# zbMATH — the first resource for mathematics

A generalized multi-point boundary value problem for second order ordinary differential equations. (English) Zbl 0910.34032
The author deals with the existence of a solution to generalized multipoint boundary value problems $x''(t)=f \bigl(t,x(t), x'(t)\bigr) +e(t),\;0<t<1,$ $x(0)=\sum^{m-2}_{i=1} a_i x(\xi_i), \quad x(1)= \sum^{n-2}_{j=1} b_jx (\tau_j)$ and $x''(t)= f\bigl(t,x(t), x'(t)\bigr) +e(t), \quad 0<t <1,$ $x(0)= \sum^{m-2}_{i=1} a_ix(\xi_i), \quad x'(1)= \sum^{n-2}_{j=1} b_jx'(\tau_j),$ where $$f:[0,1] \times \mathbb{R}^2\to \mathbb{R}$$ satisfies Carathéodory conditions, $$e\in L^1[0,1]$$, $$a_i,b_j\in\mathbb{R}$$, $$1\leq i\leq m-2$$, $$1\leq j\leq n-2$$, $$0<\xi_1<\xi_2< \cdots <\xi_{m-2}<1$$, $$0<\tau_1 <\tau_2< \cdots <\tau_{n -2} <1$$. The present work improves an earlier one due to the author et al. when all $$a_i$$’s have the same sign and all $$b_j$$’s have the same sign. In this case all $$a_i$$’s and $$b_j$$’s do not necessarily have the same sign (nonresonance case).

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text:
##### References:
  C. P. Gupta, S. K. Ntouyas, and P. Ch. Tsamatos, Existence results for multi-point boundary value problems for second order ordinary differential equations, (preprint). · Zbl 1185.34023  Il’in, 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  Il’in, V.A.; Moiseev, E.I., Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator, Differential equations, 23, 8, 979-987, (1987) · Zbl 0668.34024  Bitsadze, A.V., On the theory of nonlocal boundary value problems, Soviet math. dokl., 30, 1, 8-10, (1984) · Zbl 0586.30036  Bitsadze, A.V., On a class of conditionally solvable nonlocal boundary value problems for harmonic functions, Soviet math. dokl., 31, 1, 91-94, (1985) · Zbl 0607.30039  Bitsadze, A.V.; Samarskiĭ, A.A., On some simple generalizations of linear elliptic boundary problems, Soviet math. dokl., 10, 2, 398-400, (1969) · Zbl 0187.35501  Gupta, C.P., A second order m-point boundary value problem at resonance, Nonlinear analysis, theory, methods & applications, 24, 10, 1483-1489, (1995) · Zbl 0824.34023  Gupta, C.P., Solvability of a multi-point boundary value problem at resonance, Results in math., 28, 270-276, (1995) · Zbl 0843.34023  Gupta, C.P., Existence theorems for a second order m-point boundary value problem at resonance, Int. jour. math. & math. sci., 18, 4, 705-710, (1995) · Zbl 0839.34027  Gupta, C.P.; Ntouyas, S.K.; Tsamatos, P.Ch., On an m-point boundary value problem for second order ordinary differential equations, Nonlinear analysis, theory, methods & applications, 23, 11, 1427-1436, (1994) · Zbl 0815.34012  Gupta, C.P.; Ntouyas, S.K.; Tsamatos, P.Ch., Existence results for m-point boundary value problems, Diff. equations and dynamical systems, 2, 4, 289-298, (1994) · Zbl 0877.34019  Gupta, C.P.; Ntouyas, S.K.; Tsamatos, P.Ch., Solvability of an m-point boundary value problem for second order ordinary differential equations, Jour. math. anal. & appl., 189, 575-584, (1995) · Zbl 0819.34012  Mawhin, J., Topological degree methods in nonlinear boundary value problems, (), No. 40 · Zbl 0414.34025
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.