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Nonlinear boundary value problems involving abstract Volterra operators. (English) Zbl 0795.34057

The existence of an absolutely continuous solution to the problem \(x'(t)= (Lx)(t)+ (fx)(t)\), \(t\in [0,T]\), \(x\in \mathbb{R}^ n\), \(Ax(0)+ Bx(T)= hx\in \mathbb{R}^ n\), in which \(L\) is linear and bounded in \(L^ p\), \(f: C^ 0\to L^ p\), \(p\geq 1\) is considered. Using Schauder’s fixed point theorem the existence of at least one solution is demonstrated. The result is applied to a Sturm-Liouville problem and has relations to an earlier theorem of H. Brezis and F. Browder.

MSC:

34K10 Boundary value problems for functional-differential equations
45J05 Integro-ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
45D05 Volterra integral equations
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