On the Burton method of progressive contractions for Volterra integral equations

Abstract

In the paper [4] the author give a new method to study the existence and uniqueness of a solution on [0, ∞[ of a scalar integral equation x(t) = g(t, x(t)) + Z t0  A(t − s)f(t, s, x(s))ds, where u, v ∈ R, t ∈ [0, ∞[ imply that there exists 0 < l < 1 with |g(t, u) − g(t, v)| ≤ l |u − v| and for each b > 0 there exists Lb > 0 such that |f(t, u) − f(t, v)| ≤ Lb |u − v| , ∀t ∈ [0, b], ∀u, v ∈ R. In this paper we extend the Burton method to the case where instead of scalar equations we consider an equation in a Banach space.

 

Authors

Veronica Ilea
Babes-Bolyai University, Faculty of Mathematics and Computer Science

Diana Otrocol
Technical University of Cluj-Napoca
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Keywords

Progressive contractions; fixed points; existence; uniqueness; integrodifferential equations.

References

[1] T.A. Burton, Integral equations, transformations, and a Krasnoselskii-Schaefer type fixed point theorem, Electronic J. Qual. Theory Differ. Equ., 2016, no. 66, 1-13; DOI: 10.14232/ejqtde.2016.1.66.
[2] T.A. Burton, Existence and uniqueness results by progressive contractions for integro- differential equations, Nonlinear Dynamics and Systems Theory, 16(4)(2016), 366-371.
[3] T.A. Burton, An existence theorem for a fractional differential equation using progressive contractions, J. Fractional Calculus and Applications, 8(1)(2017), 168-172.
[4] T.A. Burton, A note on existence and uniqueness for integral equations with sum of two operators: progressive contractions, Fixed Point Theory, 20(2019), no. 1, 107-112.
[5] V. Ilea, D. Otrocol, An application of the Picard operator technique to functional integral equations, J. Nonlinear Convex Anal., 18(2017), no. 3, 405-413.
[6] N. Lungu, I.A. Rus, On a functional Volterra-Fredholm integral equation, via Picard operators, J. Math. Ineq., 3(2009), no. 4, 519-527.
[7] D. Otrocol, V. Ilea, On the qualitative properties of functional integral equations with abstract Volterra operators, Res. Fixed Point Theory Appl., Vol. 2018, Art. ID 201813, 8 pages.
[8] I.A. Rus, A class of nonlinear functional-integral equations, via weakly Picard operators, Anal. Univ. Craiova, Ser. Mat-Inf., 28(2001), 10-15.
[9] I.A. Rus, Generalized Contractions and Applications, Cluj University Press, 2001.
[10] I.A. Rus, Picard operators and applications, Sci. Math. Jpn., 58(2003), no. 1, 191-219.
[11] I.A. Rus, Abstract models of step method which imply the convergence of successive approximations, Fixed Point Theory, 9(2008), 293-297.
[12] M.A. Serban, Data dependence for some functional-integral equations, J. Appl. Math., 1(2008), no. 1, 219-234.
[13] M.A. Serban, I.A. Rus, A. Petrusel, A class of abstract Volterra equations, via weakly Picard operators technique, Math. Inequal. Appl., 13(2010), 255-269.

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Cite this paper as:

V. Ilea, D. Otrocol, On the Burton method of progressive contractions for Volterra integral equations, Fixed Point Theory, 21 (2020) no. 2, 585-594, DOI: 10.24193/fpt-ro.2020.2.41

Journal

Fixed Point Theory

Publisher Name
Print ISSN

1583-5022

Online ISSN

2066-9208

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