On some fixed point theorems on uniformly convex Banach spaces.

*(English)*Zbl 0780.47047The author introduces the concept of \(T\)-regularity and generalizes a well known fixed point theorem of F. E. Browder [Proc. Nat. Acad. Sci. USA 54, 1041-1043 (1965; Zbl 0128.358)]. Let \(X\) be a vector space and \(A\) be a subset of \(X\). \(A\) is said to be a \(T\)-regular set if and only if

(i) \(T: A\to A\); (ii) \({1\over 2}(x+Tx)\in A\), for each \(x\) in \(A\).

Main Theorem. Let \(K\) be a nonempty weakly compact \(T\)-regular subset of a uniformly convex Banach space \(X\). Further for each weakly closed \(T\)- regular subset \(F\) of \(K\), with \(\delta(F)>0\), there exists some \(\beta(F)\), \(0<\beta(F)<1\), such that \[ \| Tx-Ty\|\leq max\{\| x-y\|, \beta\delta(F)\} \] for all \(x,y\in F\). Then \(T\) has a fixed point in \(K\).

The proof is based on a Zorn’s lemma. From the concept of \(T\)-regularity and other results of the author, theorems of C. L. Outlaw [Pac. J. Math. 30, 747-750 (1969; Zbl 0174.193)] and A. A. Gillespie and B. B. Williams [J. Math. Anal. Appl. 74, 382-387 (1980; Zbl 0448.47039)] can be deduced as special cases.

(i) \(T: A\to A\); (ii) \({1\over 2}(x+Tx)\in A\), for each \(x\) in \(A\).

Main Theorem. Let \(K\) be a nonempty weakly compact \(T\)-regular subset of a uniformly convex Banach space \(X\). Further for each weakly closed \(T\)- regular subset \(F\) of \(K\), with \(\delta(F)>0\), there exists some \(\beta(F)\), \(0<\beta(F)<1\), such that \[ \| Tx-Ty\|\leq max\{\| x-y\|, \beta\delta(F)\} \] for all \(x,y\in F\). Then \(T\) has a fixed point in \(K\).

The proof is based on a Zorn’s lemma. From the concept of \(T\)-regularity and other results of the author, theorems of C. L. Outlaw [Pac. J. Math. 30, 747-750 (1969; Zbl 0174.193)] and A. A. Gillespie and B. B. Williams [J. Math. Anal. Appl. 74, 382-387 (1980; Zbl 0448.47039)] can be deduced as special cases.

Reviewer: A.A.Melentsov (Sverdlovsk)

##### MSC:

47H10 | Fixed-point theorems |

##### Keywords:

weakly compact \(T\)-regular subset of a uniformly convex Banach space; Zorn’s lemma; \(T\)-regularity
PDF
BibTeX
XML
Cite

\textit{P. Veeramani}, J. Math. Anal. Appl. 167, No. 1, 160--166 (1992; Zbl 0780.47047)

Full Text:
DOI

##### References:

[1] | Browder, F.E, Nonexpansive nonlinear operations in a Banach space, (), 1041-1043 · Zbl 0128.35801 |

[2] | Gillespie, A.A; Williams, B.B, Some theorems on fixed points in Lipschitz and Kannan type mappings, J. math. anal. appl., 74, 382-387, (1980) · Zbl 0448.47039 |

[3] | Ishikawa, S, Fixed points and iteration of a nonexpansive mapping in a Banach space, (), 65-71, No. 1 · Zbl 0352.47024 |

[4] | Kirk, W.A, A fixed point theorem for mapping which do not increase distances, Amer. math monthly, 72, 1004-1006, (1965) · Zbl 0141.32402 |

[5] | Outlaw, C.L, Mean value iteration of nonexpansive mappings in a Banach space, Pacific J. math., 30, 747-750, (1969) · Zbl 0179.19801 |

[6] | Pai, D.V; Veeramani, P, On some fixed point theorems in Banach spaces, Internat. J. math. math. sci., 5, No. 1, 113-122, (1982) · Zbl 0492.47031 |

[7] | Singh, S.P, Application of a fixed point theorem to approximation theory, J. approx. theory, 25, 88-89, (1979) · Zbl 0399.41032 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.