%0 Journal Article
%T Some Fixed Point Theorems under Weak Semicompatibility
%A A. S. Saluja
%A Mukesh Kumar Jain
%J Chinese Journal of Mathematics
%D 2014
%R 10.1155/2014/618965
%X The aim of the present paper is to prove some fixed point theorems by using the recent notion ˇ°weak semicompatibility.ˇ± The new notion is proper generalization of semicompatibility and can be applicable on commuting and compatible maps. We used compatible and absorbing mappings to prove theorems which also include (E.A.) property. 1. Introduction and New Definitions In 1995, Cho et al. [1] introduced the concept of semicompatibility and obtained the first result that established a situation in which a collection of mappings has a fixed point. They defined a pair of self-maps to be semicompatible if(a) and(b) . For some , implying holds. Singh and Jain [2] observe that (b) implies (a). Hence they defined the semicompatibility by condition (b) only. Let be a metric space and let and be two maps from into itself. and are commuting maps if for all in . To generalize the notion of commuting maps, Sessa [3] introduced the concept of weakly commuting maps. He defines and as being weakly commuting if for all . Obviously, commuting maps are weakly commuting but the converse is not true. In 1986, Jungck [4] gave more generalized commuting and weakly commuting maps called compatible maps. and are called compatible if Whenever is a sequence in such that for some , clearly, weakly commuting maps are compatible, but the implication is not reversible (see [4]). Afterwards, Jungck et al. [5] made another generalization of weakly commuting maps by introducing the concept of compatible maps of type ( ). Previous and are said to be compatible of type ( ) if in place of (1) we have the two following conditions: It is clear to see that weakly commuting maps are compatible of type ( ); from [5] it follows that the implication is not reversible. Two self-maps and of metric space are called -compatible ([6] cited from [7]) if , whenever is a sequence in such that for some in . Similarly, two self-maps and of metric space are called -compatible ([6] cited from [7]) if , whenever is a sequence in such that for some in . Two self-mappings and of a metric space are called weakly commuting [8] at a point in if for some . The two self-maps and of a metric space are called weakly commuting of type [9] if there exists some positive real number such that for all in . The two self-maps and of a metric space are called weakly commuting of type [9] if there exists some positive real number such that for all in . It may be noted that compatible mappings and can be weakly commuting of types and . Let and be two self-maps of metric space ; then will be called -absorbing [10] if there exists
%U http://www.hindawi.com/journals/cjm/2014/618965/