Some variants of contraction principle in the case of operators with Volterra property: step by step contraction principle

Year 2019, Volume: 3 Issue: 3, 111 - 120, 31.08.2019

İoan A. Rus

https://doi.org/10.31197/atnaa.604962

Cited By: 6

Abstract

Following the idea of T.A. Burton, of progressive contractions, presented in some examples (T.A. Burton, \emph{A note on existence and uniqueness for integral equations with sum of two operators: progressive contractions}, Fixed Point Theory, 20 (2019), No. 1, 107-113) and the forward step method (I.A. Rus, \emph{Abstract models of step method which imply the convergence of successive approximations}, Fixed Point Theory, 9 (2008), No. 1, 293-307), in this paper we give some variants of contraction principle in the case of operators with Volterra property. The basic ingredient in the theory of step by step contraction is $G$-contraction (I.A. Rus, \emph{Cyclic representations and fixed points}, Ann. T. Popoviciu Seminar of Functional Eq. Approxim. Convexity, 3 (2005), 171-178). The relevance of step by step contraction principle is illustrated by applications in the theory of differential and integral equations.

Keywords

Space of continuous function, operator with Volterra property, max-norm, Bielecki norm, contraction, G-contraction, fiber contraction, progressive contraction, step by step contraction, Picard operator, weakly Picard operator, , differential equation, integral equation, conjecture, fixed point

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