zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Best proximity point theorems: an exploration of a common solution to approximation and optimization problems. (English) Zbl 1245.90120
Summary: Given a non-self mapping $T:A\to B$ 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 $x\to d(x,Tx)$, 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.

MSC:
90C30Nonlinear programming
65K05Mathematical programming (numerical methods)
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
WorldCat.org
Full Text: DOI
References:
[1] Al-Thagafi, M. A.; Shahzad, N.: Best proximity sets and equilibrium pairs for a finite family of multimaps, Fixed point theor. Appl., 10 (2008) · Zbl 1169.47040 · doi:10.1155/2008/457069
[2] Al-Thagafi, M. A.; Shahzad, N.: Convergence and existence results for best proximity points, Nonlinear anal. 70, 3665-3671 (2009) · Zbl 1197.47067 · doi:10.1016/j.na.2008.07.022
[3] Al-Thagafi, M. A.; Shahzad, N.: Best proximity pairs and equilibrium pairs for Kakutani multimaps, Nonlinear anal. 70, 1209-1216 (2009) · Zbl 1225.47056 · doi:10.1016/j.na.2008.02.004
[4] Anuradha, J.; Veeramani, P.: Proximal pointwise contraction, Topol. appl. 156, 2942-2948 (2009) · Zbl 1180.47035 · doi:10.1016/j.topol.2009.01.017
[5] Cuenya, H. H.; Bonifacio, A. G.: Best proximity pairs in uniformly convex spaces, Bull. inst. Math. acad. Sin. (N.S.) 3, 391-398 (2008) · Zbl 1163.47042
[6] De La Sen, M.: Fixed point and best proximity theorems under two classes of integral-type contractive conditions in uniform metric spaces, Fixed point theor. Appl., 12 (2010) · Zbl 1206.54041 · doi:10.1155/2010/510974
[7] Di Bari, C.; Suzuki, T.; Vetro, C.: Best proximity points for cyclic Meir -- Keeler contractions, Nonlinear anal. 69, 3790-3794 (2008) · Zbl 1169.54021 · doi:10.1016/j.na.2007.10.014
[8] Eldred, A. A.; Veeramani, P.: Existence and convergence of best proximity points, J. math. Anal. appl. 323, 1001-1006 (2006) · Zbl 1105.54021 · doi:10.1016/j.jmaa.2005.10.081
[9] Eldred, A. A.; Kirk, W. A.; Veeramani, P.: Proximinal normal structure and relatively nonexpansive mappings, Studia math. 171, 283-293 (2005) · Zbl 1078.47013 · doi:10.4064/sm171-3-5
[10] Fernández-León, A.: Existence and uniqueness of best proximity points in geodesic metric spaces, Nonlinear anal. 73, 915-921 (2010) · Zbl 1196.54050 · doi:10.1016/j.na.2010.04.005
[11] Floudas, C. A.; Pardalos, P. M.: State of the art in global optimization: computational methods and applications, (1996) · Zbl 0847.00058
[12] Floudas, C. A.; Pardalos, P. M.: Frontiers in global optimization, (2003) · Zbl 1031.90001
[13] Floudas, C. A.; Pardalos, P. M.: Encyclopedia of optimization, (2009) · Zbl 1156.90001
[14] Karpagam, S.; Agrawal, S.: Best proximity point theorems for p-cyclic Meir -- Keeler contractions, Fixed point theor. Appl., 9 (2009) · Zbl 1172.54028 · doi:10.1155/2009/197308
[15] Kim, W. K.; Kum, S.; Lee, K. H.: On general best proximity pairs and equilibrium pairs in free abstract economies, Nonlinear anal. 68, 2216-2222 (2008) · Zbl 1136.91309 · doi:10.1016/j.na.2007.01.057
[16] Kirk, W. A.; Reich, S.; Veeramani, P.: Proximinal retracts and best proximity pair theorems, Numer. funct. Anal. optim. 24, 851-862 (2003) · Zbl 1054.47040 · doi:10.1081/NFA-120026380
[17] Piatek, B.: On cyclic Meir -- Keeler contractions in metric spaces, Nonlinear anal. 74, 35-40 (2011) · Zbl 1218.54047 · doi:10.1016/j.na.2010.08.010
[18] Rhoades, B. E.: Some theorems on weakly contractive maps, Nonlinear anal. 47, 2683-2693 (2001) · Zbl 1042.47521 · doi:10.1016/S0362-546X(01)00388-1
[19] Basha, S. Sadiq: Extensions of Banach’s contraction principle, Numer. funct. Anal. optim. 31, 569-576 (2010) · Zbl 1200.54021 · doi:10.1080/01630563.2010.485713
[20] Basha, S. Sadiq: Best proximity points: global optimal approximate solutions, J. global optim. 49, 15-21 (2011) · Zbl 1208.90128 · doi:10.1007/s10898-009-9521-0
[21] Basha, S. Sadiq; Shahzad, N.; Jeyaraj, R.: Common best proximity points: global optimization of multi-objective functions, Appl. math. Lett. 24, 883-886 (2011) · Zbl 1281.54018
[22] Basha, S. Sadiq; Veeramani, P.: Best approximations and best proximity pairs, Acta. sci. Math. (Szeged) 63, 289-300 (1997) · Zbl 0909.47042
[23] Basha, S. Sadiq; Veeramani, P.: Best proximity pair theorems for multifunctions with open fibres, J. approx. Theor. 103, 119-129 (2000) · Zbl 0965.41020 · doi:10.1006/jath.1999.3415
[24] Basha, S. Sadiq; Veeramani, P.; Pai, D. V.: Best proximity pair theorems, Ind. J. Pure appl. Math. 32, 1237-1246 (2001) · Zbl 1021.47027
[25] Raj, V. Sankar: A best proximity point theorem for weakly contractive non-self-mappings, Nonlinear anal. 74, 4804-4808 (2011) · Zbl 1228.54046 · doi:10.1016/j.na.2011.04.052
[26] Raj, V. Sankar; Veeramani, P.: Best proximity pair theorems for relatively nonexpansive mappings, Appl. gen. Topol. 10, 21-28 (2009) · Zbl 1213.47062 · http://agt.webs.upv.es/Volumes/V10N1/AGTV10N121.pdf
[27] Shahzad, N.; Basha, S. Sadiq; Jeyaraj, R.: Common best proximity points: global optimal solutions, J. optim. Theor. appl. 148, 69-78 (2011) · Zbl 1207.90096 · doi:10.1007/s10957-010-9745-7
[28] Srinivasan, P. S.: Best proximity pair theorems, Acta sci. Math. (Szeged) 67, 421-429 (2001) · Zbl 1012.47031
[29] Vetro, C.: Best proximity points: convergence and existence theorems for p-cyclic mappings, Nonlinear anal. 73, 2283-2291 (2010) · Zbl 1229.54066 · doi:10.1016/j.na.2010.06.008
[30] Wlodarczyk, K.; Plebaniak, R.; Banach, A.: Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces, Nonlinear anal. 70, 3332-3341 (2009) · Zbl 1182.54024 · doi:10.1016/j.na.2008.11.020
[31] Wlodarczyk, K.; Plebaniak, R.; Banach, A.: Erratum to: best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces, Nonlinear anal. 71, 3583-3586 (2009) · Zbl 1171.54311 · doi:10.1016/j.na.2008.11.020
[32] Wlodarczyk, K.; Plebaniak, R.; Obczynski, C.: Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces, Nonlinear anal. 72, 794-805 (2010) · Zbl 1185.54020 · doi:10.1016/j.na.2009.07.024