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Existence of solutions to nonlinear integrodifferential equations of Sobolev type with nonlocal condition in Banach spaces. (English) Zbl 0957.34058
The authors deal with the nonlocal Cauchy problem $$\bigl(Bu (t)\bigr)' +Au(t)=f \bigl(t,u(t) \bigr)+ \int^t_0g\bigl( t,s,u(s)\bigr) ds, \ 0<t \le a,\tag 1$$ $$u(0)+\sum^p_{k=1}c_ku(t_k)=u_0,\tag 2$$ where $A$ and $B$ are closed linear operators in a Banach space $X$ with $D(B)\subset D(A)$ and the compact $B^{-1}$, $0\le t_1<t_2<\cdots<t_p\le a$, $u_0\in X$, and $f: [t_0, t_0+a]\times X\to X$, $g:\{(s,t):0\le s\le t\le a\}\times X\to X$ are given functions. The main results are the existence of mild (under assumptions about the boundedness of $f$ and $g)$ and unique strong (under assumptions about the boundedness of $f,g$, Lipschitzian continuity of $f(\cdot,u)$ with respect to $u$ and Lipschitzian continuity of $g(t,\cdot,\cdot)$ with respect to $t)$ solutions to problem (1), (2) based on the Schauder fixed-point principle. As an example the following problem $${\partial\over \partial t}\bigl(z(t,x)-z_{xx} (t,x) \bigr)- z_{xx}(t,x)= \mu\bigl(t,z(t,x) \bigr)+\int^t_0 \eta \bigl(t,s,z (s,x)\bigr) ds,\ 0\le x\le\pi,\ 0<t\le a,$$ $$z(t,0)= z(t,\pi)=0, \quad z(0,x)+ \sum^p_{k=1} z(t_k,x)=z_0(x),$$ is considered.

34G20Nonlinear ODE in abstract spaces
34K05General theory of functional-differential equations
45J05Integro-ordinary differential equations
47J35Nonlinear evolution equations
35K90Abstract parabolic equations
Full Text: DOI
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