Common Fixed Point Theorems for Multivalued Generalized F -Suzuki-Contraction Mappings in Complete Strong b − Metric Spaces

This paper introduces a new version of multivalued generalized F -Suzuki-Contraction mapping and then establish some new common ﬁxed point theorems for these new multivalued generalized F -Suzuki-Contraction Mappings in complete strong b − Metric Spaces.


Introduction
Let X be a nonempty set and s ≥ 1 be a given real number. A mapping d : X × X → R * is said to be a b-metric if for all x, y, z ∈ X the following conditions are satisfied: The pair (X, d) is called a b-metric space with constant s. A strong b−metric is a semimetric space (X, d) if there exists s ≥ 1 for which d satisfies the following triangular inequality.
d(x, y) ≤ d(x, z) + sd(z, y), f or each x, y, z ∈ X. (1) In 1922, a mathematician Banach [1] proved a very important result regarding a contraction mapping, known as the Banach contraction principle, which states that every self-mapping T defined on a complete metric space (X, d) satisfying ∀x, y ∈ X, d(T x, T y) ≤ λd(x, y), where λ ∈ (0, 1) has a unique fixed point and for every x 0 ∈ X a sequence {T n x 0 } ∞ n=1 converges to the fixed point. Subsequently, in 1962, Edelstein [2] proved the following version of the Banach contraction principle. Let (X, d) be a compact metric space and let T : X → X be a self-mapping. Assume that for all x, y ∈ X with x y, d(x, T x) < d(x, y) =⇒ d(T x, T y) < d(x, y).
Then T has a unique fixed point in X. In 2012, Wardowski [3] introduced a new type of contractions called F-contraction and proved a new fixed point theorem concerning F-contractions.
Let (X, d) be a metric space. A mapping T : X → X is said to be an F-contraction if there exists τ > 0 such that ∀x, y ∈ X, d(T x, T y) > 0 =⇒ τ + F(d(T x, T y)) ≤ F(d(x, y)),
We denote by ζ, the set of all functions satisfying the conditions (F1) − (F3). Wardowski [3] then stated a modified version of the Banach contraction principle as follows. Let (X, d) be a complete metric space and let T : X → X be an F-contraction. Then T has a unique fixed point x * ∈ X and for every x ∈ X the sequence {T n x} ∞ n=1 converges to x * . In 2014, Hossein, P. and Poom, K. [15] defined the F-Suzuki contraction as follows and gave another version of theorem. Let (X, d) be a metric space. A mapping T : X → X is said to be an F-Suzuki-contraction if there exists τ > 0 such that for all x, y ∈ X with T x T y where F : R + → R is a mapping satisfying the following conditions: F1 F is strictly increasing, i.e. for all x, y ∈ R + such that x < y, F(x) < F(y); Let T be a self-mapping of a complete metric space X into itself. Suppose F ∈ ζ and there exists τ > 0 such that Then T has a unique fixed point x * ∈ X and for every x 0 ∈ X the sequence {T n x 0 } ∞ n=1 converges to x * .
Comsidering the definition S T x := {S y ⊆ CB(X) : ∀y ∈ T x}, we have the following result.
Theorem 2.1. Let (X, d) be a complete strong b−metric space and let T, S : X → CB(X) be multivalued generalized F-Suzuki-Contraction mappings. Then T and S has a common fixed point x * ∈ X and for every x ∈ X the sequence {T n x} ∞ n and {S n x} ∞ n converge to x * . Proof Let x 0 = x ∈ X. Let x n+1 ∈ T x n and x n+2 ∈ S x n+1 ∀n ∈ N. If there exists n ∈ N such that d(x n , T x n ) = d(x n+1 , S x n+1 ) = 0 then x n+1 = x n = x becomes a fixed point of T and S , respectively, therefore the proof is complete. Now, suppose that d(x n , T x n ) > 0 and d(x n+1 , S x n+1 ) > 0 ∀n ∈ N then the proof will be divided in to two steps.
Step one. We show that {x n } ∞ n=1 is a Cauchy sequence. Let therefore, we have that By Definition 2.3, we get Since that then by (5) and definition 2.3, we get On and therefore (6) becomes But, from (3) and the fact that ψ(d(x n+1 , x n+2 )) > 0, this is a contradiction. Thus, we conclude that By (7) and Definition 2.1(F1), we have that Therefore {d(x n , x n+1 )} is a nonnegative decreasing sequence of real numbers. Thus there exists γ ≥ 0 such that lim n→∞ d(x n , x n+1 ) = γ. From (7) as n → ∞, we have that This implies that ψ(γ) = 0 and thus γ = 0. Consequently we arrive at Now, we claim that {x n } ∞ n=1 is a Cauchy sequence. On contrary, we assume that there exists > 0 and n, m ∈ N such that, for all n ≥ n and n < n < m, It implies that By (11) and (9), we have that By triangle inequality, we have that By (9),(10), (12) and (13), we have that Similarly, we have that By (9),(10), (12) and (15), we have that Observe that By (17), we have that By (9)and (10), we select n > 0 ∈ N such that It follows that from Definition 2.3, we have, for every n ≥ n Since that By (12), (14), (16), (18) and (20), we have that Similarly By (19), (21) and (22), we have that By (23) and the fact that > 0, this is a contradiction. Hence {x n } is a Cauchy sequence in X. By completeness of (X, d), {x n } ∞ n=1 and {x n+1 } ∞ n=1 converge to some point x * ∈ X, that is, There exists increasing sequences {n k }, {n + 1 k } ⊂ N such that x n k ∈ T x * and x n+1 k ∈ S x * for all k ∈ N. Since T x * and S x * are closed and lim n→∞ d(x n k , x * ) = 0 and lim n→∞ d(x n+1 k , x * ) = 0, we get x * ∈ T x * and x * ∈ S x * .
Step two. We show that x * is a common fixed point of T and S . It suffices to show that 1 1 + s d(x n , T x n ) < d(x n , x * ) and implies and respectively. On contrary, suppose there exists m ∈ N such that By (26), we have that and therefore By (8), (26) and (27), this is a contradiction. Hence, (25) holds, and therefore and Since that and By (24) and (30), we have that By (24) and (31), we have that By (28)and (29) and by the continuity of F and ψ, we have that Hence, since T x * and S x * are closed then we have x * ∈ T x * and x * ∈ S x * , that is, x * is a fixed point of T and S .
In Theorem 2.1, when T = S = U, then we have the following result.
Corollary 2.1.1. Let (X, d) be a complete strong b−metric space and let U : X → CB(X) be a multivalued generalized F-Suzuki-Contraction mapping. Then U has a fixed point x * ∈ X and for every x ∈ X the sequence {U n x} ∞ n=1 converges to x * . In Corollary 2.1.1, when U is a single-valued then we have another new result as follows.
Corollary 2.1.2. Let (X, d) be a complete strong b−metric space and let U : X → X be a single-valued generalized F-Suzuki-Contraction mapping. Then U has a fixed point x * ∈ X and for every x ∈ X the sequence {U n x} ∞ n=1 converges to x * . In Theorem 2.1, when T and S are two single-valued then the following result holds.
Corollary 2.1.3. Let (X, d) be a complete strong b−metric space and let T, S : X → X be two single-valued generalized F-Suzuki-Contraction mappings. Then T and S have a common fixed point x * ∈ X and for every x ∈ X the sequence {T n x} ∞ n=1 and {S n x} ∞ n=1 converge to x * . In Theorem 2.1, when (X, d) is a complete b−metric space then the following new result holds.
Corollary 2.1.4. Let (X, d) be a complete b−metric space and let T, S : X → X be two single-valued generalized F-Suzuki-Contraction mappings. Then T and S have a common fixed point x * ∈ X and for every x ∈ X the sequence {T n x} ∞ n=1 and {S n x} ∞ n=1 converge to x * . In corollary 2.1.4, when T = S = U, then we have the following result.
Corollary 2.1.5. Let (X, d) be a complete b−metric space and let U : X → CB(X) be a multivalued generalized F-Suzuki-Contraction mapping. Then U has a fixed point x * ∈ X and for every x ∈ X the sequence {U n x} ∞ n=1 converges to x * . Corollary 2.1.6. Let (X, d) be a complete strong b−metric space and let U : X → CB(X) be a multivalued generalized F-Suzuki-Contraction mapping such that there exists F ∈ 0 and ψ ∈ Ψ, ∀x, y ∈ X, x y, 1 Then U has a fixed point x * ∈ X and for every x ∈ X the sequence {U n x} ∞ n=1 converges to x * . Proof from Lemma 1.1, since (2) ⇒ (32) then by the corollary 2.1.1 the result follows immediately.
Corollary 2.1.7. Let (X, d) be a complete strong b−metric space and let U : X → X be a single-valued generalized F-Suzuki-Contraction mapping such that there exists F ∈ 0 and ψ ∈ Ψ, ∀x, y ∈ X, x y, 1 s+1 d(x, U x) < d(x, y) ⇒ ψ(N(x, y)) + F(s 4 d(U x, Uy)) ≤ F(N(x, y)) in which N(x, y) = max{d(x, y), d(y, U 2 x), Then U has a fixed point x * ∈ X and for every x ∈ X the sequence {U n x} ∞ n=1 converges to x * . Proof from Lemma 1.1, since (2) ⇒ (33) then by the corollary 2.1.2 the result holds. Corollary 2.1.8. Let (X, d) be a complete strong b−metric space and let T, S : X → X be two single-valued generalized F-Suzuki-Contraction mappings such that there exists F ∈ 0 and ψ ∈ Ψ, ∀x, y ∈ X, x y, 1 s+1 d(x, T x) < d(x, y) and 1 s+1 d(y, S x) < d(y, S T x) ⇒ ψ(N(x, y))+F(s 4 H(T x, S y)) ≤ F(N(x, y)) in which N(x, y) = max{d(x, y), d(y, S T x), (d(y, T x)) + d(x, S y) 2s , (d(x, S y)) + d(S T x, S y) 2s , d(S T x, S y) + d(S y, T x), d(S T x, S y) + d(T x, x), d(T x, y)) + d(y, S y)}.
Then T and S have a common fixed point x * ∈ X and for every x ∈ X the sequence {T n x} ∞ n=1 and {S n x} ∞ n=1 converge to x * . Proof from Lemma 1.1, since (2) ⇒ (34) then by the corollary 2.1.4 the result holds.

Conclusion
Fixed point results of Piri and Kumam [11], Ahmad et al. [9], Suzuki [18] and Suzuki [19] are extended by introducing common fixed point problem for multivalued generalized F-Suzukicontraction mappings in strong b-metric spaces. In specific, Corollary 2.1.1 and corollary 2.1.2 generalize and extend the work of Ahmad et al. [9] and Kumam and Hossein [5], respectively.