Fixed point results for Geraghty quasi-contraction type mappings in dislocated quasi-metric spaces

In this paper, fixed point results for a newly introduced Geraghty quasi-contraction type mappings are proved in more general metric spaces called T-orbitally complete dislocated quasi-metric spaces. Geraghty quasi-contraction type mappings generalize, among others, Ciric’s quasi-contraction mappings and other Geraghty quasi-contractive type mappings in the literature. Fixed point results are obtained without imposing a continuity condition on the mapping, thereby further generalizing some other related work in the literature. An example is given to show the validity of results obtained.

Many extensions of Banach contraction mapping have also been investigated using different forms of contractive conditions. Asadi [8] proved some fixed point results satisfying certain contraction principles on a convex metric space. Özyurt [9] established fixed point results for extended large contraction via (c)-comparison function in a complete metric space while Özturk [10] introduced F-contraction and proved fixed point results for Fcontractive iterates in a metric space. Some interesting results using other contractive conditions include [11][12][13][14][15].
In 2000, Hitzler [16] introduced a space known as dislocated metric space in which the self distance of points is not necessarily zero and showed that the popular Banach contraction mapping is also valid in the space. Dislocated metric space has applications in semantic analysis of logical programming, electronic engineering and in topology [16]. Zeyada et al. [17] further generalized the concept of dislocated metric space and introduced the idea of dislocated quasi-metric space. In this new notion, the symmetric property is also omitted. Some other papers have been published containing fixed point results for selfmappings with different contraction conditions in metric spaces and their generalizations including dislocated metric spaces and dislocated quasi-metric spaces (see [17][18][19][20][21][22][23][24][25][26]).

Definition 1.2 ([17]
) Let X be a non-empty set and let d : X × X → R + be a function such that the following are satisfied: for all x, y, z ∈ X. Then d is called dislocated quasi-metric on X and the pair (X, d) is called a dislocated quasi-metric space.
As an improvement of α-admissible maps introduced by Samet et al. [24] and Karapínar et al. [26], Popescu [5] introduced the following concepts, which were used to prove the existence and uniqueness of fixed point results in a complete metric space.

Lemma 1.5 ([5])
Let T : X → X be a triangular α-orbital admissible mapping. Assume that there exists x 1 ∈ X such that α(x 1 , Tx 1 ) ≥ 1. Define a sequence {x n } by x n+1 = Tx n . Then, we have α(x n , x m ) ≥ 1 for all m, n ∈ N with n < m.
The following definition by Ciric [25] on quasi-contraction mappings in metric spaces are also true for dislocated quasi-metric spaces. It is clear that every complete dislocated quasi-metric space is T-orbitally complete. But the converse does not hold in general.
The purpose of this paper is to prove some fixed point results in dislocated quasi-metric space using new concepts of Geraghty quasi-contraction type self-mappings that the authors just introduced and proved fixed point results in the context of metric spaces [27]. The result is obtained by dropping the restriction of continuity and proving the existence and uniqueness of fixed point in an orbitally complete (which is a relaxation of completeness) dislocated quasi-metric space. This result generalizes many existing related work in the literature [1-5, 16, 17, 22-27].

Main results
Let denote the class of the functions φ : [0, ∞) → [0, ∞) which satisfy the following conditions: First, we state the following new mapping introduced by the authors in [27]. where , then we have the Geraghty [1] contraction mapping defined on a metric space. In addition, if β(t) = q; where q ∈ [0, 1), we have the Banach contraction mapping [2]. Inequality (2) also generalizes, among others, those of Popescu [5], Karapínar [4,28] and Cho et al. [3]. (ii) Definition 2.1 is also true for a dislocated quasi-metric space since every metric space is a dislocated quasi-metric space but the converse is not necessarily true. An example, which is inspired by Rahman and Sarwar [22], is provided to buttress this fact.
Example 2.3 Let X = R and d(x, y) = |x| for all x, y ∈ X. Let β(t) = 1 t for all t > 0. Then β ∈ F. Let φ(t) = 2t and a mapping T : X → X be defined by Then T is an α-φ-Geraghty quasi-contraction type mapping defined on a dislocated quasimetric space but not on a metric space.

Theorem 2.4
Let (X, d) be a T-orbitally complete dislocated quasi-metric space such that T : X → X is a self-mapping. Suppose α : X × X → R + is a function satisfying the following conditions: (i) T is an α-φ-Geraghty quasi-contraction type mapping.
(iii) There exists x 1 ∈ X such that α(x 1 , Tx 1 ) ≥ 1. Then T has a fixed point x * ∈ X and {T n x 1 } converges to x * .
for some 1 ≤ i ≤ n -1, then obviously T has a fixed point. Consequently, throughout the proof, we suppose that x i = x i+1 for all i ≥ 1. By Lemma 1.5, used recursively, we have By (2), for 1 ≤ j ≤ n we get where Note that, since the functions belonging to F are strictly smaller than one, inequality (2)

is a contradiction. Thus, we conclude that φ(d(T i x, T j x)) < φ(M T (T i-1 x, T j-1 x))
for all 0 ≤ i ≤ j ≤ n. Thus, the sequence {d(T i x, T j x)} is positive and decreasing. Consequently, there exists r ≥ 0 such that We claim that r = 0. Suppose, on the contrary, that r > 0. Then, from (4) we have Since β ∈ F, by definition, it implies that Using the triangle inequality, Taking the limits and using (6) we get Therefore lim m,n→∞ β(φ(M T (T n-1 x, T m-1 x))) = 1 and so lim m,n→∞ φ(M T (T n-1 x, T m-1 x)) = 0. Thus lim m,n→∞ d(T n-1 x, T m-1 x) = 0, which contradicts our assumption. Thus, the sequence {d(T i x, T j x)} is Cauchy. Since X is T-orbitally complete, there exists x * ∈ X such that lim n→∞ T n x = x * . To show that Tx * = x * , suppose that Taking limits as i tends to infinity gives So, by the definition of β ∈ F, we get (iii) The results in Karapinar [4,28] and Cho et al. [3] are also corollaries to our result. Therefore, Theorem 2.4 is an improvement and a generalization of other related work and hence an addition to the library of mappings in the literature.
The following examples validate Theorem 2.4.
Example 2.8 Let X = [0, ∞) and d(x, y) = x for all x, y ∈ X. Let β(t) = 1 1+t for all t > 0. Then β ∈ F. Let φ(t) = 2t and a mapping T : X → X be defined by One can easily see that X is a dislocated quasi-metric space but not a metric space. Also, the self-mapping T is not continuous at x = 1.
Example 2.9 Consider the set X = {{0} ∪ { 1 n : n ∈ N} ∪ N} and a dislocated quasi-metric d(x, y) = |x -y| + x, ∀x, y ∈ X. Let φ(t) = t 2 and β(t) = 1 t ∀t > 0, then β ∈ F. Define the mapping T : X → X by Also define the function α : X × X → R + by α(x, y) = 1 ∀x, y ∈ X. X is also a dislocated quasi-metric space but neither a metric space nor a dislocated metric space and the self-mapping T is not continuous.
Condition (iii) of Theorem 2.4 is also satisfied with x 1 = 1.
Therefore, all assumptions of Theorem 2.4, Theorem 2.5 are satisfied, and hence T has a unique fixed point at x * = 1.