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

Authors : İoan A RUS

Pages : 111-120

Doi:10.31197/atnaa.604962

View : 14 | Download : 11

Publication Date : 2019-08-31

Article Type : Research Paper

Abstract :Following the idea of T.A. Burton, of progressive contractions, presented in some examples insert ignore into journalissuearticles values(T.A. Burton, \emph{A note on existence and uniqueness for integral equations with sum of two operators: progressive contractions}, Fixed Point Theory, 20 insert ignore into journalissuearticles values(2019);, No. 1, 107-113); and the forward step method insert ignore into journalissuearticles values(I.A. Rus, \emph{Abstract models of step method which imply the convergence of successive approximations}, Fixed Point Theory, 9 insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(I.A. Rus, \emph{Cyclic representations and fixed points}, Ann. T. Popoviciu Seminar of Functional Eq. Approxim. Convexity, 3 insert ignore into journalissuearticles values(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 equati