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 \[ x''(t)= f(t,x(t), x'(t))+ \ell(t),\;t\in (0,1)\tag{1} \] and the boundary conditions of the form \[ x'(0)= 0,\;x(1)= \sum^{m-2}_{i=1} a_i x(\xi_i)\tag{2} \] or of the form \[ x(0)= 0,\;x(1)= \sum^{m- 2}_{i= 1} a_ix(\xi_i),\tag{3} \] where \(f:[0,1]\times \mathbb{R}^2\to\mathbb{R}\) is a continuous function, \(\ell: [0,1]\to\mathbb{R}\) 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 \[ x'(0)= 0,\;x(1)= \alpha x(\eta)\tag{4} \] and \[ x(0)= 0,\;x(1)= \alpha x(\eta),\tag{5} \] respectively, where \(\alpha\in\mathbb{R}\) 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.


34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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[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) · Zbl 0668.34025
[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, (Fitzpatrick, P. M.; Martelli, M.; Mawhin, J.; Nussbaum, R., opological methods for ordinary differential equations. opological methods for ordinary differential equations, Lecture Notes in Mathematics, 1537 (1991), Springer Verlag), 74-142 · Zbl 0798.34025
[6] Feng W. & Webb J.R.L., Solvability of \(m\); Feng W. & Webb J.R.L., Solvability of \(m\) · Zbl 0883.34020
[7] Feng W., On an \(m\); Feng W., On an \(m\)
[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), Springer Verlag) · Zbl 0326.34020
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