Some results for a finite family of uniformly L-Lipschitzian mappings in Banach spaces

The purpose of this paper is to prove a strong convergence theorem for a finite family of uniformly L-Lipschitzian mappings in Banach spaces. The results presented in the paper improve and extend some recent results in Chang [1], Cho et al. [2] Ofoedu [5], Schu [7] and Zeng [8, 9]. The present studies were supported by “the Natural Science Foundation of China (No. 10771141),” the Natural Science Foundation of Zhejiang Province (Y605191), the Natural Science Foundation of Heilongjiang Province (A0211), the Scientific Research Foundation from Zhejiang Province Education Committee (20051897).     

Generalized differentiation and bi-Lipschitz nonembedding in L1

We consider Lipschitz mappings, $f:X\to V$, where X is a doubling metric measure space which satisfies a Poincaré inequality, and V is a Banach space. We show that earlier differentiability and bi-Lipschitz nonembedding results for maps, $f:X\to {\mathbb{R}}^N$, remain valid when ${\mathbb{R}}^N$ is replaced by any separable dual space. We exhibit spaces which bi-Lipschitz embed in $L^1$, but not in any separable dual V. For certain domains, including the Heisenberg group with its Carnot–Caratheodory metric, we establish a new notion of differentiability for maps into $L^1$. This implies that the Heisenberg group does not bi-Lipschitz embed in $L^1$, thereby proving a conjecture of J. Lee and A. Naor. When combined with their work, this has implications for theoretical computer science. To cite this article: J. Cheeger, B. Kleiner, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Ishikawa type iterative algorithm for completely generalized nonlinear quasi-variational-like inclusions in Banach spaces

In this paper, we consider completely generalized nonlinear quasi-variational-like inclusions in Banach spaces and propose an Ishikawa type iterative algorithm for computing their approximate solutions by applying the new notion of $J^{\eta }$-proximal mapping given in [R. Ahmad, A.H. Siddiqi, Z. Khan, Proximal point algorithm for generalized multi-valued nonlinear quasi-variational-like inclusions in Banach spaces, Appl. Math. Comput. 163 (2005) 295–308]. We prove that the approximate solutions obtained by the proposed algorithm converge to the exact solution of our quasi-variational-like inclusions. The results presented in this paper extend and improve the corresponding results of [R. Ahmad, A.H. Siddiqi, Z. Khan, Proximal point algorithm for generalized multi-valued nonlinear quasi-variational-like inclusions in Banach spaces, Appl. Math. Comput. 163 (2005) 295–308; X.P. Ding, F.Q. Xia, A new class of completely generalized quasi-variational inclusions in Banach spaces, J. Comput. Appl. Math. 147 (2002) 369–383; N.J. Huang, Generalized nonlinear variational inclusions with non-compact valued mappings, Appl. Math. Lett. 9(3) (1996) 25–29; A. Hassouni, A. Moudafi, A perturbed algorithm for variational inclusions, J. Math. Anal. Appl. 185(3) (1994) 706–712]. Some special cases are also discussed.     

A rotation method which gives linear Lp estimates for powers of the Ahlfors–Beurling operator

In [O. Dragičević, A. Volberg, Sharp estimate of the Ahlfors–Beurling operator via averaging martingale transforms, Michigan Math. J. 51 (2) (2003) 415–435] the Ahlfors–Beurling operator T was represented as an average of two-dimensional martingale transforms. The same result can be proven for powers $T^n$. Motivated by [T. Iwaniec, G. Martin, Riesz transforms and related singular integrals, J. Reine Angew. Math. 473 (1996) 25–57], we deduce from here that $6T^n{6}_p$ are bounded from above by $Cn{p}^1$, $p^1=\max \{p,\frac{p}{p-1}\}$. We further improve this estimate to obtain the optimal behaviour of the $L^p$ norms in question.     

Rationality of the vertex algebra VL+ when L is a non-degenerate even lattice of arbitrary rank

In this paper we prove that the vertex algebra $V_L^{+}$ is rational if L is a negative definite even lattice of finite rank, or if L is a non-degenerate even lattice of a finite rank that is neither positive definite nor negative definite. In particular, for such even lattices L, we show that the Zhu algebras of the vertex algebras $V_L^{+}$ are semisimple. This extends the result of Abe from [T. Abe, Rationality of the vertex operator algebra $V_L^{+}$ for a positive definite even lattice L, Math. Z. 249 (2) (2005) 455–484] which establishes the rationality of $V_L^{+}$ when L is a positive definite even lattice of finite rank.     

Some results for asymptotically pseudo-contractive mappings and asymptotically nonexpansive mappings

Some convergence theorems of modified Ishikawa and Mann iterative sequences with errors for asymptotically pseudo-contractive and asymptotically nonexpansive mappings in Banach space are obtained. The results presented in this paper improve and extend the corresponding results in Goebel and Kirk (1972), Kirk (1965), Liu (1996), Schu (1991) and Chang et al. (to appear).

     

Viscosity approximation methods for pseudo-contractive semigroups in Banach spaces

We study the strong convergence of two viscosity iteration processes for pseudo-contractive semigroup and for ?$?$-strongly pseudo-contractive mapping in uniformly convex Banach spaces with uniformly Gâteaux differentiable norm. As special cases, we get strong convergence of two viscosity iteration processes for approximating common fixed points of nonexpansive semigroups in certain Banach spaces. The results presented in this paper extend and generalize previous results.     

Strong convergence theorems for finitely many nonexpansive mappings and applications

Let E$E$ be a uniformly convex Banach space which satisfies Opial’s condition or whose norm is Fréchet differentiable. Recently, Takahashi and Shimoji [W. Takahashi, K. Shimoji, Convergence theorems for nonexpansive mappings and feasibility problems, Math. Comput. Modelling 32 (2000) 1463–1471] introduced an iterative scheme given by finitely many nonexpansive mappings in E$E$ and proved weak convergence theorems which are connected with the problem of image recovery. In this paper we introduce a new iterative scheme which includes their iterative scheme as a special case. Under the assumption that E$E$ is a reflexive Banach space whose norm is uniformly Gâteaux differentiable and which has a weakly continuous duality mapping, we prove strong convergence theorems which are connected with the problem of image recovery. Using the established results, we consider the problem of finding a common fixed point of finitely many nonexpansive mappings.     

Strong convergence of composite iterative schemes for zeros of m-accretive operators in Banach spaces

We introduce a new composite iterative scheme to approximate a zero of an m$m$-accretive operator A$A$ defined on uniform smooth Banach spaces and a reflexive Banach space having a weakly continuous duality map. It is shown that the iterative process in each case converges strongly to a zero of A$A$. The results presented in this paper substantially improve and extend the results due to Ceng et al. [L.C. Ceng, H.K. Xu, J.C. Yao, Strong convergence of a hybrid viscosity approximation method with perturbed mappings for nonexpansive and accretive operators, Taiwanese J. Math. (in press)], Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60] and Xu [H.K. Xu, Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl. 314 (2006) 631–643]. Our work provides a new approach for the construction of a zero of m$m$-accretive operators.     

Strong Convergence Theorems for Quasi-Nonexpansive Mappings and Uniformly L-Lipschitzian Asymptotically Pseudo-Contractive Mappings in Banach Spaces

A new 𝒮-generated Ishikawa iteration with errors is proposed for a pair of quasi-nonexpansive mapping and uniformly L-Lipschitzian asymptotically pseudo-contractive mapping in real Banach spaces. We show that the proposed iterative scheme converges strongly to a common solution of quasi-nonexpansive mapping and uniformly L-Lipschitzian asymptotically pseudo-contractive mapping in real Banach spaces. A comparison table is prepared using a numeric example which shows that the proposed iterative algorithm is faster than some known iterative algorithms.     

Demiclosed principle and convergence for modified three step iterative process with errors of non-Lipschitzian mappings

A demiclosed principle is proved for asymptotically nonexpansive mappings in the intermediate sense. Moreover, it is proved that the modified three-step iterative sequence converges weakly and strongly to common fixed points of three asymptotically nonexpansive mappings in the intermediate sense under certain conditions. The results of this paper improve and extend the corresponding results of [M.O. Osilike, S.C. Aniagbosor, Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, Math. Comput. Modelling 32 (2000) 1181-1191; G.E. Kim, T.H. Kim, Mann and Ishikawa iterations with errors for non-Lipschitzian mappings in Banach spaces, Comput. Math. Appl. 42 (2001) 1565-1570; B.L. Xu, M.A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2002) 444-453; K. Nammanee, S. Suantai, The modified Noor iterations with errors for non-Lipschitzian mappings in Banach spaces, Appl. Math. Comput. 187 (2007) 669-679; K. Nammanee, M.A. Noor, S. Suantai, Convergence criteria of modified Noor iterations with errors for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 314 (2006) 320-334] and other corresponding known ones. On the other hand, we show the necessary and sufficient condition for the strong convergence of the modified three-step iterative sequence to some common fixed points of .