# Existence of asymptotically almost periodic solution of integro-differential equations

K. Sathiyanathan, and T. Nandha Gopal

Abstract:

TIn this paper, a class of semilinear integrodifferential equations of the form $$u^{\prime\prime}(t)+\alpha u^{\prime\prime\prime}(t)=\beta Au(t)+\gamma Au^\prime(t)+f(t,u(t))+\displaystyle \int^{t}_{0}g(t,s,u(s))ds, \ \ t,s \geq 0$$ satisfying $$\alpha \beta <\gamma$$ with prescribed initial conditions are studied. Using certain strongly continuous families in operator theory and fixed point theory, we have established some sufficient conditions for the existence and uniqueness of an asymptotically almost periodic solutions

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