This paper deals with the boundedness of the solutions of the following dynamic equations andon a time scale T. By using the Bellman integral inequality, we establish some sufficient conditions for bound- edness of solutions of the above equations. Our results not only unify the boundedness results for differential and difference equations but are also new for the q-difference equations.
We study the existence of traveling wave solutions for a nonlocal and non-monotone delayed reaction-diffusion equation. Based on the construction of two associated auxiliary reaction diffusion equations with monotonicity and by using the traveling wavefronts of the auxiliary equations, the existence of the positive traveling wave solutions for c 〉 c. is obtained. Also, the exponential asymptotic behavior in the negative infinity was established. Moreover, we apply our results to some reactiondiffusion equations with spatio-temporal delay to obtain the existence of traveling waves. These results cover, complement and/or improve some existing ones in the literature.
