Contractive Iterate Research Articles

Theorems of ergodic type for ρ-nonexpansive mappings in the Hilbert ball

We present two types of ergodic theorems for contractive iterations in the Hilbert ball. Both explicit and implicit iterations are discussed.

Survey of interval algorithms for ordinary differential equations

Interval algorithms compute an interval valued function in which the solution to a system of ordinary differential equations is guaranteed to lie. Interval algorithms use differential inequalities, finite difference approximations with remainders, Taylor series, defect corrections, or contractive iterations.

Asymptotic periodicity of the iterates of positive contractions on Banach lattices

Asymptotic periodicity of the iterates of positive contractions on Banach lattices

Corrections to: “Fixed-point theorems for mappings with a contractive iterate”

Corrections to: “Fixed-point theorems for mappings with a contractive iterate”

Fixed point-theorems for mappings with a contractive iterate

Fixed point-theorems for mappings with a contractive iterate

Fixed Point Theorems for Mappings with a Contractive Iterate at a Point

Let (X,d) be a complete metric space, $T:X \to X$, and $\alpha :[0,\infty )^5 \to [0,\infty )$ be nondecreasing with respect to each variable. Suppose that for the function $\gamma (t) = \alpha (t,t,t,2t,2t)$, the sequence of iterates ${\gamma ^n}$ tends to 0 in $[0,\infty )$ and ${\lim _{t \to \infty }}(t - \gamma (t)) = \infty$. Furthermore, suppose that for each $x \in X$ there exists a positive integer $n = n(x)$ such that for all $y \in X$, \[ d({T^n}x,{T^n}y) \leqslant \alpha (d(x,{T^n}x),d(x,{T^n}y),d(x,y),d({T^n}x,y),d({T^n}y,y)).\] Under these assumptions our main result states that T has a unique fixed point. This generalizes an earlier result of V. M. Sehgal and some recent results of L. Khazanchi and K. Iseki.

Fixed point theorems for mappings with a contractive iterate at a point

Let (X,d) be a complete metric space, T : X → X T:X \to X , and α : [ 0 , ∞ ) 5 → [ 0 , ∞ ) \alpha :[0,\infty )^5 \to [0,\infty ) be nondecreasing with respect to each variable. Suppose that for the function γ ( t ) = α ( t , t , t , 2 t , 2 t ) \gamma (t) = \alpha (t,t,t,2t,2t) , the sequence of iterates γ n {\gamma ^n} tends to 0 in [ 0 , ∞ ) [0,\infty ) and lim t → ∞ ( t − γ ( t ) ) = ∞ {\lim _{t \to \infty }}(t - \gamma (t)) = \infty . Furthermore, suppose that for each x ∈ X x \in X there exists a positive integer n = n ( x ) n = n(x) such that for all y ∈ X y \in X , \[ d ( T n x , T n y ) ⩽ α ( d ( x , T n x ) , d ( x , T n y ) , d ( x , y ) , d ( T n x , y ) , d ( T n y , y ) ) . d({T^n}x,{T^n}y) \leqslant \alpha (d(x,{T^n}x),d(x,{T^n}y),d(x,y),d({T^n}x,y),d({T^n}y,y)). \] Under these assumptions our main result states that T has a unique fixed point. This generalizes an earlier result of V. M. Sehgal and some recent results of L. Khazanchi and K. Iseki.

On convergence of iterates of positive contractions in Lp spaces

On convergence of iterates of positive contractions in Lp spaces

On Contractive Mappings

The Cauchy condition for convergence of a contractive iteration in a complete metric space is replaced by an equivalent functional condition. Many generalizations of the Banach-Neumann contractive mapping principle follow.

On contractive mappings

The Cauchy condition for convergence of a contractive iteration in a complete metric space is replaced by an equivalent functional condition. Many generalizations of the Banach-Neumann contractive mapping principle follow.

A Fixed Point Theorem for Semigroups of Mappings with a Contractive Iterate

A Fixed Point Theorem for Semigroups of Mappings with a Contractive Iterate

A fixed point theorem for semi-groups of mappings with a contractive iterate

In a recent paper, Felix E. Browder discussed continuous self-mappings on a metric space, satisfying a functional inequality. Browder gave sufficient conditions such that the successive approximations of any point for such mappings converge to a unique fixed point. In the present paper, Browder’s result is extended to a commutative semigroup of mappings and also to single mappings that are not necessarily continuous and satisfy a weaker form of the functional inequality.

Fixed Point Theorems for Mappings with a Contractive Iterate at a Point

In a recent paper, V. M. Sehgal discussed continuous self-mappings of a complete metric space with a contractive iterate at each point of the space. Sehgal showed that such mappings have a unique fixed point and the sequence of iterates of any point in the space converges to the fixed point. In the present paper, the result of Sehgal is generalized to mappings which are not necessarily continuous and which have a contractive iterate at each point in a (possibly proper) subset of the space.

Fixed point theorems for mappings with a contractive iterate at a point

In a recent paper, V. M. Sehgal discussed continuous self-mappings of a complete metric space with a contractive iterate at each point of the space. Sehgal showed that such mappings have a unique fixed point and the sequence of iterates of any point in the space converges to the fixed point. In the present paper, the result of Sehgal is generalized to mappings which are not necessarily continuous and which have a contractive iterate at each point in a (possibly proper) subset of the space.

A Fixed Point Theorem for Mappings with a Contractive Iterate

A Fixed Point Theorem for Mappings with a Contractive Iterate

A fixed point theorem for mappings with a contractive iterate

THEOREM. Let (X, d) be a complete metric space, and f: X->X a continuous mapping satisfying the condition: there exists a k u for each xoE0X. LEMMA. If f: X)X be any mapping satisfying the condition of the above theorem then for each xEX, r(x) =sup,n d(fp(x), x) is finite. PROOF. Let xEX and let