Abstract
In this paper, we establish two existence theorems of rotating-periodic solutions for nonlinear second order vector differential equations via the Leray–Schauder degree theory and the lower and upper solutions method. The concept “rotating-periodicity” is a kind of symmetry, which is a general version of periodicity, anti-periodicity, harmonic-periodicity, and it is also a special kind of quasi-periodicity. We also include several examples to illustrate the validity and applicability of our results.
Highlights
Periodicity is a very important property in the study of differential equations
Since Poincaré established the existence of periodic solutions to the three-body problems, there has been a rich literature body on the periodic solutions of ordinary differential equations
Chang and Li [4, 5] studied the existence of rotating-periodic solutions for second order dynamical systems by using the coincidence degree theory
Summary
Periodicity is a very important property in the study of differential equations. Since Poincaré established the existence of periodic solutions to the three-body problems, there has been a rich literature body on the periodic solutions of ordinary differential equations (see for example [9, 16, 18] and the references therein). This paper is devoted to investigating the existence of rotating-periodic solutions for the following second order vector differential equation: x + f t, x, x = 0, (1) Topological methods, in particular the degree theory, some fixed point theorems, and lower and upper solutions method, are the most relevant tools in proving the existence of solutions for boundary value problems.
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