Almost ´Ciri´c Type Contractions and Their Applications in Complex Valued b-Metric Spaces

: In this article, we present the use of a unique and common ﬁxed point for a pair of mappings that satisfy certain rational-type inequalities in complex-valued b-metric spaces. We also provide applications related to authenticity concerns in integral equations. Our results combine well-known contractions, such as the ´Ciri´c


Introduction and Preliminaries
One of the most significant and vital tools utilized by authors in the fields of nonlinear analysis, quantum physics, hydrodynamics, number theory, and economics is the Banach contraction principle [1]. This contraction has been generalized by weakening the contraction principle and enhancing the working spaces in different structures and generalized metrics, such as quasi-metric, b-metric, cone-metric, etc. For examples, see [2][3][4][5][6][7][8][9][10][11].
One of the remarkable and interesting generalizations of contraction mappings iś Cirić-type contractions (see [12]). For analyzing fixed points of self-mappings in different metrics spaces,Ćirić-type contractions offer a broader framework. A variety of results, such as the existence and uniqueness of fixed points, their stability, and the convergence of iterative procedures are investigated in the study ofĆirić-type contractions.
Similarly, a weakened form of contraction mapping, the "almost contraction", was introduced in 2004 by Berinde [13]. This contraction comprises the class of many mappings, notably Banach [14], Chatterjea [15], and Kannan [16]. However, it must be noted that unlike traditional contractions, almost contractions do not guarantee a unique fixed point.
A new concept called b-metric space was introduced in 1989 by Bakhtin [17]. Several important studies have been conducted by researchers in the field of b-metric space, including refs. [18][19][20][21]. In 2011, the metric space in complex version was firstly presented by Azam et al. [22]. Similarly, b-metric space in complex plane has been introduced in 2013 by Rao et al. [23].
Let us now recall the mentioned notions.
Definition 2 ([25]). Let ℘ 1 , ℘ 2 ∈ C. The max function for the partial order , defined on C as: Another important lemma that is helpful in justifying our new results is the following.

Definition 3 ([24]
). For the provided real number b ≥ 1 and a nonempty set Z, a functional Λ : Z × Z → C is termed as a complex valued b-metric (CVbM), if for all ℵ, ς, £ ∈ Z the necessities below fulfil: Definition 4 ([23]). Let (Z, Λ) be a CVbM space and {ℵ n } a sequence in Z and ℵ ∈ Z. (i) A sequence ℵ n in Z is convergent to ℵ ∈ Z if for every 0 ≺ c ∈ C there exists n 0 ∈ N, such that Λ(ℵ n , ℵ) ≺ c for every n > n 0 . In that case, we use the notation lim n→+∞ ℵ n = ℵ or ℵ n → ℵ as n → +∞. (ii) If for every 0 ≺ c ∈ C there exists n 0 ∈ N, such that Λ(ℵ n , ℵ n+m ) ≺ c for every n > n 0 and m ∈ N. Then, {ℵ n } is called a Cauchy sequence in (Z, Λ). (iii) If every Cauchy sequence in Z is convergent in Z, then (Z, Λ) is called a complete CVbM space.
Next, the contraction principle in [12] is to be recalled, which is the generalization of LjĆirić.
In this manuscript our aim is to combine and extend theĆirić and almost contraction conditions in the context of CVbM spaces. In addition, some examples and applications have been provided for the authenticity of our new generalization results.
We will use the following variant of the results from Miculescu and Mihail [27] (see also [28]).

Main Results
Here we present our first new result in the case of a CVbM space for a unique and common fixed point of almostĆirić-type contractions. Theorem 3. Let (C, d b ) be a complete CVbM space W, T : C → C be two continuous mappings, such that: for all z 1 , z 2 ∈ C, where 0 ≤ q < 1 s , L ≥ 0 and all elements on the right side can be compared to one another with partial order . Then, the pairs (W,T) has a unique common fixed point.
Then, by (1) and (2) we obtain We have three possible maximums. If Case I.
This implies that q ≥ 1, which is a contradiction. Case II. If Next, we have Then we find to have these three cases as below.
which is again the same contradiction. Case IIb.
From (3) and (4), for all n = 0, 1, 2, . . . we obtain For m, η ∈ N and m > η, we have Moreover, using (5) we have This implies that As a result, we have Thus, {µ η } has been proven to be a Cauchy sequence in C. Case IIc.

Case III
If Further, for the next step we obtain Then, once again, we have three cases: which is a contradiction, because we have q ≥ 1 here.
It follows from (7) and (8) that After some calculation, as completed before, we obtain Then, by (7) and (9) we obtain where 0 ≤h = qs 2−qs < 1. Then, for all η = 0,1,2, . . . , we obtain This will implies Using (11), we obtained This implies that As a result, we have Thus, µ η is a Cauchy sequence in C. We obtain µ η in all the above discussed cases as a Cauchy sequence. Because C is a complete space there, we haveḡ ∈ C, such that As we have T continuous, this implies that µ 2η+2 = Tµ 2η+1 → Tḡ as η → ∞. Since the limit is unique, we obtainḡ = Tḡ. Thus,ḡ is a common fixed point of the pair (W,T).

Uniqueness
To justify thatḡ is unique, let ∈ C be considered as another common fixed point of (W,T). Therefore, we have This implies that . This means that q ≥ 1, which causes a contradiction. Thus, =ḡ. Thus,b is unique.
be a complete CVbM space with s ≥ 1, a provided real number, and W, T : C → C be two mappings such that: for all z 1 , z 2 ∈ C, where 0 ≤ q ≤ 1 s and L ≥ 0 and all the element on the right side can be compared to one another with partial order . Then, W and T possess a unique common fixed point.
Proof. The sequence {u η } could be obtained as a Cauchy sequence using the same procedure used in Theorem 3. Because C is complete, there existsḡ ∈ C, such that d b (u η ,ḡ) → 0 as η → ∞. Because W and T omitted to have continuity, we have d b (ḡ, Wḡ) = k > 0. Then, we can estimate that so, k sqk. This implies that |k| ≤ sq|k|, which causes a contradiction. Consequently, g = Wḡ. In the same way, one can obtainḡ = Tḡ. Hence,ḡ is a common fixed point of (W,T). To justify the uniqueness ofḡ, one can use the similar approach as followed in Theorem 3.
Taking W = T we achieve the results below for, almostĆirić, type operators on CVbM spaces. Theorem 5. Let (C, d b ) be a complete CVbM space with s ≥ 1, a real number and W : C → C be a continuous mapping that fulfils: for all z 1 , z 2 ∈ C, where 0 ≤ q ≤ 1 s and L ≥ 0, and all the element on the right side can be compared to one another with partial order . Then, W possesses a unique fixed point.

Remark 1.
If operator W is omitted to be continuous, we would have a similar fixed point result.
be a complete CVbM space with s ≥ 1, coefficient, and W : C → C be a continuous mapping that fulfils: for all z 1 , z 2 ∈ C, where 0 ≤ q ≤ 1 s , L ≥ 0 n ∈ N and all the elements of the right side can be compared to one another's partial order . Then W possesses a unique fixed point.

Remark 2.
From Corollary 1, if one omits and does not consider the continuity of T, a similar result can be achieved.
Next, for almostĆirić type operators in CVbM spaces, we extend another generalization of a common fixed-point theorem. Theorem 6. Let (C, d b ) be a complete CVbM space with s ≥ 1, a provided real number, and W, T : C → C be two continuous mappings, such that: for all b 1 , b 2 ∈ C, where 0 ≤ q < 1 s , L ≥ 0 and all the elements of the right side can be compared to one another with partial order . Then, the pairs W and T possess a unique common fixed point.
To justify the uniqueness,l ∈ C is supposed to be another common fixed point of (W,T). Therefore, This implies that d b (l,ḡ) d b (l,ḡ), which causes a contradiction. Consequently,ḡ is a unique fixed point.
If the continuity of T and W is omitted in the above theorem, the below common fixed point result would be obtained. Theorem 7. Let (C, d b ) be a complete CVbM space with s ≥ 1, a provided real number, and W, T : C → C be two mappings such that for all b 1 , b 2 ∈ C where 0 ≤ q < 1 s , L ≥ 0 and all the elements of the right side can be compared to one another with partial order . Then, the pair (W,T) possesses a unique common fixed point.
Proof. It could be obtained that b η is a Cauchy sequence, using the same procedure used in Theorem 6. Because C is a complete space, there exists b * ∈ C, such that d b (b η , b * ) → 0 as n → ∞. Because we cannot consider the continuity of W and T, we obtain d b (b * , Wb * ) = k > 0. Then, we can estimate that This implies that |k| ≤ sq|k|, which causes the contradiction. Thus, b * = Wb * . Similarly, one can obtain b * = Tb * . Hence, b* is common fixed point of (W,T). To justify the uniqueness of b*, we can use the similar approach as followed proving Theorem 6.
For W = T in the previous result, we have the following result.
, a complete CVbM space with coefficient s ≥ 1, and W : C → C be a continuous mapping such that for all b 1 , b 2 ∈ C where 0 ≤ q < 1 s , L ≥ 0 and all the elements of the right side can be compared to one another with partial order . Then, W has a unique fixed point.

Remark 3.
If continuity of W is to be excluded, we can obtain the similar result. Corollary 2. Let (C, d b ) be a complete CVbM space with coefficient s ≥ 1, and W : C → C be a continuous mapping fulfilling for all b 1 , b 2 ∈ C, where 0 ≤ q ≤ 1 s , L ≥ 0, and all the element at the right side can be compared to one another with partial order . Then W possess a unique fixed point.
Proof. Considering Theorem 8, one can obtain b * ∈ C, in such a way that W η b * = b * . Then, one could obtain Then, W η b * = Wb * = b * . Therefore, the fixed point of W is unique.

Remark 4.
(i) Omitting continuity of W, we can obtain similar result from Corollary 2.
(ii) Plugging L = 0 into all the above results, one can obtain the results of [29].
Let us define two mappings where Q is a set of rational numbers andQ a set of irrational numbers.
There is no need to check the other conditions, because they fulfil the inequality (1) in Theorem 3.
(ii) If ϑ, η ∈Q, let ϑ = 1 √ 2 and η = π then Similarly, one can check (iii) and (iv). Thus, the fixed point of W and T is unique and common.

Applications to Integral-Type Contractions
In the present section, the fixed point results, derived in the above section, are implemented to prove common fixed points of some integral-type contractions. Initially, let us define altering distance function.
is an altering distance function.
Further, the first new result of this section is presented. Theorem 9. Let (C, d b ) be a complete CVbM space having s ≥ 1, a given real number, and W, T : C → C are two continuous mappings holding where all the element of M(z 1 , z 2 ) and m(z 1 , z 2 ) can be compared to one another w.r.t . Then (W,T) possess a unique common fixed point.
Proof. Considering Theorem 3, such that τ(w) = w 0h (v)dv, one can achieve the required solution.
Remark 6. The same result can be achieved, if one omit continuity of the mappings.
We deduce two fixed points theorems of integral-type results, if we take W = T, with and without continuity of W. Theorem 10. Let (C, d b ) be a complete CVbM space having s ≥ 1, a provided real number, and W : C → C be a continuous mappings that fulfil where all the elements of M(z 1 , z 2 ) and m(z 1 , z 2 ) can be compared to one another w.r.t . Then, W posses a unique common fixed point.
We would have the following common fixed-point integral-type result for the extension and generalization of almostĆirić-type contractions. Theorem 11. Let (C, d b ) be a complete CVbM space with coefficient s ≥ 1, and W, T : C → C be continuous mappings that fulfil where all the element of M(z 1 , z 2 ) and m(z 1 , z 2 ) can be compared to one another with . Then, (W,T) possesses a unique common fixed point.
Proof. Utilizing Theorem 6, such that taking ‫(ג‬t) = t 0h (v)dv one can achieve the required result.

Remark 7.
One can reach a similar conclusion if one excludes continuity and take the mappings as non-continuous.
Taking W = T, one can deduce two fixed-point theorems of integral-type results for the almostĆirić-type contractions, with and without continuous W.
where all the element of M(z 1 , z 2 ) and m(z 1 , z 2 ) can be compared to one another w.r.t . Then, W possesses a unique fixed point.
Proof. Utilizing Theorem 8, such that taking Υ(t) = t 0h (v)dv, we would achieve the result.

Conclusions
In the framework of CVbM spaces, the main goal of this publication is to combine and expand theĆirić and almost contraction conditions. Numerous applications and examples support the validity of our proposed generalization. These findings have significance for future studies in this field and provide useful insights into the behavior of mappings in complex-valued b-metric spaces.