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International Journal of Mathematics and Mathematical Sciences
Volume 28 (2001), Issue 1, Pages 41-50

Fixed point theorems for generalized Lipschitzian semigroups

1Department of Mathematics, Dong-A University, Pusan 607-714, Korea
2Govt. B. H. S. S. Gariaband, Dist. Raipur, (M. P.) 493889, India

Received 10 March 2000

Copyright © 2001 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let K be a nonempty subset of a p-uniformly convex Banach space E, G a left reversible semitopological semigroup, and 𝒮={Tt:tG} a generalized Lipschitzian semigroup of K into itself, that is, for sG, TsxTsyasxy+bs(xTsx+yTsy)+cs(xTsy+yTsx), for x,yK where as,bs,cs>0 such that there exists a t1G such that bs+cs<1 for all st1. It is proved that if there exists a closed subset C of K such that sco¯{Ttx:ts}C for all xK, then 𝒮 with [(α+β)p(αp2p11)/(cp2p1βp)Np]1/p<1 has a common fixed point, where α=lim sups(as+bs+cs)/(1-bs-cs) and β=lim sups(2bs+2cs)/(1-bs-cs).