Exponential decay in integrodifferential equations with nonlocal conditions. (English) Zbl 1263.45009

Ezzinbi, Khalil; Li, Fang; Liu, James H.; Van Minh, Nguyen

Int. J. Qual. Theory Differ. Equ. Appl. 1, No. 2, 135-141 (2007).

Summary: We study the existence, uniqueness, and exponential decay of solutions for a semi-linear integrodifferential equation with a nonlocal initial condition \[ \begin{aligned} u^\prime(t)&=Au(t)+\int^t_0 F(t-s)Au(s)ds+f(t,u(t)), \quad t\geq 0,\\u(0)&=\int^\infty_0 g(s)u(s)ds+u_0,\end{aligned} \] in a Banach space \(X\), with \(A\) the generator of a strongly continuous semigroup. The nonlocal condition can be applied in physics with better effect than the “classical” Cauchy problem \(u(0)=u_0\) since more measurements at \(t\geq 0\) are allowed. The variation of constants formula for solutions via a resolvent operator and the iteration techniques are used in the study.

MSC:

45K05	Integro-partial differential equations
34G20	Nonlinear differential equations in abstract spaces

Keywords:

integrodifferential equations; nonlocal conditions; exponential decay