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Topological degree method in functional boundary value problems at resonance. (English) Zbl 0853.34062

Sufficient conditions are obtained for the existence of solutions of second order functional differential equations of the form \(x''(t) = f(t,x(t), (Fx) (t), x'(t), (Hx') (t))\), \(t \in [0, 1]\), satisfying one of the boundary conditions \(x'(0) = 0\), \(x'(1) = 0\) or \(x(0) = x(1)\), \(x' (0) = x'(1)\), where \(f : [0,1] \times\mathbb{R}^4 \to\mathbb{R}\) and \(F,H \in {\mathcal D}\), \({\mathcal D}\) being the set of all operators \(K : C([0,1],\mathbb{R}) \to C ([0,1],\mathbb{R})\) which are continuous and bounded. The proofs of the results in this paper are based on the Mawhin continuation theorem [see J. Mawhin, Topological degree methods in nonlinear boundary value problems, AMS, Providence, R. I. (1979; Zbl 0414.34025)]. Some examples are given to illustrate the results.

MSC:

34K10 Boundary value problems for functional-differential equations

Citations:

Zbl 0414.34025
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References:

[1] Rachůnková, I., Periodic boundary value problems for second order differential equations, Acta UP Olomucensis Math., 29, 83-91 (1990) · Zbl 0752.34021
[3] Kelevedjiev, P., Existence of solutions for two-point boundary value problems, Nonlinear Analysis, 22, 217-224 (1994) · Zbl 0797.34019
[4] Granas, A.; Guenther, R.; Lee, J., Nonlinear Boundary Value Problems for Ordinary Differential Equations, Dissert. Math. (1985), Warszawa
[5] Gaines, R. E.; Mawhin, J. L., Coincidence Degree and Nonlinear Differential Equations (1977), Springer: Springer Providence · Zbl 0326.34021
[6] Mawhin, J. L., Topological Degree Methods in Nonlinear Boundary Value Problems (1979), AMS: AMS Berlin · Zbl 0414.34025
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