Convergence Criteria for Fixed Point Problems and Differential Equations

Mircea Sofonea () and Domingo A. Tarzia
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Mircea Sofonea: Laboratoire de Mathématiques et Physique, University of Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan, France
Domingo A. Tarzia: Departamento de Matemática, FCE, Universidad Austral, Paraguay 1950, Rosario S2000FZF, Argentina

Mathematics, 2024, vol. 12, issue 3, 1-19

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. We use a fixed point argument for history-dependent operators to prove the unique solvability of the problem. Then, we establish convergence criteria for both a general fixed point problem and the corresponding Cauchy problem. These criteria provide the 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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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