The purpose of this paper is to revise the definitions of Ishikawa and Mann iterative processes with errors, by using a new inequality to study the unique solution of the nonlinear strongly accretive operator equation Tx s f and the convergence problem of Ishikawa and Mann iterative sequences for strongly pseudo-contractive mappings without the Lipschitz condition. The results prew sented in this paper improve and extend the corresponding results in 4, 5, 7᎐10, x 12, 15, 16 in the more general setting. In particular, the open problem mentioned w x by Chidume in 5 has been given an affirmative answer. ᮊ 1998 Academic Press

a b s t r a c tOur current paper is devoted to studying the numerical and analytical solutions for a class of Generalized Fractional Diffusion Equations (GFDEs) with new Generalized TimeFractional Derivative (GTFD). The GTFD we propose here is defined in the Caputo sense. We consider the GFDEs on a bounded domain. The numerical solutions are obtained by using the Finite Difference Method (FDM) of full discretization. The stability of FDM is discussed and the order of convergence is evaluated numerically. Numerical experiments are given, which illustrate that the FDM is simple and effective for solving the GFDEs with different coefficients and source functions. An interesting phenomenon is that we can observe the period-like solution in GFDEs with a particular positive periodic weight function. Using the method of separation of variables, we convert the homogeneous GFDE into two ordinary differential equations, and solve them via the help of solutions of the initial value problem with Caputo derivative. In the analytical solution, we observe that the weight function in the denominator, and scale function mapping the response domain differently. Since the derivative considered in this article is new, many existing results of FDEs are generalized.

The paper carries the results on Takens-Bogdanov bifurcation obtained in [T. Faria, L.T. Magalhães, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity, J. Differential Equations 122 (1995) 201-224] for scalar delay differential equations over to the case of delay differential systems with parameters. Firstly, we give feasible algorithms for the determination of Takens-Bogdanov singularity and the generalized eigenspace associated with zero eigenvalue in R n . Next, through center manifold reduction and normal form calculation, a concrete reduced form for the parameterized delay differential systems is obtained. Finally, we describe the bifurcation behavior of the parameterized delay differential systems with T-B singularity in detail and present an example to illustrate the results.

Two models for competition of two populations in a chemostat environment with nutrient recycling are considered. In the first model, the recycling is instantaneous, whereas in the second, the recycling is delayed. For each model an equilibrium analysis is carried out, and persistence criteria are obtained. This paper extends the work done by Beretta et al. (1990) for a single species.

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In this paper, numerical solutions of Burgers equation defined by using a new Generalized Time-Fractional Derivative (GTFD) are discussed. The numerical scheme uses a finite difference method. The new GTFD is defined using a scale function and a weight function. Many existing fractional derivatives are the special cases of it. A linear recurrence relationship for the numerical solutions of the resulting system of linear equations is found via finite difference approach. Burgers equations with different fractional orders and coefficients are computed which show that this numerical method is simple and effective, and is capable of solving the Burgers equation accurately for a wide range of viscosity values. Furthermore, we study the influence of the scale and the weight functions on the diffusion process of Burgers equation. Numerical simulations illustrate that a scale function can stretch or contract the diffusion on the time domain, while a weight function can change the decay velocity of the diffusion process.MSC 2010 : Primary 26A33; Secondary 34A08.

Abstract:In this paper, we study the chaotic dynamics of a Variable-Order Fractional Financial System (VOFFS

The numerical solution of a variable-order fractional financial system is calculated by using the Adams-Bashforth-Moulton method. The derivative is defined in the Caputo variable-order fractional sense. Numerical examples show that the Adams-Bashforth-Moulton method can be applied to solve such variable-order fractional differential equations simply and effectively. The convergent order of the method is also estimated numerically. Moreover, the stable equilibrium point, quasiperiodic trajectory, and chaotic attractor are found in the variable-order fractional financial system with proper order functions.