zbMATH — the first resource for mathematics

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

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