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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)].

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
Full Text: DOI
[1] Win, V.A.; Moiseev, E.I., Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects, Differential equations, 23, 7, 803-810, (1987) · Zbl 0668.34025
[2] Gupta, C.P., Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation, J. math. anal. appl., 168, 540-551, (1992) · Zbl 0763.34009
[3] Gupta, C.P., A sharper condition for solvability of a three-point boundary value problem, J. math. anal. appl., 205, 586-597, (1997) · Zbl 0874.34014
[4] Feng, W.; Webb, J.R.L., Solvability of a three-point nonlinear boundary value problems at resonance, Nonlinear analysis, 30, 6, 3227-3238, (1997) · Zbl 0891.34019
[5] Ma, R., Positive solutions of a nonlinear three-point boundary value problem, Electron. J. diff. eqns., 1999, 34, 1-8, (1999)
[6] Lan, K.Q., Multiple solutions of semilinear differential equations with singularities, J. London math. soc., 63, 3, 690-704, (2001) · Zbl 1032.34019
[7] Lan, K.Q.; Webb, J.R.L., Positive solutions of semilinear differential equations with singularities, J. differential equations, 148, 407-421, (1998) · Zbl 0909.34013
[8] Dang, H.; Schmitt, K., Existence of positive solutions for semilinear elliptic equations in annular domains, Differential and integral equations, 7, 3, 747-758, (1994) · Zbl 0804.34021
[9] Dunninger, D.R.; Wang, H., Multiplicity of positive solutions for a nonlinear differential equation with nonlinear boundary conditions, Ann. polon. math., 69, 2, 155-165, (1998) · Zbl 0921.34024
[10] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cone, (1988), Academic Press Orlando, FL · Zbl 0661.47045
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