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Existence and nonexistence results for positive solutions of an integral boundary value problem. (English) Zbl 1104.34017

This paper is concerned with the integral boundary value problem:
\[ \begin{aligned} -(au')'+bu&= g(t)f(t,u),\quad 0<t<1,\\ (\cos\gamma_0)u(0)-(\sin\gamma_0)u'(0)&= \int_0^1u(\tau)\,d\alpha(\tau),\\ (\cos\gamma_1)u(1)+(\sin\gamma_1)u'(1)&= \int_0^1u(\tau)\,d\beta(\tau), \end{aligned} \] with \(a\in C^1([0,1],(0,+\infty))\), \(b\in C([0,1],\mathbb{R}^+)\) and \(f\in C([0,1]\times\mathbb{R}^+,\mathbb{R}^+)\), \(g\in C([0,1],\mathbb{R}^+)\). The functions \(\alpha\) and \(\beta\) are right continuous on \([0,1),\) left continuous at \(t=1,\) and nondecreasing on \([0,1],\) with \(\alpha(0)=\beta(0)=0;\) and \(\gamma_0,\,\gamma_1\in[0,\frac{\pi}{2}].\) The integrals on the right-hand side are Riemann-Stieltjies integrals of \(u\) with respect to \(\alpha\) and \(\beta,\) respectively. The problem encompasses Sturm-Liouville two-point boundary value problems and multipoint eigenvalue problems.
The author proves two existence results of at least one positive solution, one existence result of at least two positive solutions and a nonexistence result of positive solutions. A set of hypotheses on the nonlinear term \(f\) are enunciated to get existence results. The proofs are based on the properties of the Green function and on the study of the spectral radius of an auxiliary linear operator; then, the author mainly appeals to the Krein-Rutman theorem, the Krasnosel’ski fixed-point theorem in cones of Banach spaces and a lemma by the author [Nonlinear Anal. Theory Methods Appl. 62, No. 7 (A), 1251–1265 (2005; Zbl 1089.34022)].
The main theorems extend previous results obtained by R. Ma and B. Thompson [J. Math. Anal. Appl. 297, No. 1, 24–37 (2004; Zbl 1057.34011)] and Z. Yang [J. Math. Anal. Appl. 321, No. 2, 751–765 (2006; Zbl 1106.34014)].
An example is given to illustrate the existence results. Finally, the author explains how the method used in this paper may be applied to obtain positive solutions for similar boundary value problems with nonlinear integral conditions.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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References:

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