Normal structure in Banach spaces

Belluce, L. P.; Kirk, W. A.; Steiner, E. F.

doi:10.2140/pjm.1968.26.433

Cited by 63 publications

(33 citation statements)

References 6 publications

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“…That the converse of the above proposition is not true can be easily seen from the following example [13].…”

Section: Corollarymentioning

confidence: 91%

Fixed point theorems in reflexive Banach spaces

Kannan¹

1973

Proc. Amer. Math. Soc.

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Abstract.In this paper fixed point theorems are established first for mappings T, mapping a closed bounded convex subset K of a reflexive Banach space into itself and satisfying || Tx -Ty\\ g i{\\x -Tx\\ + \\y -Ty\\}, x,yeK, and then an analogous result is obtained for nonexpansive mappings giving rise to a question regarding the unification of these theorems.Let X be a reflexive Banach space and let K be a nonempty bounded closed and convex subset of X. In [10] Kirk proved the following theorem: If F be a nonexpansive mapping of K into itself i.e., \\Tx -Fj||_||x-_y||, x, y e K, and if K has normal structure, then F has a fixed point in K. This result was also proved in a uniformly convex space X by Browder [2], Göhde [4] and Goebel [5], the reflexivity of the space and the normal structure of K being consequences of the uniform convexity of X.In this paper first we establish some fixed point theorems for mappings T of K into itself which satisfy || Tx -Ty\\ ^ «I* -Tx\\ + \\y -Ty\\}, x, y e K.Mappings T of this type will be referred to as having property A over K. Such mappings have been used to study fixed point and other allied problems in [6], [7], [8], [9]. Then we obtain a theorem, analogous to the one proved for a mapping T having property A over K, for nonexpansive mappings. We conclude the paper with some observations on this last theorem.Before going to the theorems, we first recollect the following definitions. Definition 1 (Normal Structure [10]). A bounded convex set K in a Banach space X is said to have normal structure if for each convex subset S of K which contains more than one point, there exists x e S such that supvesllx-j|| <à(S), ô(S) being the diameter of S.Definition 2 [9]. A mapping F of a bounded subset A" of a Banach space X into itself is said to have property B on K [9] if for every closed

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“…That the converse of the above proposition is not true can be easily seen from the following example [13].…”

Section: Corollarymentioning

confidence: 91%

Fixed point theorems in reflexive Banach spaces

Kannan¹

1973

Proc. Amer. Math. Soc.

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“…Thus X satisfies condition (3). (3) ⇒ (1): Suppose r and ρ are given by (3). Assume that lim sup x n − x ≤ 1, x n w − → 0 and lim sup x n > 1/ρ.…”

Section: Some Banach Space Propertiesmentioning

confidence: 99%

“…Perhaps the first relevant product result was given in [3]: that the ∞ product of finitely many spaces with normal structure also has normal structure. Also Landes showed in [15] that normal structure is preserved when a general substitution space is uniformly convex.…”

Section: Permanence Propertiesmentioning

confidence: 99%

On some Banach space properties sufficient for weak normal structure and their permanence properties

Sims

Smyth

1999

Trans. Amer. Math. Soc.

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“…James [5]. This is essentially the space which has been discussed in various places in the literature, e.g., [1,2,4,5,7,8,10,15,16,19,20,21,22,23,25,26,28,39].…”

Section: Theorem 41 Suppose That X = W ⊕ Z Where W Is a Closed Submentioning

confidence: 99%