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Boundary value problems with nonlinear boundary conditions in Banach spaces. (English) Zbl 0722.34058

The existence of a solution of \(x'=A(t,x)x+f(t,x)\) which satisfies the condition \(Lx=H(x)\) is proved where X is a Banach space, \(J=[a,b]\), A(t,x) is a bounded operator defined and continuous on \(J\times X\), f(t,x) is a continuous function on \(J\times X\), L: C(J,X)\(\to X\) is a bounded linear operator and H: C(J,X)\(\to X\) is a continuous operator not necessarily linear. The proof is based on Schaefer’s fixed point theorem.

MSC:

34K10 Boundary value problems for functional-differential equations
34K30 Functional-differential equations in abstract spaces
34B15 Nonlinear boundary value problems for ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
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