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Our research introduces an innovative iterative method for approximating fixed points of contraction mappings in uniformly convex Banach spaces. To validate the stability of this iterative process, we provide a comprehensive theorem. Through detailed examples and graphical analysis, we demonstrate that our method outperforms previous approaches for contraction mappings, including those developed by Agarwal, Gursoy, Thakur, Ali, and D ∗ ∗ , using MATLAB software for implementation and comparison. Moreover, we investigate the effect of various parameters on the convergence behavior of our proposed method. By comparing it with existing iterative schemes through a specific example, we highlight the efficiency and robustness of our approach. Additionally, we establish a significant result concerning data dependence for an approximate operator, utilizing our iterative process to show how small changes in data can affect the outcome. Finally, we apply our fundamental findings to a practical problem by estimating solutions for a fractional Volterra-Fredholm integro-differential equation. This application not only illustrates the practical utility of our method but also underscores its potential for solving complex problems in applied mathematics. [ABSTRACT FROM AUTHOR]

In this paper we shall study, in terms of a metric and an order relation, the existence, uniqueness and data dependence for the solutions of integral equation x(t) = (g1(t) + t∫a K1(t, s, x(s))ds) · (g2(t) + t∫a a K2(t, s, x(s))ds), t ∈ [a, b]. Our results extend the results from I.M.Olaru An Integral Equation via Weakly Picard Operators, Fixed Point Theory, 11(2010), No. 1, pp 97-106. [ABSTRACT FROM AUTHOR]

In this paper, In this paper we prove the strong and Δ-convergence theorems of the SP*- iteration process for C-α nonexpansive mappings in CAT(0) spaces. Moreover we study the data dependence result of the proposed iteration process for contraction mappings in CAT(0) spaces. Also we provide an example that satisfies condition C-α. Further we apply to the approximate the solution of the integral equation. Our results improve and extend some recently results in the literature of fixed point theory in CAT(0) spaces. [ABSTRACT FROM AUTHOR]

This study presents a novel and efficient iterative approach to approximating the fixed points of contraction mappings in Banach spaces, specifically approximating the solutions of nonlinear fractional differential equations of the Caputo type. We establish two theorems proving the stability and convergence of the proposed method, supported by numerical examples and graphical comparisons, which indicate a faster convergence rate compared to existing methods, including those by Agarwal, Gursoy, Thakur, Ali and Ali, and D ∗ ∗ . Additionally, a data dependence result for approximate operators using the proposed method is provided. This approach is applied to achieve the solutions for Caputo-type fractional differential equations with boundary conditions, demonstrating the efficacy of the method in practical applications. [ABSTRACT FROM AUTHOR]

This research paper examines the data dependence of fixed point sets for pseudo-contractive multifunctions in partial metric spaces using the notion of $\mathcal{C}$-class functions. By building upon previous findings from the literature, this work sheds more light on some new perspectives as well as generalizations on this issue. To illustrate how the $\mathcal{C}$-class function can be applied to study the data dependence of fixed point sets for a certain pseudo-contractive multifunction, an illustrative example is given.

In the present work we take a system of two integral equations and prove the existence and uniqueness of their solution. We investigate four aspects of the problem, namely, error estimation and rate of convergence of the iteration leading to the solution, Ulam-Hyers stability, well-posedness and data dependence of the solution sets. We give some new definitions pertaining to the system we analyze here. In order to establish our results we utilize the coupled contraction mapping principle due to Bhaskar and Lakshmikantham (Nonlinear Anal. TMA 65(2006), 1379-1393) and several related results which we deduce here. [ABSTRACT FROM AUTHOR]

This research paper examines the data dependence of fixed point sets for pseudo-contractive multifunctions in partial metric spaces using the notion of C -class functions. By building upon previous findings from the literature, this work sheds more light on some new perspectives as well as generalizations on this issue. To illustrate how the C -class function can be applied to study the data dependence of fixed point sets for a certain pseudo-contractive multifunction, an illustrative example is given. [ABSTRACT FROM AUTHOR]

In this article, we re-analyse and improve the stability and data dependence results obtained by Kaur et al. [Math. Probl. Eng., 2022, Article ID 9327527] by eliminating certain restrictions imposed on control sequences by the authors. Additionally, we conceive a few weak and strong convergence results using -iterative technique to approximate the fixed point of generalized -nonexpansive mappings. However, we provide a couple of nontrivial examples to attest that the class of generalized -nonexpansive mappings and that of - nonexpansive mappings are independent. Finally, our findings are applied to approximate the solution of a certain kind of delay nonlinear Volterra integral equation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a new type of multi-valued operators and to present some basic problems of the fixed points and strict fixed points for them. Obtained results generalize, complement and extend classical results given by Ćirić (Mat. Vesnik 9 (24): 265-272, (1972)) or Nadler (Pacific J. Math. 30: 475-488 (1969)), as well as recent results given by Alecsa and PetruŞel (Anal. Univ. Vest Timisoara, LVII (1): 23-42 (2019)). [ABSTRACT FROM AUTHOR]

The purpose of this paper is to give some fixed point results for multi-valued Feng- Liu contractions in b-metric spaces, as well in b-metric spaces endowed with a graph. We will rovide existence and approximation theorems for generalized multi-valued Feng-Liu contraction. Data dependence, well-posedness and Ulam-Hyers stability for the fixed point inclusion are also studied. As an application to the coincidence problem for two multi-valued operators is given. [ABSTRACT FROM AUTHOR]

This manuscript is devoted to constructing a novel iterative scheme and reckoning of fixed points for generalized contraction mappings in hyperbolic spaces. Also, we establish Δ\Delta and strong convergence results by the considered iteration under the class of mappings satisfying condition (E). Moreover, some qualitative results of the suggested iteration, like weak w2{w}^{2}-stability and data dependence results, are discussed. Furthermore, to test the efficiency and effectiveness of the proposed iteration, practical experiments are given. To support the theoretical results, illustrative examples are presented. Finally, our results improve and generalize several classical results in the literature of fixed point iterations.

Addressing the boundary value problems of fractional-order differential equations hold significant importance due to their applications in various fields. The aim of this paper was to approximate solutions for a class of boundary value problems involving Caputo fractional-order differential equations employing the AA-iterative scheme. Moreover, the stability and data dependence results of the iterative scheme were given for a certain class of mappings. Finally, a numerical experiment was illustrated to support the results presented herein. The results presented in this paper extend and unify some well-known comparable results in the existing literature.

The aim of this paper is to exhibit a novel two-step iterative algorithm named PV algorithm to determine the fixed points of weak contractions in Banach spaces. Data dependence result is also obtained. It is proved that this PV iterative algorithm converges strongly to the fixed point of weak contractions. This iteration is almost stable for weak contraction. Furthermore, it is proved that rate of convergence of the PV iterative algorithm is faster than Picard, Ishikawa, Mann, S,normal-S, Varat, and F* algorithms. Examples are also given to support the main result. The results of this paper are original and will further enrich the existing literature. [ABSTRACT FROM AUTHOR]

In this paper, we will present existence and localization fixed point theorems and stability results for the fixed point problem involving some very general classes of operators (singlevalued and multi-valued), namely for Ćirić type operators. Then, an application to homotopy principles is given. Our results complement and extend the works in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we constructed a new and robust fixed point iterative scheme called the UO iterative scheme for the approximation of a contraction mapping. The scheme converges strongly to the fixed point of a contraction mapping. A rate of convergence result is shown with an example, and our scheme, when compared, converges faster than some existing iterative schemes in the literature. Furthermore, the stability and data dependence results are shown. Our new scheme is applied in the approximation of the solution to the oxygen diffusion model. Finally, our results are applied in the approximation of the solution to the boundary value problems using Green's functions with an example. [ABSTRACT FROM AUTHOR]

Addressing the boundary value problems of fractional-order differential equations hold significant importance due to their applications in various fields. The aim of this paper was to approximate solutions for a class of boundary value problems involving Caputo fractional-order differential equations employing the AA-iterative scheme. Moreover, the stability and data dependence results of the iterative scheme were given for a certain class of mappings. Finally, a numerical experiment was illustrated to support the results presented herein. The results presented in this paper extend and unify some well-known comparable results in the existing literature. [ABSTRACT FROM AUTHOR]

Finding the ideal circumstances for a mapping to have a fixed point is the fundamental goal of fixed point theory. These criteria can also be used for the structure under investigation. One of this theory’s most well-known theorems, Banach’s fixed point theorem, has been expanded adopting various methods, making it possible to conduct numerous research studies. Thanks to the Jungck-Contraction Theorem, which has been proven through commutative mappings, many fixed point theorems have been obtained using classical fixed point iteration methods and newly defined methods. This study aims to investigate the convergence, stability, convergence rate, and data dependency of the new four-step fixed-point iteration method. Nontrivial examples are also included to support some of the results herein.

A class of nonlinear integral equations with delay, related to infectious diseases, is studied. Making use of some tools from operators theory, we deal with the well-posedness in an adequate functional space, approximation of solution, estimates of lower/upper solutions and the data dependence of solutions.

The purpose of this article is to study A ∗ iterative algorithm in hyperbolic space. We prove the weak w 2 -stability, data dependence and convergence results of the proposed iterative algorithm for contractive-like mappings in hyperbolic spaces. Furthermore, we study several strong and ▵ -convergence analysis for fixed points of generalized Reich–Suzuki nonexpansive-type mappings. Some new numerical examples are provided to compare the efficiency and applicability of the proposed iterative algorithm over existing iterative algorithms. As an application, we use the proposed iterative method to approximate the solution of a delay nonlinear Volterra integral equation in hyperbolic spaces. We also furnished an example which validate the mild conditions in the application results. Our results are new and improve several results in the current literature. [ABSTRACT FROM AUTHOR]

We consider a frictionless contact model whose constitutive law and contact condition are described by means of subdifferential inclusions. For this model, we deliver a variational formulation based on two bipotentials. Our formulation envisages the computation of a three-field unknown consisting of the displacement vector, the stress tensor and the normal stress on the contact zone, the contact being described by a generalized Winkler condition. Subsequently, we obtain existence and uniqueness results. Some properties of the solution are also discussed, focusing on the data dependence. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iterative process and show that our iteration scheme is faster than other existing iteration schemes with the help of numerical examples. Next, we have established convergence, stability, and data dependency results for the approximation of fixed points of the Contractive-like mapping in the framework of uniformly convex Banach space. [ABSTRACT FROM AUTHOR]

This article analyzes the JK iteration process with the class of mappings that are essentially endowed with a condition called condition (E). The convergence of the iteration toward a fixed point of a specific mapping satisfying the condition (E) is obtained under some possible mild assumptions. It is worth mentioning that the iteration process JK converges better toward a fixed point compared to some prominent iteration processes in the literature. This fact is confirmed by a numerical example. Furthermore, it has been shown that the iterative scheme JK is stable in the setting of generalized contraction. The data dependence result is also established. Our results are new in the iteration theory and extend some recently announced results of the literature.

A class of nonlinear integral equations with delay, related to infectious diseases, is studied. Making use of some tools from operators theory, we deal with the well-posedness in an adequate functional space, approximation of solution, estimates of lower/upper solutions and the data dependence of solutions. [ABSTRACT FROM AUTHOR]

In this article, we propose the modified AH iteration process in Hyperbolic spaces to approximate the fixed points of mappings enriched with condition $ (E) $. The data dependence result of the proposed iteration process is studied for almost contraction mappings. Further, we obtain several new strong and $ \vartriangle $-convergence results of the proposed iteration algorithm for the class of mappings enriched with the condition $ (E) $. Also, we illustrate the efficiency of our results over existing results in literature with the aid of some numerical examples. Finally, we use our main results to find the solution of nonlinear integral equation with two delays.

In this paper we formulate a setvalued fixed point problem by combining four prevalent trends of fixed point theory. We solve the problem by showing that the set of fixed points is nonempty. Further we have a data dependence result pertaining to the problem and also a stability result for the fixed point sets. The main result is extended to metric spaces with a graph. The results are obtained without the use of metric completeness assumption which is replaced by some other conditions suitable for solving the fixed point problem. There are some consequences of the main result. The main result is illustrated with an example.

The purpose of this article is to study the convergence, stability and data dependence results of an iterative method for contractive-like mappings. The concept of stability considered in this study is known as $ w^2 $-stability, which is larger than the simple notion of stability considered by several prominent authors. Some illustrative examples on $ w^2 $-stability of the iterative method have been presented for different choices of parameters and initial guesses. As an application of our results, we establish the existence, uniqueness and approximation results for solutions of a nonlinear Volterra integral equation with delay. Finally, we provide an illustrative example to support the application of our results. The novel results of this article extend and generalize several well known results in existing literature.

In this paper, we modify the K F -iteration process into hyperbolic metric spaces where the symmetry condition is satisfied and establish the weak w 2 -stability and data dependence results for contraction mappings. We also prove some Δ -convergence and strong convergence theorems for generalized (α , β) -nonexpansive type 1 mappings. Finally, we offer a numerical example of generalized (α , β) -nonexpansive type 1 mappings and show that the K F -iteration process is more effective than some other iterations. Our results generalize and improve several relevant results in the literature. [ABSTRACT FROM AUTHOR]

In this paper we present some suggestions for unifying the terminology of the mappings appearing in fixed point iterative methods for the case when the setting is a metric space. We consider the following concepts: contraction type mapping, contractive type mapping and nonexpansive type mapping, for which some problems are formulated. [ABSTRACT FROM AUTHOR]

In this article, we propose the modified AH iteration process in Hyperbolic spaces to approximate the fixed points of mappings enriched with condition (E) . The data dependence result of the proposed iteration process is studied for almost contraction mappings. Further, we obtain several new strong and △ -convergence results of the proposed iteration algorithm for the class of mappings enriched with the condition (E) . Also, we illustrate the efficiency of our results over existing results in literature with the aid of some numerical examples. Finally, we use our main results to find the solution of nonlinear integral equation with two delays. In this article, we propose the modified AH iteration process in Hyperbolic spaces to approximate the fixed points of mappings enriched with condition . The data dependence result of the proposed iteration process is studied for almost contraction mappings. Further, we obtain several new strong and -convergence results of the proposed iteration algorithm for the class of mappings enriched with the condition . Also, we illustrate the efficiency of our results over existing results in literature with the aid of some numerical examples. Finally, we use our main results to find the solution of nonlinear integral equation with two delays. [ABSTRACT FROM AUTHOR]

In this paper, we prove that a three-step iterative algorithm converges strongly to the solution of a functional Volterra integral equation of the second kind. We also show the solution presented in the convergence theorem can obtain by removing a strong restriction imposed on the control sequence. Finally, we analyse the data-dependence result by using this iterative algorithm, and to support obtained result we give an example. [ABSTRACT FROM AUTHOR]

We have developed a Jungck version of the DK iterative scheme called the Jungck–DK iterative scheme. Our analysis focuses on the convergence and stability of the Jungck–DK scheme for a pair of non-self-mappings using the more general contractive condition. We demonstrate that this iterative scheme converges faster than all other leading Jungck-type iterative schemes. To further illustrate its effectiveness, we provide an example to verify the rate of convergence and prove the data dependence result for the Jungck–DK iterative scheme. Finally, we calculate the escape criteria for generating Mandelbrot and Julia sets for polynomial functions and present visually appealing images of these sets by our modified iteration. [ABSTRACT FROM AUTHOR]

In this paper, we utilize an inequality involving a coupled multivalued mapping and a singlevalued mapping to obtain a coupled coincidence point theorem. We discuss special conditions under which coupled common fixed point theorems are obtained. The result combines together several ideas prevailing in fixed point theory related studies. There are several corollaries and an illustrative example. In the second part of the paper, we establish a multivalued coupled coincidence point result by putting together several ideas. A contraction through an implicit relation for the coupled mapping is assumed in the main theorem for quadruples of points determined by certain functions. The coupled mapping is also required to satisfy certain dominated conditions which are defined in this paper. In Secs. 3 and 4, we study the data dependence and stability of coupled coincidence point sets, respectively. For describing set distance we use the Hausdorff metric. The work is in the domain of setvalued analysis. [ABSTRACT FROM AUTHOR]

In this paper we establish a coupled fixed point theorem of a coupled multivalued mapping defined on a complete metric space. In our result we use a new contractive inequality. There are rational terms in the expression of the inequality. The contractive condition on the nonlinear map is assumed only to hold on the elements that are related by a binary relation. The main theorem has several corollaries and is illustrated with example. The main result is deduced in metric spaces. Its consequences are discussed for α- admissible mapping as well as in metric spaces with partial order and graph respectively.

In this paper we will study existence, uniqueness and data dependence of the fixed points of multi-valued operators on a set endowed with two metrics. The case of multi-valued graph contractions is considered. Then, an extension to a more general contraction type condition is also given. [ABSTRACT FROM AUTHOR]

The purpose of this article is to study the convergence, stability and data dependence results of an iterative method for contractive-like mappings. The concept of stability considered in this study is known as w2-stability, which is larger than the simple notion of stability considered by several prominent authors. Some illustrative examples on w2-stability of the iterative method have been presented for different choices of parameters and initial guesses. As an application of our results, we establish the existence, uniqueness and approximation results for solutions of a nonlinear Volterra integral equation with delay. Finally, we provide an illustrative example to support the application of our results. The novel results of this article extend and generalize several well known results in existing literature. [ABSTRACT FROM AUTHOR]

In this paper, a system of nonlinear delay integral equations related to the spread of certain infectious diseases is investigated. Namely, using Perov's fixed-point theorem, we obtain sufficient conditions for which the system admits a unique positive solution, and provide a numerical algorithm that converges to this solution. Moreover, we establish some comparison theorems, and study the data dependence of solutions. [ABSTRACT FROM AUTHOR]

In this paper, we have provided some fixed point results for self-mappings fulfilling generalized contractive conditions on altered metric spaces. In addition, some applications of the main results to continuous data dependence of the fixed points of operators defined on these spaces were shown. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new mixed-type iterative algorithm for approximation of common fixed points of two multivalued almost contractive mappings and two multivalued mappings satisfying condition (E) in hyperbolic spaces. We consider new concepts of weak w 2 -stability and data dependence results involving two multivalued almost contractive mappings. We provide examples of multivalued almost contractive mappings to show the advantage of our new iterative algorithm over some exiting iterative algorithms. Moreover, we prove several strong ∆-convergence theorems of our new algorithm in hyperbolic spaces. Furthermore, with another novel example, we carry out a numerical experiment to compare the efficiency and applicability of a new iterative algorithm with several leading iterative algorithms. The results in this article extend and improve several existing results from the setting of linear and CAT(0) spaces to hyperbolic spaces. Our main results also extend several existing results from the setting of single-valued mappings to the setting of multivalued mappings. [ABSTRACT FROM AUTHOR]

In this paper, we consider several problems related to the so-called multi-valued Feng–Liu–Subrahmanyan contractions in complete metric spaces. Existence of the fixed points and of the strict fixed points, as well as data dependence and stability properties for the fixed point problem, are discussed. Some results are presented, under appropriate conditions, and some open questions are pointed out. Our results extend recent results given for multi-valued graph contractions and multi-valued Subrahmanyan contractions. [ABSTRACT FROM AUTHOR]

In this paper, our primary result is on the existence of non-empty fixed point sets for a new multivalued function defined here. An admissibility condition is also postulated and used in the main theorem. In two separate sections, we present data dependence and stability results for the non-empty fixed-point sets of these mappings. Some consequences of the main theorem are discussed. The analysis is in the most general setting of metric spaces. An illustrative example is discussed, which shows that our existence theorem effectively extends some results in this line of research.

The purpose of this paper is to introduce a new iterative algorithm to approximate the fixed points of almost contraction mappings and generalized α-nonexpansive mappings. Also, we show that our proposed iterative algorithm converges weakly and strongly to the fixed points of almost contraction mappings and generalized α-nonexpansive mappings. Furthermore, it is proved analytically that our new iterative algorithm converges faster than one of the leading iterative algorithms in the literature for almost contraction mappings. Some numerical examples are also provided and used to show that our new iterative algorithm has better rate of convergence than all of S, Picard-S, Thakur and M iterative algorithms for almost contraction mappings and generalized α-nonexpansive mappings. Again, we show that the proposed iterative algorithm is stable with respect to T and data dependent for almost contraction mappings. Some applications of our main results and new iterative algorithm are considered. The results in this article are improvements, generalizations and extensions of several relevant results existing in the literature.

In this paper we give some fixed point theorems for nonself generalized contractions on a large Kasahara space, which generalize some results given by I.A. Rus and M.-A. S¸erban (I.A. Rus, M.-A. S¸erban, Some fixed point theorems for nonself generalized contractions, Miskolc Math. Notes, 17(2016), no.2, 1021-1031) and by S. Reich and A.J. Zaslavski (S. Reich, A.J. Zaslavski, A note on Rakotch contractions, Fixed Point Theory, 9(2008), no.1, 267-273) in complete metric spaces. We prove our results without using the completeness of the metric structure. [ABSTRACT FROM AUTHOR]

In this paper, the concept of perturbed metric was introduced within the metric spaces and some fixed point results were established for self-mappings satisfying such contractive conditions, using Picard operators technique and generalized contractions. Moreover, some applications of the main result to continuous data dependence of the fixed points of Picard operators defined on these spaces were presented. Also, the main result of this paper was applied to study the existence and uniqueness of the solution for an integral equation which models an epidemiological problem. [ABSTRACT FROM AUTHOR]

We approximate the fixed point of the Suzuki's generalized nonexpansive mappings via the Picard-Ishikawa hyprid iterative process, recently introduced by Okeke [18]. We prove some weak and strong convergence theorems of this type of mappings in the setting of uniformly convex Banach spaces. We apply our results in finding the solution of a mixed type Volterra-Fredholm functional nonlinear integral equation in Banach spaces. Finally, we give several numerical examples to validate our analytical results. Our results extend and improve several known results in literature, including the results of Okeke [18], Ullah and Arshad [30] and Craciun and Serban [7] among others. [ABSTRACT FROM AUTHOR]