## About the solvability of ordinary differential equations with asymptotic boundary conditions.(English)Zbl 0587.34013

The following statement is proved: Let J be a real (not necessarily compact) interval, $$g: J\times R^ n\times R^ n\to R^ n$$ a Caratheodory function, $$s\subset C(J,R^ n)$$ and for a certain subset Q of $$C(J,R^ n)$$ the following conditions are satisfied: a) for any $$q\in Q$$ the boundary value problem $$\dot x($$t)$$=g(t,x(t),q(t))$$ (t$$\in J)$$, $$x\in s$$ admits a unique solution $$x=T(q)$$ (or has nonempty set of solutions T(q)); b) T(Q) is bounded in $$C(J,R^ n)$$; c) there exists a locally integrable function $$a: J\to R$$ such that $$| g(t,x(t),q(t))| \leq a(t)$$ (t$$\in J)$$. Then T(Q) is a relatively compact subset of $$C(J,R^ n)$$. Conditions under which T(q) is continuous (resp. the multivalent operator $$T: Q\to s$$ is upper semicontinuous with compact values) are also given. A further theorem with a proof based on this one gives conditions which guarantee the existence of at least one solution of the problem $$\dot x($$t)$$=f(t,x(t))$$, $$x\in s$$. This theorem is generalized for the equation $x^{(n)}(t)+\sum^{n-1}_{1}a_ i(t,x(t),...,x^{(n-1)}(t))\cdot x^{(i)}(t)=f(t,x(t),...,x^{(n-1)}(t)).$ Also the following statement holds: Consider $x''(t)+a(t,x)x'(t)+b(t,x(t))=0\quad (t\geq 0);\quad x(0)=1;\quad x(t)>0,\quad x'(t)<0,\quad t>0$ for a,b: [0,$$\infty)\times [0,1]\to R$$ continuous. Assume that for any continuous h: [0,$$\infty)\to [0,1]$$ the function b(t,h(t)) is not eventually vanishing and nonnegative. Then the considered problem has at least one solution. Some instructive examples are also discussed.
Reviewer: St.Fenyö

### MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations