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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

B u ( t ) ' +Au(t)=ft , u ( t )+ 0 t gt , s , u ( s )ds,0<ta,(1)
u(0)+ k=1 p c k u(t k )=u 0 ,(2)

where A and B are closed linear operators in a Banach space X with D(B)D(A) and the compact B -1 , 0t 1 <t 2 <<t p a, u 0 X, and f:[t 0 ,t 0 +a]×XX, g:{(s,t):0sta}×XX 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(·,u) with respect to u and Lipschitzian continuity of g(t,·,·) with respect to t) solutions to problem (1), (2) based on the Schauder fixed-point principle. As an example the following problem

tz (t,x) - z xx (t,x)-z xx (t,x)=μt , z ( t , x )+ 0 t ηt , s , z ( s , x )ds,0xπ,0<ta,
z(t,0)=z(t,π)=0,z(0,x)+ k=1 p z(t k ,x)=z 0 (x),

is considered.

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