# 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 points: Global optimal approximate solutions. (English) Zbl 1208.90128
Summary: Let $A$ and $B$ be non-empty subsets of a metric space. As a non-self mapping ${T : A \longrightarrow B}$ does not necessarily have a fixed point, it is of considerable interest to find an element $x$ in $A$ that is as close to $Tx$ in $B$ as possible. In other words, if the fixed point equation $Tx = x$ has no exact solution, then it is contemplated to find an approximate solution $x$ in $A$ such that the error $d(x, Tx)$ is minimum, where $d$ is the distance function. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, called best proximity points, to the fixed point equation $Tx = x$ when there is no exact solution. As the distance between any element $x$ in $A$ and its image $Tx$ in $B$ is at least the distance between the sets $A$ and $B$, a best proximity pair theorem achieves global minimum of $d(x, Tx)$ by stipulating an approximate solution $x$ of the fixed point equation $Tx = x$ to satisfy the condition that $d(x, Tx) = d(A, B)$. The purpose of this article is to establish best proximity point theorems for contractive non-self mappings, yielding global optimal approximate solutions of certain fixed point equations. Besides establishing the existence of best proximity points, iterative algorithms are also furnished to determine such optimal approximate solutions.

##### MSC:
 90C20 Quadratic programming
##### Keywords:
fixed point; contractive mapping
Full Text:
##### References:
 [1] Al-Thagafi M.A., Shahzad N.: Convergence and existence results for best proximity points. Nonlinear Anal. 70(10), 3665--3671 (2009) · Zbl 1197.47067 · doi:10.1016/j.na.2008.07.022 [2] Al-Thagafi M.A., Shahzad N.: Best proximity pairs and equilibrium pairs for Kakutani multimaps. Nonlinear Anal. 70(3), 1209--1216 (2009) · Zbl 1225.47056 · doi:10.1016/j.na.2008.02.004 [3] Al-Thagafi, M.A., Shahzad, N.: Best proximity sets and equilibrium pairs for a finite family of multimaps, Fixed Point Theory Appl., Art. ID 457069, 10 pp (2008) · Zbl 1169.47040 [4] Anthony Eldred 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 [5] Anthony Eldred A., Kirk W.A., Veeramani P.: Proximinal normal structure and relatively nonexpanisve mappings. Studia Math. 171(3), 283--293 (2005) · Zbl 1078.47013 · doi:10.4064/sm171-3-5 [6] Di Bari C., Suzuki T., Vetro C.: Best proximity points for cyclic Meir-Keeler contractions. Nonlinear Anal. 69(11), 3790--3794 (2008) · Zbl 1169.54021 · doi:10.1016/j.na.2007.10.014 [7] Edelstein M.: On fixed and periodic points under contractive mappings. J. London Math. Soc. 37, 74--79 (1962) · Zbl 0113.16503 · doi:10.1112/jlms/s1-37.1.74 [8] Fan K.: Extensions of two fixed point theorems of F. E. Browder. Math. Z. 112, 234--240 (1969) · Zbl 0185.39503 · doi:10.1007/BF01110225 [9] Karpagam, S., Agrawal, S.: Best proximity point theorems for p-cyclic Meir-Keeler contractions, Fixed Point Theory Appl., Art. ID 197308, 9 pp (2009) · Zbl 1172.54028 [10] Kim W.K., Kum S., Lee K.H.: On general best proximity pairs and equilibrium pairs in free abstract economies. Nonlinear Anal. 68(8), 2216--2222 (2008) · Zbl 1136.91309 · doi:10.1016/j.na.2007.01.057 [11] 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 [12] Prolla, J.B.: Fixed point theorems for set valued mappings and existence of best approximations, Numer. Funct. Anal. Optim. 5, 449--455 (1982--1983) · Zbl 0513.41015 [13] Reich S.: Approximate selections, best approximations, fixed points and invariant sets. J. Math. Anal. Appl. 62, 104--113 (1978) · Zbl 0375.47031 · doi:10.1016/0022-247X(78)90222-6 [14] Sadiq Basha S., Veeramani P.: Best approximations and best proximity pairs. Acta Sci. Math. (Szeged) 63, 289--300 (1997) · Zbl 0909.47042 [15] Sadiq Basha S., Veeramani P.: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 103, 119--129 (2000) · Zbl 0965.41020 · doi:10.1006/jath.1999.3415 [16] Sadiq Basha S., Veeramani P., Pai D.V.: Best proximity pair theorems. Indian J. Pure Appl. Math. 32, 1237--1246 (2001) · Zbl 1021.47027 [17] Sehgal V.M., Singh S.P.: A generalization to multifunctions of Fan’s best approximation theorem. Proc. Am. Math. Soc. 102, 534--537 (1988) · Zbl 0672.47043 [18] Sehgal V.M., Singh S.P.: A theorem on best approximations. Numer. Funct. Anal. Optim. 10, 181--184 (1989) · Zbl 0635.41022 · doi:10.1080/01630568908816298 [19] Srinivasan P.S.: Best proximity pair theorems. Acta Sci. Math. (Szeged) 67, 421--429 (2001) · Zbl 1012.47031 [20] Vetrivel V., Veeramani P., Bhattacharyya P.: Some extensions of Fan’s best approximation theorem. Numer. Funct. Anal. Optim. 13, 397--402 (1992) · Zbl 0763.41026 · doi:10.1080/01630569208816486 [21] 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(9), 3332--3341 (2009) · Zbl 1182.54024 · doi:10.1016/j.na.2008.04.037