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Solution sets and boundary value problems in Banach spaces. (English) Zbl 0799.34069

Let \(X\) be a Banach space, \(a,b\in \mathbb{R}\), \(f\in C([a,b]\times X,X)\), \(g\in C([a,b]\times X^ 2,X)\) such that \(g(.,x,x)= f(.,x)\) for all \(x\in X\); \(S,\Omega,Q \subset C([a,b],X)\), \(\Omega\) open bounded convex set, \(Q\) closed bounded convex set, \(S_ 1\) closed \(\subset S\cap Q\); \(K: \Omega\times Q\to C([a,b],X)\) such that \(\psi(K(U,C))\leq h\psi(U)\) for any compact \(C\subset Q\), \(U\subset\Omega\) and some \(h\in [0,1[\) where \(\psi: 2^{C([a,b],X)}\to [0,+\infty[\) is such that \(\psi(\overline {\text{co}U})= \psi(U)\) for all \(U\subset C([a,b],X)\), \(\psi(U_ 1)\leq \psi(U_ 2)\) if \(U_ 1\subset U_ 2\subset C([a,b], X)\), \(\psi(C)=0\) iff \(C\) is relatively compact in \(C([a,b],X)\) and such that \(\text{ind}(K(.,q), \Omega)\neq 0\) for some \(q\in Q\) (ind being an index suitably defined); let \(\Sigma(q)= \{x\in Q\): \(x= K(x,q)\}\). If \(\Sigma(q)\) is a set of isolated points, \(\Sigma(q)\subset S_ 1\), \(x'= g(.,x,q)\) for all \(q\in Q\), \(x\in \Sigma(q)\), then there exists \(x\in S\) such that \(x'= f(.,x)\).
Reviewer: G.Bottaro (Genova)

MSC:

34G20 Nonlinear differential equations in abstract spaces
34B15 Nonlinear boundary value problems for ordinary differential equations

Keywords:

Banach space
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