A Common Fixed Point Theorem for Weakly Compatible Multi-Valued Mappings Satisfying Strongly Tangential Property

The concept of compatibility was been introduced and used by G. Jungck [8] to prove the existence of a common fixed point, this notion generalizes the weakly commuting, further there are various type of compatibility, compatibility of type (A),of type (B),of type (C) and of type (P) for two self mappings f and g of metric space (X, d) was introduced respectively in [10], [18],[17] and [16] as follows: the pair{f, g} is compatible of type (A) if


Introduction
The concept of compatibility was been introduced and used by G. Jungck [8] to prove the existence of a common fixed point, this notion generalizes the weakly commuting, further there are various type of compatibility, compatibility of type (A),of type (B),of type (C) and of type (P) for two self mappings f and g of metric space (X, d) was introduced respectively in [10], [18], [17] and [16] as follows: the pair{f, g} is compatible of type (A) if lim n→∞ d(f gx n , g 2 x n ) = 0 and lim n→∞ d(gf x n , f 2 x n ) = 0, f and g are compatible of type (B) if In 1996 , Jungck [11] introduced a concept which generalizes the all above type of compatibility and it is weaker than them: two self mappings of metric space (X, d) into itself are to be weakly compatible if they are commute at their coincidence points, i.e if f u = gu for some u ∈ X, then f gu = gf u.Let (X, d) be a metric space, CB(X) is the set of all non-empty bounded closed subsets of X.For all A, B ∈ CB(X) the metric of Hausdorff defined by: For all a ∈ A, we have d(a, B) ≤ H(A, B).

Preliminaries
H. Kaneko and S. Sessa [13] extended the concept of compatibility to the setting of single and set-valued maps as follows: Let f : X → X and S : X → CB(X) two single and set-valued mappings, the pair {f, S} is to be compatible if for all x ∈ X, f Sx ∈ CB(X) and Jungck and Rhoades [8] generalised the concept of weak compatibility to setting of single and set valued mappings: Definition 2.1.Two single mapping f : X → X and set valued mapping S : X → CB(X) of metric space (X, d) are said to be weakly compatible if they commute at their coincidence point, i.e if f u ∈ Su for some u ∈ X, then f Su = Sf u.
Recently, Al-Thagafi and Shahzad [4] introduced the notion of occasionally weakly compatible maps in metric spaces: Two self mappings f and g of a metric space (X, d) are to be occasionally weakly compatible (owc) if and only if there is a point u ∈ X such that f u = gu and f gu = gf u.
Notice that the weak compatibility implies occasional weakly compatibility, the converse may be not.Later, Abbas and Rhoades [1] extended the occasionally weakly compatible mappings to the setting of single and set-valued mappings: Definition 2.2.Two mappings f : X → X and S : X → CB(X) are said to be owc if and only if there exists some point u in X such that f u ∈ Su and f Sz ⊆ Sf z.
Example 2.2.Let X = [0, ∞) and d is the euclidian metric, we define f and S as follows: , then f and S are owc.
Pathak and Shahzad [16] introduced the concept of tangential property as follows: Let f, g : X → X two self mappings of metric space (X, d), a point z ∈ X is said to be a weak tangent point to (f, g) if there exist two sequences In 2011, W. Sintunavarat and P. Kumam [25] extended the last notion for single and multi valued maps: Definition 2.3.Let f, g : X → X be single mappings and S, T : X → B(X) two multi-valued mappings on metric space (X, d), the pair {f, g} is said to be tangential with respect to {S, T } if there exists two sequences {x n }, {y n } in X such that S. Chauhan, M. Imdad, E. Karapinar and B. Fisher [6] introduced a generalization to the last notion by adding another condition as follows: Definition 2.4.Let f, g : X → X be single valued mappings and S, T : X → CB(X) two multi-valued mappings on metric space (X, d), the pair {f, g} is said to be strongly tangential with respect to {S, T } if there exists two sequences {x n }, {y n } in X such that Example 2.3.Let ([0, 4] and d the euclidian metric, we define f, g, S and T by: We have f (X) = [1, 3] and g(X) = [0, 5], then f (X) ∩ g(X) = [1,3].Consider two sequences {x n }, {y n } which defined for all n ≥ 1 by: Clearly that , then {f, g} is strongly tangential with respect to {S, T }.
If in Definition 2.4 we have S = T and f = g we get to the following definition: Definition 2.5.Let f : X → X and S : X → B(X) two mappings on metric space (X, d), f is said to be strongly tangential with respect to S if Example 2.4.Let ([0, 2] with the euclidian metric, f and S defined by: Consider two sequence {x n }, {y n } which defined for all n ≥ 1 by: x n = 1 n , y n = 1 + 1 n , we have: and Let Φ be the set of all upper semi continuous functions φ : R 5 + → R + satisfying the conditions: (φ 1 ): φ is non decreasing in each coordinate variable.(φ 2 ): For any t > 0, ψ(t) = max φ(0, t, 0, 0, t), φ(0, 0, t, t, 0), φ(t, 0, 0, t, t) < t.
The aim of this paper is to prove the existence of a common fixed point for weakly compatible single and set valued mappings in metric space, which satisfying a contractive condition of integral type by using the strongly tangential property, our results generalize and extend some previous results.

Main results
Theorem 3.1.Let f, g : X → X, be single valued mappings and S, T : X → CB(X) multi-valued mappings of metric space (X, d) such for all x, y in X we have: (1) where φ ∈ Φ and ϕ : R + → R + is a Lebesgue-integrable function which is summable on each compact subset of R + , non-negative, and such that for each ε > 0, ε 0 ϕ(t)dt > 0. Suppose that the two pairs {f, S}, {g, T } are weakly compatible and {f, g} is strongly tangential with respect to {S, T }, then f, g, S and T have a unique common fixed point in X.
Proof.Suppose {f, g} is strongly tangential with respect to {S, T }, then there exists two sequences {x n }, {y n } such and z ∈ f (X) ∩ g(X), then there exists u, v ∈ X such z = f u = gv, now we claim z ∈ Su, if not by using (1) we get which is a contradiction with (φ 2 ), then d(z, Su) = 0 and so z ∈ Su.
We claim z = gv ∈ T v, if not and using (1) we get: which is a contradiction, then z ∈ T v.
Since {f, S} is weakly compatible and f u ∈ Su, then f Su = Sf u and so f z ∈ Sz, as well as {g, T } we obtain gz ∈ T z.Now, we claim z = f z, if not by using (1) we get: which is a contradiction, then z = f z ∈ Sz which implies that z is common fixed point of f and S.
Similarly, we claim z = gz, if not by using (1) we get: which is a contradicts (φ 2 ), then z = gz ∈ T z, consequently z is common fixed point of f, g, S and T .
For the uniqueness, suppose there is an other point w satisfying w = f w = gw ∈ Sw = T , if w = z by using (1) we get: which is a contradiction, then z = w.
If S = T and f = g, we obtain the following corollary: Corollary 3.1.Let f : X → X, and S : X → CB(X) be single and set valued mappings of metric space (X, d) such: where φ ∈ Φ and ϕ : R + → R + is a Lebesgue-integrable function which is summable on each compact subset of R + , non-negative, and such that for each ε > 0, ε 0 ϕ(t)dt > 0, if f is strongly tangential with respect to S and {f, S} is weakly compatible, then f and S have a unique common fixed point.Corollary 3.2.Let f, g : X → X, and S, T : X → CB(X) be single and set valued mappings of metric space (X, d) such: a∈A d(a, B), sup b∈B d(b, A) , where d(a, B) = inf b∈B d(a, b) and (CB(X), H) is a metric space.
) d t , letting n → ∞, since d(z, f z) ≤ H(M, Sz) and applying the triangle inequality we get d(f z, M ) ≤ d(f z, z) + d(z, M ) = d(f z, z), then:

,
where φ ∈ Φ and ϕ : R + → R + is a Lebesgue-integrable function which is summable on each compact subset of R + , non-negative, and such that for each ε > 0, ε 0 ϕ(t) d t > 0. and a, b, c are nonnegative real numbers such a + b + b < 1 and p ∈ N , if {f, g} is strongly tangential with respect to {S, T } and the two pairs {f, g}, {S, T } are weakly compatible, then f, g, S and T have a unique common fixed point.