## Banach Journal of Mathematical Analysis

- Banach J. Math. Anal.
- Volume 11, Number 3 (2017), 497-512.

### On the existence of at least a solution for functional integral equations via measure of noncompactness

Calogero Vetro and Francesca Vetro

#### Abstract

In this article, we use fixed-point methods and measure of noncompactness theory to focus on the problem of establishing the existence of at least a solution for the following functional integral equation

$$u\left(t\right)=g(t,u(t\left)\right)+{\int}_{0}^{t}G(t,s,u(s\left)\right)\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}t\in [0,+\infty [,$$ in the space of all bounded and continuous real functions on ${\mathbb{R}}_{+}$, under suitable assumptions on $g$ and $G$. Also, we establish an extension of Darbo’s fixed-point theorem and discuss some consequences.

#### Article information

**Source**

Banach J. Math. Anal., Volume 11, Number 3 (2017), 497-512.

**Dates**

Received: 18 June 2016

Accepted: 28 July 2016

First available in Project Euclid: 19 April 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bjma/1492618126

**Digital Object Identifier**

doi:10.1215/17358787-2017-0003

**Mathematical Reviews number (MathSciNet)**

MR3679893

**Zentralblatt MATH identifier**

06754300

**Subjects**

Primary: 47H08: Measures of noncompactness and condensing mappings, K-set contractions, etc.

Secondary: 45N05: Abstract integral equations, integral equations in abstract spaces 54H25: Fixed-point and coincidence theorems [See also 47H10, 55M20]

**Keywords**

Banach space functional integral equation measure of noncompactness

#### Citation

Vetro, Calogero; Vetro, Francesca. On the existence of at least a solution for functional integral equations via measure of noncompactness. Banach J. Math. Anal. 11 (2017), no. 3, 497--512. doi:10.1215/17358787-2017-0003. https://projecteuclid.org/euclid.bjma/1492618126