# 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

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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

1583-5022

2066-9208