Nonlinear Operator Theory and Its ApplicationsView this Special Issue
Editorial | Open Access
Juan Martinez-Moreno, Dhananjay Gopal, Vijay Gupta, Edixon Rojas, Satish Shukla, "Nonlinear Operator Theory and Its Applications", Journal of Function Spaces, vol. 2018, Article ID 9713872, 2 pages, 2018. https://doi.org/10.1155/2018/9713872
Nonlinear Operator Theory and Its Applications
Nonlinear operator theory falls within the general area of nonlinear functional analysis, an area which has been of increasing research interest in recent years. Nonlinear operator theory applies to diverse nonlinear problems in many areas such as differential equations, nonlinear ergodic theory, game theory, optimization problems, control theory, variational inequality problems, equilibrium problems, and split feasibility problems.
This special issue reflects both the state-of-the-art theoretical research and important recent advances in applications.
Concerning this special issue, ten papers have been accepted and published with twenty-five different authors. Five manuscripts come from China with fourteen authors. Other papers come from Chile, Saudi Arabia, Turkey, Japan, and Poland.
The selected and published papers are the following items.
One paper proposes stochastic convex semidefinite programs (SCSDPs) to handle uncertain data in applications. For these models, S. Chen et al. design an efficient inexact stochastic approximation (SA) method and prove the convergence, complexity, and robust treatment of the algorithm and apply it for solving SCSDPs where the subproblem in each iteration is only solved approximately and show that it enjoys the similar iteration complexity as the exact counterpart if the subproblems are progressively solved to sufficient accuracy.
Another paper extends a number of existing results on $b$-metric spaces. For it, an existence and uniqueness of new contractive operators combining admissible and simulation functions are proved for complete $b$-metric spaces by A. S.S. Alharbi et al.
The Monge-Ampère equations are a type of important fully nonlinear elliptic equations. In the third paper, W. Shen establishes the global bifurcation results from the trivial solutions axis and from infinity for some Monge-Ampère equations and some applications are given.
The main aim of the fourth paper is to investigate the Mobius gyrovector spaces which are open balls centered at the origin in a real Hilbert space with the Mobius addition, the Mobius scalar multiplication, and the Poincaré metric introduced by Ungar. In particular, for an arbitrary point, K. Watanabe obtains the unique closest point in any closed gyrovector subspace, by using the ordinary orthogonal decomposition and shows that each element has the orthogonal gyroexpansion with respect to any orthogonal basis in a Mobius gyrovector space. Finally, a concrete procedure to calculate the gyrocoefficients of the orthogonal gyroexpansion is presented.
One of the papers studies a nonlocal fourth-order elliptic equation of Kirchhoff type with dependence on the gradient and Laplacian. Y. Ru et al. show that there exists a $>0$ such that the problem has a nontrivial solution for some cases through an iterative method based on the mountain pass lemma and truncation method previously developed by Figuereido, Girard, and Matzeu.
A paper also studies fixed-point results in the setting of $b$-metric spaces. In this case, E. Karapinar et al. present generalized $(,)$-Meir-Keeler type contractions and, for them, establish a fixed-point result that improves, generalizes, and unifies many existing famous results in the corresponding literature. Two examples are presented to illustrate main results.
In another paper, by using two fixed-point theorems on cone, Q. Sun et al. discuss the existence results of positive solutions for a boundary value problem of fractional differential equation with integral boundary conditions.
The purpose of T. Xiong et al. in one of the papers is to introduce and study a class of new two-step viscosity iteration approximation methods for finding fixed points of set-valued nonexpansive mappings in $ CAT(0)}$ Spaces. By means of some properties and characteristic to $ CAT(0)}$ Spaces, and using Cauchy-Schwarz inequality and Xu’s inequality, strong convergence theorems of the new two-step viscosity iterative process for set-valued nonexpansive and contraction operators in complete $ CAT(0)}$ Spaces are provided.
Another paper’s author, Tomonari Suzuki, by introducing the concept of $$-semicompleteness in semimetric spaces, extends Caristi’s fixed-point theorem to $$-semicomplete semimetric spaces. Via this extension, $$-semicompleteness is characterized and Banach contraction principle generalized.
In one of the papers, the existence and uniqueness of weak solutions for the boundary value problem modelling the stationary case of the bioconvective low problem are proved. The bioconvective model is a boundary value problem for a system of four equations: the nonlinear Stokes equation, the incompressibility equation, and two transport equations. The unknowns of the model are the velocity of the fluid, the pressure of the fluid, the local concentration of microorganisms, and the oxygen concentration. A. Coronel et al. derive some appropriate a priori estimates for the weak solution, which implies the existence, by application of Gossez theorem, and the uniqueness by standard methodology of comparison of two arbitrary solutions.
Conflicts of Interest
As Guest Editorial team of special issue named “Nonlinear Operator Theory and Its Applications” in Journal of Function Spaces, we declare that there are no conflicts of interest or private agreements with companies regarding our work for this special issue. We have no financial relationships through employment and consultancies, either stock ownership or honoraria, with industry.
We want to thank all the authors of these works, which provide a wide view of some of the most recent topics in the field. Also, we acknowledge with thanks the work done by the reviewers who collaborated to make this special issue possible. Our gratitude goes also to the editors of the journal for the support and help with the preparation of this special issue.
Copyright © 2018 Juan Martinez-Moreno 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.