N. Hussain

Common fixed-point results in best approximation theory

A common fixed-point generalization of the results of Dotson, Tarafdar, and Taylor is obtained which in turn extends a recent theorem by Jungck and Sessa to locally convex spaces. As applications of our work, we improve and unify well-known results on fixed points and common fixed points of best approximation.

Applications of the Best Approximation Operator to *-Nonexpansive Maps in Hilbert Spaces

Abstract The notion of a *-nonexpansive multivalued map is different from that of a continuous map. We give some Ky Fan type best approximation theorems for *-nonexpansive mappings defined on closed convex unbounded subsets of a Hilbert space. As applications of our theorems, we derive fixed point results under many boundary conditions. Approximating sequences to the fixed points are also constructed.

Convergence theorems for nonself asymptotically nonexpansive mappings

In this paper, we prove some strong and weak convergence theorems using a modified iterative process for nonself asymptotically nonexpansive mappings in a uniformly convex Banach space. This will improve and generalize the corresponding results in the existing literature. Finally, we will state that our theorems can be generalized to the case of finitely many mappings.

commuting maps and invariant approximations

We obtain common fixed point results for generalized -nonexpansive -commuting maps. As applications, various best approximation results for this class of maps are derived in the setup of certain metrizable topological vector spaces.

RANDOM FIXED POINTS AND RANDOM APPROXIMATIONS IN NONCONVEX DOMAINS

Stochastic generalizations of some fixed point theorems on a class of nonconvex nsets in a locally bounded topological vector space are established. As applications, Brosowski-Meinardus type theorems about random invariant approximation are obtained. This work extends or provides stochastic versions of several well known results.

Random Coincidence Point Theorem in Fréchet Spaces with Applications

Abstract We proved a random coincidence point theorem for a pair of commuting random operators in the setup of Fréchet spaces. As applications, we obtained random fixed point and best approximation results for *-nonexpansive multivalued maps. Our results are generalizations or stochastic versions of the corresponding results of Shahzad and Latif [Shahzad, N.; Latif, A. A random coincidence point theorem. J. Math. Anal. Appl. 2000, 245, 633–638], Khan and Hussain [Khan, A.R.; Hussain, N. Best approximation and fixed point results. Indian J. Pure Appl. Math. 2000, 31 (8), 983–987], Tan and Yaun [Tan, K.K.; Yaun, X.Z. Random fixed point theorems and approximation. Stoch. Anal. Appl. 1997, 15 (1), 103–123] and Xu [Xu, H.K. On weakly nonexpansive and *-nonexpansive multivalued mappings. Math. Japon. 1991, 36 (3), 441–445].

RANDOM FIXED POINTS FOR ,-NONEXPANSIVE RANDOM OPERATORS

The notion of a .-nonexpansive multivalued map is different from that of a continuous map. In this paper we prove some fixed point theorems for .-nonexpansive multivalued random operators in the setup of Banach spaces and Frchet spaces. Our work generalizes, refines and improves the

A NOTE ON KAKUTANI TYPE FIXED POINT THEOREMS

We present Kakutani type fixed point theorems for certain semigroups of self maps by relaxing conditions on the underlying set, family of self maps, and the mappings themselves in a locally convex space setting.

AN EXTENSION OF A THEOREM OF SAHAB, KHAN, AND SESSA

A fixed point theorem of Fisher and Sessa is generalized to locally convex spaces and the new result is applied to extend a recent theorem on invariant approxi- mation of Sahab, Khan, and Sessa.

Some best approximation results in locally convex spaces

In this note we obtain generalization of well known results of carbone and Conti, Sehgal and Singh and Tanimoto concerning the existence of best approximation and simultaneous best approximation of continuous funcitons from the set up of a normed space to the case of a Hausdorff locally convex space.