A note on Ceng-Wang-Yao’s result [Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities,Math. Meth. Oper. Res. (2008) 67: 375–390]

In this paper, we give a short and simple proof of the recent result of Ceng et al. (Math Methods Oper Res 67(3):375–390, 2008).     

On the modulus of -convexity

In this paper, we prove that the moduli of W^*-convexity, introduced by Ji Gao [J. Gao, The W^*-convexity and normal structure in Banach spaces, Appl. Math. Lett. 17 (2004) 1381–1386], of a Banach space X and of the ultrapower of X itself coincide whenever X is super-reflexive. Moreover, we improve a sufficient condition for uniform normal structure of the space and its dual. This generalizes and strengthens the main results of [J. Gao, The W^*-convexity and normal structure in Banach spaces, Appl. Math. Lett. 17 (2004) 1381–1386].     

Another look at Wang’s new method for solving split common fixed-point problems without priori knowledge of operator norms

In this paper, we give a simple proof of Wang’s recent result concerning split common fixed-point problems (F. Wang, J Fixed Point Theory Appl 19(4): 2427–2436, 2017). Moreover, we provide a more general sufficient condition than Wang’s for the weak convergence to a solution of a split common fixed-point problem.     

Halpern’s iteration in Banach spaces

We prove strong convergence theorems for a sequence which is generated by Halpern’s iteration. We also apply our result for finding zeros of an accretive operator. Our result improves the recent result of Aoyama et al. [K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007) 2350-2360] by removing some assumptions on the parameters. Finally we discuss the new sufficient condition studied by Song [Y. Song, A new sufficient condition for the strong convergence of Halpern type iterations. Appl. Math. Comput. 198 (2) (2008) 721-728; Y. Song, New strong convergence theorems for nonexpansive nonself-mappings without boundary conditions. Comput. Math. Appl. 56 (6) (2008) 1473-1478] and correct the main result of Song and Chai [Y. Song, X. Chai, Halpern iteration for firmly type nonexpansive mappings, Nonlinear Anal. 71 (10) (2009) 4500-4506].     

Remarks on sufficient conditions for fixed points of multivalued nonexpansive mappings

In recent papers [S. Dhompongsa, T. Domínguez-Benavides, A. Kaewcharoen, A. Kaewkhao, B. Panyanak, The Jordan–von Neumann constants and fixed points for multivalued nonexpansive mappings, J. Math. Anal. Appl. 320 (2006) 916–927; S. Dhompongsa, A. Kaewcharoen, A. Kaewkhao, The Domínguez–Lorenzo condition and multivalued nonexpansive mappings, Nonlinear Anal. 64 (2006) 958–970], two sufficient conditions, namely the Domínguez–Lorenzo condition and property (D$D$), for fixed points of multivalued nonexpansive mappings are introduced. The authors also give some sufficient conditions for the Domínguez–Lorenzo condition and property (D$D$). In their proofs, it seems reasonable to use the James and von Neumann–Jordan constants. With slight modifications, significant improvements are obtained. Some new estimates are tight in Hilbert spaces.     

Weak convergence theorems for a countable family of Lipschitzian mappings

This paper is concerned with convergence of an approximating common fixed point sequence of countable Lipschitzian mappings in a uniformly convex Banach space. We also establish weak convergence theorems for finding a common element of the set of fixed points, the set of solutions of an equilibrium problem, and the set of solutions of a variational inequality. With an appropriate setting, we obtain and improve the corresponding results recently proved by Moudafi [A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 9 (2008) 37–43], Tada–Takahashi [A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133 (2007) 359–370], and Plubtieng–Kumam [S. Plubtieng and P. Kumam, Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings. J. Comput. Appl. Math. (2008) doi:10.1016/j.cam.2008.05.045]. Some of our results are established with weaker assumptions.     

The von Neumann-Jordan constant, weak orthogonality and normal structure in Banach spaces

We give some sufficient conditions for normal structure in terms of the von Neumann-Jordan constant, the James constant and the weak orthogonality coefficient introduced by B. Sims. In the rest of the paper, the von Neumann-Jordan constant and the James constant for the Bynum space are computed, and are used to show that our results are sharp.

     

Another look at Cesàro sequence spaces

We consider the Cesàro sequence space cesp as a closed subspace of the infinite ?p-sum of finite dimensional spaces. We easily obtain many known results, for example, cesp has property (β) of Rolewicz, uniform Opial property, and weak uniform normal structure. We also consider some generalized Cesàro sequence spaces. Finally, we compute the von Neumann-Jordan and James constants of the two-dimensional Cesàro sequence space when 1<p?2.     

Sufficient conditions for uniform normal structure of Banach spaces and their duals

In this paper, we present some sufficient conditions for the uniform normal structure of Banach spaces and their dual spaces. More precisely, many known conditions implying the uniform normal structure of Banach spaces are shown to imply uniform normal structure of their dual spaces as well.     

The characteristic of convexity of a Banach space and normal structure

We present some sufficient conditions for normal structure of Banach spaces and their dual spaces in terms of the characteristic of convexity, the James constant, and the coefficient of weak orthogonality. Many known results are improved and strengthened. We also show that some of our results are sharp.