Some New Results of Interpolative Hardy–Rogers and Ćirić–Reich–Rus Type Contraction

In this paper, we present new concepts on completeness of Hardy–Rogers type contraction mappings in metric space to prove the existence of fixed points. Furthermore, we introduce the concept of g-interpolative Hardy–Rogers type contractions in b-metric spaces to prove the existence of the coincidence point. Lastly, we add a third concept, interpolative weakly contractive mapping type, Ćirić–Reich–Rus, to show the existence of fixed points. (ese results are a generalization of previous results, which we have reinforced with examples.


Introduction and Preliminaries
e theory of fixed points has known a lot of evolution. It has been given great merit and concern, thanks to its many uses in several fields of mathematics, such as differential equations, graph theory, and nonlinear analysis [1][2][3][4][5][6][7][8]. Besides, the emergence of the fixed theorem with Banach [4] in 1922 on complete normed space was followed by several improvements and generalizations of this theorem on two levels: the first level is related to the applications used, and the second to the spaces used in them. It first knew improvements with Kannan [9] in 1968, and later with other researchers such as Rus,Ćirić, Reich, Hardy, and Rogers. Afterwards, it took another turning with Karapinar [10] in 2018 in a new version, which has made several researchers pursue this field (see [11][12][13][14][15][16][17][18][19]). us, the concept has been applied in various spaces: metric space, b-metric space, rectangular b-metric spaces, and the Branciari distance. More recently, Errai et al. [14] have inserted g-interpolation overĆirić-Reich-Rus type contraction. ey have also introduced the concept of interpolative weakly contractive mapping, which makes us use these two concepts in this paper: the first concept on Hardy-Rogers type contraction and the second onĆirić-Reich-Rus type contraction as a generalization of the previous findings, reinforced by various examples.
is leads us to come up with some remarks. Before starting, we will take some basic concepts that we will use in this article.
Definition 1 (see [20,21]). Let s ≥ 1 be a given real number and 5 be a nonempty set. A function d: 5 × 5 ⟶ R + is a b-metric if the following conditions are met for all υ, y, z ∈ 5 e pair (5, d) is called a b-metric space. It is worth mentioning that b-metric spaces are a broader category than metric spaces. e definitions of b-convergent and b-Cauchy sequences, as well as b-complete b-metric spaces, are defined in the same way as usual metric spaces (see, e.g., [22]).
For the interesting examples and properties of b-metric, see the following papers [23][24][25] as examples.
(2) A b-convergent sequence has a unique limit.
(3) In general, a b-metric is not continuous.
To prove our results, the fact in the previous remark necessitates the following lemma regarding b-convergent sequences: Lemma 1 (see [26]). Let (5, d) be a b-metric space with s ≥ 1 , and assume that υ n and y n are b-convergent to υ, y, respectively, so we have 1 s 2 d(υ, y) ≤ lim inf n⟶∞ d υ n , y n ≤ lim sup n⟶∞ d υ n , y n ≤ s 2 d(υ, y).
Proof. Let υ n be the sequence defined by υ 0 � υ and υ n+1 � Tυ n for all integer n. If there exists n 0 such that υ n 0 � υ n 0 +1 , then υ n 0 is a fixed point of T. e proof is complete. Suppose that υ n+1 ≠ Tυ n for all n ≥ 0.
By substituting the values υ � υ n−1 and y � υ n in (5), we have Using the fact ξ(t) < t for each t > 0, we obtain From (7), we obtain us, which implies By (12), we obtain Assume that there exists a real ρ(n) such that From (12), we deduce which gives which is convergent, so υ n is Cauchy sequence in (5, d), and then it converges to some ] ∈ 5. Suppose that ] ≠ T], we find by (5): Passing the limit as n ⟶ +∞, we get 20) and the function ξ(t) � 2/7t 2 for all t ∈ [0, ∞).
Hence, in all cases, we have for all u, v ∈ [0, 3]\ 1/3 { }. As a result, all the conditions of eorem 1 are fulfilled, and T has a fixed point, u � 1/3. Example 2. Let 5 � a, q, m, r be endowed with the metric given in the following chart (Table 1).
Consider the self mapping T on 5 as T: a q m r q q a r . (25) Take We have for all u, v ∈ 5\ q, r . en, T has two fixed points, which are q and r.
If there exists υ ∈ 5 such that d(x, Tx) < 1, then T has a fixed point in 5. (5, d, s) be a b-metric space and T, g: 5 ⟶ 5 be self mappings on 5. We say that T is a g-interpolative Hardy-Rogers type contraction if there exist ψ ∈ Ψ and α, η, ω ∈ (0, 1) such that
(2) g5 is closed en, T and g have a coincidence point in 5.
By Lemma 2, we deduce that Tυ n is a b-Cauchy sequence, and consequently, gυ n is also a b-Cauchy sequence. Let z ∈ 5 such that, lim n⟶∞ d Tυ n , z � lim n⟶∞ d gυ n+1 , z � 0.
And, since z ∈ g5, there exists u ∈ 5 such that z � gu. We claim that u is a coincidence point of g and T. us, if we assume that gu ≠ Tu, we obtain Consequently, which is a contradiction. is implies that en, u is a coincidence point in 5 of T and g. For all u, v ∈ 5\ r, m { }, it is obvious that g, T fulfills (28). Furthermore, r and m are two coincidence points of g and T.
Hence, the following issues are discussed.
Fourth case: if υ, y ∈ (2, +∞) and υ ≠ y, we have Using the property of ψ, we get For all υ, y ∈ 5\ 1 { }, it is obvious that g, T fulfills (28). Furthermore, one is a coincidence point of g and T.
e two previous examples lead us to the following remark.
Taking n ⟶ ∞ in inequality (60), we obtain We deduce that c � 0. Hence, erefore, υ n is a Cauchy sequence. Suppose not, then there exists a real number ε > 0, for any k ∈ N, ∃m k ≥ n k ≥ k such that d υ m k , υ n k ≥ ε and d υ m k −1 , υ n k < ε.

Remark 3.
If T is an interpolative weakly contractive mapping typeĆirić-Reich-Rus, T accepts a fixed point that is not necessarily a single one.
for all υ, y ∈ 5 and υ ≠ Tυ, y ≠ Ty, then T has a fixed point.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.