Some common fixed point theorems for weakly compatible mappings in uniform spaces (Q532100)

The paper deals with common fixed point theorems for weakly compatible mappings on uniform spaces. Theorem 1. Let \((X,u)\) be a Hausdorff uniform space and let \(I,J:X\to X\) be single valued mappings and \( S,T:X\to 2^X\) be multi-valued mappings such that \(\cup S(X)\subset J(X)\), \(\cup T(X)\supset I(X)\). If one of \(I(X)\), \(J(X)\) is complete, \(\{S,T\} \) and \(\{T,J\}\) are weakly compatible and for all \(x,y\in X\), \(i\in I\) \[ \delta_i(Sx,Ty)\leq F(d_i(Ix,Jy),\delta_i(Ix,Sx),\delta_i(Jy,Ty),d_i(Ix,Ty),d_i(Jy,Sx)) \] where \(F\) is a contractive function, then there exists \(z\in X\) such that \[ z=\{Tz\}=\{Jz\}=Sz=Tz. \] The above result is extended for families of weakly compatible mappings. An application to locally convex spaces is presented.