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Inequalities in Banach spaces with applications. (English) Zbl 0757.46033

The norm in a Hilbert space H satisfies the identity:

λx+(1-λ)y 2 =λx 2 +(1-λ)y 2 -λ(1-λ)x-y 2 ,forallx,yinHand0λ1·

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.


MSC:
46B20Geometry and structure of normed linear spaces
47J20Inequalities involving nonlinear operators
46C99Inner product spaces, Hilbert spaces