## 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

Zbl 0414.34025
Full Text:

### 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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