On interpolative Hardy-Rogers contractive of Suzuki type mappings


 In this paper, we obtain a fixed point theorem ω- ψ-interpolative Hardy-Rogers contractive of Suzuki type mappings. In the following, we present an example to illustrate the new theorem is applicable. Subsequently, some results are given. These results generalize several new results present in the literature.


Introduction
The xed point theory is one of the main research areas in nonlinear analysis. In metric spaces, this theory began with the Banach xed point theorem. Banach [7] introduced Banach xed point theorem known as the "Banach contraction principle". This principle states that: Theorem 1.1. [7] Let (X, d) be a complete metric space and let T be a mapping of X into itself such that d(Tx, Ty) ≤ αd(x, y).
where α ∈ [ , ) and x, y ∈ X. Then T has a unique xed point.
There were many generalizations of this theorem. One of the generalizations was given by Kannan [20], which characterize the completeness of underlying metric spaces. Theorem 1.2. [20] Let (X, d) be a complete metric space. A mapping T : X → X is said to be a Kannan contraction if there exists λ ∈ [ , ) such that d(Tx, Ty) ≤ λ(d(x, Tx) + d(y, Ty)) for all x, y ∈ X\ Fix(T). Then, T posses a unique xed point.
Kannan's xed point theorem [20] has been generalized in di erent ways by many authors. In 2018, Karapınar [18] used the interpolative approach to de ne the generalized Kannan-type contraction and proved a xed point result on it.
A mapping T : X → X on (X, d) a complete metric space such that where κ ∈ [ , ) and α ∈ ( , ), for each x, y ∈ X\ Fix(T). Thus T has a xed point in X.
One another of the most interesting the Banach contraction principle generalizations of it was given by Hardy-Rogers [11]. for each x, y ∈ X\ Fix(T). Then, the mapping T has a unique xed point in X.
Following this, Karapınar et al. [12] introduced the following notion of interpolative Hardy-Rogers type contraction.
Theorem 1.4. [12] Let (X, d) be a complete metric space. The mapping T : X → X is called an interpolative Hardy-Rogers type contraction if there exist λ ∈ [ , ) and positive reals β, α, γ > , with β + α + γ < , such that for each x, y ∈ X\ Fix(T). Then the mapping T has a xed point in X.
Some interesting results in this concept may be found in the work of [2,4,6,10,15].
In the following, we recollect the notions of ω -orbital admissible mappings. The concept of ω -orbital admissible mappings was introduced by Popescu as a clari cation of the concept of α -admissible mappings of Samet et al. [24].
De nition 1.5. [22] Let T be a self map de ned on X and ω : X × X → [ , ∞) be a function. T is said to be an ω -orbital admissible if for all x ∈ X, we have: In our theorem, the following condition has often been considered in order to avoid the continuity of the involved contractive mappings. (R) A spaces (X, d) is de ned ω-regular, if {xn} is a sequence in X which xn → t ∈ X as n → ∞ and satis es ω (xn , x n+ ) ≥ for each n and then, we have ω (xn , t) ≥ . The existence results of xed points of this sense maps have been extensively studied, see [3,5,9,14,19].
Popescu [21] introduced two generalizations of a result given by Bogin [8] for a class of non-expansive mappings on complete metric spaces. The aim of his work was to replace the non-expansiveness condition with the weaker C-condition investigated by Suzuki [26]. Karapınar [13] introduced the de nition of a nonexpansive mapping satisfying the C-condition. Subsequently, the existence of xed points of maps satisfying the C-condition has been extensively studied; see [16,17,25,27,28]. We state rst the de nition of a nonexpansive map and a map satisfying the C-condition on a metric space.

De nition 1.6. A mapping T on a metric space
for all x, y ∈ X.

De nition 1.7. A mapping T on a metric space (X, d) satis es the C -condition if
for each x, y ∈ X.

Main Results
First, let's start with the de nition of ω-ψ-interpolative Hardy-Rogers contractive of Suzuki type.

De nition 2.1. Let (X, d) be a metric space. The mapping T : X → X is called an ω-ψ-interpolative Hardy-
for each x, y ∈ X\ Fix(T). Proof. Let x ∈ X satisfy ω(x , Tx ) ≥ . Let {xn} be the sequence constructed by T n (x ) = xn for each positive integer n. Assume that xn = xn + for some n ∈ N, so xn = Txn that means xn is a xed point of T. Then, we can suppose that xn ≠ x n+ for each positive integer n.
Similarly, continuing this process, By substituting the assign x = x n− and y = Tx n− = xn in (1), we obtain then, assume that for every n ∈ N, then, which is a contradiction, hence we get that for all n ∈ N, Then, the positive sequence {d(x n− , xn)} is non-increasing sequence with positive terms so, we attain that there exists ȷ ≥ such that limn→∞ d(x n− , xn) = ȷ. Accordingly, we get Further, by (3) [d(xn , Thence, by repeating this condition, we can write, We claim that {xn} is a fundamental sequence in (X, d). Then, we shall use the triangle inequality with (7), we can nd Letting n → ∞ in (8) holds. In case that (10) holds, we get If the second condition, (11) holds, we have Therefore, letting n → ∞ in (12) and (13), we get that d(t, Tt) = , that is t = Tt.
In case that the assumption ii. is true, that is the mapping T is continuous, If the last assumption, iii. holds, as above, we have T t = lim n→∞ T xn = lim n→∞ x n+ = t and we want to show that, also Tt = t. Assuming on the contrary, that t ≠ Tt, since which is a contradiction. Consequently, t = Tt, that is, t is a xed point of the mapping T.

Example 2.3. Let X = [ , ]
and d : X × X → [ , +∞) be the usual metric on R. The mapping T : X → X be de ned as Further, let ω : , if x = , y =

, otherwise
The mapping T is not continuous but, since T = we have that T is a continuous mapping. Let the function ψ ∈ Ψ de ned as ψ(t) = t and we choose β = , α = and γ = . Then, we have to check that (1) For all other cases (1) holds, since ω(x, y) = .
As a result, the assumptions of Theorem 2.2 is satis ed, also the mapping T has a xed point, that is x = .
Proof. In Theorem 2.2 is su cient to get ω(x, y) = for proof.