Extensions of Banach’s contraction principle. (English) Zbl 1200.54021

For a metric space \((X,d)\) and \(A,B\subset X,\) let \(d(A,B)=\inf\{d(a,b) :a\in A,B\in B\}\), \(A_0=\{a\in A : \exists b\in B, d(a,b)=d(A,B)\}\), and \(B_0=\{b\in B : \exists a\in A,\;d(a,b)=d(A,B)\}\). The set \(B\) is called approximatively compact with respect to \(A\) if for every \(x\in A\), every sequence \((y_n)\) in \(B\) such that \(d(x,y_n)\to d(x,B)\) contains a subsequence converging to some point of \(B\). A mapping \(T:A\to B\) is called a proximal contraction if there exists \(\alpha, \, 0\leq \alpha<1,\) such that for every \((x,y)\in A\times B\) for which there exists \((u,v)\in A\times B\) with \(d(u,Tx)=d(v,Ty)=d(A,B),\) the following inequality holds \[ d(u,Tx)+d(Tx,Ty)+d(v,Ty)\leq \alpha d(x,y). \] The author shows by an example (a mapping \(T:[0,1]\to[2,3]\)) that a proximal contraction need not be continuous. Let \((X,d)\) be a complete metric space, \(\, A,B \subset X\) nonempty closed such that \(B\) is approximatively compact with respect to \(A\) and \(A_0,B_0\) are nonempty. Then for every proximal contraction \(T:A\to B\) with \(T(A_0)\subset T(B_0),\) there is a unique point \(x\in A\) such that \(d(x,Tx)=d(A,B)\) (Theorem 3.1). In Theorem 3.3, a similar result is proved for two mappings \(T:A\to B,\, S:B\to A,\) where \(T\) is a proximal contraction and \(S\) is nonexpansive.


54H25 Fixed-point and coincidence theorems (topological aspects)
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[1] DOI: 10.1016/j.na.2007.10.014 · Zbl 1169.54021
[2] DOI: 10.4064/sm171-3-5 · Zbl 1078.47013
[3] DOI: 10.1016/j.jmaa.2005.10.081 · Zbl 1105.54021
[4] DOI: 10.1007/BF01110225 · Zbl 0185.39503
[5] Karpagam S., Fixed Point Theory Appl. (2009) · Zbl 1172.54028
[6] DOI: 10.1016/j.na.2007.01.057 · Zbl 1136.91309
[7] DOI: 10.1081/NFA-120026380 · Zbl 1054.47040
[8] DOI: 10.1080/01630568308816149 · Zbl 0513.41015
[9] DOI: 10.1016/0022-247X(78)90222-6 · Zbl 0375.47031
[10] Sadiq Basha S., J. Glob. Optim. (2010)
[11] Sadiq Basha S., Acta. Sci. Math. (Szeged) 63 pp 289– (1997)
[12] Sadiq Basha S., Indian J. Pure Appl. Math. 32 pp 1237– (2001)
[13] DOI: 10.1006/jath.1999.3415 · Zbl 0965.41020
[14] Sehgal V.M., Proc. Amer. Math. Soc. 102 pp 534– (1988)
[15] DOI: 10.1080/01630568908816298 · Zbl 0635.41022
[16] Srinivasan P.S., Acta Sci. Math. (Szeged) 67 pp 421– (2001)
[17] Al-Thagafi M.A., Fixed Point Theory Appl. (2008)
[18] DOI: 10.1016/j.na.2008.02.004 · Zbl 1225.47056
[19] DOI: 10.1016/j.na.2008.07.022 · Zbl 1197.47067
[20] DOI: 10.1080/01630569208816486 · Zbl 0763.41026
[21] DOI: 10.1016/j.na.2008.04.037 · Zbl 1182.54024
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