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Zbl 0757.46033
Xu, Hong-Kun
Inequalities in Banach spaces with applications.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 16, No.12, 1127-1138 (1991). ISSN 0362-546X

The norm in a Hilbert space $H$ satisfies the identity: $$\Vert\lambda x+(1-\lambda)y\Vert\sp 2=\lambda\Vert x\Vert\sp 2+(1-\lambda)\Vert y\Vert\sp 2-\lambda(1-\lambda)\Vert x-y\Vert\sp 2,\text{ for all } x,y\text{ in } H\text{ and } 0\leq\lambda\leq 1.$$ In this paper, inequalities in uniformly convex or uniformly smooth Banach spaces which are analogous to the above identity are established. Applications to the existence of fixed points for uniformly Lipschitzian mappings in $p$- uniformly convex Banach spaces and to the estimation of the modulus of continuity of certain metric projections are also given.
[J.R.Holub (Blacksburg)]
MSC 2000:
*46B20 Geometry and structure of normed spaces
47J20 Inequalities involving nonlinear operators
46C99 Inner product spaces, Hilbert spaces

Keywords: inequalities in uniformly convex or uniformly smooth Banach spaces; existence of fixed points for uniformly Lipschitzian mappings in $p$- uniformly convex Banach spaces; modulus of continuity; metric projections

Cited in: Zbl 1017.47048 Zbl 0997.46009

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