On the fixed point of -weak and generalized -weak contraction mappings

Let ( X , d ) be a complete metric space, and T : X ⟶ X be a ( ψ − φ ) -weak or generalized ( ψ − φ ) -weak contraction mapping, where ψ , φ : [ 0 , + ∞ ) ⟶ [ 0 , + ∞ ) are two mappings with ψ − 1 ( 0 ) = φ − 1 ( 0 ) = 0 , lim n → ∞ t n = 0 , if lim n → ∞ φ ( t n ) = 0 and ψ is continuous or ψ is monotone nondecreasing with φ ( a ) > ψ ( a ) − ψ ( a − ) for all a > 0 . Then T 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].