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A boundary value problem on a half-line for differential equations with indefinite weight. (English) Zbl 1244.34045
The authors study the boundary value problem consisting of the differential equation \[ (a(t)\phi(x'))'= b(t)F(x), ~t\geq 0, \] and the conditions \[ x(0)= c> 0, ~x(t)> 0, ~0<\lim_{t\to\infty} x(t)<\infty, ~\lim_{t\to\infty} x'(t)= 0, \] where \(\phi\) is an increasing odd homeomorphism, \(F\) a continuous non-decreasing function on \(\mathbb{R}\) such that \(F(u) > 0\) for \(u\neq 0\) and \(a\) and \(b\) are continuous functions for \(t\geq 0\) satisfying certain conditions. The existence of solutions is based on the Tychonov fixed point theorem applied to the operator \(T\) defined by \[ T(u)(t)= C+\int^t_0 \phi^*\Biggl({1\over a(s)} \Biggl(\int^\infty_s b_-F(u(r))\,dr- \int^\infty_s b_+(r) F(u(r))\,dr\Biggr)\Biggr)\,ds, \] where \(\phi^*\) is the inverse function of \(\phi\). The properties of solutions depend on the variation of \(\phi^*\) at \(u= 0\), where by variation of a positive continuous function \(g\) on \((0,\vartheta)\) is understood the limit \(\lim_{u\to 0+} {g(\lambda u)\over g(u)}\), \(\lambda> 0\). If the limit is \(\lambda^p\) then \(g\) is called regularly varying of index \(p> 0\). If it is \(1\), then \(g\) is called slowly varying. Finally, \(g\) is said to be rapidly varying if the limit is \(0\) for \(0<\lambda< 1\) and \(\infty\) for \(\lambda> 1\). The cases of regular variation, slow variation and rapid variation of \(\phi^*\) are considered.

34B40 Boundary value problems on infinite intervals for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations