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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.

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