Abstract
In this paper, we study the existence of anti-periodic solutions for a second-order ordinary differential equation. Using the interaction of the nonlinearity with the Fucik spectrum related to the anti-periodic boundary conditions, we apply the Leray-Schauder degree theory and the Borsuk theorem to establish new results on the existence of anti-periodic solutions of second-order ordinary differential equations. Our nonlinearity may cross multiple consecutive branches of the Fucik spectrum curves, and recent results in the literature are complemented and generalized.
Highlights
1 Introduction and main results In this paper, we study the existence of anti-periodic solutions for the following secondorder ordinary differential equation:
–x under various boundary conditions was required as a nonresonance condition to obtain the solvability of equation ( . )
During the last thirty years, anti-periodic problems of nonlinear differential equations have been extensively studied since the pioneering work by Okochi [ ]
Summary
–x under various boundary conditions was required as a nonresonance condition to obtain the solvability of equation ( . ). ). Recall that the Fučík spectrum of –x with an anti-periodic boundary condition is the set of real number pairs (λ+, λ–) ∈ R such that the problem. We are interested in the nonresonance condition on for the solvability of involving the Fučík spectrum of –x under the anti-periodic boundary condition. During the last thirty years, anti-periodic problems of nonlinear differential equations have been extensively studied since the pioneering work by Okochi [ ]. For recent developments involving the existence of anti-periodic solutions, one can see [ – ] and the references therein. Denote by the Fucík spectrum of the operator –x under the anti-periodic boundary condition. It is seen that the set can be seen as a subset of the Fucík spectrum of –x under the corresponding Dirichlet boundary condition; one can see the definition of the set i+ , i ∈ N, or Figure in [ ].
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