On the fixed point of (ψ−φ)(ψ−φ)-weak and generalized (ψ−φ)(ψ−φ)-weak contraction mappings

Let (X,d)(X,d) be a complete metric space, and T:X⟶XT:X⟶X be a (ψ−φ)(ψ−φ)-weak or generalized (ψ−φ)(ψ−φ)-weak contraction mapping, where ψ,φ:[0,+∞)⟶[0,+∞)ψ,φ:[0,+∞)⟶[0,+∞) are two mappings with ψ−1(0)=φ−1(0)=0ψ−1(0)=φ−1(0)=0, limn→∞tn=0, if limn→∞φ(tn)=0 and ψψ is continuous or ψψ is monotone nondecreasing with φ(a)>ψ(a)−ψ(a−)φ(a)>ψ(a)−ψ(a−) for all a>0a>0. Then TT has a unique fixed point. Our results extend the previous results given by Rhoades (2001) [3], Dutta and Choudhury (2008) [4], Doric (2009) [5] and Popescu (2011) [6].