## Journal of Integral Equations and Applications

- J. Integral Equations Applications
- Volume 31, Number 2 (2019), 149-164.

### Solvability and uniform local attractivity for a Volterra equation of convolution type

Luciano Abadias, Edgardo Alvarez, Józef Banaś, and Carlos Lizama

#### Abstract

We show the existence of uniformly locally attractive solutions for a nonlinear Volterra integral equation of convolution type with a general kernel. We use methods and techniques of fixed point theorems and properties of measure of noncompactness. We extend earlier results obtained in the context of integral equations of fractional order. We give new insights about a new and striking relation between the size of data and the fractional order $\alpha >0$ of the kernel $k(t)=t^{\alpha -1}/\Gamma (\alpha )$.

#### Article information

**Source**

J. Integral Equations Applications, Volume 31, Number 2 (2019), 149-164.

**Dates**

First available in Project Euclid: 23 September 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.jiea/1569225671

**Digital Object Identifier**

doi:10.1216/JIE-2019-31-2-149

**Mathematical Reviews number (MathSciNet)**

MR4010582

**Zentralblatt MATH identifier**

07118799

**Subjects**

Primary: 45D05: Volterra integral equations [See also 34A12]

Secondary: 47H08: Measures of noncompactness and condensing mappings, K-set contractions, etc. 34A08: Fractional differential equations

**Keywords**

Nonlinear Volterra integral equation fixed-point theorem measure of noncompactness uniformly locally attractive solution

#### Citation

Abadias, Luciano; Alvarez, Edgardo; Banaś, Józef; Lizama, Carlos. Solvability and uniform local attractivity for a Volterra equation of convolution type. J. Integral Equations Applications 31 (2019), no. 2, 149--164. doi:10.1216/JIE-2019-31-2-149. https://projecteuclid.org/euclid.jiea/1569225671