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Existence theorems for a second order \(m\)-point boundary value problem at resonance. (English) Zbl 0839.34027
The paper deals with the differential equation (1) \(x''(t) = f(t,x(t)\), \(x'(t)) + e(t)\), \(0 < t < 1\), where \(f \in \text{Car} ([0,1] \times \mathbb{R}^2)\), with the \(m\)-point boundary condition \((m \geq 3)\) (2) \(x'(0) = 0\), \(x(1) = \sum^{m - 2}_{i = 1} a_i x(\xi_i)\), where \(a_i \geq 0\), \(\xi_i \in (0,1)\), \(i = 1,2, \dots, m - 2\), with \(\sum^{m - 2}_{i = 1} a_i = 1\), \(0 < \xi_1 < \xi_2 < \cdots < \xi_{m - 2} < 1\). The problem (1), (2) is at resonance in the sense that the associated linear homogeneous boundary value problem has nontrivial solutions. The author proves the existence of a solution to (1), (2), provided \(f\) satisfies certain sign conditions and a linear growth condition with coefficients having sufficiently small norms. The proofs are based on Mawhin’s version of the Leray-Schauder continuation theorem. The results complete the earlier ones concerning the nonresonance case of (1), (2) proved by the author together with S. K. Ntouyas and P. Ch. Tsamatos in [J. Math. Anal. Appl. 189, No. 2, 575-584 (1995; Zbl 0819.34012)].

MSC:
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
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