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A new approach for solving nonlinear BVP’s on the half-line for second order equations and applications. (English) Zbl 1349.34065
In this interesting paper, the author develops a new approach for solving nonlinear boundary value problems (BVPs) on non-compact intervals. This approach is illustrated in the case of the BVP \[ \begin{cases} (a(t)\Phi(x'))'=f(t,x,x')+\lambda g(t,x,x'),\quad t\geq 0,\\ x \in \mathfrak{B},\end{cases} \] where \(\Phi\) is an increasing homeomorphism, \(a\) is a continuous function, \(f,g\) are continuous functions, \(\lambda\) is a parameter and \(\mathfrak{B}\) is a subset of \(C[0,\infty)\). The condition \(x \in \mathfrak{B}\) is allowed to be global, in the sense that it involves the behaviour of the solution \(x\) on the whole interval \([0,\infty)\), one particular example being the condition \(\int_0^{\infty} x(t)dt=0\).
The methodology here is to prove the solvability of two auxiliary BVPs, one on a compact interval and another one on a non-compact interval, and to utilize some continuity arguments and phase-space analysis.

MSC:
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
34B09 Boundary eigenvalue problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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