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.

MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations
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References:

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