TY  - JOUR
T1  - Some variants of contraction principle in the case of operators with Volterra property: step by step contraction principle
AU  - Rus, İoan A.
PY  - 2019
DA  - August
DO  - 10.31197/atnaa.604962
JF  - Advances in the Theory of Nonlinear Analysis and its Application
JO  - ATNAA
PB  - Erdal KARAPINAR
WT  - DergiPark
SN  - 2587-2648
SP  - 111
EP  - 120
VL  - 3
IS  - 3
LA  - en
AB  - 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.
KW  - Space of continuous function
KW  - operator with Volterra property
KW  - max-norm
KW  - Bielecki norm
KW  - contraction
KW  - G-contraction
KW  - fiber contraction
KW  - progressive contraction
KW  - step by step contraction
KW  - Picard operator
KW  - weakly Picard operator
KW  - 
KW  - differential equation
KW  - integral equation
KW  - conjecture
KW  - fixed point
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UR  - https://doi.org/10.31197/atnaa.604962
L1  - https://dergipark.org.tr/en/download/article-file/788789
ER  -