Many different kinds of methods for the existence of traveling waves (exact form of solutions) haven been developed, such as Painleve expansion method [6, 7], bilinear method [8, 9], symmetry methods [10].
Also, a method for the existence of solutions for Eqs.
Wei [25], developed the upper and lower solutions method for the existence of positive solutions of the following coupled system of BVPs: (1.1).
The Banach contraction theorem formulated for a complete metric space is one of the most simple and, at the same time, the most important method for the existence and uniqueness of solution of nonlinear problems arising in mathematics and its applications to engineering and natural sciences [22 25].
Based on the proposed switching method, sufficient conditions for existence of the state feedback controllers are presented via the switched non-parallel distributed compensation (non-PDC) scheme.
Hence, the boundary value problems studied in this paper are more general, and it is important to develop certain methods for investigating the existence results of them.
One of the standard methods for showing the existence of a specific type of solutions of systems of difference equations is application of fixed point theorems.
As to methods for studying the existence and uniqueness of periodic solutions, such as the Lyapunov second method, fixed point theorem, coincidence degree theory, the Yoshizawa type theorem, and the Massera type theorem, see [1, 28 30] and references therein.
It is not homogeneous and this feature turns out to impose some technical difficulties in applying usual elliptic methods for obtaining the existence and regularity of weak solutions of problems involving this operator.
The concept of local nondeterminism was first introduced by Berman [31] to unify and extend his methods for studying the existence and joint continuity of local times of real-valued Gaussian processes.
