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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