Let ((X, d)) be a complete metric space, and let (T: Xto X) be a mapping satisfying the inequality psibigl(d(Tx,Ty bigr)leqpsibigl(d x,y bigr -varphibigl(d(x,y bigr -varphibigll x,yin X, where (psi,varphi:Bbb{R}^toBbb{R}^) are both continuous and xonotone nondecreasing functions with (psi(t)=varphi(t)=0) if and only if (t=0).
Let T, S be two self mappings in a metric space ((X, d)) satisfying psibigl(d(Tx,Ty bigr)leqpsibigl(d(Sx,Sy bigr -varphibigl(d(Sx,Sy bigr -varphibiglll x,yin X, where (psi,varphi:Bbb{R}^toBbb{R}^) are both continuous and monotone nondecreaSx,Syfunctions with (psi(t)=varphi(t)=0) if and only if (t=0).
Let ( X, d ) be a complete metric space and let T : X → X be a self-mapping satisfying the inequality ψ ( d ( T x, T y ) ) ≤ ψ ( d ( x, y ) ) − φ ( d ( x, y ) ). for all x, y ∈ X, where ψ, φ : [ 0, ∞ [ → ∞ are both continuous and monotone nondecreasing functions with ψ ( t ) = φ ( t ) = 0 if and only if t = 0. Then T has a unique fixed point. Theorem A.9 (Luong and Thuan [42]).
Theorem 1.4 Let f be a mapping from a complete metric space ( X, d ) into itself satisfying ψ ( d ( f x, f y ) ) ≤ ψ ( d ( x, y ) ) − φ ( d ( x, y ) ), ∀ x, y ∈ X, where ψ, φ : R + → R + are both continuous and monotone nondecreasing functions with ψ ( t ) = φ ( t ) = 0 if and only if t = 0. Then f has a unique fixed point a ∈ X such that lim n → ∞ f n x = a for each x ∈ X.
Therefore, is continuous and nondecreasing.
(a) is continuous and nondecreasing.
φ is continuous and nondecreasing, and.
There exist continuous and nondecreasing such that.
There exists and continuous and nondecreasing such that.
The nonnegative functions are Lipschitz continuous and nondecreasing,.
