## Abstract and Applied Analysis

### Singular Initial Value Problem for a System of Integro-Differential Equations

#### Abstract

Analytical properties like existence, uniqueness, and asymptotic behavior of solutions are studied for the following singular initial value problem: ${g}_{i}(t){y}_{i}^{\prime }(t)={a}_{i}{y}_{i}(t)(1+{f}_{i}(t,\mathbf{y}(t),{\int }_{{0}^{+}}^{t}{K}_{i}(t,s,\mathbf{y}(t),\mathbf{y}(s))ds))$,   ${y}_{i}({0}^{+})=0$,   $t\in (0, {t}_{0}]$, where $\mathbf{y}=({y}_{1}, \dots , {y}_{n})$,   ${a}_{i}>0$,   $i=1, \dots , n$ are constants and ${t}_{0}>0$. An approach which combines topological method of T. Ważewski and Schauder's fixed point theorem is used. Particular attention is paid to construction of asymptotic expansions of solutions for certain classes of systems of integrodifferential equations in a right-hand neighbourhood of a singular point.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 918281, 18 pages.

Dates
First available in Project Euclid: 5 April 2013

https://projecteuclid.org/euclid.aaa/1365168320

Digital Object Identifier
doi:10.1155/2012/918281

Mathematical Reviews number (MathSciNet)
MR3004875

Zentralblatt MATH identifier
1258.45005

#### Citation

Šmarda, Zdeněk; Khan, Yasir. Singular Initial Value Problem for a System of Integro-Differential Equations. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 918281, 18 pages. doi:10.1155/2012/918281. https://projecteuclid.org/euclid.aaa/1365168320

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