On Pata–Suzuki-Type Contractions

: In this manuscript, we introduce two notions, Pata–Suzuki Z -contraction and Pata Z -contraction for the pair of self-mapping g , f in the context of metric spaces. For such types of contractions, both the existence and uniqueness of a common ﬁxed point are examined. We provide examples to illustrate the validity of the given results. Further, we consider ordinary differential equations to apply our obtained results.


Introduction and Preliminaries
One of the interesting approach to extending existing fixed point results is to involve an auxiliary function into the hypotheses of theorems. In this paper, we consider the notion of the simulation function that is defined by Khojasteh et al. [1].
The main goal of this paper is to combine the notion of simulation functions, the concept of C-distance and Pata type contraction so that the obtained notions (namely, Pata-Suzuki Z-contraction and Pata Z-contraction) unify, extend and generalize several existing results in the literature of fixed point theory.
To winnow out the trivial cases, throughout the proof, we suppose that ν m+1 = ν m for all m ∈ N. Indeed, if we suppose, on the contrary, that ν m 0 +1 = ν m 0 for some m 0 ∈ N, then we conclude a common fixed point of f and g without any effort. Without loss of generality we may assume ν 2n 0 +1 = ν 2n 0 .
In Theorem 3, to provide C-condition, we need to suppose that both g and f are continuous. We realize that in case of removing C-condition, we relax the continuity conditions on g and f . In the following, we introduce Pata Z-contraction which is more relaxed than Pata-Suzuki Z-contraction Definition 4. A pair (g, f ), defined on a (X, d) , is said to be a Pata Z-contraction if for every ε ∈ [0, 1] and all ν, ω ∈ X, fulfills where where ζ ∈ Z, α ≥ 1, Λ ≥ 0, and β ∈ [0, α] are constants, and, This is the second main results of this paper.
Theorem 4. If a pair (g, f ), on a (X * , d), forms a Pata Z-contraction, then g, f have a common fixed point ν * ∈ X.
Notice that in Pata-Suzuki Z-contraction we need to satisfy the C-condition ( 1 2 d(ν, gν) ≤ d(ν, ω)), but in Pata Z-contraction, we do not need to check it. Therefore, we can repeat the proof of Theorem 3 by ignoring the C-condition.
Therefore ν * = f ν * . The uniqueness of the common fixed point of g and f is derived from the proof Theorem 3.
Proof. It is sufficient to take g = f in Theorem 3.

Application to Ordinary Differential Equations
We consider the following initial boundary value problem of second order differential equation: where f : [0, 1] × R → R is a continuous function.
Applying Corollary 1, we obtain that g has a unique fixed point in C[0, 1], which is a solution of integral equation.

Conclusions
In this paper, we combine and extend Pata type contractions and Suzuki type contraction via simulation function. The success of V. Pata [9] is to define an auxiliary distance function u = d(u, a) where a is an arbitrary but fixed point. This is based on the fact that most of the proofs in metric fixed point theory are established on the Picard sequence: For a self-mapping f on a metric space X and arbitrary point "a" (renamed as "a 0 "). Then, a 1 = Ta 0 , a n = f a n−1 for all positive integers.
In Banach's proof (and also, in many other metric fixed point theorems) for any point "a", this sequence converges to the fixed point of T. Under this setting, V.Pata, suggest such auxiliary distance function (initiated from an arbitrary point "a" ) to refine Banach's fixed point theorem, like the construction of Picard operator.
In this short note, we employ the approach of Pata in a more general case to generalize and unify several existing results in the literature. For this purpose, we have use simulation functions. We also emphasize that the simulation functions are very wide, see, e.g., [2][3][4][5][6]. Thus, several consequences of our results can be listed by using the examples that have been introduced in [2][3][4][5][6]. Similarly, we can generalize more inequalities on metric and normed spaces.