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Inequalities in Banach spaces with applications. (English) Zbl 0757.46033
The norm in a Hilbert space $$H$$ satisfies the identity: $\|\lambda x+(1-\lambda)y\|^ 2=\lambda\| x\|^ 2+(1-\lambda)\| y\|^ 2-\lambda(1-\lambda)\| x-y\|^ 2,\text{ for all } x,y\text{ in } H\text{ and } 0\leq\lambda\leq 1.$ 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.

##### MSC:
 46B20 Geometry and structure of normed linear spaces 47J20 Variational and other types of inequalities involving nonlinear operators (general) 46C99 Inner product spaces and their generalizations, Hilbert spaces
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