A common fixed point theorem for multi-valued θ_{δ} contractions via subsequential continuity

The main objective of this paper is to present a common fixed point theorem for two pairs of single and set valued mappings via subsequential continuity and \delta- compatibility. To illustrate the validity of our results, an example is provided and we give also an application for a system of integral inclusions of Volterra type.


Introduction
Fixed point theory is one of the important tools in the study of several problems in non linear analysis, physics, economics,.... Starting from Banach principle, some results and generalizations were given in this way. A common …xed point theorem generally involves conditions on commutativity, continuity and contractive condition of the given mappings, with completeness, or closedness of the underlying space or subspaces, along with conditions on suitable containment amongst the ranges of involved mappings. Sessa [20] has weakened the notion of commuting mappings to weakly commuting, later Jungck [14] introduced the concept of compatibility for a pair of self maps, which was extended to hybrid pair of mappings by Kaneko and Sessa [16]. Afterwards Jungck et al. [15] have furnished an extension to compatible mappings notion, called weak compatibility in the setting of single-valued and 1474 A. ALI, S. M AHIDEB, S. BELOUL multi-valued mappings. Recently, Bouhadjera and Godet Tobie [7] introduced subsequential continuity which is weaker than the reciprocal continuity introduced by Pant [19]. In fact every non-vacuously pair of reciprocally continuous maps is naturally subsequentially continuous. However subsequentially continuous mappings are neither sequentially continuous nor reciprocally continuous. Quite recently, Beloul et al. [5] extended the notion of subsequential continuity to the context of set value maps in order to establish a common …xed point by using Hausdor¤ distance, while there is a function called -distance which de…ned by Fisher [10], although -distance is not a metric like the Hausdor¤ distance, but shares most of the properties of a metric, some results on -distance can be found in [1,4,6]. Common …xed point theorem commonly require commutativity, continuity, completeness together with a suitable condition on containment of ranges of involved maps beside an appropriate contraction condition. Thus, research in this …eld is aimed at weakening one or more of these conditions. In this paper we will utilize a -contraction introduced by Jleli and Samet [12] and -distance to establish a strict coincidence and a strict common …xed point of a -compatible and subsequentially hybrid pair of mappings, without continuity or reciprocal continuity, weak reciprocal continuity, completeness and containment of ranges.

Preliminaries
Let (X; d) be a metric space, B(X) is the set of all non-empty bounded subsets of X.

De…nition 1.
[20] Two mappings S : X ! B(X) and f : X ! X are to be weakly commuting on X if f Sx 2 B(X) and for all x 2 X: (Sf x; f Sx) maxf (f x; Sx); diam(f Sx)g: whenever fx n g is a sequence in X such that f Sx n 2 B(X); lim n!1 Sx n = fzg, and lim n!1 f x n = z, for some z 2 X: where lim n!1 f x n = lim n!1 gx n = t, for some t in X.
Later, Singh and Mishra [21] generalized the concept of reciprocal continuity to the setting of single and set-valued maps as follows. for some t 2 X.
The pair (f; g) of self mappings is said to be subsequentially continuous if there exists a sequence fx n g in X such that lim n!1 f x n = lim n!1 gx n = z; for some z 2 X and lim n!1 f gx n = f z; lim n!1 gf x n = gz: [5] Let f : X ! X and S : X ! CB(X) two single and set-valued mappings respectively, the hybrid pair (f; S) is to be subsequentially continuous if there exists a sequence fx n g such that for some z 2 X and lim n!1 f Sx n = f M; lim n!1 Sf x n = Sz: Notice that continuity or reciprocal continuity implies subsequential continuity, but the converse may be not.
We consider a sequence fx n g such that for each n 1 we have: On the other hand, consider a sequence fy n g which de…ned for all n 1 by: then f and S are never reciprocally continuous.
Example 9. For all i 2 f1; 2; 3g, the following functions are elements of .
De…nition 10. [12] Let (X; d) be a metric space and T : X ! X be a mapping.
Theorem 11. [12] Let (X; d) be a complete metric space and let T : X ! X be an -contraction. Then T has a unique …xed point in X.

Main results
In this section, we introduce a multivalued -contraction and prove a common …xed point theorem for hybrid pair mappings with -distance. Now we extend the last de…nition for two pairs of hybrid pair, in order to establish a common …xed point for set valued and single valued mapping in metric space, without continuity and completeness of space, we use only subsequential continuity with -compatibility.
Theorem 14. Let f; g : X ! X be single valued mappings and S; T : X ! B(X) be multi-valued mappings of metric space (X; d). If the two pairs (f; S) and (g; T ) are subsequentially continuous and -compatible. Then the pair (f; S) as well as (g; T ) has a strict coincidence point. Moreover, f; g; S and T have a common strict …xed point provided that there exists k 2 (0; 1) such that for all x; y in X we have: T gy n = T t: The pair (g; T ) is -compatible, implies that lim n!1 (gT y n ; T gy n ) = (gN; T t) = 0: Then gN = T t and T t is a singleton, i.e, T t = fgtg and t is a strict coincidence point of g and T . Now, we claim f z = gt, if not so (Sz; T t) > 0, otherwise which is a contradiction. Then by using (1), we get: which contradicts that z 6 = f z, and so we have ( (Sx n ; T t)) [ (R(x n ; t))] k : Letting n ! 1, we get M (x n ; t) ! d(z; f z) and so we have: which is a contradiction. Hence z is a …xed point for f and S. We will show z = t, if not by taking x = x n and y = y n in (1), (Sx n ; T y n ) > 0, if not letting n ! 1, we get: which is a contradiction, so we have: Taking n ! 1, we get: which is a contradiction. Hence z = t and consequently z is a common …xed point for f; g; S and T . For the uniqueness, let w be another …xed point, by and using (1), (Sz; T w) > 0, if not d(z; t) (Sz; T t) = 0, which is a contradiction. Then we have: (d(z; w)) < ( (Sz; T w)) [ (d(z; w))] k < (d(z; w)); which is a contradiction. Then z is unique.
If f = g and S = T we obtain the following corollary: Corollary 15. Let f : X ! X be a single valued mapping and S : X ! B(X) be a multi-valued mapping of metric space (X; d). Suppose that the pair (f; S) is subsequentially continuous, as well is -compatible and there exist 2 and k 2 [0; 1) such that for all x; y in X we have: Consider a sequence fx n g for all n 1 such that x n = 1 1 n , it is clear that lim (e R(x;y) ) 9 10 : hence f and T satisfy (1), therefore 1 is the unique common strict …xed point of f and S.

Application to integral inclusions
In this section, we apply the obtained results to assert the existence of solution for a system of integral inclusions. Consider the following integral inclusions system's.
where g is a continuous function on Clearly X = C([0; 1]) with convergence uniform metric's d 1 (x; y) = sup x2X jx(t) y(t)j is a complete metric space. De…ne two set valued mappings: s; x 2 (s))dsg: Assume that;  Denote I X the identity operator on X.
From condition (4), the two pairs (I X ; S) and (I X ; T ) are subsequentially continuous as well as -compatible.