Approximate Fixed Point Sequence Research Articles

On Annihilated Points and Approximate Fixed Points of General Higher-Order Nonexpansive Mappings

In this paper, we extend the results obtained by Ezearn on annihilated points for his higher-order nonexpansive mappings to the context of general higher-order nonexpansive mappings. Precisely in his thesis, Ezearn introduced the concept of annihilated points, which extends the notion of fixed points, and it is only meaningful in the context of higher-order nonexpansive mappings and gave some mild conditions when the annihilated points could exist in strictly convex Banach spaces. In the last direction, we also extend Ezearn’s result on the approximate fixed point sequence for higher-order nonexpansive mappings to general higher-order nonexpansive mappings.

Fixed point theorems for suzuki nonexpansive mappings in banach spaces

In this paper, we investigate the existence of fixed points for Suzuki nonexpansive mappings in the setting of Banach spaces using the asymptotic center technique. We also establish the convergence of regular approximate fixed point sequence to the fixed points of Suzuki nonexpansive mappings. Examples are also given to illustrate the results. Our theorems generalize several results in the literature.

Alternative proofs of some classical metric fixed point theorems by using approximate fixed point sequences

The notion of approximate fixed point sequence, emphasized in Chidume (Geometric properties of Banach spaces and nonlinear iterations. Lecture Notes in Mathematics, 1965. Springer-Verlag London, Ltd., London, 2009), is a very useful tool in proving convergence theorems for fixed point iterative schemes in the class of nonexpansive-type mappings. In the present paper, our aim is to present simple and unified alternative proofs of some classical fixed point theorems emerging from Banach contraction principle, by using a technique based on the concepts of approximate fixed point sequence and graphic contraction.

Inertial Iteration Scheme for Approximating Fixed Points of Lipschitz Pseudocontractive Maps in Arbitrary Real Banach Spaces

"We study a perturbed inertial Krasnoselskii-Mann-type algorithm and prove that the algorithm is an approximate fixed point sequence for Lipschitz pseudocontractive maps in arbitrary real Banach spaces. Strong convergence results are then established for our inertial iteration scheme for approximation of fixed points of Lipschitz pseudocontractive maps and solutions of certain important accretive-type operator equations in certain real Banach spaces. Implementation of our algorithm is illustrated using numerical examples in both finite and infinite dimensional Banach spaces. Our results improve rate of convergence and extend several related recent results. "

A Krasnoselskii–Ishikawa Iterative Algorithm for Monotone Reich Contractions in Partially Ordered Banach Spaces with an Application

Iterative algorithms have been utilized for the computation of approximate solutions of stationary and evolutionary problems associated with differential equations. The aim of this article is to introduce concepts of monotone Reich and Chatterjea nonexpansive mappings on partially ordered Banach spaces. We describe sufficient conditions for the existence of an approximate fixed-point sequence (AFPS) and prove certain fixed-point results using the Krasnoselskii–Ishikawa iterative algorithm. Moreover, we present some interesting examples to highlight the superiority of our results. Lastly, we provide both weak and strong convergence results for such mappings and consider an application of our results to prove the existence of a solution to an initial value problem.

Remarks on the approximate fixed point sequence of $$(\alpha ,\beta )$$-maps

Remarks on the approximate fixed point sequence of $$(\alpha ,\beta )$$-maps

Some results on (c)-mapping

Compared with [1], in this paper, We will give first some sufficient conditions under which a (c)-mapping possesses an approximate fixed point sequence (AFPS). And then, we will prove that (c)-mapping has a fixed point. Finally, we will check some special properties of the fixed point sets of these mappings, such as closedness, convexity.

Approximate Fixed Point Sequences of an Evolution Family on a Metric Space

In this article, we study the approximate fixed point sequence of an evolution family. A family E=Ux,y;x≥y≥0 of a bounded nonlinear operator acting on a metric space M,d is said to be an evolution family if Ux,x=I and Ux,yUy,z=Ux,z for all x≥y≥z≥0. We prove that the common approximate fixed point sequence is equal to the intersection of the approximate fixed point sequence of two operators from the family. Furthermore, we apply the Ishikawa iteration process to construct an approximate fixed point sequence of an evolution family of nonlinear mapping.

Approximations of Fixed Points in the Hadamard Metric Space CATp(0)

In this paper, we consider the recently introduced C A T p ( 0 ) , where the comparison triangles belong to ℓ p , for p ≥ 2 . We first establish an inequality in these nonlinear metric spaces. Then, we use it to prove the existence of fixed points of asymptotically nonexpansive mappings defined in C A T p ( 0 ) . Moreover, we discuss the behavior of the successive iteration introduced by Schu for these mappings in Banach spaces. In particular, we prove that the successive iteration generates an approximate fixed point sequence.

Some characterizations of Reich and Chatterjea type nonexpansive mappings

We introduce two types of mappings, namely Reich type nonexpansive and Chatterjea type nonexpansive mappings, and derive some sufficient conditions under which these two types of mappings possess an approximate fixed point sequence (AFPS). We obtain the desired AFPS using the well-known $$Sch\ddot{a}efer$$ iteration method. Along with these, we check some special properties of the fixed point sets of these mappings, such as closedness, convexity, remotality, unique remotality, etc. We also derive a nice interrelation between AFPS and maximizing sequence for both types of mappings. Finally, we will get some sufficient conditions under which the class of Reich type nonexpansive mappings reduces to that of nonexpansive maps.

Approximate fixed points of α-nonexpansive mappings

In this paper, we define the class of (α,β)-nonexpansive mappings which is properly larger than the class of α-nonexpansive mappings and prove that every (α,β)-nonexpansive mapping T:C→C has an approximate fixed point sequence, where C is a nonempty bounded subset of a Banach space X, α>0 and β≥0. This, in particular, gives an affirmative answer to the open question posed by Ariza-Ruiz and et al. concerning the existence of an approximate fixed point sequence for α-nonexpansive mappings, Ariza-Ruiz et al. (2016) [4].

On common approximate fixed points of monotone nonexpansive semigroups in Banach spaces

AbstractIn this paper, we investigate the common approximate fixed points of monotone nonexpansive semigroups of nonlinear mappings $\{T(t)\}_{t \geq0}$ { T ( t ) } t ≥ 0 , i.e., a family such that $T(0)x=x$ T ( 0 ) x = x , $T(s+t)x=T(s)\circ T(t)x $ T ( s + t ) x = T ( s ) ∘ T ( t ) x , where the domain is a Banach space. In particular we prove that under suitable conditions, the common approximate fixed points are the same as the common approximate fixed points set of two mappings from the family. Then we give an algorithm of how to construct an approximate fixed point sequence of the semigroup in the case of a uniformly convex Banach space.

Convergence theorems for finite family of a general class of multi-valued strictly pseudo-contractive mappings

Let H be a real Hilbert space, K a nonempty subset of H, and $T:K\rightarrow \mathit{CB}(K)$ a multi-valued mapping. Then T is called a generalized k-strictly pseudo-contractive multi-valued mapping if there exists $k\in[0,1)$ such that, for all $x,y\in D(T)$ , we have $D^{2}(Tx,Ty)\leq\|x-y\|^{2}+kD^{2}(Ax,Ay)$ , where $A:=I-T$ , and I is the identity operator on K. A Krasnoselskii-type algorithm is constructed and proved to be an approximate fixed point sequence for a common fixed point of a finite family of this class of maps. Furthermore, assuming existence, strong convergence to a common fixed point of the family is proved under appropriate additional assumptions.

Approximate Fixed Point Sequences of Nonlinear Semigroups in Metric Spaces

AbstractIn this paper, we investigate the common approximate fixed point sequences of nonexpansive semigroups of nonlinear mappings {T1}t≥0, i.e., a family such that T0(x) = x, Ts+t = Ts(Tt(x)), where the domain is a metric space (M; d). In particular, we prove that under suitable conditions the common approximate fixed point sequences set is the same as the common approximate fixed point sequences set of two mappings from the family. Then we use the Ishikawa iteration to construct a common approximate fixed point sequence of nonexpansive semigroups of nonlinear mappings.

A Remark on the Theorem of Ishikawa

Given a Lipschitz pseudocontractive mapping T from a closed convex and bounded subset K of a real Hilbert space H onto itself, and an arbitrary x1 ∈ K, a Krasnolselskii-type sequence defined by xn+1 = (1− λ)xn + λTyn, yn = (1− λ)xn + λTxn is proved to be an approximate fixed point sequence of T , for a suitable λ ∈ (0, 1). Under some suitable compactness assumptions on K or on T , the sequence converges strongly to a fixed point of T . The algorithm is simple and natural, and the theorems presented here improve the theorem of Ishikawa [1] and other similar results in the literature.

Krasnoselskii-type algorithm for family of multi-valued strictly pseudo-contractive mappings

A Krasnoselskii-type algorithm is constructed and the sequence of the algorithm is proved to be an approximate fixed point sequence for a common fixed point of a suitable finite family of multi-valued strictly pseudo-contractive mappings in a real Hilbert space. Under some mild additional compactness-type condition on the operators, the sequence is proved to converge strongly to a common fixed point of the family.

Fixed-Point Theorems for Mean Nonexpansive Mappings in Banach Spaces

We define a mean nonexpansive mappingTonXin the sense thatTx-Ty≤ax-y+bx-Ty,a,b≥0,a+b≤1. It is proved that mean nonexpansive mapping has approximate fixed-point sequence, and, under some suitable conditions, we get some existence and uniqueness theorems of fixed point.

Reflexivity is equivalent to the perturbed fixed point property for cascading nonexpansive maps in Banach lattices

Using a theorem of Domínguez Benavides and the Strong James’ Distortion Theorems, we prove that if a Banach space is a Banach lattice, or has an unconditional basis, or is a symmetrically normed ideal of operators on an infinite-dimensional Hilbert space, then it is reflexive if and only if it has an equivalent norm that has the fixed point property for cascading nonexpansive mappings. This new class of mappings strictly includes nonexpansive mappings.Also, using a theorem of Mil’man and Mil’man, we show that for the above classes of Banach spaces, reflexivity is equivalent to the fixed point property for affine cascading nonexpansive mappings.Further, we show that a cascading nonexpansive mapping on a closed, bounded, convex set in a Banach space always has an approximate fixed point sequence.

Krasnoselskii-type algorithm for fixed points of multi-valued strictly pseudo-contractive mappings

Let q>1 and let K be a nonempty, closed and convex subset of a q-uniformly smooth real Banach space E. Let T:K→CB(K) be a multi-valued strictly pseudo-contractive map with a nonempty fixed point set. A Krasnoselskii-type iteration sequence { x n } is constructed and proved to be an approximate fixed point sequence of T, i.e., lim n → ∞ d( x n ,T x n )=0. This result is then applied to prove strong convergence theorems for a fixed point of T under additional appropriate conditions. Our theorems improve several important well-known results.MSC:47H04, 47H06, 47H15, 47H17, 47J25.

Convergence Theorems for Fixed Points of Multivalued Strictly Pseudocontractive Mappings in Hilbert Spaces

LetKbe a nonempty, closed, and convex subset of a real Hilbert spaceH. Suppose thatT:K→2Kis a multivalued strictly pseudocontractive mapping such thatF(T)≠∅. A Krasnoselskii-type iteration sequence{xn}is constructed and shown to be an approximate fixed point sequence ofT; that is,limn→∞d(xn,Txn)=0holds. Convergence theorems are also proved under appropriate additional conditions.