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

Xu, Hong-Kun

Nonlinear Anal., Theory Methods Appl. 16, No. 12, 1127-1138 (1991).

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.

Reviewer: J.R.Holub (Blacksburg)

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

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

References:

[1]	Reich, S., An iterative procedure for constructing zeros of accretive sets in Banach spaces, Nonlinear Analysis, 2, 85-92 (1978) · Zbl 0375.47032
[2]	Poffald, E. I.; Reich, S., An incomplete Cauchy problem, J. math. Analysis Applic., 113, 514-543 (1986) · Zbl 0599.34078
[3]	Lim, T. C., Fixed point theorems for uniformly Lipschitzian mappings in \(L^p\) spaces, Nonlinear Analysis, 7, 555-563 (1983) · Zbl 0533.47049
[5]	Prus, B.; Smarzewski, R., Strongly unique best approximation and centers in uniformly convex spaces, J. math. Analysis Applic., 121, 10-21 (1987) · Zbl 0617.41046
[6]	Smarzewski, R., Strongly unique minimization of functionals in Banach spaces with applications to theory of approximations and fixed points, J. math. Analysis Applic., 115, 155-172 (1986) · Zbl 0593.49004
[7]	Zalinescu, C., On uniformly convex functions, J. math. Analysis Applic., 95, 344-374 (1983) · Zbl 0519.49010
[8]	Beauzamy, B., Introduction to Banach Spaces and Their Geometry (1985), North-Holland: North-Holland Amsterdam · Zbl 0585.46009
[10]	You, Z. Y., Nonlinear Analysis (1990), Xi’an Jiaotong University Press: Xi’an Jiaotong University Press Xi’an, Shaaxi Province, People’s Republic of China
[11]	Casini, E.; Maluta, E., Fixed points of uniformly Lipschitzian mappings in spaces with uniformly normal structure, Nonlinear Analysis, 9, 103-108 (1985) · Zbl 0526.47034
[12]	Goebel, K.; Kirk, W. A., A fixed point theorem for transformations whose iterates have uniform Lipschitz constant, Studia math., 47, 135-140 (1973) · Zbl 0265.47044
[13]	Lifschitz, E. A., Fixed point theorems for operators in strongly convex spaces, Voronez Gos. Univ. Trudy Math. Fak., 16, 23-28 (1975), (In Russian.)
[14]	Xu, H. K., Fixed point theorems for uniformly Lipschitzian semigroups in uniformly convex spaces, J. math. Analysis Applic., 152, 391-398 (1990) · Zbl 0722.47050
[15]	Lim, T. C., On the normal structure coefficient and the bounded sequence coefficient, Proc. Am. math. Soc., 88, 262-264 (1983) · Zbl 0541.46017
[17]	Geobel, K.; Reich, S., Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1984), Marcel Dekker: Marcel Dekker New York
[18]	Bjornestal, B. O., Local Lipschitz continuity of the metric projection operator, (Approximation Theory, Banach Center Publ. (1979), PWN: PWN Warsaw), 43-53, No. 4 · Zbl 0428.41024
[19]	Abatzoglou, T., Continuity of metric projection in uniformly convex and uniformly smooth Banach spaces, J. approx. Theory, 39, 299-307 (1983) · Zbl 0543.41033
[20]	Bynum, W. L., Normal structure coefficients for Banach spaces, Pacif. J. math., 86, 427-436 (1980) · Zbl 0442.46018
[21]	Reich, S., Constructive techniques for accretive and monotone operators, (Applied Nonlinear Analysis (1979), Academic Press: Academic Press New York), 335-345