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

Yıl 2019, Cilt: 3 Sayı: 3, 111 - 120, 31.08.2019

İoan A. Rus

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

Cited By: 6

Öz

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.

Anahtar Kelimeler

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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