Common fixed points under strict conditions

In this contribution, three new concepts called reciprocally continuous, strictly subweakly compatible and strictly subreciprocally continuous single and multivalued mappings are given for obtention some common fixed point theorems in a metric space. Our results improve and complement the results of Aliouche and Popa, Azam and Beg, Deshpande and Pathak, Kaneko and Sessa, Popa and others.


Introduction and preliminaries
Let (X , d) be a metric space. We denote by CL(X ) (resp., CB(X )) the nonempty closed (resp., closed and bounded) subsets of X and H the Hausdorff metric on CL(X ) (resp., CB(X )) Now, let f and g be two self-mappings of a metric space (X , d). In 1982, Sessa [11] gave the weaker concept of the commutativity, namely the weakly commuting notion. f and g are weakly commuting if for all x ∈ X .
In 1986, Jungck [5] gave a generalization of the weak commutativity by giving the notion of compatible mappings. He defined f and g to be compatible if lim n→∞ d(f gx n , gf x n ) = 0, whenever {x n } is a sequence in X such that lim n→∞ f x n = lim n→∞ gx n = t for some t ∈ X .
Weakly commuting mappings are compatible. However, compatible mappings need not be weakly commuting (see example 2.2 of [13]).
In 1996, Jungck [6] generalized the above notion by introducing the concept of weakly compatible mappings. He defined f and g to be weakly compatible if they commute at their coincidence points; i.e., if f u = gu for some u ∈ X , then f gu = gf u.
If f and g are compatible then they are obviously weakly compatible but as shown in example 2.52 of [1] the converse is not true.
In their paper [8], Kaneko and Sessa extended the definition of compatibility to include multivalued mappings in the following way: mappings f : X → X and F : To generalize the above notion, Jungck and Rhoades [7] gave the concept of weakly compatible mappings. f : X → X and F : X → CB(X ) are said to be weakly compatible if they commute at their coincidence points; i.e., if Recall that a point t ∈ X is called a strict coincidence point (resp. strict common fixed point) of mappings f : X → X and F : X → CB(X ) if F t = {f t} (resp. F t = {f t} = {t}).

Main results
Our first objective in this contribution is to generalize the above definition by introducing the concept of strictly subweakly compatible single and multivalued mappings. The example below shows that there exist sswc mappings which are not weakly compatible. .
We have f F x ∈ CB(X ). Consider the sequence {x n } in X defined by x n = 2 + 1 n for n = 1, 2, . . . , we have lim , therefore f and F are not weakly compatible.
In 1999, Pant [9] introduced the concept of reciprocally continuous singlevalued mappings as a generalization of continuous mappings: f and g are reciprocally continuous if and only if lim In 2002, Singh and Mishra [12] introduced the concept of reciprocal continuity for single and multivalued mappings as follows.
Definition 2. The mappings F : X → CL(X ) and f : X → X are reciprocally continuous on X (resp., at t ∈ X ) if and only if f F x ∈ CL(X ) for each x ∈ X ( resp., f F t ∈ CL(X )) and lim Motivated by Pant, Singh and Mishra, we give the following notion of reciprocally continuous single and multivalued mappings which is different from the above definition and represents our second objective.
Our third objective here is to extend the concept of reciprocally continuous mappings of Pant and the above one to the setting of single and multivalued mappings. The next example shows that there exist ssrc mappings which are not continuous.
Example 2. Let X = R. Define f : X → X and F : X → CB(X ) by It is clear to see that f and F are discontinuous at x = 0. Consider the sequence {x n } in X defined by x n = 1 n for n = 1, 2, . . . We have therefore f and F are ssrc. Now, we are ready to present and prove our main result. Theorem 1. Let (X , d) be a metric space. Let f , g : X → X and F , G : X → CB(X ) be single and multivalued mappings respectively such that f and F as well as g and G are reciprocally continuous and sswc or ssrc and compatible. Let ϕ : R 6 + → R be a lower semi continuous function satisfying: (ϕ 1 ): ϕ is nonincreasing in variables t 5 and t 6 , (ϕ 2 ): ϕ(u, u, 0, 0, u, u) > 0 for all u > 0 and the inequality for all x and y in X , then, f , g, F and G have a strict common fixed point in X .
Proof. Since f and F as well as g and G are reciprocally continuous and sswc or ssrc and compatible then, there exist two sequences {x n } and {y n } in X such that lim   Letting n tends to infinity and taking in account that ϕ is lower semi continuous, we get which contradicts ϕ 2 . Then f t = t.
Next, we prove that z = t. Indeed, by (1) we have When n tends to infinity, we get which contradicts (ϕ 2 ), therefore z = t. Consequently, t is a strict common fixed point of f , g, F and G.
Corollary 1. Let (X , d) be a metric space. Let f , g : X → X and F , G : X → CB(X ) be single and multivalued mappings respectively such that f and F as well as g and G are reciprocally continuous and sswc or ssrc and compatible. Assume that where m ∈ (0, 1), or where m 2 ∈ (0, 1), or where a, b, c > 0 and a + b + c < 1.
Then f , g, F and G have a strict common fixed point.

Remark 1.
(1) Our main result improves the main result of Popa [10]. (2) By the above corollary and (2) for f = g and F = G, we obtain an extension of the main result of Kaneko and Sessa [8]. Popa [2], Deshpande and Pathak [4] because, in our work, we have not continuity, neither completeness nor inclusion, and we did not impose a lot of conditions on the four mappings.
Example 3. Let X = [0, 2] endowed with the Euclidean metric, define mappings f , g, F and G as follows: i.e., g and G are ssrc and compatible.
By taking m = 4 5 in corollary 1, we show that inequality (2) is satisfied. We have the following cases: (1) For x, y ∈ [0, 1], we have H(F x, Gy) = 0, obviously inequality (2)  then all hypotheses of corollary 1 are satisfied and the point 1 is a strict common fixed point for f , g, F and G.
Corollary 2. Let f , g : X → X be single valued mappings and let F , G : X → CB(X ) be multivalued mappings on a metric space (X , d) such that the pairs f and F as well as g and G are ssrc and compatible for all x, y ∈ X . Then the pair of mappings (f, F ) and (g, G) has a strict coincidence point. Moreover f , g, F and G have a strict common fixed point in X provided that mappings satisfy where φ : R 5 + → R + is an upper semi continuous function such that φ(0) = 0 and φ(t, 0, 0, t, t) < t for each t > 0.