On continuity and compactness of some nonlinear operators associated with differential equations in noncompact intervals. (English) Zbl 0563.34018

The authors consider boundary value problems: (1.1) \(\dot x=f(t,x(t))\), \(t\in J\), (1.2) \(x\in S\), where J is a given real interval, possibly unbounded, \(f: J\times R^ n\to R^ n\), continuous, S a given (nonempty) subset of the Fréchet space \(C(J,R^ n)\). It is assumed that f is the ”restriction to the diagonal” of a continuous function \(g: J\times R^ n\to R^ n\), i.e. \(g(t,c,c)=f(t,c)\), and that there exists a subset Q of \(C(J,R^ n)\) such that for any \(q\in Q\) the boundary value problem (2.1) \(\dot x(t)=g(t,x(t),q(t))\) \(t\in J\), (2.2) \(x\in S\), admits a unique solution (a nonempty set of solutions). To deduce the existence of a solution for (1.1), (1.2) it is sufficient to prove the existence of a fixed point for the operator \(T: Q\to S\subset C(J,R^ n)\), which associates to any \(q\in Q\) the unique solution (the set of solutions) \(x=T(q)\) of (2.1), (2.2). Continuity and compactness results for the operator T are given when T is single valued and when T is multivalued. The main existence result is the following: Theorem: ”Consider the boundary value problems (1.1), (1.2) and (2.1), (2.2) with the previous assumptions. Assume there exists a closed convex subset Q of \(C(J,R^ n)\) and a bounded closed subset \(S_ 1\) of \(S\cap Q\) which make the problem (2.1) \(\dot x=g(t,x(t),q(t))\), \(t\in J\), (2.3) \(x\in S_ 1\) uniquely solvable (solvable with a convex set of solutions) for each \(q\in Q\). Then (1.1), (1.2) admits a solution”. Some examples are given to illustrate how such method can be used.


34B15 Nonlinear boundary value problems for ordinary differential equations
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