### A class of second order BVPs on infinite intervals.

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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,...

We consider the equation $$-{y}^{\text{'}}\left(x\right)+q\left(x\right)y(x-\varphi \left(x\right))=f\left(x\right),\phantom{\rule{1.0em}{0ex}}x\in \mathbb{R},$$ where $\varphi $ and $q$ ($q\ge 1$) are positive continuous functions for all $x\in \mathbb{R}$ and $f\in C\left(\mathbb{R}\right)$. By a solution of the equation we mean any function $y$, continuously differentiable everywhere in $\mathbb{R}$, which satisfies the equation for all $x\in \mathbb{R}$. We show that under certain additional conditions on the functions $\varphi $ and $q$, the above equation has a unique solution $y$, satisfying the inequality $$\parallel {y}^{\text{'}}{\parallel}_{C\left(\mathbb{R}\right)}+{\parallel qy\parallel}_{C\left(\mathbb{R}\right)}\le c{\parallel f\parallel}_{C\left(\mathbb{R}\right)},$$ where the constant $c\in (0,\infty )$ does not depend on the choice of $f$.