## On some modification of the Levinson operator and its application to a three-point boundary value problem.(English)Zbl 0756.34024

The author considers the three-point boundary value problem $$x''=f(t,x)$$, $$x(0)=x(a)=x(2a)$$, $$a\in\mathbb{R}$$, $$a\neq 0$$, where $$f$$ is a continuous function. Under the assumption $$\liminf_{| x|\to\infty}f(t,x)\cdot\text{sgn} x>0$$ he proves the existence of a solution. The method of proofs is based on the Borsuk antipodal theorem for some modified Levinson operator.

### MSC:

 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text:

### References:

 [1] Poincaré H.: Les méthodes nouvelles de la mécanique céleste. Gauthirs-Villars, Paris (1892). · Zbl 0651.70002 [2] Levinson N.: On the existence of periodic solutions of second order differential equations with a forcing term. J. Math. Phys. 22 (1943), 41-48. · Zbl 0061.18909 [3] Krasnoselskii M.A.: Translation operator along the trajectories of differential equations. Nauka, Moscow (1966) [4] Mawhin J.: Topological degree methods in nonlinear boundary value problems. CBMS Reg. Conf. Math., No.40, AMS, Providence (1979). · Zbl 0414.34025 [5] Hille E.: On the Landau-Kallman-Rota inequality. J. Approx. Theory 6 (1972), 117-122. · Zbl 0238.47007 [6] Mawhin J.: Boundary value problems at resonance for vector second order nonlinear differential equations. Sém. Math. Appl. Méc., No.103, Nov., Louvain (1977). [7] Bebernes J.W.: A simple alternative problem for finding periodic solutions of second order ordinary differential systems. Proceed. Amer. Math. Soc. 42, 1 (1974), 21-127. · Zbl 0286.34055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.