Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces

“…Typical examples of such spaces are the Lebesque L p , the sequence p and the Sobolev W m p spaces for 1 < p < ∞. In particular, for 1 < p ≤ 2, these spaces are p-uniformly smooth and for 2 ≤ p < ∞, they are 2-uniformly smooth (see, e.g., [15]). …”

Section: Preliminariesmentioning

confidence: 99%

“…If E is a real q-uniformly smooth Banach space, then (see, e.g., [15]) the following geometric inequality holds:…”

Section: Preliminariesmentioning

confidence: 99%

Iterative approximation of fixed points of Lipschitz pseudocontractive maps

Chidume

2001

Proc. Amer. Math. Soc.

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Abstract. Let E be a q-uniformly smooth Banach space possessing a weakly sequentially continuous duality map (e.g., p, 1 < p < ∞). Let T be a Lipschitzian pseudocontractive selfmapping of a nonempty closed convex and bounded subset K of E and let ω ∈ K be arbitrary. Then the iteration sequence {zn} defined by z 0 ∈ K, z n+1 = (1 − µ n+1 )ω + µ n+1 yn; yn = (1 − αn)zn + αnT zn, converges strongly to a fixed point of T , provided that {µn} and {αn} have certain properties. If E is a Hilbert space, then {zn} converges strongly to the unique fixed point of T closest to ω.

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“…Similarly, whenever the modulus of smoothness of the norm · is of power type s, there exists, according to [45,Remark 5,p. 208], some constant L > 0 such that, for all x, y ∈ X, (2.5)…”

Section: Introductionmentioning

confidence: 99%

Prox-regular sets and epigraphs in uniformly convex Banach spaces: Various regularities and other properties

Bernard

Thibault

Zlateva

2011

Trans. Amer. Math. Soc.

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Abstract. We continue the study of prox-regular sets that we began in a previous work in the setting of uniformly convex Banach spaces endowed with a norm both uniformly smooth and uniformly convex (e.g., L p , W m,p spaces). We prove normal and tangential regularity properties for these sets, and in particular the equality between Mordukhovich and proximal normal cones. We also compare in this setting the proximal normal cone with different Hölderian normal cones depending on the power types s, q of moduli of smoothness and convexity of the norm. In the case of sets that are epigraphs of functions, we show that J-primal lower regular functions have prox-regular epigraphs and we compare these functions with Poliquin's primal lower nice functions depending on the power types s, q of the moduli. The preservation of prox-regularity of the intersection of finitely many sets and of the inverse image is obtained under a calmness assumption. A conical derivative formula for the metric projection mapping of prox-regular sets is also established. Among other results of the paper it is proved that the Attouch-Wets convergence preserves the uniform r-prox-regularity property and that the metric projection mapping is in some sense continuous with respect to this convergence for such sets.

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Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces

Cited by 256 publications

References 11 publications

Phase retrieval problems in x-ray physics

Phase retrieval problems in x-ray physics

Iterative approximation of fixed points of Lipschitz pseudocontractive maps

Prox-regular sets and epigraphs in uniformly convex Banach spaces: Various regularities and other properties

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