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 [].

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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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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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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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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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.

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)).

**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.

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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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:

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).

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

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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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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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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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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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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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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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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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.