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global solution; nonlocal boundary value problem; noncompact interval; continuous dependence of solution; fixed point theorem; principal solution
We present a new approach to solving boundary value problems on noncompact intervals for second order differential equations in case of nonlocal conditions. Then we apply it to some problems in which an initial condition, an asymptotic condition and a global condition is present. The abstract method is based on the solvability of two auxiliary boundary value problems on compact and on noncompact intervals, and uses some continuity arguments and analysis in the phase space. As shown in the applications, Kneser-type properties of solutions on compact intervals and a priori bounds of solutions on noncompact intervals are key ingredients for the solvability of the problems considered, as well as the properties of principal solutions of an associated half-linear equation. The application of this method leads to some new existence results, which complement and extend some previous ones in the literature.
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