## Abstract and Applied Analysis

### Existence of Mild Solutions for a Semilinear Integrodifferential Equation with Nonlocal Initial Conditions

#### Abstract

Using Hausdorff measure of noncompactness and a fixed-point argument we prove the existence of mild solutions for the semilinear integrodifferential equation subject to nonlocal initial conditions ${u}^{\prime }(t)=Au(t)+{\int }_{0}^{t}B(t-s)u(s)ds+f(t,u(t))$, $t\in [0,1]$, $u(0)=g(u)$, where $A:D(A)\subseteq X\to X$, and for every $t\in [0,1]$ the maps $B(t):D(B(t))\subseteq X\to X$ are linear closed operators defined in a Banach space $X$. We assume further that $D(A)\subseteq D(B(t))$ for every $t\in [0,1]$, and the functions $f:[0,1]{\times}X\to X$ and $g:C([0,1];X)\to X$ are $X$-valued functions which satisfy appropriate conditions.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 647103, 15 pages.

Dates
First available in Project Euclid: 7 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1399486652

Digital Object Identifier
doi:10.1155/2012/647103

Mathematical Reviews number (MathSciNet)
MR3004930

Zentralblatt MATH identifier
1261.34062

#### Citation

Lizama, Carlos; Pozo, Juan C. Existence of Mild Solutions for a Semilinear Integrodifferential Equation with Nonlocal Initial Conditions. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 647103, 15 pages. doi:10.1155/2012/647103. https://projecteuclid.org/euclid.aaa/1399486652

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