Iterative Stability Analysis for Generalized α -Nonexpensive Mappings with Fixed Points

: This article introduces a novel iterative process, denoted as F ⋆ , designed for the class of generalized α -Nonexpensive mappings. The study establishes strong and weak convergence theorems within the context of Banach spaces, supported by carefully chosen assumptions. The convergence results contribute to the theoretical foundation of iterative processes in functional analysis. The presented framework is applied to address nonlinear integral equations, showcasing the versatility and applicability of the proposed F ∗ for the class of generalized iteration process. Additionally, the article includes numerical examples that not only validate the theoretical findings but also provide insights into the practical utility of the developed methodology.


Introduction
When the presence of a solution for a given operator is affirmed, ordinary analytical approaches often fall short of obtaining such solutions.To surmount this challenge, resorting to approximation methods becomes imperative.One of the key findings in fixed point (FP) theory is the Banach contraction theorem, which Polish mathematician Stefan Banach presented in 1922 [1].The Banach contraction theorem relies on the convergence of a fundamental iterative process known as the sequence of successive approximations or the Picard iteration process.This theorem addresses an FP problem for a contraction mapping defined on a complete metric space [2].It has evolved into a crucial tool for demonstrating the existence and approximation of solutions to nonlinear functional equations, including those arising in differential, integral, and partial differential equations.While certain scenarios ensure the existence of a solution to the FP problem, determining the exact solution may be infeasible.In these situations, there is a great desire to approximate the solution to the given problem, which prompts the creation of numerous iterative procedures.In the domain of nonexpansive mappings, denoted as NonExp mappings, the assurance of Picard iteration process convergence to a FP is not guaranteed for specific classes of NonExp mappings [2].Consequently, alternative iterative approaches are employed, incorporating different procedural steps and parameter sets.Several iterative processes, such as those introduced by Mann [3], Ishikawa [4], Noor [5], S iteration by Agarwal et al. [6], Abbas [7], and Picard-S [8], have undergone extensive investigation.
In 2018, Ullah and Arshad [9] introduced the M iterative scheme for Suzuki mappings, showcasing its faster convergence compared to the aforementioned processes.In 2008, Suzuki [10] proved analogous FP theorems in Banach space, denoted as (BS), showing that the class of maps meeting condition (C) is weaker than the idea of NonExp mappings.Browder in 1965 [11] focused on uniformly convex BS, while Kirk [12], in a reflexive BS, established FPs for NonExp mappings.More recently, Ullah et al. [13] applied the M iteration scheme to find FPs of generalized α-Nonexpensive mappings, denoted as GNZ-α-NonExp mappings, in BS.
In 2020, Ali and Ali [14] introduced a novel iteration process, the F iterative process, for generalized contractions.They demonstrated the stability and superior convergence rate of this F iterative scheme compared to other iteration processes in the context of generalized contractions.
[2] established a connection between the F iteration process and the class of GNZ-α-NonExp mappings.Under suitable assumptions, they derived both strong and weak convergence outcomes within the framework of BS.The following iteration process (1) for uniformly convex Banach space, denoted as UCBS, as given in [2], is described as follows: where Ω is GNZ-α-NonExp mapping and α t ∈ (0, 1).
In this paper, we modify the iterative process given in [2] for GNZ-α-NonExp mappings.For this purpose a new function x t = Ωx t is defined in the above iterative process (1).By using this new function, we observe early convergence with high accuracy.This finding is supported by providing two different examples.The obtained numerical results are presented in Tables 1 and 2. These results are compared with the previous iterative processes.Another interesting property of the above process is that if the number of functions similar to x t is increased, then rapid convergence along with high accuracy can be obtained.However, for simplicity, we consider only one function x t in our study.Noor [5] 1 6.9 6.9 6.9 6.9 6.9 6.9 6.9 2 6.04921875In this manuscript, we have six sections.Section 1, includes the introduction of our research, and Section 2 is related to the preliminaries and results, which are helpful to construct the main result of the paper.Section 3 is the main part of the paper where we define our main scheme and the strong and weak convergence results of the scheme, whereas Sections 4 and 5 are about the useability of our results.Here, we apply our findings for the solution of non-linear integral equation.In the last section, we conclude our findings.

Preliminaries
This section deals with the preliminaries and results, which help construct the main results of this article.

Definition 1 ([2]
). Suppose that T is a BS and S ⊆ T is a nonempty subset.Let Ω : S → S be a mapping.Then Ω is said to be ). Suppose that T is a BS and S ⊆ T is a nonempty subset.Let Ω : S −→ S be a mapping.

Definition 3 ([15]
).A BS T is said to satisfy the Opial condition (OpCd), if for every sequence w t ⊆ T that weakly converges to j ∈ T , the following condition holds: In the case of UCBS, A(S, {w t }) has a cardinality equal to one [6,16].Pant and Shukla [17] described the following facts about GNZ-α-NonExp mappings.

2.
If Ω is a GNZ-α-NonExp mapping with a nonempty FP, then for any d ′ ∈ S and d ⋆ as the When Ω is a GNZ-α-NonExp mapping, the set F ⋆ Ω is closed.Additionally, if the fundamental space T is strictly convex and S is convex, then F ⋆ Ω is convex.
We have an interesting property of a UCBS [18].

Lemma 1 ([18]
). Suppose that T is any UCBS for which the r, α t , s are defined as for some q ≥ 0. Then lim t→∞ ||w t − x t || = 0.

Generalized Iterative Scheme for α-NonExp Mappings
Suppose that S is a UCBS and Ω : S → S is a GNZ-α-NonExp mapping.Then a modified iteration process for the present study is described as where Ω is GNZ-α-NonExp mapping and α t ∈ (0, 1).
In the mentioned procedure, a novel function x t = Ωx t is introduced.It is noted that attaining early convergence with enhanced accuracy is possible by augmenting the number of functions akin to x t utilized in the aforementioned approach.However, for the sake of simplicity in the current scenario, we focus on a single function x t .We term this innovative iterative method as F ⋆ , representing a generalized version of the scheme proposed by Junaid [2].
In this section, the results related to the strong and weak convergence for the new iterative scheme F ⋆ are described.
Lemma 2. Suppose that T is any UCBS and S ⊆ T is closed, nonempty and convex.
Let Ω : S −→ S be a GNZ-α-NonExp mapping satisfying F ⋆ Ω ̸ = ∅ and {w t } be a sequence of F ⋆ iterates (2).Then Using Proposition 1(2), we have Consequently, From Lemma 1, we have By applying the lim and using (4), we have lim sup Since d ⋆ is in the set F ⋆ Ω , and by using Proposition 1(2), we get the following: From Lemma 1, we have From ( 5) and ( 6), we have By ( 3) and ( 1), we have By Lemma 1, we get lim Conversely, we will demonstrate the non-emptiness of the set F ⋆ Ω under the conditions that {w t } is bounded and approaches zero as lim t→∞ ||Ωw t − w t || = 0.If we apply the Proposition 1(4), we have We observe that Ωd ⋆ ∈ A(S, {w t }).In the case of UCBS, the set A(S, {w t }) has cardinality equal to one.This concludes that Ωd ⋆ = d ⋆ .Thus, the set F ⋆ Ω is nonempty.
The weak and strong convergences of F ⋆ iteration (2) are established as follows.
Theorem 2. Consider any UCBS T satisfying the OpCd, where S ⊆ T is a closed, nonempty, and convex subset.If Ω : S → S is a GNZ-α-NonExp mapping with F ⋆ Ω ̸ = ∅, and {w t } represents a sequence of F ⋆ iterates (2), then {w t } weakly converges to a FP of Ω.
Proof.By Theorem 1, {w t } is bounded in S. Since T is a UCBS, T will be reflexive; hence, the bounded sequence {w t } admits a weakly convergent sub sequence {w t r } with weak limit, namely, w 1 ∈ S. If we apply Theorem 1 on this subsequence, we obtain lim t→∞ ||w t r − Ωw t r || = 0. Thus, by Proposition 1(5), one has w 1 ∈ F ⋆ Ω .If we prove that w 1 is a weak limit for {w t }, then the proof will be finished.We prove this by contradiction.Suppose w 1 is not a weak limit for {w t }; that is, another subsequence {w t s } of {w t } exists that admits a weak limit w 2 ∈ S. The same calculations give w 2 ∈ F Ω .Now we know that J admits the OpCd.Thus, one has lim The above estimate suggests a contradiction and, hence, we must accept that w 1 = w 2 .Accordingly, {w t } converges weakly to w 1 ∈ F ⋆ Ω .
Theorem 3. Assuming that T is a UCBS satisfying the OpCd, and S ⊆ T is a convex, nonempty, and compact subset.If Ω : S → S is a GNZ-α-NonExp mapping with F ⋆ Ω ̸ = ∅, and {w t } denotes a sequence of F ⋆ iterates (2), then {w t } exhibits strong convergence to a FP of Ω.
Proof.Due to the convexity of S, we have {w t } ⊆ S. Theorem 5. Assume that T is a UCBS with the OpCd, and S ⊆ J is a closed, nonempty, and convex subset.Let Ω : S −→ S be a mapping satisfying condition (C), with F ⋆ Ω ̸ = ∅ and {w t } representing a sequence of F ⋆ iterates (2).Then, {w t } exhibits strong convergence to a FP of Ω.
Proof.From (1), we have lim inf By using the definition of condition (I) [2], we have Applying ( 7) and ( 8), we have By Theorem 4, {w t } is strongly convergent to an FP of Ω.
To support the main results, we provide two examples by Proposition 1(1) of GNZ-α-NonExp mapping, which is provided with C. By using these examples, we implement our new proposed method to obtain numerical results.These results are compared with other iteration schemes that are used for GNZ-α-NonExp mappings.Our aim is to establish that Ω is a GNZ-α-NonExp mapping with α = 1 2 , while it does not fall under the category of Suzuki mappings.This example, therefore, extends beyond the class of Suzuki mappings.
Case I: If d ′ , d ′′ = 13, then we have Case II: If d ′ , d ′′ ≤ 13, then we have Case III: If d ′ = 13 and d ′′ < 13, then we have Example 2. Consider the closed and bounded subset S = [6,15] within the Banach space R.
In this example, we prove again that Ω is a GNZ-α-NonExp mapping with α = 1 4 , but not a Suzuki mapping.This example thus exceeds the class of Suzuki mappings.
Case I: If d ′ = 15 = d ′′ , then we have Case II: If d ′ , d ′′ ≤ 15, then we have Then the integral Equation ( 4) possesses a solution.Clearly, Ω satisfies the condition (C) and by Definition 2, Ω is a GNZ-α-NonExp mapping.All the conditions given in Theorem 5 are satisfied.Thus, Ω has an FP that is the solution of the nonlinear integral Equation (4).

Experimental Results
We have a nonlinear integral equation of the form The equation represents a relationship between the derivative of an unknown function d(t) and the integral of a function w(s, d(s)) over a given interval [c, d], weighted by the kernel function K(t, s).The function f (t) represents a driving force.
To solve this equation, we will employ the FP Iteration method.We start with an initial approximation d 0 (t) and update it iteratively by using the following formula: where n represents the iteration number.
In the example, we will consider specific functions for f (t), w(s, d), and K(t, s), and the interval [c, d].We will solve (9) by using the FP Iteration method with an initial approximation d 0 (t).We will evaluate the solution at various points within the interval [c, d].
Table 3 provided in the example shows the approximation, exact value, and error at each iteration.Each row corresponds to a specific iteration, with the approximation denoted as d n (t), the exact value denoted as d(t), and the calculated error being the absolute difference between the approximation and the exact value |d n (t) − d(t)|.Furthermore, a graphical representation will be given to visualize the convergence behavior of the iterative scheme.The graph will show the exact solution and the iteratively computed approximations d n (t) for each iteration.
Overall, the example will demonstrate the process of solving a nonlinear integral equation by using the FP Iteration method and provide insights into the convergence behavior through the table and graph.
where f (t) and w(s, d) are continuous functions, and K(t, s) is a kernel function.To solve (10), we can use an iterative scheme called the FP Iteration method.We start with an initial approximation d 0 (t) and iteratively update it by using the following formula: We will solve (10) by using the FP Iteration method with an initial approximation d 0 (t) = t.We will evaluate the solution at various points in the interval [0, 1].Table 3 shows the approximation, exact value, and error at each iteration while Table 4 shows the solution function d(t) and the iteratively computed approximations d n (t) for each iteration.

Conclusions
This paper presents several novel contributions: (i) We introduce a new iterative scheme, denoted as F * , specifically designed for GNZ-α-NonExp mappings.Our findings indicate that the inclusion of an additional function

Figure 2 .Figure 3 .
Figure 2. Graphical representation of the approximation and exact values given in Table3.

Table 1 .
Convergence comparison of F * scheme (2) for GNZ-α-NonExp mapping given in Example 1 with other schemes with respect to the number of iterations.

Table 2 .
Convergence comparison of F * scheme (2) for GNZ-α-NonExp mapping given in Example 2 with other schemes with respect to the number of iterations.
Suppose that T is a BS and S ⊆ T is nonempty and {w t } ⊆ T is a bounded sequence.For a fix j ∈ T the following assumptions hold: 1.at the point j of asymptotic radius of bounded sequence {w t } by R(j, {w t }) = lim sup Definition 4 ([2]).t→∞ ||j − w t ||; 2. the connection of S with asymptotic radius of bounded sequence {w t } by R(S, {w t }) = inf{r(j, {w t }) : j ∈ S}; 3. the connection of asymptotic center of bounded sequence {w t } with S by A(S,

Table 3 .
Approximation, exact, and error values for different iterations.

Table 4 .
Solution function d(t) and the iteratively computed approximations d n (t) for different iterations., we can plot the graph of the solution function d(t) and the iteratively computed approximations d n (t) for each iteration.Figures 2 and 3 provide a visual representation of the convergence behavior for each iteration. Finally