## Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces.(English)Zbl 0561.34048

Assume that $$J=[a,b]$$, E is a Banach space and $$\alpha$$ is the measure of noncompactness in E. The author considers the problem $(1)\quad x''=f(t,x,x'),\quad c_ 1x(a)-d_ 1x'(a)=w_ 1,\quad c_ 2x(b)+d_ 2x'(b)=w_ 2,$ assuming that (1) $$f: J\times E^ 2\to E$$ satisfies Caratheodory conditions; (2) $$h: J\to R_+$$ is integrable and $$d: 2^ E\times 2^ E\to R_+$$ is such that $$d(X,Y)<p\alpha (X)+q\alpha (Y)$$ whenever $$\alpha (X)+\alpha (Y)>0$$; (3) for any bounded $$X,Y\in 2^ E$$ and $$\epsilon >0$$ there exists a closed $$J_{\epsilon}\subset J$$ such that $$mes(J\setminus J_{\epsilon})<\epsilon$$ and $$\alpha (f(T\times X\times Y))\leq \sup_{t\in T}h(t)d(X,Y)$$ for each closed $$T\subset J_{\epsilon}$$. He proves that if $p\sup_{t\in J}\int^{b}_{a}| G(t,s)| h(s)ds+q \sup_{t\in J}\int^{b}_{a}| (dG/\partial t)(t,s)| h(s)ds\leq 1,$ where G is the corresponding Green function, then there exists a solution of (1).

### MSC:

 34G20 Nonlinear differential equations in abstract spaces 34B15 Nonlinear boundary value problems for ordinary differential equations 34B27 Green’s functions for ordinary differential equations
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### References:

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