Some observations on generalized non-expansive mappings with an application

Ali, Faeem; Ali, Javid; Nieto, Juan J.

doi:10.1007/s40314-020-1101-4

Cited by 22 publications

(17 citation statements)

References 31 publications

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“…A nontrivial example of a nonlinear Volterra delay integro-differential equation which satisfies all the mild conditions used in obtaining our result has been provided. We have also seen that the class of delay differential equation studied in [4,9,12,23,[25][26][27] is a special case of the class nonlinear Volterra delay integro-differential equation considered in this article. Hence, our results generalize, improve and unify the corresponding results in [4,9,12,15,[23][24][25][26][27][28] and several others in the existing literature.…”

Section: Discussionmentioning

confidence: 85%

“…x(t) = ψ(t), t ∈ [−r, 0], (6.10) which is the initial value problem for a nonlinear Volterra integro-differential equation. The approximation of solution the problem (6.9)-(6.10) has been studied by several authors for ℘ 1 (t, s, x(s), x(t − r)) = 0 (see for example [4,9,12,23,[25][26][27] and the references there in). Hence, our result in Theorem 8 generalizes the corresponding results in [4,9,12,23,[25][26][27] and several others.…”

Section: Applicationmentioning

confidence: 99%

“…Case (c): If x, y ∈(3,4], then we haved(Ty, x) + αd(Ty, x) + (1 − 2α) x − y ≥ 0 = H (Tx, Ty).Hence, T is a multivalued generalized 1/3-nonexpansive mapping.Next we show that T does not satisfy condition (C). Now, take x = 29/10 and y = 19/6…”

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confidence: 99%

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A Robust Iterative Approach for Solving Nonlinear Volterra Delay Integro–differential Equations

Ofem

Udofia

Igbokwe

2021

UMJ

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This paper presents a new iterative algorithm for approximating the fixed points of multivalued generalized $\alpha$–nonexpansive mappings. We study the stability result of our new iterative algorithm for a larger concept of stability known as weak $w^2$–stability. Weak and strong convergence results of the proposed iterative algorithm are also established. Furthermore, we show numerically that our new iterative algorithm outperforms several known iterative algorithms for multivalued generalized $\alpha$–nonexpansive mappings. Again, as an application, we use our proposed iterative algorithm to find the solution of nonlinear Volterra delay integro-differential equations. Finally, we provide an illustrative example to validate the mild conditions used in the result of the application part of this study. Our results improve, generalize and unify several results in the existing literature.

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Section: Discussionmentioning

confidence: 85%

Section: Applicationmentioning

confidence: 99%

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A Robust Iterative Approach for Solving Nonlinear Volterra Delay Integro–differential Equations

Ofem

Udofia

Igbokwe

2021

UMJ

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“…It will be useful to note the following. Next, we consider the following cases: Case 1: If p = (p 1 , p 2 , p 3 ), q = (q 1 , q 2 , q 3 ) ∈ [0, 4) × [0, 4) × [0, 4), then using (3.13), we have 4,8], then using (3.14), we obtain 4,8] and q = (q 1 , q 2 , q 3 ) ∈ [0, 4) × [0, 4) × [0, 4), then using (3.13), we obtain…”

Section: Rate Of Convergencementioning

confidence: 99%

A new iterative approximation scheme for Reich–Suzuki-type nonexpansive operators with an application

et al. 2022

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In this article, we propose a faster iterative scheme, called the AH iterative scheme, for approximating fixed points of contractive-like mappings and Reich–Suzuki-type nonexpansive mappings. We show that the AH iterative scheme converges faster than a number of existing iterative schemes for contractive-like mappings. The $w^{2}$ w 2 -stability result of the new iterative scheme is established and a supportive example is provided to illustrate the notion of $w^{2}$ w 2 -stability. Then, we prove weak and several strong convergence results of our new iterative scheme for fixed points of Reich–Suzuki-type nonexpansive mappings. Further, we carry out a numerical experiment to illustrate the efficiency of our new iterative scheme. As an application, we use our main result to prove the existence of a solution of a mixed-type nonlinear integral equation. An illustrative example to validate the result in our application is also given. Our results extend and generalize several related results in the existing literature.

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“…(e.g., see Ali et al 2020;Khan 2013;Thakur et al 2016;Katchang and Kumam 2010;Maingè and Mȃruşter 2011). The following iterative algorithms are called (Picard 1890;Mann 1953;Ishikawa 1974), S (Agrawal et al 2007), normal-S (Sahu 2011), and Varat (Sintunavarat and Pitea 2016) algorithms, respectively, for the self-mapping G defined on Y :…”

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confidence: 99%

Convergence, stability, and data dependence of a new iterative algorithm with an application

Ali

2020

Comp. Appl. Math.

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The purpose of this article is to introduce a new two-step iterative algorithm, called F * algorithm, to approximate the fixed points of weak contractions in Banach spaces. It is also showed that the proposed algorithm converges strongly to the fixed point of weak contractions. Furthermore, it is proved that F * iterative algorithm is almost-stable for weak contractions, and converges to a fixed point faster than Picard, Mann, Ishikawa, S, normal-S, and Varat iterative algorithms. Moreover, a data dependence result is obtained via F * algorithm. Some numerical examples are presented to support the main results. Finally, the solution of the nonlinear quadratic Volterra integral equation is approximated by utilizing our main result. The results of the paper are new and extend several relevant results in the literature. Keywords F * iterative algorithm • Weak contraction • Fixed points • Numerically stable • Data dependence • Nonlinear quadratic Volterra integral equation Mathematics Subject Classification 47H05 • 47H09 • 47H10 1 Introduction and preliminaries Throughout this article, we assume that Z + is the set of nonnegative integers, Y a nonempty, closed and convex subset of a Banach space X , and F(G), the set of fixed points of the self-mapping G defined on Y. The iterative approximation of fixed points of linear and nonlinear mappings is one of the most significant tools in the fixed point theory that has many applications in different fields like Engineering, Differential equations, Integral equations, etc. Hence, a large number of researchers introduced and studied many iterative algorithms for certain classes of mappings Communicated by Apala Majumdar.

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Some observations on generalized non-expansive mappings with an application

Cited by 22 publications

References 31 publications

A Robust Iterative Approach for Solving Nonlinear Volterra Delay Integro–differential Equations

A Robust Iterative Approach for Solving Nonlinear Volterra Delay Integro–differential Equations

A new iterative approximation scheme for Reich–Suzuki-type nonexpansive operators with an application

Convergence, stability, and data dependence of a new iterative algorithm with an application

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