Abstract. In this paper, by using double fixed point theorem and a new fixed point theorem, some sufficient conditions for the existence of at least two and at least three positive solutions of an nth-order boundary value problem with integral boundary conditions are established, respectively. We also give two examples to illustrate our main results.Keywords: boundary value problems, fixed point theorem, positive solutions, integral boundary conditions.

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In this paper, by using fixed point index theory, we study the existence of positive solutions for nonlinear second-order m-point impulsive boundary value problems on time scales. As an application, we give an example to demonstrate our results.

In the present study, we investigate the existence of Li-Yorke chaos in the dynamics of shunting inhibitory cellular neural networks (SICNNs) on time scales. It is rigorously proved by taking advantage of external inputs that the outputs of SICNNs exhibit Li-Yorke chaos. The theoretical results are supported by simulations, and the controllability of chaos on the time scale is demonstrated by means of the Pyragas control technique. This is the first time in the literature that the existence as well as the control of chaos are provided for neural networks on time scales.