Existence and iterative approximations of solutions for mixed quasi-variational-like inequalities in Banach spaces

In this paper, we introduce and investigate a new class of mixed quasi-variational-like inequalities in reflexive Banach spaces. By applying a minimax inequality due to Ding–Tan and a lemma due to Chang, we establish some existence and uniqueness results of solution for the mixed quasi-variational-like inequality. Next, by using a KKM theorem due to Fan and an auxiliary principle technique due to Cohen, we suggest two iterative algorithms and study the convergence criteria of iterative sequences generated by the iterative algorithms. Our results extend, improve and unify several known results in the literature.     

On the semilocal convergence of inexact Newton methods in Banach spaces

We provide two types of semilocal convergence theorems for approximating a solution of an equation in a Banach space setting using an inexact Newton method [I.K. Argyros, Relation between forcing sequences and inexact Newton iterates in Banach spaces, Computing 63 (2) (1999) 134–144; I.K. Argyros, A new convergence theorem for the inexact Newton method based on assumptions involving the second Fréchet-derivative, Comput. Appl. Math. 37 (7) (1999) 109–115; I.K. Argyros, Forcing sequences and inexact Newton iterates in Banach space, Appl. Math. Lett. 13 (1) (2000) 77–80; I.K. Argyros, Local convergence of inexact Newton-like iterative methods and applications, Comput. Math. Appl. 39 (2000) 69–75; I.K. Argyros, Computational Theory of Iterative Methods, in: C.K. Chui, L. Wuytack (Eds.), in: Studies in Computational Mathematics, vol. 15, Elsevier Publ. Co., New York, USA, 2007; X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math. 25 (2) (2007) 231–242]. By using more precise majorizing sequences than before [X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math. 25 (2) (2007) 231–242; Z.D. Huang, On the convergence of inexact Newton method, J. Zheijiang University, Nat. Sci. Ed. 30 (4) (2003) 393–396; L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982; X.H. Wang, Convergence on the iteration of Halley family in weak condition, Chinese Sci. Bull. 42 (7) (1997) 552–555; T.J. Ypma, Local convergence of inexact Newton methods, SIAM J. Numer. Anal. 21 (3) (1984) 583–590], we provide (under the same computational cost) under the same or weaker hypotheses: finer error bounds on the distances involved; an at least as precise information on the location of the solution. Moreover if the splitting method is used, we show that a smaller number of inner/outer iterations can be obtained.     

Strong convergence theorems of an iterative scheme for strictly pseudocontractive mappings in Banach spaces

The article establishes the sufficient and necessary conditions for the strong convergence of the Ishikawa iterative scheme with random errors in the sense of Xu for a strictly pseudocontractive mapping in an arbitrary real Banach space under certain assumptions. It affirmatively answers the open question that put forth by Zeng. The results in this paper extend and improve the corresponding results of Browder, Petryshyn, Rhoades, Osilike and Zeng.     

Viscosity approximation methods for a family of finite nonexpansive mappings in Banach spaces

Viscosity approximation methods for a family of finite nonexpansive mappings are established in Banach spaces. The main theorems extend the main results of Moudafi [Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000) 46–55] and Xu [Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291] to the case of finite mappings. Our results also improve and unify the corresponding results of Bauschke [The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 202 (1996) 150–159], Browder [Convergence of approximations to fixed points of nonexpansive mappings in Banach spaces, Archiv. Ration. Mech. Anal. 24 (1967) 82–90], Cho et al. [Some control conditions on iterative methods, Commun. Appl. Nonlinear Anal. 12 (2) (2005) 27–34], Ha and Jung [Strong convergence theorems for accretive operators in Banach spaces, J. Math. Anal. Appl. 147 (1990) 330–339], Halpern [Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73 (1967) 957–961], Jung [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005) 509–520], Jung et al. [Iterative schemes with some control conditions for a family of finite nonexpansive mappings in Banach space, Fixed Point Theory Appl. 2005 (2) (2005) 125–135], Jung and Kim [Convergence of approximate sequences for compositions of nonexpansive mappings in Banach spaces, Bull. Korean Math. Soc. 34 (1) (1997) 93–102], Lions [Approximation de points fixes de contractions, C.R. Acad. Sci. Ser. A-B, Paris 284 (1977) 1357–1359], O’Hara et al. [Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces, Nonlinear Anal. 54 (2003) 1417–1426], Reich [Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980) 287–292], Shioji and Takahashi [Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (12) (1997) 3641–3645], Takahashi and Ueda [On Reich's strong convergence theorems for resolvents of accretive operators, J. Math. Anal. Appl. 104 (1984) 546–553], Wittmann [Approximation of fixed points of nonexpansive mappings, Arch. Math. 59 (1992) 486–491], Xu [Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2) (2002) 240–256], and Zhou et al. [Strong convergence theorems on an iterative method for a family nonexpansive mappings in reflexive Banach spaces, Appl. Math. Comput., in press] among others.     

Strong convergence of viscosity approximation methods for finding zeros of accretive operators in Banach spaces

In this paper, we introduce a composite iterative scheme by viscosity approximation method for finding a zero of an accretive operator in Banach spaces. Then, we establish strong convergence theorems for the composite iterative scheme. The main theorems improve and generalize the recent corresponding results of Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51-60], Qin and Su [X. Qin, Y. Su, Approximation of a zero point of accretive operator in Banach spaces, J. Math. Anal. Appl. 329 (2007) 415-424] and Xu [H.K. Xu, Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl. 314 (2006) 631-643] as well as Aoyama et al. [K. Aoyama, Y Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in Banach spaces, Nonlinear Anal. 67 (2007) 2350-2360], Benavides et al. [T.D. Benavides, G.L. Acedo, H.K. Xu, Iterative solutions for zeros of accretive operators, Math. Nachr. 248-249 (2003) 62-71], Chen and Zhu [R. Chen, Z. Zhu, Viscosity approximation fixed points for nonexpansive and m-accretive operators, Fixed Point Theory and Appl. 2006 (2006) 1-10] and Kamimura and Takahashi [S. Kamimura, W. Takahashi, Approximation solutions of maximal monotone operators in Hilberts spaces, J. Approx. Theory 106 (2000) 226-240].     

Strong convergence of viscosity approximation methods with strong pseudocontraction for Lipschitz pseudocontractive mappings

In this paper, for a Lipschitz pseudocontractive mapping T, we study the strong convergence of iterative schemes generated by

Strong convergence theorems for strictly asymptotically pseudocontractive mappings in Hilbert spaces

The purpose of this paper is by using CSQ method to study the strong convergence problem of iterative sequences for a pair of strictly asymptotically pseudocontractive mappings to approximate a common fixed point in a Hilbert space. Under suitable conditions some strong convergence theorems are proved. The results presented in the paper are new which extend and improve some recent results of Acedo and Xu [Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal., 67(7), 2258??271 (2007)], Kim and Xu [Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal., 64, 1140??152 (2006)], Martinez-Yanes and Xu [Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal., 64, 2400??411 (2006)], Nakajo and Takahashi [Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl., 279, 372??79 (2003)], Marino and Xu [Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces. J. Math. Anal. Appl., 329(1), 336??46 (2007)], Osilike et al. [Demiclosedness principle and convergence theorems for k-strictly asymptotically pseudocontractive maps. J. Math. Anal. Appl., 326, 1334??345 (2007)], Liu [Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Anal., 26(11), 1835??842 (1996)], Osilike et al. [Fixed points of demi-contractive mappings in arbitrary Banach spaces. Panamer Math. J., 12 (2), 77??8 (2002)], Gu [The new composite implicit iteration process with errors for common fixed points of a finite family of strictly pseudocontractive mappings. J. Math. Anal. Appl., 329, 766??76 (2007)].     

Convergence of iterative algorithms for multivalued mappings in Banach spaces

We show strong and weak convergence for Mann iteration of multivalued nonexpansive mappings T$T$ in a Banach space. Furthermore, we give a strong convergence of the modified Mann iteration which is independent of the convergence of the implicit anchor-like continuous path _zt∈tu+(1−t)T_zt$z_t\in tu+(1-t)T{z}_t$.     

Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces

In this paper, we introduce two iterative sequences for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a Banach space. Then we study the strong and weak convergence of the sequences.     

Strong convergence of composite iterative schemes for a countable family of nonexpansive mappings in Banach spaces

In this paper we propose a new modified viscosity approximation method for approximating common fixed points for a countable family of nonexpansive mappings in a Banach space. We prove strong convergence theorems for a countable family nonexpansive mappings in a reflexive Banach space with uniformly Gateaux differentiable norm under some control conditions. These results improve and extend the results of Jong Soo Jung [J.S. Jung, Convergence on composite iterative schemes for nonexpansive mappings in Banach spaces, Fixed Point Theory and Appl. 2008 (2008) 14 pp., Article ID 167535]. Further, we apply our result to the problem of finding a zero of an accretive operator and extend the results of Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51-60], Ceng, et al. [L.-C. Ceng, A.R. Khan, Q.H. Ansari, J.-C, Yao, Strong convergence of composite iterative schemes for zeros of m-accretive operators in Banach space, Nonlinear Anal. 70 (2009)1830-1840] and Chen and Zhu [R. Chen, Z. Zhu, Viscosity approximation methods for accretive operator in Banach space, Nonlinear Anal. 69 (2008) 1356-1363].     

Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups

The Mann iterations for nonexpansive mappings have in general only weak convergence in a Hilbert space. We modify an iterative method of Mann's type introduced by Nakajo and Takahashi [Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379] for nonexpansive mappings and prove strong convergence of our modified Mann's iteration processes for asymptotically nonexpansive mappings and semigroups.     

Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces

We study the convergence of two iterative algorithms for finding common fixed points of finitely many Bregman strongly nonexpansive operators in reflexive Banach spaces. Both algorithms take into account possible computational errors. We establish two strong convergence theorems and then apply them to the solution of convex feasibility, variational inequality and equilibrium problems.     

Strong convergence theorems obtained by a generalized projections hybrid method for families of mappings in Banach spaces

Let C be a nonempty, closed and convex subset of a uniformly convex and smooth Banach space and let {T_n} be a family of mappings of C into itself such that the set of all common fixed points of {T_n} is nonempty. We consider a sequence {x_n} generated by the hybrid method by generalized projection in mathematical programming. We give conditions on {T_n} under which {x_n} converges strongly to a common fixed point of {T_n} and generalize the results given in [12], [14], [13] and [11].     

On the semilocal convergence of efficient Chebyshev-Secant-type methods

We introduce a three-step Chebyshev-Secant-type method (CSTM) with high efficiency index for solving nonlinear equations in a Banach space setting. We provide a semilocal convergence analysis for (CSTM) using recurrence relations. Numerical examples validating our theoretical results are also provided in this study.     

A convergence theorem for Newton-like methods in Banach spaces

Summary A convergence theorem for Newton-like methods in Banach spaces is given, which improves results of Rheinboldt [27], Dennis [4], Miel [15, 16] and Moret [18] and includes as a special case an updated (affine-invariant [6]) version of the Kantorovich theorem for the Newton method given in previous papers [35, 36]. Error bounds obtained in [34] are also improved. This paper unifies the study of finding sharp error bounds for Newton-like methods under Kantorovich type assumptions.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041 and by the Ministry of Education, Japan     

Differential inequalities and equations in Banach spaces with a cone

A class of second-order abstract dissipative evolution differential operators D$D$ with 0∈kerD$0\in kerD$ is shown for which the fact that a non-zero t?u(t)$t?u\left(t\right)$ belongs to a cone and −Du$-Du$ to a dual cone may hold only on time intervals whose length is less than or equal to a defined number. Then oscillatory functions are dealt with in the framework of Banach spaces with a cone and conditions for the existence of a uniform oscillatory time for solutions of the equation Du=0$Du=0$ are given.     

Strong convergence theorems for an infinite family of nonexpansive mappings in Banach spaces

In an infinite-dimensional Hilbert space, the normal Mann’s iteration algorithm has only weak convergence, in general, even for nonexpansive mappings. In order to get a strong convergence result, we modify the normal Mann’s iterative process for an infinite family of nonexpansive mappings in the framework of Banach spaces. Our results improve and extend the recent results announced by many others.     

Existence of solutions for generalized variational inequality problems in Banach spaces

In this paper, we prove the existence of solutions of generalized variational inequality for upper semicontinuous multivalued mappings with compact contractible values over compact convex subsets in a reflexive Banach space with a Fréchet differentiable norm. Moreover, we give some conditions that guarantee the existence of solutions of generalized variational inequality for upper semicontinuous multivalued mappings with compact contractible values over unbounded closed convex subsets. The result obtained in this paper improves and extends the recent ones announced by Yu and Yang [J. Yu, H. Yang, Existence of solutions for generalized variational inequality problems, Nonlinear Anal., 71 (2009) e2327-e2330] and many others.     

Some new extragradient iterative methods for variational inequalities

In this paper, we suggest and analyze some new extragradient iterative methods for finding the common element of the fixed points of a nonexpansive mapping and the solution set of the variational inequality for an inverse strongly monotone mapping in a Hilbert space. We also consider the strong convergence of the proposed method under some mild conditions. Several special cases are also discussed. Results proved in this paper may be viewed as improvement and refinement of the previously known results.