Best proximity point theorems: An exploration of a common solution to approximation and optimization problems

Abstract Given a non-self mapping T : A → B in the setting of a metric space, this work concentrates on the resolution of the non-linear programming problem of globally minimizing the real valued function x → d ( x , Tx ) , thereby yielding an optimal approximate solution to the equation Tx = x . An iterative algorithm is also presented to compute a solution of such problems. As a sequel, it is possible to compute an optimal approximate solution to some non-linear equations.

[1]  S. Sadiq Basha,et al.  Extensions of Banach's Contraction Principle , 2010 .

[2]  P. Veeramani,et al.  Proximal pointwise contraction , 2009 .

[3]  P. Pardalos,et al.  State of the art in global optimization: computational methods and applications , 1996 .

[4]  Calogero Vetro,et al.  Best proximity points: Convergence and existence theorems for p-cyclic mappings , 2010 .

[5]  Existence and uniqueness of best proximity points in geodesic metric spaces , 2010 .

[6]  A. Anthony Eldred,et al.  Existence and convergence of best proximity points , 2006 .

[7]  S. Sadiq Basha,et al.  Best Proximity Pair Theorems for Multifunctions with Open Fibres , 2000 .

[8]  W. A. Kirk,et al.  Proximinal Retracts and Best Proximity Pair Theorems , 2003 .

[9]  S. Sadiq Basha,et al.  Best proximity pair theorems , 2001 .

[10]  Robert Plebaniak,et al.  Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces , 2009 .

[11]  Robert Plebaniak,et al.  Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces , 2010 .

[12]  S. Sadiq Basha,et al.  Best proximity points: global optimal approximate solutions , 2011, J. Glob. Optim..

[13]  Best proximity pair theorems for relatively nonexpansive mappings , 2009 .

[14]  Panos M. Pardalos,et al.  Frontiers in Global Optimization , 2012 .

[15]  Robert Plebaniak,et al.  Erratum to: “Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces” [Nonlinear Anal. (2008), doi: 10.1016/j.na.2008.04.037] , 2009 .

[16]  M. Sen.  Fixed Point and Best Proximity Theorems under Two Classes of Integral-Type Contractive Conditions in Uniform Metric Spaces , 2010 .

[17]  Panos M. Pardalos,et al.  State of the Art in Global Optimization , 1996 .

[18]  B. Piatek.  On cyclic Meir–Keeler contractions in metric spaces , 2011 .

[19]  M. de La Sen.  Fixed Point and Best Proximity Theorems under Two Classes of Integral-Type Contractive Conditions in Uniform Metric Spaces , 2010 .

[20]  B. Rhoades,et al.  Some theorems on weakly contractive maps , 2001 .

[21]  Naseer Shahzad,et al.  Common Best Proximity Points: Global Optimal Solutions , 2011, J. Optim. Theory Appl..

[22]  William A. Kirk,et al.  Proximal normal structure and relatively nonexpansive mappings , 2005 .

[23]  Sushama Agrawal,et al.  Best Proximity Point Theorems for p-Cyclic Meir-Keeler Contractions , 2009 .

[24]  V. Sankar Raj,et al.  A best proximity point theorem for weakly contractive non-self-mappings , 2011 .

[25]  Panos M. Pardalos,et al.  Encyclopedia of Optimization , 2006 .

[26]  Naseer Shahzad,et al.  Best proximity pairs and equilibrium pairs for Kakutani multimaps , 2009 .

[27]  Naseer Shahzad,et al.  Best Proximity Sets and Equilibrium Pairs for a Finite Family of Multimaps , 2008 .

[28]  Sangho Kum,et al.  On general best proximity pairs and equilibrium pairs in free abstract economies , 2008 .

[29]  Calogero Vetro,et al.  Best proximity points for cyclic Meir–Keeler contractions , 2008 .

[30]  Naseer Shahzad,et al.  Common best proximity points: Global optimization of multi- objective functions , 2011, Appl. Math. Lett..

[31]  Naseer Shahzad,et al.  Convergence and existence results for best proximity points , 2009 .