In this paper, the existence and multiplicity of nontrivial solutions are obtained for nonlinear fractional differential systems with p‐Laplacian by combining the properties of fractional calculus with critical point theory. Firstly, we present a result that a class of p‐Laplacian fractional differential systems exists infinitely many solutions under the famous Ambrosetti‐Rabinowitz condition. Then, a criterion is given to guarantee that the fractional systems exist at least 1 nontrivial solution without satisfying Ambrosetti‐Rabinowitz condition. Our results generalize some existing results in the literature.

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In this paper, by using variational methods and a critical point theorem due to Bonanno and Marano, the existence of at least three weak solutions is obtained for a class of p-Laplacian type nonlinear fractional coupled systems depending on two parameters. Two examples are given to illustrate the applications of our main results.

The stochastic resonance is investigated in the generalized Langevin equation with exponential memory kernel subjected to the joint action of internal noise, external noise and external sinusoidal forcing. The system is converted into three-dimensional Markovian Langevin equations. Furthermore, using the Shapiro-Loginov formula and the Laplace transformation technique, the exact expressions of the first moment and the steady response amplitude are obtained. The research results show that with the variations of external sinusoidal force frequency and the parameters of memory kernel and external noise, the system presents bona-fide stochastic resonance, conventional stochastic resonance and stochastic resonance in a broad sense under the condition of Routh-Hurwitz stability. In addition, the stochastic resonance can be weakened as the memory time increases. Moreover, the numerical results of power spectrum of system are in agreement with the analytic results.

Matrix Riccati differential equations arise in many different areas and are particular important within the field of control theory. In this paper we consider numerical integration for large-scale systems of stiff matrix Riccati differential equations. We show how to apply exponential Rosenbrock-type integrators to get approximate solutions. Two typical exponential integration schemes are considered. The implementation issues are addressed and some low-rank approximations are exploited based on high quality numerical algebra codes. Numerical comparisons demonstrate that the exponential integrators can obtain high accuracy and efficiency for solving large-scale systems of stiff matrix Riccati differential equations.