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.
The Rayleigh equation with two deviating argumentsx′′(t)+f(x'(t))+g1(t,x(t-τ1(t)))+g2(t,x(t-τ2(t)))=e(t)is studied. By using Leray-Schauder index theorem and Leray-Schauder fixed point theorem, we obtain some new results on the existence of periodic solutions, especially for the existence of nontrivial periodic solutions to this equation. The results are illustrated with two examples, which cannot be handled using the existing results.
We present a survey on the existence of periodic solutions of singular differential equations. In particular, we pay our attention to singular scalar differential equations, singular damped differential equations, singular impulsive differential equations, and singular differential systems.
In this paper, based on some biological meanings and a model which was proposed by Lefever and Garay (1978), a nonlinear delay model describing the growth of tumor cells under immune surveillance against cancer is given. Then, boundedness of the solutions, local stability of the equilibria and Hopf bifurcation of the model are discussed in details. The existence of periodic solutions explains the restrictive interactions between immune surveillance and the growth of the tumor cells.
Periodic Solutions and Nontrivial Periodic Solutions for a Class of Rayleigh-Type Equation with Two Deviating Arguments
