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Convergence Criteria for Fixed Point Problems and Differential Equations
Version 1
: Received: 17 November 2023 / Approved: 17 November 2023 / Online: 21 November 2023 (10:37:08 CET)
A peer-reviewed article of this Preprint also exists.
Sofonea, M.; Tarzia, D.A. Convergence Criteria for Fixed Point Problems and Differential Equations. Mathematics 2024, 12, 395. Sofonea, M.; Tarzia, D.A. Convergence Criteria for Fixed Point Problems and Differential Equations. Mathematics 2024, 12, 395.
Abstract
We consider a Cauchy problem for differential equations in a Hilbert space $X$. The problem is stated in a time interval $I$, which can be finite or infinite. Weuse a fixed point argument for history-dependent operators to prove the unique solvability of the problem. Then, we state and prove convergence criteria for both a general fixed point problem and thecorresponding Cauchy problem. These criteria provide necessary and sufficient conditions on a sequence $\{u_n\}$ which guarantee its
convergence to the solution of the corresponding problem, in the space of both continuous and continuously differentiable functions. We then specify our results in the study of a particular differential equation governed by two nonlinear operators. Finally, we provide an application in viscoelasticity and give a mechanical interpretation of the corresponding convergence result.
Keywords
differential equation; Cauchy problem; fixed point; history-dependent operator; convergence criterion; viscoelastic constitutive law
Subject
Computer Science and Mathematics, Analysis
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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