Fixed Point Sequence Research Articles

Fixed point results for almost ($ \zeta -\theta _{\rho } $)-contractions on quasi metric spaces and an application

<abstract><p>This research paper investigated fixed point results for almost ($ \zeta-\theta _{\rho } $)-contractions in the context of quasi-metric spaces. The study focused on a specific class of ($ \zeta -\theta _{\rho } $)-contractions, which exhibit a more relaxed form of contractive property than classical contractions. The research not only established the existence of fixed points under the almost ($ \zeta -\theta _{\rho } $)-contraction framework but also provided sufficient conditions for the convergence of fixed point sequences. The proposed theorems and proofs contributed to the advancement of the theory of fixed points in quasi-metric spaces, shedding light on the intricate interplay between contraction-type mappings and the underlying space's quasi-metric structure. Furthermore, an application of these results was presented, highlighting the practical significance of the established theory. The application demonstrated how the theory of almost ($ \zeta -\theta _{\rho } $)-contractions in quasi-metric spaces can be utilized to solve real-world problems.</p></abstract>

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. "

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.

Convergence of proximal splitting algorithms in operatorname{CAT}(kappa) spaces and beyond

In the setting of operatorname{CAT}(kappa) spaces, common fixed point iterations built from prox mappings (e.g. prox-prox, Krasnoselsky–Mann relaxations, nonlinear projected-gradients) converge locally linearly under the assumption of linear metric subregularity. Linear metric subregularity is in any case necessary for linearly convergent fixed point sequences, so the result is tight. To show this, we develop a theory of fixed point mappings that violate the usual assumptions of nonexpansiveness and firm nonexpansiveness in p-uniformly convex spaces.

Strong and pure fixed point properties of mappings on normed spaces

In this paper, the concept of strong fixed point property of a mapping is defined. The relationship between an approximating fixed point sequence and compactness is studied. The notion of a center set for a mapping is introduced. Some necessary and sufficient conditions for existence of mappings admitting a center set are presented. The results proved herein extend, generalize and improve some well-known comparable results.

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.

Hyperbolic spaces and directional contractions

The axiomatic approach to metric convexity goes back to the pioneering work of Karl Menger in 1928. This is an overview of this concept and the role it plays in metric fixed point theory especially in conjunction with spaces possessing a “hyperbolic” type structures. These include the CAT(0) spaces, hyperconvex metric spaces, and [Formula: see text]-trees. Much of the discussion involves the existence of “approximate” fixed point sequences for mappings satisfying weak contractive conditions. Applications of a well-known fixed point theorem due to Caristi are also included. These involve fixed and approximate fixed points for mappings satisfying local “directional” contractive and non-expansive conditions. Convexity plays a role in this part of the discussion as well. While the paper is semi-expository in nature, some detailed proofs appear here for the first time. Also the concept of a weak [Formula: see text]-directional contraction introduced in Sec. 8 appears to be new. Several suggestions for further research are also discussed.

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.

Faster multicollision attack on Davies-Meyer hash function scheme implementing Simeck32/64 block cipher algorithm

Davies-Meyer is one scheme among 12 compression functions found through systematic analysis by Preneel et al. to be provably secure under black-box analysis. But in the development, this scheme proved to be vulnerable to fixed-point attack. With this vulnerability, it is possible to implement one of attack in an iterated hash function that exploits fixed-point weakness named faster multicollision attack. Implementing Simeck32/64 as an underlying block cipher, the attack induced by firstly searching for fixed-point collisions. To accomplish this finding stage based on Yuval’s birthday attack, a sum of 2.2n/2 different fixed-point sequences are needed. Based on that, two sets of 216 different inputs are generated by modifying 16 bits of least significant bits from each pair of five input samples to find a collision between them. The final result makes an outstanding fact, with 4.194.304 total collisions obtained from five samples and 16 different IV values that already produced before in fixed-point collision finding stage. These facts conclude that Davies-Meyer scheme is not resistance against faster multicollision attack because of its fixed-point weakness.

Necessary conditions for linear convergence of iterated expansive, set-valued mappings

We present necessary conditions for monotonicity, in one form or another, of fixed point iterations of mappings that violate the usual nonexpansive property. We show that most reasonable notions of linear-type monotonicity of fixed point sequences imply {\em metric subregularity}. This is specialized to the alternating projections iteration where the metric subregularity property takes on a distinct geometric characterization of sets at points of intersection called {\em subtransversality}. Our more general results for fixed point iterations are specialized to establish the necessity of subtransversality for consistent feasibility with a number of reasonable types of sequential monotonicity, under varying degrees of assumptions on the regularity of the sets.

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].

Convergence theorems to common fixed points of multi-valued ρ-quasi-nonexpansive mappings in modular function spaces

We constructed a Mann-type iteration process for a finite family of multivalued mappings in modular function spaces and the sequence of the algorithm is proved to be a common ρ-approximate fixed point sequence. Using this algorithm we also proved the ρ-converge theorems to common fixed points of finite family of mappings. By doing so we extended very recent results of Zegeye et al.

Fixed point iteration for a countable family of multi-valued strictly pseudo-contractive-type mappings

This paper introduces a new averaged algorithm for finding a common fixed point of a countably infinite family of generalized k-strictly pseudocontractive multi-valued mappings. The new iterative sequence introduced is proved to be an approximating fixed point sequence for common fixed points of a countably infinite family of this class of mappings. Furthermore, under some mild assumptions, strong convergence theorems are also proved for this class of mappings. The method of proof used here is new and enables to overcome many strong restrictions appearing in contemporary literature. The stated theorems improve and generalize many recent works in iterative scheme for multi-valued mappings.

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.