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Balachandran, K.; Park, D. G.; Marshal Anthoni, S.

Existence of solutions of abstract nonlinear second-order neutral functional integrodifferential equations. (English) Zbl 1054.45006

Comput. Math. Appl. 46, No. 8-9, 1313-1324 (2003).
Sufficient conditions for the existence of mild solutions for abstract second-order neutral functional integrodifferential equations are established by using the theory of strongly continuous cosine families of operators and the Schaefer theorem. The authors deal with the nonlinear second-order neutral functional integro-differential equation having the form \[ (d/dt)[x'(t)- g(t,x_t)]= Ax(t)+ \int^t_0 F\left(t,s,x_s,x'(s), \int^s_0 f\bigl(s,\tau,x_\tau, x'(\tau)\bigr)d \tau\right)ds,\;\tag{1} \] \(t\in(0,T)\), satisfying the following initial conditions (2) \(x_0=\varphi\), \(x'(0)= y_0\). Here it is assumed that \(A\) is the infinitesimal generator of the strongly continuous cosine family of bounded linear operator in a Banach space \(X\). Under some assumptions it is proved that the problem (1)–(2) has at least one mild solution. Some interesting applications are also presented.
Reviewer: J. Banaś (Rzeszów)

Cited in 1 Review
Cited in 10 Documents

MSC:

45N05 Abstract integral equations, integral equations in abstract spaces
45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations

Keywords:

neutral functional integrodifferential equation; strongly continuous cosine operators; Schaefer fixed-point theorem; Banach space; mild solutions
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\textit{K. Balachandran} et al., Comput. Math. Appl. 46, No. 8--9, 1313--1324 (2003; Zbl 1054.45006)
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References:

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