence of stationary points in the set-valued case provided approximate stationary point sequences exist.

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Existence and Approximation of Fixed Points for Set-Valued Mappings

Existence and Approximation of Fixed Points for Set-Valued Mappings

Taking into account possibly inexact data, we study both existence and approximation of fixed points for certain set-valued mappings of contractive type. More precisely, we study the existence of convergent iterations in the presence of computational errors for two classes of set-valued mappings. The first class comprises certain mappings of contractive type, while the second one contains mappings satisfying a Caristi-type condition.

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Fixed point of set valued graph contractive mappings

Fixed point of set valued graph contractive mappings

We begin with the following theorem that gives the existence of a common fixed point (not necessarily unique) in metric spaces endowed with a graph for the set-valued mappings. Further, we assume that (X, d) is a complete metric space and G is a directed graph such that E(G) is symmetric.

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