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Many of our daily-life problems can be written in the form of an optimization problem. Therefore, solution methods are needed to solve such problems. Due to the complexity of the problems, it is not always easy to find the exact solution. However, approximate solutions can be found. The theory of the best approximation is applicable in a variety of problems arising in nonlinear functional analysis and optimization. This book highlights interesting aspects of nonlinear analysis and optimization together with many applications in the areas of physical and social sciences including engineering. It is immensely helpful for young graduates and researchers who are pursuing research in this field, as it provides abundant research resources for researchers and post-doctoral fellows. This will be a valuable addition to the library of anyone who works in the field of applied mathematics, economics and engineering.



Best Proximity Points

Ky Fan’s best approximation theorems, best proximity pair theorems, and best proximity point theorems have been studied in the literature when the fixed point equation \(Tx = x\) does not admit a solution. This chapter contains some basic results on best proximity points of cyclic contractions and relatively nonexpansive maps. An application of a best proximity point theorem to a system of differential equations has been discussed. Though it is not possible to include all the available interesting results in best proximity points, an attempt has been made to introduce some results involving best proximity points and references of related work have been indicated.
P. Veeramani, S. Rajesh

Semi-continuity Properties of Metric Projections

This chapter presents some selected results regarding semi-continuity of metric projections onto closed subspaces of normed linear spaces. Though there are several significant results relevant to this topic, only a limited coverage of the results is undertaken, as an extensive survey is beyond our scope. This exposition is divided into three parts. The first one deals with results from finite dimensional normed linear spaces. The second one deals with results connecting semi-continuity of metric projection maps and duality maps. The third one deals with subspaces of finite codimension of infinite dimensional normed linear spaces.
V. Indumathi

Convergence of Slices, Geometric Aspects in Banach Spaces and Proximinality

Some geometric properties of Banach spaces and proximinality properties in best approximation theory are characterized in terms of convergence of slices. The paper begins with some basic geometric properties of Banach spaces involving slices and their geometric interpretations. Two notions of convergence of sequence sets, called Vietoris convergence and Hausdorff convergence, with their characterizations are presented. It is observed that geometric properties such as uniform convexity, strong convexity, Radon-Riesz property, and strong subdifferentiability of the norm can be characterized in terms of the convergence of slices with respect to the notions mentioned above. Proximinality properties such as approximative compactness and strong proximinality of closed convex subsets of a Banach space are also characterized in terms of convergence of slices.
P. Shunmugaraj

Measures of Noncompactness and Well-Posed Minimization Problems

This chapter presents facts concerning the theory of well-posed minimization problems. We recall some classical results obtained in the framework of the theory but focus mainly on the detailed presentation of the application of the theory of measures of noncompactness to investigations of the well-posedness of minimization problem.
Józef Banaś

Well-Posedness, Regularization, and Viscosity Solutions of Minimization Problems

This chapter is divided into two parts. The first part surveys some classical notions for well-posedness of minimization problems. The main aim here is to synthesize some known results in approximation theory for best approximants, restricted Chebyshev centers, and prox points from the perspective of well-posedness of these problems. The second part reviews Tikhonov regularization of ill-posed problems. This leads us to revisit the so-called viscosity methods for minimization problems using the modern approach of variational convergence. Lastly, some of these results are particularized to convex minimization problems, and also to ill-posed inverse problems.
D. V. Pai

Best Approximation in Nonlinear Functional Analysis

An introduction to best approximation theory and fixed point theory are presented. Several known fixed point theorems are given. Ky Fan’s best approximation is studied in detail. The study of approximating sequences followed by convergence of the sequence of iterative process is studied. An introduction to variational inequalities is also presented.
S. P. Singh, M. R. Singh

Hierarchical Minimization Problems and Applications

In this chapter, several iterative methods for solving fixed point problems, variational inequalities, and zeros of monotone operators are presented. A generalized mixed equilibrium problem is considered. The hierarchical minimization problem over the set of intersection of fixed points of a mapping and the set of solutions of a generalized mixed equilibrium problem is considered. A new unified hybrid steepest descent-like iterative algorithm for finding a common solution of a generalized mixed equilibrium problem and a common fixed point problem of uncountable family of nonexpansive mappings is presented and analyzed.
D. R. Sahu, Qamrul Hasan Ansari

Triple Hierarchical Variational Inequalities

In this chapter, we give a survey on hierarchical variational inequality problems and triple hierarchical variational inequality problems. By combining hybrid steepest descent method, Mann’s iteration method, and projection method, we present a hybrid iterative algorithm for computing a fixed point of a pseudo-contractive mapping and for finding a solution of a triple hierarchical variational inequality in the setting of real Hilbert space. We prove that the sequence generated by the proposed algorithm converges strongly to a fixed point which is also a solution of this triple hierarchical variational inequality problem. On the other hand, we also propose another hybrid iterative algorithm for solving a class of triple hierarchical variational inequality problems concerning a finite family of pseudo-contractive mappings in the setting of real Hilbert spaces. Under very appropriate conditions, we derive the strong convergence of the proposed algorithm to the unique solution of this class of problems.
Qamrul Hasan Ansari, Lu-Chuan Ceng, Himanshu Gupta

Split Feasibility and Fixed Point Problems

In this survey article, we present an introduction of split feasibility problems, multisets split feasibility problems and fixed point problems. The split feasibility problems and multisets split feasibility problems are described. Several solution methods, namely, CQ methods, relaxed CQ method, modified CQ method, modified relaxed CQ method, improved relaxed CQ method are presented for these two problems. Mann-type iterative methods are given for finding the common solution of a split feasibility problem and a fixed point problem. Some methods and results are illustrated by examples.
Qamrul Hasan Ansari, Aisha Rehan

Isotone Projection Cones and Nonlinear Complementarity Problems

A brief introduction of complementarity problems is given. We discuss the notion of *-isotone projection cones and analyze how large is the class of these cones. We show that each generating *-isotone projection cone is superdual. We prove that a simplicial cone in \(R^{m}\) is *-isotone projection cone if and only if it is coisotone (i.e., it is the dual of an isotone projection cone. We consider the solvability of complementarity problems defined by *-isotone projection cones. The problem of finding nonzero solution of these problems is also presented.
M. Abbas, S. Z. Németh


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