In this paper, we study the Sobolev’s spaces on time scales and their properties. As applications, we present a recent approach via variational methods and the critical point theory to obtain the existence of solutions for a class of second order Hamiltonian systems on time scales. By establishing a proper variational setting, three existence results for systems under consideration are obtained. Finally, three examples are presented to illustrate the feasibility and effectiveness of the existence results.

Sobolev’s spaces on time scales and its applications to a class of second order Hamiltonian systems on time scales

In this paper, we consider Clifford-valued fuzzy cellular neural networks with time-varying delays. In order to avoid the inconvenience caused by the non-commutativity of the multiplication of Clifford numbers, we first decompose the considered n-dimensional Clifford-valued systems into $$2^{m}n$$-dimensional real-valued systems. Then by using the Banach fixed point theorem and a proof by contradiction, we establish sufficient conditions ensuring the existence, the uniqueness and the global exponential stability of $$S^{p}$$-almost periodic solutions for the considered neural networks. Finally, we give an example to illustrate the effectiveness of the obtained results. Our results are new even when the considered neural networks degenerates to real-valued, complex-valued and quaternion-valued neural networks.

Hongtao Zhang

Papers

Existence of positive periodic solutions for functional differential equations with impulse effects on time scales