New Fixed Point Theorems for Admissible Hybrid Maps

Fixed point hypothesis has been a considerable area of research for mathematics and other sciences for the last century. It is the basis of functional analysis in mathematics, which is one of the critical topics of mathematics. The first concept of fixed point theory is knowing to appear in the work of Liouville in 1837 and Picard in 1890. But the main fixed point theorem was introduced by Banach [1]. The theorem is named after Banach. There are many generalizations of Banach theorem in the literature. In 1968, one of the most famous generalizations due to know, Kannan [2] introduced a new and useful contraction using Banach’s theorem. Suzuki [3] introduced important extensions of Banach’s main theorem, which we refer to [4–6]. In one of these studies [7], the researchers investigated a new extensive result by using simulation function. On the other hand, in [8], by using other auxiliary functions, called the Wardowski functions, they observed a contraction that combines both linear and nonlinear contractions. We also mention that in [9], the author obtained a fixed point theorem without the Picard operator. For more interesting results, see, e.g., [10–19]. In addition, Banach’s fixed point theorem is a significant mean in the theory of metric spaces. The metric concept has been generalized from different angles. One of the significant generalizes is defined b-metric which was defined as follows. Definition 1 (see [20, 21]). Let L be a (nonempty) set and s ≥ 1 a real number. A function b : L ×L ⟶ 1⁄20,∞Þ is a b -metric space on L if following conditions are satisfied:


Introduction and Preliminaries
Fixed point hypothesis has been a considerable area of research for mathematics and other sciences for the last century. It is the basis of functional analysis in mathematics, which is one of the critical topics of mathematics. The first concept of fixed point theory is knowing to appear in the work of Liouville in 1837 and Picard in 1890. But the main fixed point theorem was introduced by Banach [1]. The theorem is named after Banach. There are many generalizations of Banach theorem in the literature. In 1968, one of the most famous generalizations due to know, Kannan [2] introduced a new and useful contraction using Banach's theorem. Suzuki [3] introduced important extensions of Banach's main theorem, which we refer to [4][5][6]. In one of these studies [7], the researchers investigated a new extensive result by using simulation function. On the other hand, in [8], by using other auxiliary functions, called the Wardowski functions, they observed a contraction that combines both linear and nonlinear contractions. We also mention that in [9], the author obtained a fixed point theorem without the Picard operator. For more interesting results, see, e.g., [10][11][12][13][14][15][16][17][18][19]. In addition, Banach's fixed point theorem is a significant mean in the theory of metric spaces. The metric concept has been generalized from different angles. One of the significant generalizes is defined b-metric which was defined as follows.
We recollect some basic notions that are used in our study.
Karapinar [26] introduced interpolation Kannan-type contraction generalized from the famous Kannan fixed point theorem by using interpolative operator. In the following, the common fixed point theorem for this contraction was obtained [27]. In [28], the authors extended the results in [26] by introducing the interpolative Reich-Rus-Ćirić contractive in a general framework, in the setting of partial metric space. In addition, the interpolative Hardy-Rogers-type contractive was defined and discussed in [28]. The contraction, defined in [29], was generalized in [30] by involving the admissibility into the contraction inequality. Furthermore, in [31], hybrid contractions were considered. Indeed, the notion of hybrid contraction here refers to combination of interpolative (nonlinear) contraction and linear contraction. For more interesting papers, see [32][33][34].
In 2019, inspired by interpolative contraction, researchers [35] obtained and published a hybrid contractive that integrates Reich-Rus-Ćirić-type contractive and interpolative-type mappings. In particular, this approach was applied for Pata-Suzuki-type contraction in [36]. On the other hand, by using hybrid contraction, a solution for a Volterra fractional integral equation was proposed in [37]. Furthermore, the hybrid contractions were discussed in a distinct abstract space, namely, Branciari-type distance space, in [38]. Another advance was recorded in [39] where the authors investigated the Ulam-type stability of this consideration. In addition, new hybrid contractions were developed in b-metric spaces [40]. As a result, as can be seen in the literature review, many papers were published on the subject of interpolative contraction and hybrid contraction inspired by it. The contractions are a current study topic for fixed point theory. Therefore, the results of the study contribute to the existing literature. Now we give the idea of α-admissibility defined by Samet et al. [41] and Karapnar and Samet [42].
The mapping of w-orbital admissibility was presented by Popescu [43] as a modification of α-admissibility as follows: The following condition has often been considered on account of refraining from the continuity of the concerned contractive mappings. Definition 6. A space ðL, b, sÞ is defined w-regular, if fr q g is a sequence in L such that αðr q , r q+1 Þ ≥ 1 for all q ∈ ℕ and r q ⟶ r ∈ L as q ⟶ ∞; then, there exists a subsequence fr qðpÞ g of fr q g such that wðr qðpÞ , rÞ ≥ 1 for all p.
The framework of this study is organized into four sections. After the first introduction section, in Section 2, we introduced the definitions, theorems, and some results on the Ćirić-Rus-Reich-Suzuki-type hybrid. In Section 3, we give an application Ulam-Hyers-type stability to show the well-posedness for our fixed point theorem. Finally, in the last section, the conclusions are drawn.

Main Results
We begin with the definition of the Ćirić-Rus-Reich-Suzukitype hybrid contraction: for each r, v ∈ L, where a ≥ 0 and ρ i ≥ 0, i = 1,2,3, such that Theorem 8. Let ðL, b, sÞ be a complete b-metric space and w -orbital admissible map also wðr 0 , Mr 0 Þ ≥ 1 for some r 0 ∈ L. Given that M : L ⟶ L be a CRRS-type hybrid contraction satisfying one of the following conditions: Thereupon, M admits a fixed point in L.

Journal of Function Spaces
Proof. We install an iterative sequence fr q g of points such that M q ðr 0 Þ = r q for q = 0,1,2, ⋯ and r 0 ∈ L with wðr 0 , Mr 0 Þ ≥ 1. If r q 0 = r q 0+1 for some integers q 0 , then r q 0 = Mr q 0 . Thus, suppose that r q ≠ r q+1 , as M is w-orbital admissible, then wðr 0 , Mr 0 Þ = wðr 0 , r 1 Þ ≥ 1 implies that wðr 1 , Mr 1 Þ = w ðr 1 , r 2 Þ ≥ 1. Continuing this process, we get Condition 1: a > 0, by taking χ a M ðr, vÞ choosing r = r q−1 and v = Mr q−1 = r q in (3) we get where χ a M r q−1 , Mr q−1 Whereupon, we deduce that If we have given that bðr q , r q+1 Þ ≥ bðr q−1 , r q Þ, then, accompanying that ψ is nondecreasing with (9), we get which is a contradiction. Thus, we obtain b r q , r q+1 As a result, from (9), we will turn into b r q , r q+1 and by similarly this process, we obtain that for any q ∈ ℕ. We argue that fr q g is a fundamental sequence in ðL, b , sÞ. Then, let q, l ∈ ℕ such that l > q and using the triangle inequality with (13), we take By using Lemma 3, the series ∑ ∞ q=0 s q ψ q ðbðr 1 , r 0 ÞÞ is convergent where H t = ∑ t q=0 s q ψ q ðbðr 0 , r 1 ÞÞ, the above inequality finds and q, l ⟶ ∞, we obtain b r q , r l À Á ⟶ 0: Thus, fr q g is a fundamental sequence. Accompanying this together with the fact that the space ðL, b, sÞ is complete, it will imply that there exists p ∈ L such that We argue that p is a fixed point of M. If the suppose ðh 1 Þ takes, we get wðr q , pÞ ≥ 1, and we assert that for every q ∈ ℕ. Since, if we have given that then, by using conditions of b-metric spaces ðL, b, sÞ, since the sequence fbðr q , r q+1 Þg is decreasing, we write that b r q , r q+1 3 Journal of Function Spaces a contradiction. Therefore, for all q ∈ ℕ, either provides. In the condition that (21) takes, then by (3), we get If the second condition, (22) holds, we obtain Thereupon, taking q ⟶ ∞ in (23) and (24), which is contraction. Therefore, we get that bðp, MpÞ = 0 that is p = Mp: If the presume ðh 2 Þ is correct, and the map M is continuous, we get In case that last supposition, ðh 3 Þ holds, from above, we write M 2 p = lim q⟶∞ M 2 r q = lim q⟶∞ r q+2 = p, we want to show that Mp = p. Let us pretend otherwise, that is, p ≠ Mp from a contradiction. Eventually, p = Mp.
Condition 2: if a = 0, in the equation χ a M ðr, vÞ taking r = r q−1 and v = Mr q−1 = r q in (3) we write From (30), we find b r q , r q+1 and from ϱ 1 + ϱ 2 + ϱ 3 = 1, we attain that bðr q , r q+1 Þ < bðr q−1 , r q Þ for every q ∈ ℕ. Using (30), we take and as in condition 1, we can prove that Since the equal methods as in the case of a > 0, we clearly prove that fr q g is a fundamental sequence in a complete b -metric space. Additionally, for p ∈ L so, lim q⟶∞ bðr q , pÞ = 0 also we assert that p = Mp. In the meanwhile, ðL, b, sÞ is w-regular; thus, as fr q g confirm (5), and wðr q , r q+1 Þ ≥ 1 for each q ∈ ℕ, we obtain wðr q , pÞ ≥ 1. Moreover, as in the proof of condition 1, we know that either holds, for each q ∈ ℕ. If (34) is taken, we conclude that Let us assume that inequality (35) is satisfied, then Journal of Function Spaces Then, getting to the limit, we conclude that bðp, MpÞ = 0, and p = Mp: Now, the continuity of M implies p = Mp (from condition 1). Therefore, supposition ðh 3 Þ lead to M 2 p = lim q⟶∞ M 2 r q = lim q⟶∞ r q+2 = p. We will prove that Mp = p. Let's presume otherwise, that is, p ≠ Mp Thus, which is contradiction. In the case that a = 0, then, from (4) we get that a contradiction. Eventually, p = p * , so p is a unique fixed point of M.
also, M 2 = r/10, we get that M 2 is continuous but M is not continuous, where L = ½0, 2.