Positive solutions for a nonlocal boundary-value problem with increasing response. (English) Zbl 0971.34008

The authors consider the ordinary differential equation \[ (p(t) x')'+ q(t) f(x) =0,\quad \text{a.e. }t\in [ 0,1], \] with the initial condition \(x(0) =0\) and the nonlocal boundary condition \[ x'(1) =\int_{\eta}^{1}x'(s) dg(s), \quad \eta \in (0,1).\tag \(*\) \] They assume that \(f\) is 1.) increasing and positive and 2.) there exist real positive numbers \(u,v\) such that \(f(u) \geq \rho u\) and \(f(v)<\theta v\), where \(\rho \), \(\theta\) are prescribed positive numbers.
They prove the existence of at least one positive solution by making use of the classical Krasnoselskii fixed-point theorem.
As a corollary, they deduce existence results when \(f\) satisfies either
3.) \(\limsup_{x\to 0^{+}}f(x)/x=+\infty\) and \(\limsup_{x\to \infty}f(x)/x=0\) or
4.) \(\limsup_{x\to \infty}f(x)/x=0\) and \(\limsup_{x\to 0^{+}}f(x)/x=+\infty \)
which are more restrictive than 2.) when 1.) is satisfied. It is worth mentioning that in the literature, it is condition 3.) or 4.) (or similar) which constitute the main assumption on \(f\) when boundary conditions are assumed, rather than the nonlocal condition \((*)\). In this direction one can refer to V. Anuradha, D. D. Hai and R. Shivaji [Proc. Am. Soc. 124, No. 3, 757-763 (1996; Zbl 0857.34032)], the authors [Electron. J. Differ. Equ. 2000, Paper No. 49 (2000; Zbl 0957.34022)] and I. Addou [ibid. 1999, Paper No. 21 (1999; Zbl 0923.34026)].


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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