Study on Pata E-contractions

In this paper, we introduce the notion of an α–ζ̃––Pata contraction that combines well-known concepts, such as the Pata contraction, the E-contraction and the simulation function. Existence and uniqueness of a fixed point of such mappings are investigated in the setting of a complete metric space. An example is stated to indicate the validity of the observed result. At the end, we give an application on the solution of nonlinear fractional differential equations.


Introduction
In 2015, Khojasteh et al. [1] initiated the concept of simulation functions. We denote by Z the family of all above simulation functions. Let (X , d ) be a metric space and α : X × X → [0, ∞) be a function. A mapping h : X → X is called α-orbital admissible if the following condition holds:

2)
Definition 1.2 A set X is said to be regular with respect to a given function α : X × X → [0, ∞) if for each sequence {ν n } in X such that α(ν n , ν n+1 ) ≥ 1 for all n and ν n → ν ∈ X as n → ∞, then α(ν n , ν) ≥ 1 for all n.
The notion of α-admissible Z-contractions with respect to a given simulation function was merged and used by Karapinar in [2]. Using this new type of contractive mappings, he investigated the existence and uniqueness of a fixed point in standard metric spaces.
Remark 1.5 The continuity condition in Theorem 1.4 can be replaced by the "regularity" condition, which is considered in Definition 1.2.
We will consider the following set of functions: and we denote ν = d (ν, ν 0 ), for an arbitrary but fixed ν 0 ∈ X .
Definition 1.7 Let (X , d ) be a metric space. We say that h : X → X is a Pata type Zamfirescu mapping if for all ν, ω ∈ X , ψ ∈ Z and for every ε ∈ [0, 1], h, it satisfies the following inequality: and (1.8) In this paper, we combine the concepts of simulation functions and α-admissibility to give a generalized Pata type fixed point result. At the end, we present an application on fractional calculus.

Theorem 2.3 Every α-ζ -E-Pata contraction h on a complete metric space
(iii) either h is continuous, or the set X is regular. If in addition we assume that the following condition is satisfied: Proof Let u 0 ∈ X be a point such that α(u 0 , hu 0 ) ≥ 1. On account of the assumption that h is a triangular α-orbital admissible mapping, we derive that and iteratively we find Again, iteratively, one writes Starting from this point u 0 ∈ X , we build an iterative sequence {u n } where u n = hu n-1 = h n u 0 for n = 1, 2, 3, . . . . We can presume that any two consequent terms of this sequence are distinct. Indeed, if, on the contrary, there exists i 0 ∈ N such that then u i 0 is a fixed point. To avoid this, we will assume in the following that for all n ∈ N We mention that (2.4) can be rewritten as for any n ∈ N. In the sequel, we will denote d (ν, u 0 ) = ν for all ν ∈ X.
Let us go back now and prove that γ = 0 (where γ = lim n→∞ γ n ). In view of (2.10) and the fact that the sequence {γ n } is non-increasing, one writes E(u n-1 , u n ) ≤ 2γ nγ n+1 .

Recall that
Taking into account that h is an α-ζ -E * contraction, keeping in mind (2.6) and using (ζ 1 ), we have We have (2.13) Letting n → ∞ in the previous inequality, we obtain which is equivalent to When ε → 0, we get γ ≤ 0. Therefore, As a next step, we claim that {u n } is a Cauchy sequence. On the contrary, assuming that the sequence is not Cauchy, it follows from Lemma 1.9 that there exist e > 0 and subsequences {u n l } and {u m l } such that (1.7) and (1.8) hold. Replacing ν = u n l and ω = u m l in (2.1), we have 0 ≤ζ α(u n l , u m l )d (hu n l , hu m l ), (1ε)E + S(u n l , u m l ) Denoting by a l = d (u n l +1 , u m l +1 ) and b l = (1ε)E(u n l , u m l ) + S(u n l , u m l ), by Lemma 1.9, it follows that a l → e and lim sup Thus, passing to the limit as l → ∞ in (2.16), we get Furthermore, i.e., That is, e = 0. Therefore, {u n } is a Cauchy sequence in the complete metric space. For this reason, there exists ν * ∈ X such that u n → ν * , as n → ∞. Furthermore, in the case that h is a continuous mapping, we get hν * = ν * , that is, ν * is a fixed point of h. Now, suppose that X is regular. From (2.1), one writes Using the regularity of X and (ζ 1 ), we get d (u n ,hu n )+d (ν * ,hν * )+|d (u n ,hu n )-d (ν * ,hν * )| 2 , d (u n ,hν * )+d (ν * ,hu n )+|d (u n ,hu n )-d (ν * ,hν * )| 2 Taking into account the boundedness of the sequence {κ n }, we have On the other hand, Letting n → ∞ in the inequality (2.19), we find which is equivalent to Obviously, we obtain for ε = 0 that d (ν * , hν * ) ≤ 0, so ν * = hν * . Thus, ν * is a fixed point of h. Finally, to prove the uniqueness of the fixed point, we suppose that there exist two fixed points ν * , ω * ∈ Fix X (h) such that ν * = ω * . We have Taking into account (iv), we obtain which leads to d ν * , ω * ≤ ε λ-1 ψ(ε) 1 + 2 ν * + 2 ω * β .

An application on a fractional boundary value problem
In this section, we ensure the existence of a solution of a nonlinear fractional differential equation (for more related details, see [17][18][19][20][21][22][23]). Denote by X = C[0, 1] the set of all continuous functions defined on [0, 1]. We endow X with the metric given as Consider the fractional differential equation Here, c D μ corresponds for the Caputo fractional derivative of order μ, given as D μ f (t) = 1 (nμ) 1 0 (ts) n-μ-1 f n (s) ds, (3.3) where n -1 < μ < n and n = [μ] + 1, and I μ f is the Riemann-Liouville fractional integral of order μ of a continuous function f , defined by In [24], it is showed that the problem (3.1) and (3.2) can be written in the following integral form:

Conclusion and remarks
Our results merged from and generalized several existing results in the related literature. First of all, as underlined in Remark 2.2, the main result of [16] is a consequence of our given theorem. On the other hand, by choosing the auxiliary functions in a proper way, we may state a long list of corollaries. More precisely, by choosing the mapping α in a proper way, we can get the analogue of our result in the setting of partially ordered metric spaces, or in the set-up of cyclic mappings. Note that, if we take α(x, y) = 1 for all x, y, we get the standard fixed point theorems in the context of complete metric spaces; see [25][26][27][28][29]. In addition, by choosing the appropriate simulation function, one can get several more results; see [30][31][32][33][34][35].