Fixed Point Results for an Almost Generalized α-Admissible Z- Contraction in the Setting of Partially Ordered b-Metric Spaces

In this paper, we introduce an almost generalized α-admissible Z-contraction with the help of a simulation function and study fixed point results in the setting of partially ordered b-metric spaces. The presented results generalize and unify several related fixed point results in the existing literature. Finally, we verify our results by using two examples. Moreover, one of our fixed point results is applied to guarantee the existence of a solution of an integral equation.


Introduction
Metric fixed point theory is a vivid topic, which furnishes useful methods and notions for dealing with various problems. In particular, we refer to the existence of solutions of mathematical problems reducible to equivalent fixed point problems. Thus, we recall that Banach contraction principle [1] is at the foundation of this theory. Due to its usefulness, Banach contraction principle has been extended and generalized in various spaces using different conditions either by modifying the basic contractive condition or by generalizing the ambient spaces or both. For some extensions of Banach contraction principle in metric spaces, see References [2][3][4][5][6][7][8][9][10][11][12].
The existence of fixed point in partially ordered sets has been considered by Turinici in ordered metrizable uniform spaces [11]. The applications of fixed point results in partially ordered metric spaces were studied by Ran and Reurings [13] to solve matrix equations and by Nieto and Rodríguez-López [14] to obtain solutions of certain partial differential equations with periodic boundary conditions. Many researchers have focused on different contractive conditions in complete metric spaces endowed with a partial order and obtained many fixed point results in such spaces. For further works in this direction, see References [3,[15][16][17][18][19]. The concept of metric spaces has been generalized in many directions. The notion of a b-metric space was introduced by Bakhtin in [20] and later extensively used by Czerwik in [21,22]. Since then, several papers have been published on the fixed point theory of various classes of single-valued and multivalued operators in (ordered) bmetric spaces. For further works in this direction, see References [2,4,[23][24][25][26][27].
Khojasteh et al. [28] introduced the notion of Z-contraction and studied existence and uniqueness of fixed points for Z-contraction type operators. This class of Z-contractions unifies large types of nonlinear contractions existing in the literature. Afterwards, Karapinar [25] originated the concept of α-admissible Z-contraction. For more works in this line of research, see References [3,4,9,12,25]. Recently, Melliani et al. [9] introduced a new concept of α-admissible almost type Z-contraction and proved the existence of fixed points for admissible almost type Z-contractions in a complete metric space.
Inspired and motivated by the works of [9,25], the purpose of this paper is to introduce a new class of mappings, namely, an almost generalized α-admissible Z-contraction, and prove the existence and uniqueness of fixed points for such mappings in the setting of partially ordered b-metric spaces.

Preliminaries
In this section, we present some notions, definitions, and theorems used in the sequel.
Throughout this paper, we shall use ℝ and ℝ + to represent the set of real numbers and the set of nonnegative real numbers, respectively.
Definition 1 (see [21]). Given a nonempty set X: A function d : X × X ⟶ ℝ + is called b-metric if there is a real number s ≥ 1 such that for all x, y, z ∈ X, the following conditions hold: The triplet ðX, d, sÞ is called a b-metric space.
Definition 2 (see [29]). Let ðX, dÞ be a b-metric space. Then, a sequence fx n g in X is said to be (a) b-convergent if there exists x ∈ X such that dðx n , xÞ ⟶ 0 as n ⟶ ∞: In this case, we write lim Remark 3 (see [29]). In a b-metric space ðX, dÞ, the following assertions hold: (1) (R1) A convergent sequence has a unique limit.
(4) (R4) In general, a b-metric does not induce a topology on X Definition 4. A partially ordered set (poset) is a system ðX, ≼Þ, where X is nonempty set and ≼ is a binary relation of X satisfying (i) x≼x (reflexivity).
(ii) if x≼y and y≼x, then x = y (antisymmetry).
(iii) if x≼y and y≼z, thenx≼z (transitivity) for all x, y, z ∈ X Definition 5. Let X be a nonempty set. Then, ðX, d,≼Þ is called partially ordered b-metric spaces if (i) ðX, dÞ is a b-metric space and (ii) ðX, ≼Þ is a partially ordered set. Now, we give an example to show that a b-metric is not necessarily metric.
Definition 6 (see [18]). Let ðX, ≼Þ be a partially ordered set and T : X ⟶ X is a self-mapping; we say T is monotone nondecreasing with respect to ≼ if for x, y ∈ X, Definition 7. Let (X, ≼) be a partially ordered set and x, y ∈ X; then, x and y are said to be comparable elements of X if x≼y or y≼x: Theorem 8 (see [30]). Let ðX, dÞ be a complete metric space and T : X ⟶ X be a map satisfying for all x, y ∈ X where α, β ≥ 0 with α + β < 1: Then, T has a unique fixed point.
If there exist ζ ∈ Z and α : for all x, y ∈ X, then T is called an α-admissible Z-contraction with respect to ζ.
Theorem 18 (see [25]). Let ðX, dÞ be a complete metric space and T be an α-admissible Z-contraction with respect to ζ. Suppose that Then, there exists u ∈ X such that u = Tu.

Main Result
In this section, we present our main findings.
Definition 19. Let ðX, d,≼Þ be a partially ordered b-metric space with parameter s ≥ 1. Let T : X ⟶ X and α : X × X ⟶ ℝ + be the maps. Assume that there exist a simulation function ζ and a constant L ≥ 0 such that for all x, y ∈ X with x≼y where Then, T is called an almost generalized α-admissible Z -contraction with respect to ζ. Theorem 20. Let ðX, d,≼Þ be a complete partially ordered bmetric space and T be an almost generalized α-admissible Z -contraction with respect to ζ. Suppose that (i) T is nondecreasing and continuous (ii) There exists x 0 ∈ X such that x 0 ≼Tx 0 and αðx 0 , Then, T has a fixed point.

Abstract and Applied Analysis
Define an iterative sequence fx n g in X as follows: for every n ≥ 0.
It follows that Furthermore, using Equation (15) and since s ≥ 1, we have the following: It follows that dðx n , x n+1 Þ < dðx n−1 , x n Þ: Hence, the sequence fdðx n−1 , x n Þg is nonincreasing and bounded below. Accordingly, there exists r ≥ 0 such that Assuming r > 0 and using Equation (20), we get the following: Letting t n = fαðx n−1 , x n Þdðx n , x n+1 Þg, s n = fdðx n−1 , x n Þg, and using ζ 3 , we obtain the following: which is a contradiction. Thus, we have the following: Now, we need to show that fx n g is a Cauchy sequence in X.
Suppose fx n g is not a Cauchy sequence in X, then there exists an ε > 0 such that Abstract and Applied Analysis where fm k g and fn k g are two sequences of positive integers with n k > m k > k for all positive integers k: Moreover, m k is chosen as the smallest integer satisfying Equation (25). Thus, we have the following: By applying the triangle inequality and using Equations (25) and (26), we get the following: Now, taking the upper limit as k ⟶ ∞ in Equation (27) and using Equation (24), we get the following: Similarly, applying the triangle inequality twice, we get the following: Furthermore, Taking the upper limit as k ⟶ ∞ in Equations (29) and (30) and combining, we get the following: Again, Furthermore, Taking the upper limit as k ⟶ ∞ in Equations (32) and (33), using Equation (31), and combining, we get the following: Similarly, we can show that Now, for simplicity, we denote the following expressions as follows: which give Mðx m k −1 , x n k −1 Þ = max fa k , b k , c k g and mðx m k −1 , x n k −1 Þ = min fd k , e k g: Using Equations (24), (34), and (35), and the property of lim sup, we can see that Since T is triangular α-orbital admissible, we have the following: Using Equation (38), we have the following: Using Equation (9) with x = x m k −1 and y = x n k −1 , we obtain the following: It follows that Combining Equations (39) and (41), we get the following: If Mðx m k −1 , x n k −1 Þ = b k , then after rearranging, collecting like terms, and applying the triangle inequality, Equation (42) becomes the following: 5 Abstract and Applied Analysis Taking the upper limit as k ⟶ ∞ in Equation (43) and using Equation (31), we get the following: which is a contradiction due to Equation (9). Hence, Using Equations (32) and (45), we get the following: Using Equations (45) and (46) together with Equations (24), (34), and (35) and applying the property of lim sup, it follows that Taking the lower limit in Equation (39) as k ⟶ ∞ and using Equation (25), we get the following: Taking the upper limit in Equation (40) and using Equations (37), (47), and (48), we get the following: which is a contradiction. Hence, fx n g is a Cauchy sequence in a complete bmetric space X. So, there exists u ∈ X such that Since T is continuous, we obtain the following: Thus, u is a fixed point of T.
Theorem 21. Let ðX, d,≼Þ be a complete partially ordered bmetric space and T is an almost generalized α-admissible Z -contraction with respect to ζ. Suppose that (i) T is nondecreasing (ii) There exists x 0 ∈ X such that x 0 ≼Tx 0 and αðx 0 , Tx 0 Þ ≥ 1 (iii) T is triangular α-orbital admissible (iv) There exists a nondecreasing sequence fx n g in X such that x n ⟶ x with x n ≼x and αðx n , xÞ ≥ 1 Then, T has a fixed point.
Proof. Following the proof of Theorem 20, we know that the sequence fx n g defined by x n+1 = Tx n for all n ≥ 0 converges to some u ∈ X: Hence, by (iii), we have the following: for all n: By (i), there exists x 0 ∈ X with x 0 ≼Tx 0 and since T is a nondecreasing map, we have the following: Thus, fx n g is a nondecreasing sequence that converges to u.
It follows that x n ≼u, for all n ∈ N: Now, we show that u = Tu. Applying Equation (9) with x = x n and y = u, we get the following: where Abstract and Applied Analysis From Equation (54), it follows that Using Equations (52) and (56), we have the following: Since lim n⟶∞ dðx n+1 , TuÞ = lim n⟶∞ ðMðx n , uÞ + L•mðx n , uÞÞ = dðu, TuÞ, we have the following: If dðu, TuÞ = 0, then u = Tu, then we are done. Assume dðu, TuÞ ≠ 0: Letting q n = fαðx n , uÞdðx n+1 , TuÞg and p n = fMðx n , uÞg and using ζ 3 , we obtain the following: which is a contradiction. Thus, we have dðu, TuÞ = 0; that is, u = Tu: Theorem 22. In addition to the hypotheses of Theorem 20 or Theorem 21, suppose that for every x, y ∈ X, there exists u ∈ X such that u≼x and u≼y and αðu, TuÞ ≥ 1: Then, T has a unique fixed point.
Proof. Referring to Theorem 20 or Theorem 21, the sets of fixed points of T are nonempty. Now, we shall show the uniqueness of fixed point. To prove the uniqueness of the fixed point, assume that there exist z 1 , z 2 ∈ X such that z 1 = Tz 1 and z 2 = Tz 2 with z 1 ≠ z 2 : Assume that there exists u 0 ∈ X such that u 0 ≼z 1 and u 0 ≼z 2 , then as in the proof of Theorem 20, we define the sequence such that for all n ≥ 0: Using the hypotheses of Theorem 22, Equation (60), and proceeding inductively, we get the following: Due to the monotone property of T, we have the following: T n u 0 = u n ≼z 1 = T n z 1 , T n u 0 = u n ≼z 2 = T n z 2 : Since z 1 , z 2 , u n ∈ X for all n ≥ 0, then z 1 = u m for some positive integer m, and hence, z 1 = Tz 1 = Tu n = u n+1 for all n ≥ m. It follows that u n ⟶ z 1 as n ⟶ ∞.
Using Equation (61) and the fact that u n ⟶ z 1 as n ⟶ ∞, we have αðu n , z 1 Þ ≥ 1 for all n ≥ 0: Now, suppose that z 1 ≠ u n for all n ≥ 0, so u n ≼z 1 for all n ≥ 0, then dðu n , z 1 Þ ≠ 0 for all n ≥ 0: Applying Equation (9) with x = u n and y = z 1 , we get the following: Hence, the sequence fdðu n , z 1 Þg is nonincreasing and bounded below. Accordingly, there exists r ≥ 0 such that lim n⟶∞ dðu n , z 1 Þ = r: Assume r > 0: Using Equation (69), we get the following: Letting b n = fαðu n , z 1 Þdðu n+1 , z 1 Þg and a n = fdðu n , z 1 Þg, and using ζ 3 , we obtain the following: which is a contradiction. Thus, we have lim n⟶∞ dðu n , z 1 Þ = 0; that is, u n ⟶ z 1 as n ⟶ ∞.
Similarly u n ⟶ z 2 as n ⟶ ∞.
Due to the uniqueness of the limit, it implies that z 1 = z 2 : Thus, T has a unique fixed point. Now, we give corollaries of our main theorem, Theorem 20.
If we take αðx, yÞ = 1 for all x, y ∈ X in Theorem 20, then we have the following result. for all x, y ∈ X with x≼y where Mðx, yÞ and mðx, yÞ are the same as in Theorem 20. Also, assume that the following conditions hold; (i) T is nondecreasing and continuous (ii) There exists x 0 ∈ X such that x 0 ≼Tx 0 Then, T has a fixed point. Similarly, we can deduce the following results.

Corollary 24.
Let ðX, d,≼Þ be a complete partially ordered bmetric space with parameter s ≥ 1. Let T : X ⟶ X and α : X × X ⟶ ½0,∞Þ be the maps. Suppose there exists a simulation function ζ such that for all x, y ∈ X with x≼y where Mðx, yÞ is defined in Theorem 20. Also, assume that the following conditions hold; (i) T is nondecreasing and continuous (ii) There exists x 0 ∈ X such that x 0 ≼Tx 0 and αðx 0 , Tx 0 Þ ≥ 1 (iii) T is triangular α-orbital admissible Then, T has a fixed point. for all x, y ∈ X with x≼y where Mðx, yÞ is defined in Theorem 20. Also, assume that the following conditions hold; (i) T is nondecreasing and continuous (ii) There exists x 0 ∈ X such that x 0 ≼Tx 0 Then, T has a fixed point. Now, we provide an example in support of Theorem 20.
Example 3. Let X = f0, 1, 2, 3, 4g and let d : X × X ⟶ ℝ + be defined by the following: for all x, y ∈ X. Hence, d is a complete b-metric space with parameter s = 2: Define a partial order on X as follows: Then, ðX, ≼Þ is a partially ordered set. We define the maps T : X ⟶ X and α : X × X ⟶ ℝ + as follows: Clearly, T is a continuous, nondecreasing, and triangular α-orbital admissible map.