- ISSN: 2587-2648
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Ioan A. Rus
Department of Mathematics, Babeș-Bolyai University, 1 Kogălniceanu Street, 400084, Cluj-Napoca, Romania
Following the idea of T.A. Burton on progressive contractions presented in several examples (Fixed Point Theory, 20 (2019), No. 1, 107–113) and the forward step method (Fixed Point Theory, 9 (2008), No. 1, 293–307), this paper presents new variants of the contraction principle in the case of operators with the Volterra property. The main concept underlying the step-by-step contraction approach is the notion of G-contraction (Ann. T. Popoviciu Seminar of Functional Eq. Approx. Convexity, 3 (2005), 171–178). The significance of the step-by-step contraction principle is illustrated through its 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, fixed point, Picard operator, weakly Picard operator, differential equation, integral equation, conjecture.
Rus, İ. A. (2019). Some variants of contraction principle in the case of operators with Volterra property: step by step contraction principle. Advances in the Theory of Nonlinear Analysis and its Application, 3(3), 111-120.