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Solvability of three point boundary value problems at resonance. (English) Zbl 0891.34019
The authors investigate the $m$-point boundary value problem consisting of the differential equation $$x''(t)= f(t,x(t), x'(t))+ \ell(t),\ t\in (0,1)\tag1$$ and the boundary conditions of the form $$x'(0)= 0,\ x(1)= \sum^{m-2}_{i=1} a_i x(\xi_i)\tag2$$ or of the form $$x(0)= 0,\ x(1)= \sum^{m- 2}_{i= 1} a_ix(\xi_i),\tag3$$ where $f:[0,1]\times \bbfR^2\to\bbfR$ is a continuous function, $\ell: [0,1]\to\bbfR$ belongs to $L'[0, 1]$, the numbers $a_i$, $i=1,2,\dots, m-2$ have the same sign, and the numbers $\xi_i$ satisfy $0<\xi_1< \xi_2\cdots\xi_{m- 2}< 1$. In the case $m=3$ the conditions (2) and (3) simplify to the conditions $$x'(0)= 0,\ x(1)= \alpha x(\eta)\tag4$$ and $$x(0)= 0,\ x(1)= \alpha x(\eta),\tag5$$ respectively, where $\alpha\in\bbfR$ and $\eta\in (0,1)$. Using the coincidence degree theory developed by J. Mawhin, the authors show that the question of existence and uniqueness of solutions to the problems (1), (2) and (1), (3) can be reduced to the investigation of the simpler problems (1), (4) and (1), (5) respectively.

34B10Nonlocal and multipoint boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
Full Text: DOI
[1] Ilin, 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)
[2] Gupta, C.P.; Ntouyas, S.K.; Tsamatos, P.Ch.: Solvability of an m-point boundary value problem for second order ordinary differential equations. J. math. Anal. appl. 189, 575-584 (1995) · Zbl 0819.34012
[3] Gupta, C.P.; Ntouyas, S.K.; Tsamatos, P.Ch.: On an m-point boundary-value problem for second-order ordinary differential equations. Nonlinear analysis, TMA 23, 1427-1436 (1994) · Zbl 0815.34012
[4] 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
[5] Mawhin, J.: Topological degree and boundary value problems for nonlinear differential equations. Lecture notes in mathematics 1537, 74-142 (1991)
[6] Feng W. & Webb J.R.L., Solvability of m-point boundary value problems with nonlinear growth, to appear. · Zbl 0883.34020
[7] Feng W., On an m-point boundary value problem, these Proceedings.
[8] Gupta, C.P.: A second order m-point boundary value problem at resonance. Nonlinear analysis, TMA 24, 1483-1489 (1995) · Zbl 0824.34023
[9] Gaines, R.E.; Mawhin, J.: Coincidence degree and nonlinear differential equations. Lecture notes in mathematics 568 (1977) · Zbl 0339.47031