×

zbMATH — the first resource for mathematics

Existence theorems for a second order \(m\)-point boundary value problem. (English) Zbl 0884.34024
The paper is devoted to the study of the \(m\)-point boundary value problem \[ x''= f(t,x(t),x'(t))+ e(t),\;t\in(0,t),\;x'(0)= 0,\;x(1)= \sum^{m-2}_{i=1} a_ix(\xi_i),\tag{1} \] where \(f: [0,1]\times \mathbb{R}^2\to\mathbb{R}\) and \(e:[0, 1]\to\mathbb{R}\) are continuous, \(0<\xi_1<\xi_2<\cdots< \xi_{m-2}<1,\) and the real numbers \(a_i\) have the same sign.
The author considers both the case \(a= \sum^{m-2}_{i=1} a_i\neq 1\) and the resonance case \(a= 1\), where the associated linear homogeneous BVP has nontrivial solutions. The existence of a solution for (1) is proved in both cases. The sign conditions are required for \(f\) only, and no growth restrictions. The proof is based on the nonlinear alternative of Leray-Schauder type.

MSC:
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Boucherif, A., Nonlinear multipoint boundary value problems, Nonlinear anal., 10, 957-964, (1986) · Zbl 0607.34014
[2] Granas, A.; Guenther, R.B.; Lee, J.W., Nonlinear boundary value problems for ordinary differential equations, Dissert. math. warszawa, (1985) · Zbl 0476.34017
[3] Gupta, C.P.; Ntouyas, S.K.; Tsamatos, P.Ch., Solvability of anm, J. math. anal. appl., 189, 575-584, (1995) · Zbl 0819.34012
[4] Gupta, C.P., Existence theorems for a second orderm, Internat. J. math. math. sci., 18, 705-710, (1995) · Zbl 0839.34027
[5] Il’in, V.A.; Moiseev, E., Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects, Differential equations, 23, 803-810, (1987) · Zbl 0668.34025
[6] Kelevedjiev, P., Existence of solutions for two-point boundary value problem, Nonlinear anal., 22, 217-224, (1994) · Zbl 0797.34019
[7] Ruyun, Ma, Existence theorem for a second order three-point boundary value problems, J. Math. Anal. Appl. · Zbl 0989.34009
[8] O’Regan, D., Boundary value problems for second and higher order differential equations, Proc. amer. math. soc., 113, 761-775, (1991) · Zbl 0742.34023
[9] Rachunkova, I.; Staněk, S., Topological degree method in functional boundary value problems, Nonlinear anal., 27, 153-166, (1996) · Zbl 0856.34075
[10] Rachunkova, I.; Staněk, S., Topological degree method in functional boundary value problems at resonance, Nonlinear anal., 27, 271-285, (1996) · Zbl 0853.34062
[11] Rodrigvez, A.; Tineo, A., Existence theorems for the Dirichlet problem without growth restrictions, J. math. anal. appl., 135, 1-7, (1988) · Zbl 0674.34016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.