editorial advance in nonlinear analysis: algorithm...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013, Article ID 154975, 2 pageshttp://dx.doi.org/10.1155/2013/154975

EditorialAdvance in Nonlinear Analysis: Algorithm, Convergence,and Applications

Yisheng Song,1,2 Rudong Chen,3 Guoyin Li,4 and Changsen Yang1

1 College of Mathematics and Information Science, Henan Normal University, Henan 453007, China2Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong3Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China4Department of Applied Mathematics, University of New South Wales, Sydney, NSW 2052, Australia

Correspondence should be addressed to Yisheng Song; [email protected]

Received 15 September 2013; Accepted 15 September 2013

Copyright © 2013 Yisheng Song et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We are glad to achieve this special issue. This special issuewas opened in late November of 2012 and closed in earlyApril of 2013. There were 72 submissions in total and 27 ofthemwere accepted for publication after strict reviews, whichgave important developments in nonlinear analysis and itsapplications. The accepted rate of papers is 37.5 percent.

The topics of the accepted 27 articles cover the area fromtheory to real applications. This special issue mainly concen-trates on some latest developments in nonlinear analysis andits applications with a particular emphasis. With the help oflinear and nonlinear functional analysis, various algorithmsand their analyzed techniques of convergence presented somenewmethods of solving some nonlinear problems.Moreover,some of them gave real applications and numerical examples.These are our main aims of sponsoring this special issue.

The article by Y. Ren et al. studies the common fixed pointtheorems for nonlinear contractions in dislocated metricspaces. The article by F. Gu and Y. Yin proved a commonfixed point theorem for three pairs of weakly compatibleself-maps satisfying a new contractive condition in theframework of a generalized metric space by means of thecommon (E, A) property.The article by M. De la Sen showedfixed points and best proximity points of a class of the so-called generalized point-dependent (𝐾, 𝜆)-hybrid 𝑝-cyclicself-mappings relative to a Bregman distance and obtainedthe weak convergence results of certain average sequences.

The article by Y. Dang and Y. Gao proposed an inertialiterative algorithm for solving the split common fixed-point problem for quasinonexpansive mappings and provedthe asymptotical convergence of the algorithm under some

suitable conditions.The article byN.Hussain et al. introduceda new iterative scheme called Jungck-CR iterative schemeand studied the stability and strong convergence of thisiterative scheme for a pair of nonself-mappings. The articleby S. Wang and S. M. Kang introduced the concept ofBregman asymptotically quasinonexpansive mappings andconstructed an iterative scheme to find a common elementof the set of solutions of an equilibrium problem and theset of common fixed points of a countable family of Breg-man asymptotically quasinonexpansivemappings in reflexiveBanach spaces. The article by Y. Wang and W. Xuan somestrong convergence theorems for a common fixed point of afinite family of relatively nonexpansive mappings by using anew hybrid iterative method in mathematical programmingand the generalized projection method in a Banach space.

The article by H. Zegeye and N. Shahzad introducedan iterative process and proved that the iterative sequenceconverged strongly to solutions of a certain variationalinequity problem for 𝜂-inverse strongly accretive mappingsin the set of common fixed points of finite family of strictlypseudocontractive mappings in Banach spaces. The otherarticle by the same authors studied an iterative process whichconverges strongly to a zero of a finite sum of monotonemappings. The article by H. Du obtained the strong conver-gence of modified Mann-Halpern algorithms for finding thefixed points of pseudocontractive mappings. The article byJ. S. Jung showed strong convergence of an iterative sequencedefined by a continuous pseudocontractive mapping and acontinuous bounded strongly pseudocontractive mapping.

2 Abstract and Applied Analysis

The article by Y.-Y. Chen and S.-Q. Du considered threekinds of nonlinear conjugate gradient methods with Wolfetype line search for unstrained optimization problems andshowed the global convergence results and the numericalresults. The other article by the same authors continued tostudy a new smoothing nonlinear conjugate gradient methodfor nonsmooth equations with finitely manymaximum func-tions. The article by Y. Qiu et al. presented a hybrid spectralconjugate gradient method for large-scale unconstrainedoptimization, which possesses a self-adjusting property andestablished global convergence result under the standardWolfe conditions. The article by G. Yu et al. combined mul-tivariate spectral gradient method with projection scheme topresent an adaptive prediction-correctionmethod for solvinglarge-scale nonlinear systems of monotone equations.

The article by M. F. Abad et al. considered two iterativemethods of order four and five for solving nonlinear systemsof equations and made the numerical comparisons withother existing second- and fourth-order schemes to solvethe nonlinear system of equations of the Global PositioningSystem and some academic nonlinear systems. The articleby C. Mu et al. dealt with the extinction and nonextinctionproperties of the fast diffusion equation of homogeneousDirichlet boundary condition and showed that the solutioneither extinct or nonextinct in finite time depends strongly onthe initial data and the first eigenvalue the diffusion equationof homogeneous Dirichlet boundary condition.

The article by Z. Wang et al. studied an autonomy systemwith time-delayed feedback by using the theory of functionaldifferential equation andHassard’s method and gave the con-ditions onwhich zero equilibriumexists andHopf bifurcationoccurs. The article by J. Banas and S. Dudek considered thesolvability of some nonlinear functional integral equations inthe Banach algebra of real functions that defined, continuous,and bounded on the real half-axis.The article by H. Yuan andCheng Song introduced the stability and convergence of two-step Runge-Kutta methods with compound quadrature for-mula for solving nonlinear Volterra delay integrodifferentialequations.

The article by R. Chen et al. reported the approximatesplit equality problem (ASEP) in the framework of infinite-dimensional Hilbert spaces, used the regularization methodto establish a single-step iteration for solving the ASEP,and showed that the sequence generated by such algorithmstrongly converges to the minimum-norm solution of theASEP.The article by L. Y. Shi et al. provedΔ-convergence the-orems of iterative sequences for asymptotically nonexpansivemappings in CAT(0) spaces.

The article by M. Senol et al. applied the previouslydeveloped new perturbation-iteration algorithm to differen-tial equation systems and gave an iteration algorithm. Thearticle by D. Li et al. studied the existence and monotoneiteration of solutions for a third-order four-point boundaryvalue problem with 𝑝-Laplacian.The article by G. Dong et al.proved the existence, uniqueness, and comparison principleof the minimizer for the variational problem and establishedthe existence, uniqueness, and long-time behavior of theassociated evolution equation.

The article by C. Yang and Y. Liu discussed some operatorinequalities on chaotic order about several operators, whichare generalization of Furuta inequality and showmonotonic-ity of related Furuta type operator function. The article byX. Hu and S. Zhang presented a new extension of Serrin’slower semicontinuity theorem and proved that the variationalfunctional defined on 𝑊1,1loc (Ω) is lower semicontinuous withrespect to the strong convergence.

Yisheng SongRudong Chen

Guoyin LiChangsen Yang

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