The Connection Between the Metric and Generalized Projection Operators in Banach Spaces

In this paper we study the connection between the metric projection operator PK ： B →K, where B is a reflexive Banach space with dual space B^＊ and K is a non-empty closed convex subset of B, and the generalized projection operators ∏K ： B → K and πK ： B^＊ → K. We also present some results in non-reflexive Banach spaces.

CONVERGENCE ANALYSIS OF A MONOTONE PROJECTION ALGORITHM IN REFLEXIVE BANACH SPACES

In this article, fixed points of generalized asymptotically quasi-Ф-nonexpansive mappings and equilibrium problems are investigated based on a monotone projection algo- rithm. Strong convergence theorems are established without the aid of compactness in the framework of reflexive Banach spaces.

Generalized projective synchronization of a class of chaotic （hyperchaotic） systems with uncertain parameters

In this paper is investigated the generalized projective synchronization of a class of chaotic （or hyperchaotic） systems, in which certain parameters can be separated from uncertain parameters. Based on the adaptive technique, the globally generalized projective synchronization of two identical chaotic （hyperchaotic） systems is achieved by designing a novel nonlinear controller. Furthermore, the parameter identification is realized simultaneously. A sufficient condition for the globally projective synchronization is obtained. Finally, by taking the hyperchaotic L system as example, some numerical simulations are provided to demonstrate the effectiveness and feasibility of the proposed technique.

Generalized projective synchronization for the chaotic Lorenz system and the chaotic Chen system

Projective synchronization and generalized projective synchronization have recently been observed in the coupled chaotic systems. In this paper, a new synchronization, called ＂generalized projective synchronization＂, is reported in the chaotic Lorenz system and the chaotic Chen one.

Generalized projective synchronization of chaotic systems via adaptive learning control

In this paper, a learning control approach is applied to the generalized projective synchronisation （GPS） of different chaotic systems with unknown periodically time-varying parameters. Using the Lyapunov--Krasovskii functional stability theory, a differential-difference mixed parametric learning law and an adaptive learning control law are constructed to make the states of two different chaotic systems asymptotically synchronised. The scheme is successfully applied to the generalized projective synchronisation between the Lorenz system and Chen system. Moreover, numerical simulations results are used to verify the effectiveness of the proposed scheme.

Generalized projective synchronization of the fractional-order chaotic system using adaptive fuzzy sliding mode control

An adaptive fuzzy sliding mode strategy is developed for the generalized projective synchronization of a fractional- order chaotic system, where the slave system is not necessarily known in advance. Based on the designed adaptive update laws and the linear feedback method, the adaptive fuzzy sliding controllers are proposed via the fuzzy design, and the strength of the designed controllers can he adaptively adjusted according to the external disturbances. Based on the Lya- punov stability theorem, the stability and the robustness of the controlled system are proved theoretically. Numerical simu- lations further support the theoretical results of the paper and demonstrate the efficiency of the proposed method. Moreover, it is revealed that the proposed method allows us to manipulate arbitrarily the response dynamics of the slave system by adjusting the desired scaling factor λi and the desired translating factor ηi, which may be used in a channel-independent chaotic secure communication.

Adaptive generalized matrix projective lag synchronization between two different complex networks with non-identical nodes and different dimensions

The adaptive generalized matrix projective lag synchronization between two different complex networks with non-identical nodes and different dimensions is investigated in this paper. Based on Lyapunov stability theory and Barbalat＇s lemma, generalized matrix projective lag synchronization criteria are derived by using the adaptive control method. Furthermore, each network can be undirected or directed, connected or disconnected, and nodes in either network may have identical or different dynamics. The proposed strategy is applicable to almost all kinds of complex networks. In addition, numerical simulation results are presented to illustrate the effectiveness of this method, showing that the synchronization speed is sensitively influenced by the adaptive law strength, the network size, and the network topological structure.

Generalized Projective Synchronization Between Rssler System and New Unified Chaotic System

Based on symbolic computation system Maple and Lyapunov stability theory,an active control method isused to projectively synchronize two different chaotic systems—Lorenz-Chen-Lü system(LCL)and Rssler system,which belong to different dynamic systems.In this paper,we achieve generalized projective synchronization between thetwo different chaotic systems by directing the scaling factor onto the desired value arbitrarily.To illustrate our result,numerical simulations are used to perform the process of the synchronization and successfully put the orbits of drivesystem(LCL)and orbits of the response system(Rssler system)in the same plot for understanding intuitively.

Dynamic analysis of a new chaotic system with fractional order and its generalized projective synchronization

Based on the stability theory of the fractional order system, the dynamic behaviours of a new fractional order system are investigated theoretically. The lowest order we found to have chaos in the new three-dimensional system is 2.46, and the period routes to chaos in the new fractional order system are also found. The effectiveness of our analysis results is further verified by numerical simulations and positive largest Lyapunov exponent. Furthermore, a nonlinear feedback controller is designed to achieve the generalized projective synchronization of the fractional order chaotic system, and its validity is proved by Laplace transformation theory.

Adaptive generalized projective synchronization of two different chaotic systems with unknown parameters

This paper presents a general method of the generalized projective synchronization and the parameter identification between two different chaotic systems with unknown parameters. This approach is based on Lyapunov stability theory, and employs a combination of feedback control and adaptive control. With this method one can achieve the generalized projective synchronization and realize the parameter identifications between almost all chaotic （hyperchaotic） systems with unknown parameters. Numerical simulations results are presented to demonstrate the effectiveness of the method.

Generalized projective synchronization of fractional-order complex networks with nonidentical nodes

This paper investigates the synchronization problem of fractional-order complex networks with nonidentical nodes. The generalized projective synchronization criterion of fractional-order complex networks with order 0 〈 q 〈 1 is obtained based on the stability theory of the fractional-order system. The control method which combines active control with pinning control is then suggested to obtain the controllers. Furthermore, the adaptive strategy is applied to tune the control gains and coupling strength. Corresponding numerical simulations are performed to verify and illustrate the theoretical results.

Generalized projective synchronization via the state observer and its application in secure communication

Based on the improved state observer and the pole placement technique, by adding a constant which extends the scope of use of the original system, a new design method of generalized projective synchronization is proposed. With this method, by changing the projective synchronization scale factor, one can achieve not only complete synchronization, but also anti-synchronization, as well as arbitrary percentage of projective synchronization, so that the system may attain arbitrary synchronization in a relatively short period of time, which makes this study more meaningful. By numerical simulation, and choosing appropriate scale factor, the results of repeated experiments verify that this method is highly effective and satisfactory. Finally, based on this method and the relevant feedback concept, a novel secure communication project is designed. Numerical simulation verifies that this secure communication project is very valid, and moreover, the experimental result has been greatly improved in decryption time.

A new three-dimensional chaotic system and its modified generalized projective synchronization

Based on the Chen chaotic system, this paper constructs a new three-dimensional chaotic system with higher order nonlinear term and studies the basic dynamic behaviours of the system. The modified generalized projective synchronization has been observed in the coupled new three-dimensional chaotic system with unknown parameters. Furthermore, based on Lyapunov stability theory, it obtains the control laws and adaptive laws of parameters to make modified generalized projective synchronization of the coupled new three-dimensional chaotic systems. Numerical simulation results are presented to illustrate the effectiveness of this method.

Generalized projective synchronization between two chaotic gyros with nonlinear damping

In this paper, the chaotic generalized projective synchronization of a controlled, noised gyro with an expected gyro is investigated by a simple control law. Based on the theory of discontinuous dynamical systems, the necessary and sufficient conditions for such a synchronization are achieved. From such conditions, non-synchronization, partial and full synchronizations between the two coupled gyros are discussed. The switching scenarios between desynchronized and synchronized states of the two dynamical systems are shown. Numerical simulations are illustrated to verify the effectiveness of this method.

A novel fractional-order hyperchaotic complex system and its synchronization

A novel fractional-order hyperchaotic complex system is proposed by introducing the Caputo fractional-order derivative operator and a constant term to the complex simplified Lorenz system. The proposed system has different numbers of equilibria for different ranges of parameters. The dynamics of the proposed system is investigated by means of phase portraits, Lyapunov exponents, bifurcation diagrams, and basins of attraction. The results show abundant dynamical characteristics. Particularly, the phenomena of extreme multistability as well as hidden attractors are discovered. In addition, the complex generalized projective synchronization is implemented between two fractional-order hyperchaotic complex systems with different fractional orders. Based on the fractional Lyapunov stability theorem, the synchronization controllers are designed, and the theoretical results are verified and demonstrated by numerical simulations. It lays the foundation for practical applications of the proposed system.

Strong Convergence for a Countable Family of Total Quasi-φ-asymptotically Nonexpansive Nonself Mappings in Banach Space

The purpose of this article is to introduce a class of total quasi-Ф- asymptotically nonexpansive nonself mappings. Strong convergence theorems for common fixed points of a countable family of total quasi-Ф-asymptotically nonexpan- sive mappings are established in the framework of Banach spaces based on modified Halpern and Mann-type iteration algorithm. The main results presented in this article extend and improve the corresponding results of many authors.

Strong convergence theorem for relatively nonexpansive mapping and inverse-strongly-monotone mapping in a Banach space

In this paper, we introduce an iterative sequence for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping in a Banach space. Then, we show that the sequence converges strongly to a common element of the two sets. Our results improve and extend the corresponding results reported by many others.

MULTI-VALUED TOTALLY QUASI-φ-ASYMPTOTICALLY NONEXPANSIVE SEMI-GROUPS AND STRONG CONVERGENCE THEOREMS IN BANACH SPACES

The purpose of this article is first to introduce the concept of multi-valued to- tally Quasi-φ-asymptotically nonexpansive semi-groups, which contains many kinds of semi- groups as its special cases, and then to modify the Halpern-Mann-type iteration algorithm for multi-valued totally Quasi-cS-asymptotically nonexpansive semi-groups to have the strong convergence under a limit condition only in the framework of Banach spaces. The results presented in this article improve and extend the corresponding results announced by many authors recently.

A GENERALIZED GRADIENT PROJECTION METHOD FOR OPTIMIZATION PROBLEMS WITH EQUALITY AND INEQUALITY CONSTRAINTS ABOUT ARBITRARY INITIAL POINT

In this paper, we give a new generalized gradient projection algorithm for nonlinear optimization problems with arbitrary initial point. This new algorithm has some important advantages as follows: (1) The algorithm does not require initial feasible point; (2) It can deal with nonlinear equality and inequality constraints problems; (3) The structure of our algorithm is very simple;(4) Under some mild assumptions, it has global convergence.

A 2.79 COMPETITIVE ON-LINE ALGORITHM FOR TWO PROCESSOR REAL-TIME SYSTEMS WITH UNIFORM VALUE DENSITY

In this paper, a new superlinearly convergent algorithm is presented for optimization problems with general nonlinear equality and inequality constraints. Comparing with other methods for these problems, the algorithm has two main advantages. First, it doesn’t solve any quadratic programming (QP), and its search directions are determined by the generalized projection technique and the solutions of two systems of linear equations. Second, the sequential points generated by the algorithm satisfy all inequality constraints and its step length is computed by the straight line search. The algorithm is proved to possess global and superlinear convergence.