Existence of solutions of quasilinear integrodifferential equations with nonlocal condition. (English) Zbl 0976.45011

The authors prove the existence and uniqueness of mild and classical solutions of the quasilinear abstract differential equation \[ du(t)/dt + Au(t)=f (t, u(t))\quad (t\in (0, a]) \] with the nonlocal condition \[ u (0) + g(t_1, t_2,\dots , t_p, u(t_1),\dots , u(t_p))=u_0 \] where \(-A\) is the infinitesimal generator of a \(C_0\)-semigroup in a Banach space \(X\), \(u_0\in X\) and \(f : [0, a] \times X\rightarrow X, f : [0, a]^p \times X^p\rightarrow X\) are given functions, and \[ 0\leq t_1\leq t_2\leq \dots \leq t_p\leq a. \] The main results of the paper are obtained by using \(C_0\)-semigroups and the Banach fixed point theorem.


45N05 Abstract integral equations, integral equations in abstract spaces
45K05 Integro-partial differential equations
34G20 Nonlinear differential equations in abstract spaces
45G10 Other nonlinear integral equations
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