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This article proposes two control strategies to realize matrix projective synchronization (MPS) for a class of discrete-time complex dynamical networks (DTCDNs). It is worth pointing out that our network model is composed of nodes with different state dimensions and its outer coupling configuration matrix can be asymmetric. In addition, since a small percentage of nodes that play more important role than others usually exist in a network, thus, one important node, which is named as crucial node in this paper, is taken into account in our network model. Besides, the commonality between each node and the crucial node is also precisely expressed. Further, based on the crucial node and the commonality and associated with the Lyapunov stability theory, two control strategies are addressed to realize the exponential matrix projective synchronization (EMPS) and asymptotic matrix projective synchronization (AMPS), respectively. It should be stressed that only the crucial node is controlled and only the state of the crucial node is applied to design the EMPS and AMPS controllers. This is promising to reduce the control cost as much as possible and to improve the feasibility of our MPS strategies. The effectiveness and feasibility of our MPS schemes are rigorously proved in theory as well as verified by two numerical examples.

This paper investigates the matrix projective synchronization (MPS) problem for time-varying disturbed complex dynamical networks (CDNs) with uncertain nonlinear structures and different dimensional nodes. In order to well describe the practical networks, the unavoidable unknown nonlinear structure and the inevitable uncertain disturbance of each node are considered into our network model, which is different from the previous works. Besides, it is worth pointing out that the outer coupling configuration matrix, which represents the coupling strength and the topological structure, is not restricted by the dissipatively coupled condition in this paper. The definition of MPS is also introduced for the networks with different dimensional nodes. Moreover, several MPS schemes are respectively put forward for our network model according to the norm bound of the uncertain nonlinear structures and disturbances being unknown or not. Finally, two proper examples associated with numerical simulations are given to verify the effectiveness of our theoretical results.

We consider nonlinear, nonhomogeneous elliptic Dirichlet equations driven by the sum of a p‐Laplacian and a Laplacian (so‐called (p, 2)‐equation). We are concerned with both cases 1 2. In the first one, the reaction f(z,x) is linear grow near ±∞ and resonant with respect to a nonprincipal nonnegative eigenvalue. In the second case, the reaction f(z,·) is (p−1)‐superlinear near ±∞ and has z‐dependent zeros of constant sign. Using variational methods together with flow invariance arguments, we establish the existence of nodal solutions.

This paper investigates the generalized outer synchronization (GOS) between two non-dissipatively coupled complex dynamical networks (CDNs) with different time-varying coupling delays. Our drive-response networks also possess nonlinear inner coupling functions and time-varying outer coupling configuration matrices. Besides, in our network models, the nodes in the same network are nonidentical and the nodes in different networks have different state dimensions. Asymptotic generalized outer synchronization (AGOS) and exponential generalized outer synchronization (EGOS) are defined for our CDNs. Our main objective in this paper is to design AGOS and EGOS controllers for our drive-response networks via the open-plus-closed-loop control technique. Distinguished from most existing literatures, it is the partial intrinsic dynamics of each node in response network that is restricted by the QUAD condition, which is easy to be satisfied. Representative simulation examples are given to verify the effectiveness and feasibility of our GOS theoretical results in this paper.

In this paper we study the existence of multiple sign-changing solutions for the following nonlocal Kirchhoff-type boundary value problem: $$\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\int _{\Omega }|\nabla u|^2{ dx}\right) \triangle {u}=\lambda |u|^{p-1}u,&{}\quad \text{ in }\quad \Omega ,\\ u=0,&{} \quad \text{ on }\quad \partial \Omega . \\ \end{array}\right. \end{aligned}$$ Using a new method, we prove that this problem has infinitely many sign-changing solutions and has a least energy sign-changing solution for $$p\in (3,5)$$ . Few existence results of multiple sign-changing solutions are available in the literature. This new method is that, by choosing some suitable subsets which separate the action functional and on which the functional is bounded, so that we can use genus and the method of invariant sets of descending flow to construct the minimax values of the functional. Our work generalize some results in literature.

We consider a nonlinear Neumann problem driven by a nonhomogeneous differential operator and an indefinite potential. Using variational methods together with flow invariance arguments, we show that the problem has at least one nodal solution. The result presented in this paper gives an answer to the open question raised by Papageorgiou and Radulescu (2016).

This paper investigates the stabilization problem for time-varying and disturbed complex dynamical networks (CDNs) with different-dimensional nodes and uncertain nonlinearities. To be consistent with the properties of real-world networks, both the disturbances of our networks and the nonlinear structures of the nodes permit are completely unknown but bounded. Furthermore, the norm bounds of the uncertain nonlinearities and disturbances (NBUND) are applied to design the stabilization controllers. When the NBUND are known in advance, some decentralized state feedback controllers are proposed to stabilize our networks. And when they are unknown, adaptive decentralized stabilization schemes are brought forward for our network models. The effectiveness and feasibility of our theoretical results are verified by two simulation examples.

In this paper, we consider a type of nth order functional difference systems depending on a parameter λ. By using the fixed point index theory, we prove that there exists a positive number \({\widetilde{\lambda}}\) separating \({(0,+ \infty) \backslash \widetilde{\lambda}}\) into two disjoint subintervals Λ1 and Λ2 such that the system has zero, at least one or at least two positive periodic solutions according to \({\lambda \in \Lambda_1, \lambda = \widetilde{\lambda}}\), or \({\lambda \in \Lambda_2}\). This work improves and extends some recent results in the literature for the first order systems. In addition, we obtain conditions for a general system to have nontrivial periodic solutions. The results are illustrated with an example.