An Extension of Generalized ψ , φ-Weak Contractions

We prove a fixed-point theorem for a class of maps that satisfy generalized (𝜓,𝜑)-weak contractions depending on a given function. An example is given to illustrate our extensions.


Introduction
Because fixed-point theory has a wide array of applications in many areas such as economics, computer science, and engineering, it plays evidently a crucial role in nonlinear analysis.One of the cornerstones of this theory is the Banach fixed-point theorem, also known as the Banach contraction mapping theorem 1 , which can be stated as follows.
Let T : X → X be a contraction on a compete metric space X; d ; that is, there is a nonnegative real number k < 1 such that d T x , T y ≤ kd x, y for all x, y ∈ X.Then the map T admits one and only one point x * ∈ X such that Tx * x * .Moreover, this fixed point is the limit of the iterative sequence x n 1 T x n for n 0, 1, 2, . .., where x 0 is an arbitrary starting point in X.This theorem attracted a lot of attention because of its importance in the field.Many authors have started studying on fixed-point theory to explore some new contraction mappings to generalize the Banach contraction mapping theorem.In particular, Boyd and Wong 2 introduced the notion of Φ-contractions.In 1997 Alber and Guerre-Delabriere 3 defined the ϕ-weak contraction which is a generalization of Φ-contractions see also 4-8 .On the other hand, the notion of T -contractions introduced and studied by the authors of the interesting papers in 9-11 .Following this trend, we explore in this paper another extension of ψ, ϕ -weak contractions in the context of T -contractions.

Preliminaries
Let X, d be a metric space.Boyd and Wong 2 introduced the notion of Φ-contraction as follows.A map T : X → X is called a Φ-contraction if there exists an upper semicontinuous function Φ : 0, ∞ → 0, ∞ such that d Tx, Ty ≤ Φ d x, y 2.1 for all x, y ∈ X.The concept of the ϕ-weak contraction was defined by Theorem 2.4.Let X, d be a complete metric space, and let T, S : X → X be self-mappings satisfying where ∞ is a continuous and nondecreasing function with ψ t 0 if and only if t 0, and ϕ : 0, ∞ → 0, ∞ is a lower semicontinuous function with ϕ t 0 if and only if t 0. Then f, g have a unique common fixed point.
The notion of the T -contraction is defined in 10, 11 as follows.
Definition 2.5.Let T and S be two self-mappings on a metric space X, d .The mapping S is said to be a T -contraction if there exists k ∈ 0, 1 such that d TSx, TSy ≤ kd Tx, Ty , ∀x, y ∈ X. 2.9 It can be easily seen that if T is the identity map, then the T -contraction coincides with the usual contraction.

International Journal of Mathematics and Mathematical Sciences
The aim of this work is to give a proper extension of D − oricorić's result of using the concept of T -contraction, that is, the contraction depending on a given function.We will show the existence of a common fixed point for a class of certain maps.

Main Results
We start this section by recalling the following two classes of functions.
Let Ψ denote the set of all functions ψ : 0, ∞ → 0, ∞ which satisfy  Proof.We will follow the lines in the proof of the main result in 13 .By injection of T , we easily check that M Tx, Ty 0 if and only if x y is a common fixed point of f and g.Let x 0 ∈ X.We define two iterative sequences {x n } and {y n } in the following way: M Tx n , Tx n−1 r.

3.11
By the lower semicontinuity of ϕ, we have

3.12
Taking the upper limits as n → ∞ on either side of M Tx n , Tx n−1 0.

3.16
Now, we claim that {y n } is a Cauchy sequence.Since lim n → ∞ d y n , y n 1 0, it is sufficient to prove that {y 2n } is a Cauchy sequence.Suppose on the contrary that {y 2n } is not a Cauchy sequence.Then, there exist ε > 0 and subsequences {y 2n k } and {y 2m k } of {y 2n } such that n k is the smallest index for which 3.17 This means that d y 2m k , y 2n k −2 < ε.

3.18
From 3.18 and the triangle inequality, we get

3.19
Letting k → ∞ and using 3.15 , we get

3.20
By the fact 3.21 and using 3.15 and 3.20 , we obtain lim Moreover, from and combining with 3.15 and 3.22 , we conclude that Now, by the definition of M Tx, Ty and from 3.10 , 3.15 , and 3.20 -3.24 , we can deduce that

3.25
Due to 3.1 , we have

3.26
Letting k → ∞ and using 3.22 and 3.25 , we have It is a contradiction to ϕ t > 0 for every t > 0. This proves that {y n } is a Cauchy sequence.Since X is a complete metric space, there exists u ∈ X such that lim n → ∞ y n u.Since T is sequentially convergent, we can deduce that {x n } converges to v ∈ X.By the continuity of T , we infer that

3.35
This implies that d Tv, Tw 0, or Tv Tw.Since T is injective, we have w v.The theorem is proved.Remark 3.2. 1 In Theorem 3.1, if we choose Tx x for all x ∈ X, then we get Theorem 2.4.
2 In Theorem 3.1, if we fix ψ t t for all t, then we obtain another extension of Theorem 2.3.
3 In Theorem 3.1, if we choose f g, then we get the uniqueness and existence of fixed point of generalized ϕ-weak T -contractions.
The following example shows that Theorem 3.1 is a proper extension of Theorem 2.4.
Example 3.3.Let X 1, ∞ and d be the usual metric in X.Consider the maps f x g x 4 √ x.It is easy to see that 16 is the unique fixed point of f and g.We claim that f and g are not generalized ϕ-weak contraction.Indeed, if there exist lower semicontinuous functions ψ, ϕ : 0, ∞ → 0, ∞ with ψ t > 0, ϕ t > 0 for t ∈ 0, ∞ and ϕ 0 ψ 0 0, such that

Theorem 3 . 1 .
Let X, d be a complete metric space and T : X → X an injective, continuous, and sequentially convergent mapping.Let f, g : X → X be self-mappings.If there exist ψ ∈ Ψ and ϕ ∈ Φ such that ψ d Tfx, Tgy ≤ ψ M Tx, Ty − ϕ M Tx, Ty , 3.1 for all x, y ∈ X, where M Tx, Ty max d Tx, Ty , d Tx, Tfx , d Ty, Tgy , d Tx, Tgy d Ty, Tfx 2 , 3.2 then f, g have a unique common fixed point.

Dutta and Choudhury proved an extension of Rhoades. Theorem 2.2. Let X, d be a complete metric space, and let T : X → X be a self-mapping satisfying
Alber and Guerre-Delabriere 3 as a generalization of Φ-contraction under the setting of Hilbert spaces and obtained fixed-point results.A map T : X → X is a ϕ-weak contraction, if there exists a function ϕ : 0, ∞ → 0, ∞ such that Combining the theorems above with the results of Dutta and Choudhury 12 , D − oricorić 13 obtained the following theorem.
d Tx, Ty ≤ d x, y − ϕ d x, y 2.2for all x, y ∈ X provided that the function ϕ satisfies the following condition: , ψ 3 t t 2 belong to Ψ and ϕ 1 t min{t, 1}, ϕ 2 t ln 1 t belong to Φ.We are ready to state our main theorem that is a proper extension of Theorem 2.4.
≤ ψ M Tx n , Tx n−1 − ϕ M Tx n , Tx n−1 ≤ ψ M Tx n , Tx n−1 , n , y n 1 > d y n−1 , y n ≥ 0 then M Tx n , Tx n−1 d y n , y n 1 , hence ψ d y n , y n 1 ≤ ψ d y n , y n 1 − ϕ d y n , y n 1 3.7and which contradicts with d y n , y n 1 > 0 and the property of ϕ.Thus, it follows from 3.5 that d y n 1 , y n ≤ M Tx n , Tx