## 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

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[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.

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[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.

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## About this paper

##### 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

##### Print ISSN

1583-5022

##### Online ISSN

2066-9208