The structure of the generalized reflective function of three-degree polynomial differential systems is considered in this paper. The generated results are used for discussing the existence of periodic solutions of these systems.
Two criteria are constructed to guarantee the existence of periodic solutions for a second-order -dimensional differential system by using continuation theorem. It is noticed that the criteria established are found to be associated with the system’s damping coefficient, natural frequency, parametrical excitation, and the coefficient of the nonlinear term. Based on the criteria obtained, we investigate the periodic motions of the simply supported at the four-edge rectangular thin plate system subjected to the parametrical excitation. The effectiveness of the criteria is validated by corresponding numerical simulation. It is found that the existent range of periodic solutions for the thin plate system increases along with the increase of the ratio of the modulus of nonlinear term’s coefficient and parametric excitation term, which generalize and improve the corresponding achievements given in the known literature.
We provide sufficient conditions for the existence of periodic solutions of the polynomial third order differential systemx.=-y+εP(x,y,z)+h1(t), y.=x+εQ(x,y,z)+h2(t), and z.=az+εR(x,y,z)+h3(t), whereP,Q, andRare polynomials in the variablesx,y, andzof degreen, hi(t)=hi(t+2π)withi=1,2,3being periodic functions,ais a real number, andεis a small parameter.
In this paper, the concept of a new ?-generalized quasi metric space is
introduced. A number of well-known quasi metric spaces are retrieved from
?-generalized quasi metric space. Some general fixed point theorems in a
?-generalized quasi metric spaces are proved, which generalize, modify and
unify some existing fixed point theorems in the literature. We also give
applications of our results to obtain fixed points for contraction mappings
in the domain of words and to prove the existence of periodic solutions of
delay differential equations.
AbstractWe consider periodic solutions of the following problem associated with the fractional Laplacian:
{(-\partial_{xx})^{s}u(x)+\partial_{u}F(x,u(x))=0} in {\mathbb{R}}.
The smooth function {F(x,u)} is periodic about x and is a double-well potential with respect to u with wells at {+1} and -1 for any {x\in\mathbb{R}}.
We prove the existence of periodic solutions whose periods are large integer multiples of the period of F about the variable x by using variational methods.
An estimate of the energy functional, Hamiltonian identity and Modica-type inequality for periodic solutions are also established.
Abstract
In this paper, firstly, we study the structural form of reflective integral for a given system. Then the sufficient conditions are obtained to ensure there exists the reflective integral with these structured form for such system. Secondly, we discuss the necessary conditions for the equivalence of such systems and a general three-dimensional differential system. And then, we apply the obtained results to the study of the behavior of their periodic solutions when such systems are periodic systems in t.
On the Generalized Reflective Function of the Three-Degree Polynomial Differential System