editorial iterative methods and applications 2014 · 2019. 7. 31. · lipschitz strongly...

EditorialIterative Methods and Applications 2014

Giuseppe Marino,1 Filomena Cianciaruso,1 Claudio H. Morales,2

Luigi Muglia,1 and D. R. Sahu3

1Dipartimento di Matematica, Universita della Calabria, Arcavacata, 87036 Rende, Italy2Department of Mathematics, University of Alabama in Huntsville, USA3Department of Mathematics, Banaras Hindu University, Varanasi, India

Correspondence should be addressed to Giuseppe Marino; [email protected]

Received 17 February 2015; Accepted 17 February 2015

Copyright © 2015 Giuseppe Marino et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

An important branch of nonlinear analysis theory, applied inthe study of nonlinear phenomena, in engineering, physics,and life sciences, is related to the existence of fixed points ofnonlinear mappings, to the approximation of fixed points ofnonlinear operators and of zeros of nonlinear operators, andto the approximation of solutions of variational inequalities.

This special issue is focused on the latest achievements onthese topics and the related applications.

The aim is to present newest and extended coverage ofthe fundamental ideas, concepts, and important results onthe topics as follows: (i) new iterative schemes to approximatefixed points of nonlinear mappings, common fixed points ofnonlinear mappings, or semigroups of nonlinear mappings,(ii) iterative approximations of zeros of accretive-type opera-tors, (iii) iterative approximations of solutions of variationalinequalities problems or split feasibility problems and appli-cations, (iv) optimization problems and their algorithmicapproaches, (v) methods for the global continuation of fixedpoint curves in engineering problems, and (vi) fixed point ofnonlinear operators in cone metric spaces with applicationsand fixed points of nonlinear operators in ordered metricspaces with applications.

We invited the authors to present their original articlesthat will stimulate the continuing efforts in developing newresults in the previously mentioned areas.

F. Zhang et al. introduce a new iterative scheme forfinding a common fixed point of two countable families ofmultivalued quasi-nonexpansive mappings and prove a weakconvergence theorem under the suitable control conditionsin a uniformly convex Banach space.

S. M. Kang et al. establish the strong convergence for thehybrid 𝑆-iterative scheme associated with nonexpansive andLipschitz strongly pseudocontractive mappings in Banachspaces.

R. Wangkeeree and P. Yimmuang first consider an aux-iliary problem for the generalized mixed vector equilibriumproblem with a relaxed monotone mapping and prove theexistence and uniqueness of the solution for the auxiliaryproblem. Then they introduce a new iterative scheme forapproximating a common element of the set of solutions of ageneralizedmixed vector equilibrium problemwith a relaxedmonotone mapping and the set of common fixed points of acountable family of nonexpansive mappings.

Y. Tang et al. prove strong convergence theorems for acommon element of the set of fixed points of a finite familyof pseudocontractive mappings and the set of solutions of afinite family of monotonemappings.The common element isthe unique solution of a certain variational inequality.

L.-C.Ceng et al. introduce a new relaxed viscosity approx-imation method with regularization and prove the strongconvergence of the method to a common fixed point offinitely many nonexpansive mappings and a strict pseudo-contraction that also solves a convex minimization problemand a suitable equilibrium problem.

Recently, S. Takahashi and W. Takahashi proposed aniterative algorithm for finding common solutions of gener-alized equilibrium problems governed by inverse stronglymonotone mappings and of fixed points problems for nonex-pansivemappings. H. Zhang and F.Wang provide a result that

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2015, Article ID 207976, 2 pageshttp://dx.doi.org/10.1155/2015/207976

2 Journal of Applied Mathematics

allows for the removal of one condition ensuring the strongconvergence of the algorithm.

S. Cao introduces and analyzes the viscosity approxi-mation algorithm for solving the split common fixed pointproblem for the strictly pseudononspreading mappings inHilbert spaces.

Y.Wang presents a viscositymethod for hierarchical fixedpoint problems to solve variational inequalities, where theinvolved mappings are nonexpansive nonself-mappings.

H. Zegeye and N. Shahzad introduce an iterative pro-cess which strongly converges to a common fixed point offinite family of uniformly continuous asymptotically 𝑘

𝑖-strict

pseudocontractive mappings in the intermediate sense for𝑖 = 1, 2, . . . , 𝑁.

C.-X. Li et al. present some comparison theorems for thespectral radius of double splittings of differentmatrices undersuitable conditions, which are superior to the correspondingresults in the recent papers.

P. Duan proposes an implicit iterative scheme and anexplicit iterative scheme for finding a common element ofthe set of solutions of system of equilibrium problems anda constrained convex minimization problem by the generaliterative methods in Hilbert spaces.

H. Sun and Z. Xue demonstrate that the proof of maintheorem of B. E. Rhoades and Stefan M. Soltuz in J. Math.Anal.Appl. 283 (2003), 681–688, is incorrect. The authorsprovide the correct version of this result concerning theequivalence between the convergences of Ishikawa andManniterations for uniformly 𝐿-Lipschitzian asymptotically pseu-docontractive maps.

S. M. Kang et al. study the convergence of implicitPicard iterative sequences for strongly accretive and stronglypseudocontractive mappings.

G. Chiaselotti et al. use a discrete dynamical model withthree evolution rules in order to analyze the structure of apartially ordered set of signed integer partitions whose mainproperties are actually not known.Thismodel is related to thestudy of some extremal combinatorial sum problems.

Applying the Taylor and multiplication rule of two gen-eralized polynomials, H.-K. Liu develops a series solution oflinear homogeneous 𝑞-difference equations.

X. Mu and Y. Zhang, among the others, present a rank-two feasible direction algorithm based on the semidefi-nite programming relaxation of the binary quadratic pro-gramming. The proposed algorithm restricts the rank ofmatrix variable to be two in the semidefinite programmingrelaxation and yields a quadratic objective function withsimple quadratic constraints. Moreover a feasible directionalgorithm is used to solve the nonlinear programming andthe convergent analysis and time complexity of the method isgiven.

M. De la Sen investigates the limit properties of distancesand the existence and uniqueness of fixed points, best prox-imity points, and existence and uniqueness of limit cycles, towhich the iterated sequences converge, and of single-valued,and so-called, contractive precyclic self-mappings.

X.-F. Zhang et al. propose three kinds of preconditionersto accelerate the generalized AOR (GAOR) method for the

linear system from the generalized least squares problem.Theconvergence and comparison results are also obtained.

L. Bergamaschi and A. Martınez propose a parallelpreconditioner for the Newtonmethod in the computation ofthe leftmost eigenpairs of large and sparse symmetric positivedefinite matrices.

J. Park et al. apply a numerical procedure introducedby Choi and Jang in 2012 for the numerical analyzing ofstatic deflection of an infinite beam on a nonlinear elasticfoundation.

G. Chiaselotti et al. introduce the concept of fundamentalsequence for a finite graded poset 𝑋 which is also a discretedynamical model.

J. Iqbal and M. Arif study symmetric successive over-relaxation (SSOR) method for absolute complementarityproblems.

Acknowledgments

The editors would like to thank the authors for their interest-ing contributions, as well as the Staff and the Editorial Officeof the Journal for their valuable support.

Giuseppe MarinoFilomena CianciarusoClaudio H. Morales

Luigi MugliaD. R. Sahu

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