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Positive solutions for nonlinear \(m\)-point boundary value problems of Dirichlet type via fixed-point index theory. (English) Zbl 1070.34039
This paper deals with the \(m\)-point boundary value problem \[ u''(t)+a(t)u'(t)+b(t)u(t)+g(t)f(u(t))=0, \quad t \in [0,1], \] \[ u(0)=0, u(1)=\sum^{m-2}_{i=1}\alpha_i u(\xi_i). \] It is assumed that \(a, b : [0,1] \to \mathbb R\) are continuous, with \(b(t)\leq 0 \), and \(\xi_i\in \;]0,1[\), \(\alpha_i \in \; ]0,+\infty[\) are such that \(0<\sum^{m-2}_{i=1}\alpha_i\phi_1(\xi_i)<1\), where \(\phi_1 \) is the solution of \[ u''(t)+a(t)u'(t)+b(t)u(t) =0, \quad t \in [0,1], \quad u(0)=0, u(1)=1. \] Further, \(g : [0,1] \to [0,+\infty[\) is continuous, with \(\int_0^1 g >0\), and \(f : [0,+\infty[ \; \to [0,+\infty[\) is continuous too. Under suitable conditions on the quotient \(f(u)/u\), the authors prove the existence of at least one or at least two positive solutions. The proof is based on some results on Hammerstein integral equations obtained by K. Lan and J. R. L. Webb [J. Differ. Equations 148, 407–421 (1998; Zbl 0909.34013)] and K. Q. Lan [J. Lond. Math. Soc., II. Ser. 63, No.3, 690–704 (2001; Zbl 1032.34019)].

MSC:
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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