Solvability of \(m\)-point boundary value problems with nonlinear growth. (English) Zbl 0883.34020

Let \(f:[0,1] \times \mathbb{R}^2\to \mathbb{R}\) be a continuous function and \(e(.) \in L^1 (0,1)\). Let \(\xi_i\in (0,1)\), \(a_i\in \mathbb{R}\) with all the \(a_i\)’s having the same sign, \(i=1,2, \dots, m-2\), and \(0<\xi_1< \xi_2< \cdots <\xi_{m-2} <1\).
This paper is concerned with the existence of solutions of the second order ordinary differential equation \[ x''(t)= f\bigl(t,x(t),x'(t) \bigr)+e(t) \quad t\in (0,1) \tag{1} \] subject to one of the following nonlocal boundary conditions \[ x'(0)=0, \quad x(1) =\sum^{m-2}_{i=1} a_ix(\xi_i) \tag{2} \]
\[ x(0)=0, \quad x(1)= \sum^{m-2}_{i=1} a_i x(\xi_i). \tag{3} \] Conditions for the existence of a solution for the above \(m\)-point boundary value problems are given.
The authors assume that \(f(t,x,p)= g(t,x,p)+ h(t,x,p)\) and deal with both the nonresonant and the resonant case, using the coincidence degree theory of Mawhin.


34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI


[1] Constantin, A., On a two-point boundary value problem, J. Math. Anal. Appl., 193, 318-328 (1995) · Zbl 0836.34021
[2] Feng, W.; Webb, J. R.L., Solvability of three-point boundary value problems at resonance, Nonlinear Anal. (1997) · Zbl 0891.34019
[3] Gupta, C. P.; Ntouyas, S. K.; Tsamatos, P. Ch., On an \(m\), Nonlinear Anal., 23, 1427-1436 (1994) · Zbl 0815.34012
[4] Gupta, C. P.; Ntouyas, S. K.; Tsamatos, P. Ch., Solvability of an \(m\), J. Math. Anal. Appl., 189, 575-584 (1995) · Zbl 0819.34012
[5] 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
[6] Gupta, C. P., A note on a second order three-point boundary value problem, J. Math. Anal. Appl., 186, 277-281 (1994) · Zbl 0805.34017
[7] Il’in, V.; Moiseev, E., Nonlocal boundary value problems 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
[8] Marano, S. A., A remark on a third-order three-point boundary value problem, Bull. Austral. Math. Soc., 49, 1-5 (1994) · Zbl 0808.34019
[10] Mawhin, J., Topological degree and boundary value problems for nonlinear differential equations, (Fitzpatrick, P. M.; Martelli, M.; Mawhin, J.; Nussbaum, R., Topological Methods for Ordinary Differential Equations. Topological Methods for Ordinary Differential Equations, Lecture Notes in Mathematics, 1537 (1991), Springer-Verlag: Springer-Verlag New York/Berlin), 74-142 · Zbl 0798.34025
[11] Petryshyn, W. V.; Yu, Z. S., Solvability of Neumann BV problems for nonlinear second-order ODEs which need not be solvable for the highest-order derivative, J. Math. Anal. Appl., 91, 244-253 (1983) · Zbl 0513.34020
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.