Approximating fixed points of a pair of contractive type mappings in generalized convex metric spaces

In this paper, we consider an Ishikawa type iteration process with errors, which converges to the unique common fixed point of a pair of contractive type mappings in complete generalized convex metric spaces. Furthermore, we obtain the corresponding results in Banach spaces. Our results extend and generalize many known results.     

Explicit averaging cyclic algorithm for common fixed points of asymptotically strictly pseudocontractive maps

Weak and strong convergence theorems for the approximation of common fixed points of a finite family of asymptotically strictly pseudocontractive mappings are proved in Hilbert spaces using an explicit averaging cyclic algorithm.     

A common fixed point theorem for compatible quasi contractive self mappings in metric spaces

A common fixed point theorem for weakly commuting quasi contractive self mappings with contracting orbital diameters in metric spaces [V. Berinde, A common fixed point theorem for quasi contractive type mappings, Ann. Univ. Sci. Budapest. 46 (2003) 81-90] is extended to the more general class of compatible quasi contractive self mappings. Our result does extend and generalize numerous related results in literature.     

A modified projection method with a new direction for solving variational inequalities

In this paper, we propose a new projection method for solving variational inequality problems, which can be viewed as an improvement of the method of Li et al. [M. Li, L.Z. Liao, X.M. Yuan, A modified projection method for co-coercive variational inequality, European Journal of Operational Research 189 (2008) 310-323], by adopting a new direction. Under the same assumptions as those in Li et al. (2008), we establish the global convergence of the proposed algorithm. Some preliminary computational results are reported, which illustrated that the new method is more efficient than the method of Li et al. (2008).     

A NUMERICAL STUDY OF UNIFORM SUPERCONVERGENCE OF LDG METHOD FOR SOLVING SINGULARLY PERTURBED PROBLEMS

In this paper, we consider the local discontinuous Galerkin method （LDG） for solving singularly perturbed convection-diffusion problems in one- and two-dimensional settings. The existence and uniqueness of the LDG solutions are verified. Numerical experiments demonstrate that it seems impossible to obtain uniform superconvergence for numerical fluxes under uniform meshes. Thanks to the implementation of two-type different anisotropic meshes, i.e., the Shishkin and an improved grade meshes, the uniform 2p ＋ i-order superconvergence is observed numerically for both one-dimensional and twodimensional cases.     

带广义边界条件的四阶抛物型方程的混合间断时空有限元法

构造具有广义边界条件的四阶线性抛物型方程的混合间断时空有限元格式,利用混合有限元方法将高阶方程降阶,利用空间连续而时间允许间断的时空有限元方法离散方程,证明了离散解的存在唯一性,稳定性和收敛性,并给出数值算例验证了方法的有效性.     

Mann iteration of weak convergence theorems in Banach space

In this paper, by using Mann＇s iteration process we will establish several weak convergence theorems for approximating a fixed point of k-strictly pseudocontractive mappings with respect to p in p-uniformly convex Banach spaces. Our results answer partially the open question proposed by Marino and Xu, and extend Reich＇s theorem from nonexpansive mappings to k-strict pseudocontractive mappings.     

An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings

In this paper, we propose an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudo-contraction mapping in the setting of real Hilbert spaces. We establish some weak and strong convergence theorems of the sequences generated by our proposed scheme. Our results combine the ideas of Marino and Xu’s result [G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007) 336–346], and Takahashi and Takahashi’s result [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515]. In particular, necessary and sufficient conditions for strong convergence of our iterative scheme are obtained.     

Periodic solution and its stability of a delayed Beddington‐DeAngelis type predator‐prey system with discontinuous control strategy

This paper investigates the periodic solution of a delayed Beddington‐DeAngelis (BD) type predator‐prey model with discontinuous control strategy. Firstly, the regularity and visibility analysis of the delayed predator‐prey model is carried out by using the principle of differential inclusion. Secondly, the positiveness and boundeness of the solution is discussed by employing the comparison theorem. Based on the boundary conditions of the model and the Mawhin‐like coincidence theorem, it is shown that the solution of the delayed BD system is asymptotically stable in finite time. Furthermore, it is found that there exists at least one periodic solution of the nonautonomous delayed predator‐prey model by using the principle of topological degree and set value mapping. Specially, when the nonautonomous delayed BD system degenerates into an autonomous system, some criteria are obtained to guarantee the convergence behavior of the harvesting solutions for the corresponding autonomous delayed BD system. Finally, numerical examples are given to demonstrate the applicability and effectiveness of main results. It is worthy to point out that the discontinuous control strategy is superior to the continuous harvesting policies adopted in existing literature.     

Convergence analysis of modified iterative approaches in geodesic spaces with curvature bounded above

The objective of this article is to establish a new modified iteration process for nonexpansive mappings in complete CAT(κ) spaces. We prove strong and Δ‐convergence theorems of the proposed method in such spaces under some standard conditions. Furthermore, numerical experiments of non‐trivial examples are also provided to show performance and comparison speed of convergence with many previously known methods. Our main result extended and improved the corresponding recent results announced by many researchers.