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This document is a retraction notice published in the Proceedings of the American Mathematical Society. The author, Khadime Salame, apologizes to the mathematical community and withdraws three papers related to fixed point theory. The papers attempted to address the question of whether left amenable semitopological semigroups have a certain fixed point property, but each paper contains errors. The author acknowledges these errors and suggests that Lau's conjecture should be regarded as an open question. The author hopes that this question will be resolved in the future. [Extracted from the article]

In this paper, we developed a new type iterative scheme to approximate a common solution of split monotone variational inclusion, variational inequality and fixed point problems for an infinite family of nonexpansive mappings in the framework of Hilbert spaces. Further, we proved that the sequence generated by the proposed iterative method converges strongly to a common solution of split monotone variational inclusion, variational inequality and fixed point problems. Furthermore, we give some consequences of the main result. Finally, we discuss a numerical example to demonstrate the applicability of the iterative algorithm. The result presented in this paper unifies and extends some known results in this area. [ABSTRACT FROM AUTHOR]

Picard iteration is on the basis of a great number of numerical methods and applications of mathematics. However, it has been known since the 1950s that this method of fixed-point approximation may not converge in the case of nonexpansive mappings. In this paper, an extension of the concept of nonexpansiveness is presented in the first place. Unlike the classical case, the new maps may be discontinuous, adding an element of generality to the model. Some properties of the set of fixed points of the new maps are studied. Afterwards, two iterative methods of fixed-point approximation are analyzed, in the frameworks of b-metric and Hilbert spaces. In the latter case, it is proved that the symmetrically averaged iterative procedures perform well in the sense of convergence with the least number of operations at each step. As an application, the second part of the article is devoted to the study of fractal mappings on Hilbert spaces defined by means of nonexpansive operators. The paper considers fractal mappings coming from φ -contractions as well. In particular, the new operators are useful for the definition of an extension of the concept of α -fractal function, enlarging its scope to more abstract spaces and procedures. The fractal maps studied here have quasi-symmetry, in the sense that their graphs are composed of transformed copies of itself. [ABSTRACT FROM AUTHOR]

In the present paper, under certain assumptions, we establish the convergence of iterates for self-mappings of complete metric spaces with graphs which are of a contractive type. The class of mappings considered in the paper contains the so-called cyclical mappings introduced by W. A. Kirk, P. S. Srinivasan and P. Veeramani in 2003 and the class of monotone nonexpansive operators. Our results hold in the case of a symmetric graph. [ABSTRACT FROM AUTHOR]

Addressing the boundary value problems of fractional-order differential equations hold significant importance due to their applications in various fields. The aim of this paper was to approximate solutions for a class of boundary value problems involving Caputo fractional-order differential equations employing the AA-iterative scheme. Moreover, the stability and data dependence results of the iterative scheme were given for a certain class of mappings. Finally, a numerical experiment was illustrated to support the results presented herein. The results presented in this paper extend and unify some well-known comparable results in the existing literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose an iterative algorithm, which is based on the Mann iterative method for solving simultaneously common fixed point problem with a finite family of demicontractive mappings and systems of variational inequalities involving an infinite family of strongly accretive operators. Under suitable assumptions, we prove the strong convergence of this algorithm in Banach spaces. Application to systems of constrained convex minimization problem is provided to support our main results. The results of this paper improve and extend results of [M. Eslamian, C. R. Math. Acad. Sci. Paris, 355 (2017), 1168–1177], and of many others. [ABSTRACT FROM AUTHOR]

We devise an iterative algorithm incorporating inertial techniques to approximate the shared solution of a generalized equilibrium problem, a fixed point problem for a finite family of relatively nonexpansive multivalued mappings, and a variational inequality problem. Our discussion encompasses the strong convergence of the proposed algorithm and highlights specific outcomes derived from our theorem. Additionally, we provide a computational analysis to underscore the significance of our findings and draw comparisons. The results presented in this paper serve to extend and unify numerous previously established outcomes in this particular research domain. [ABSTRACT FROM AUTHOR]

This paper presents and examines a newly improved linear technique for solving the equilibrium problem of a pseudomonotone operator and the fixed point problem of a nonexpansive mapping within a real Hilbert space framework. The technique relies two modified mildly inertial methods and the subgradient extragradient approach. In addition, it can be viewed as an advancement over the previously known inertial subgradient extragradient approach. Based on common assumptions, the algorithm’s weak convergence has been established. Finally, in order to confirm the efficiency and benefit of the proposed algorithm, we present a few numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the strong convergence of the iteration scheme given by Akutsah et al., for mean nonexpansive mappings in uniformly convex Banach space endowed with directed graph. Also, we show fastness of Akutsah et al. iteration scheme with some well-known iteration schemes with the help of numerical examples. We, also present an application of fixed point theory in solution of integral equation. [ABSTRACT FROM AUTHOR]

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The purpose of this paper is to introduce a three step iterative algorithm which include a general viscosity explicit method for approximating a common solution of fixed point problem of an infinite family of ki-demimetric mapping and a directed operator in the framework of real Hilbert space. Furthermore, we prove a strong convergence theorem for approximating a common solution of the aforementioned problems. We also show that our iterative algorithm holds for an infinite family of L-Lipschitzian and quasi-pseudocontractive mapping together with a directed operator. The iterative algorithm presented in this article is design in such a way that it solves some variational inequality problem and no compactness condition is impose on our scheme and mapping. Finally, we give applications of our main result to variational inclusion and equilibrium problems. Our result complements and extends some related result in literature. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discuss some results concerning the demiclosedness principle of generalized, nonexpansive mappings in uniformly convex spaces. Further, we present some new fixed-point theorems for generalized nonexpansive mappings in different settings of Banach spaces. [ABSTRACT FROM AUTHOR]

This paper proposes a new inertial subgradient extragradient method for solving equilibrium problems with pseudomonotone and Lipschitz-type bifunctions and fixed point problems for nonexpansive mappings in real Hilbert spaces. Precisely, we prove that the sequence generated by proposed algorithm converges strongly to a common solution of equilibrium problems and fixed point problems. We use an effective self-adaptive step size rule to accelerate the convergence process of our proposed iterative algorithm. Moreover, some numerical results are given to show the effectiveness of the proposed algorithm. The results obtained in this paper extend and improve many recent ones in the literature. [ABSTRACT FROM AUTHOR]

In order to approximate the common solution of quasi-nonexpansive fixed point and pseudo-monotone variational inequality problems in real Hilbert spaces, this paper presented three new modified sub-gradient extragradient-type methods. Our algorithms incorporated viscosity terms and double inertial extrapolations to ensure strong convergence and to speed up convergence. No line search methods of the Armijo type were required by our algorithms. Instead, they employed a novel self-adaptive step size technique that produced a non-monotonic sequence of step sizes while also correctly incorporating a number of well-known step sizes. The step size was designed to lessen the algorithms' reliance on the initial step size. Numerical tests were performed, and the results showed that our step size is more effective and that it guarantees that our methods require less execution time. We stated and proved the strong convergence of our algorithms under mild conditions imposed on the control parameters. To show the computational advantage of the suggested methods over some wellknown methods in the literature, several numerical experiments were provided. To test the applicability and efficiencies of our methods in solving real-world problems, we utilized the proposed methods to solve optimal control and image restoration problems. [ABSTRACT FROM AUTHOR]

In this paper, we first introduced a new class of generalized non-expansive mappings, which was larger than the class satisfying the condition Bγ,μ. Also, we proposed a new iterative process to approximate the fixed point of the mapping we introduced in this work, then we prove convergence theorems for these mappings by using our iteration process. Lastly, a numerical example was given to show the efficiency of this new iteration process. Our results were the extension and generalization of many known results in the literature in fixed point theory. [ABSTRACT FROM AUTHOR]

This paper presents the stability theorem for the T-Picard iteration scheme and establishes the existence and uniqueness theorem for fixed points concerning T-mean nonexpansive mappings within b-metric-like spaces. The outcome of our fixed point theorem substantiated the existence and uniqueness of solutions to the Fredholm-Hammerstein integral equations defined on time scales. Additionally, we provided two numerical examples from distinct time scales to support our findings empirically. [ABSTRACT FROM AUTHOR]

This paper presents and analyzes a new viscosity approximation method with the inertial technique for finding a common fixed point of a countable family of nonexpansive mappings and then its strong convergence theorem is established under some suitable conditions. As a consequence, we employ our proposed algorithm for solving some convex bilevel optimization problems and then apply it for solving regression of a graph of cosine function and classification of some noncommunicable diseases by using the extreme learning machine model. We perform a comparative analysis with other algorithms to demonstrate the performance of our approach. Our numerical experiments confirm that our proposed algorithm outperforms other methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose a modified general inertial Mann algorithm and prove that it generates a sequence which converges weakly to a fixed point of a nonexpansive mapping in Hilbert spaces. Moreover, by using the viscosity method, we introduce a general inertial viscosity algorithm and prove that it generates a sequence which converges strongly to a common fixed point of a countable family of nonexpansive operators. We also derive schemes for solving constrained convex optimization, monotone inclusion, and nonsmooth convex optimization problems. Finally, we apply one of our proposed algorithms to solve image restoration problem. [ABSTRACT FROM AUTHOR]

The concept of symmetry has played a major role in Hilbert space setting owing to the structure of a complete inner product space. Subsequently, different studies pertaining to symmetry, including symmetric operators, have investigated real Hilbert spaces. In this paper, we study the solutions to multiple-set split feasibility problems for a pair of finite families of β -enriched, strictly pseudocontractive mappings in the setup of a real Hilbert space. In view of this, we constructed an iterative scheme that properly included these two mappings into the formula. Under this iterative scheme, an appropriate condition for the existence of solutions and strong and weak convergent results are presented. No sum condition is imposed on the countably finite family of the iteration parameters in obtaining our results unlike for several other results in this direction. In addition, we prove that a slight modification of our iterative scheme could be applied in studying hierarchical variational inequality problems in a real Hilbert space. Our results improve, extend and generalize several results currently existing in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose a hybrid Halpern-based extragradient algorithm for finding a common solution of a pseudomonotone equilibrium problem and a fixed point problem of a nonexpansive mapping. We prove the strong convergence of the proposed algorithm under some mixed conditions. Some numerical examples are provided to illustrate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper attempts to prove the Lipschitz continuity of the resolvent operator associated with a (P , η) -accretive mapping and compute an estimate of its Lipschitz constant. This is done under some new appropriate conditions that are imposed on the parameter and mappings involved in it; with the goal of approximating a common element of the solution set of a system of generalized variational-like inclusions and the fixed point set of a total asymptotically nonexpansive mapping in the framework of real Banach spaces. A new iterative algorithm based on the resolvent operator technique is proposed. Under suitable conditions, we prove the strong convergence of the sequence generated by our proposed iterative algorithm to a common element of the two sets mentioned above. The final section is dedicated to investigating and analyzing the notion of a generalized H(.,.)-accretive mapping introduced and studied by Kazmi et al. (Appl Math Comput 217:9679–9688, 2011). In this section, we provide some comments based on the relevant results presented in their work. [ABSTRACT FROM AUTHOR]

In this paper, we obtain an equivalent condition to the existence of a common fixed point of a given family of nonexpansive mappings defined on a Hadamard space. Moreover, if the space is bounded, we show that the generating process of the approximate sequence by a specific projection method will stop in finite steps if there is no common fixed point. It is a significant advantage to reveal the nonexistence of a common fixed point in a finite time. [ABSTRACT FROM AUTHOR]

We explore the intermixed method for finding a common element of the intersection of the solution set of a mixed variational inequality and the fixed point set of a nonexpansive mapping. We point out that Khuangsatung and Kangtunyakarn's statement [J. Inequal. Appl. 2023:1, 2023] regarding the resolvent utilized in their paper is not correct. To resolve this gap, we employ the epigraphical projection and the product space approach. In particular, we obtain a strong convergence theorem with a weaker assumption. [ABSTRACT FROM AUTHOR]

In this paper, we constructed a novel fixed point iterative scheme called the Modified-JK iterative scheme. This iteration process is a modification of the JK iterative scheme. Our scheme converged weakly to the fixed point of a nonexpansive mapping and strongly to the fixed point of a mapping satisfying condition (E). We provided some examples to show that the new scheme converges faster than some existing iterations. Stability and data dependence results were proved for this iteration process. To substantiate our results, we applied our results to solving delay differential equations. Furthermore, the newly introduced scheme was applied in approximating the solution of a class of third order boundary value problems (BVPs) by embedding Green's functions. Moreover, some numerical examples were presented to support the application of our results to BVPs via Green's function. Our results extended and generalized other existing results in literature. [ABSTRACT FROM AUTHOR]

The purpose of this paper was to prove that a novel algorithm with an inertial approach, used to generate an iterative sequence, strongly converges to a fixed point of a nonexpansive mapping in a real uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Furthermore, zeros of accretive mappings were obtained. The proposed algorithm has been implemented and tested via numerical simulation in MATLAB. The simulation results showed that the algorithm converges to the optimal configurations and shows the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we consider a faster iterative method for approximating the fixed points of generalized α -nonexpansive mappings. We prove several weak and strong convergence theorems of the considered method in mild conditions within the control parameters. In order to validate our findings, we present some nontrivial examples of the considered mappings. Furthermore, we show that the class of mappings considered is more general than some nonexpansive-type mappings. Also, we show numerically that the method studied in our article is more efficient than several existing methods. Lastly, we use our main results to approximate the solution of a delay fractional differential equation in the Caputo sense. Our results generalize and improve many well-known existing results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iterative algorithm of inertial form for approximating the common solution of Split Generalized Mixed Equilibrium Problem (SGMEP) and Hierarchical Fixed Point Problem (HFPP) in real Hilbert spaces. Motivated by the subgradient extragradient method, we incorporate the inertial technique to accelerate the convergence of the proposed method. Under standard and mild assumption of monotonicity and lower semicontinuity of the SGMEP and HFPP associated mappings, we establish the strong convergence of the iterative algorithm. Some numerical experiments are presented to illustrate the performance and behaviour of our method as well as comparing it with some related methods in the literature. [ABSTRACT FROM AUTHOR]

This article examines the process for solving the fixed-point problem of Bregman strongly nonexpansive mapping as well as the variational inequality problem of the pseudomonotone operator. Within the context of p-uniformly convex real Banach spaces that are also uniformly smooth, we introduce a modified Halpern iterative technique combined with an inertial approach and Tseng methods for finding a common solution of the fixed-point problem of Bregman strongly nonexpansive mapping and the pseudomonotone variational inequality problem. Using our iterative approach, we develop a strong convergence result for approximating the solution of the aforementioned problems. We also discuss some consequences of our major finding. The results presented in this paper complement and build upon many relevant discoveries in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study the generalized multiple-set split feasibility problem including the common fixed-point problem for a finite family of generalized demimetric mappings and the monotone inclusion problem in 2-uniformly convex and uniformly smooth Banach spaces. We propose an inertial Halpern-type iterative algorithm for obtaining a solution of the problem and derive a strong convergence theorem for the algorithm. Then, we apply our convergence results to the convex minimization problem, the variational inequality problem, the multiple-set split feasibility problem and the split common null-point problem in Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the split common fixed point problem for a finite family of demimetric mappings and a finite family of Bregman relatively nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces. We prove a strong convergence theorem of Halpern's type iteration for finding a solution of the split common fixed point problem. The iterative scheme does not require prior knowledge of operator norm. The obtained result of this paper complements many recent results in this direction. [ABSTRACT FROM AUTHOR]

In this paper, we study [m,C]-expansive operators and [m,Ĉ]-expansive operators on a Banach space. More precisely, after exploring their properties as operators, we examine the spectral properties of these two operator classes. [ABSTRACT FROM AUTHOR]

In this paper, we modified proximal point algorithm with some convex combination technique to approximate a minimizer, equilibrium point and a common fixed point of a family of nonexpansive mappings in Hilbert spaces. We establish a weak convergence theorem under some mild conditions. Moreover, we also provide a numerical example to illustrate the convergence behavior of the proposed iterative method. [ABSTRACT FROM AUTHOR]

We define and study enriched Suzuki mappings in Hadamard spaces. The results obtained here are extending fundamental findings previously established in related research. The extension is realized with respect to at least two different aspects: the setting and the class of involved operators. More accurately, Hilbert spaces are particular Hadamard spaces, while enriched Suzuki nonexpansive mappings are natural generalizations of enriched nonexpansive mappings. Next, enriched Suzuki nonexpansive mappings naturally contain Suzuki nonexpansive mappings in Hadamard spaces. Besides technical lemmas, the results of this paper deal with (1) the existence of fixed points for enriched Suzuki nonexpansive mappings and (2) Δ and strong (metric) convergence of Picard iterates of the α -averaged mapping, which are exactly Krasnoselskij iterates for the original mapping. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new three-step iterative scheme for finding the common solutions of the variational inequality using the technique of updating the solution. We suggest, iterative algorithms involving three-steps for the predictor-corrector method of variational inequality in real Hilbert spaces H. Our results include the Takahashi and Toyoda, extra gradient, Mann and Noor iterations as special cases. We also investigate the convergence criteria of the three-step iterative scheme. As special cases, the earlier findings are included in our results, which can be seen as an advancement and improvement over the previous investigation. This is a new refinement in our existing literature and previously known algorithms. A numerical example is given to illustrate the efficiency and performance of the proposed self-adaptive scheme. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose an inertial-type AA-viscosity algorithm for approximating the common solutions of the split variational inclusion problem, the generalized equilibrium problem and the common fixed-point problem of nonexpansive mappings. The strong convergence of an iterative sequence obtained through the proposed method is proved under some mild assumptions. Consequently, approximations of the solution of the split feasibility problem, the relaxed split feasibility problem, the split common null point problem and the split minimization problem are given. The applicability of our proposed algorithm has been illustrated with the help of a numerical example. Our iterative method was then compared graphically with different comparable methods in the existing literature. [ABSTRACT FROM AUTHOR]

Proximal mappings are essential in splitting algorithms for both convex and noncon-vex optimization. In this paper, we show that proximal mappings of every prox-bounded function are nonexpansive if and only if they are firmly nonexpansive if and only if they are averaged if and only if the function is convex. Lipschitz proximal mappings of prox-bounded functions are also characterized via hypoconvex or strongly convex functions. Our results generalize a recent result due to Rockafellar [ABSTRACT FROM AUTHOR]

A new iterative algorithm is proposed for finding a common solution of an equilibrium problem and a fixed point problem. Then, a strong convergence theorem is proved. As a consequence, they can be obtained some strong convergence theorems for an equilibrium problem and a split common fixed point problem. The obtained theorems can be applied to solve an equilibrium problem and a split common null point problem. The results presented in this paper extend and improve some corresponding ones in the literature. Finally, a numerical example is given to show the validity of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, two generalized algorithms for solving the variational inequality problem in Banach spaces are proposed. Then the strong convergence of the sequences generated by these algorithms will be proved under suitable conditions. Finally, using MATLAB software, we provide some numerical examples to illustrate our results. [ABSTRACT FROM AUTHOR]

We give a brief account on a basic result (Lemma 3.2) which is a very useful tool in proving various convergence theorems in the framework of the iterative approximation of fixed points of demicontractive mappings in Hilbert spaces. This Lemma relates the class of quasi-nonexpansive mappings, by one hand, and the class of k-demicontractive mappings (quasi k-strict pseudocontractions), on the other hand and essentially states that the class of demicontractive mappings, which strictly includes the class of quasi-nonexpansive mappings, can be embedded in the later by means of an averaged perturbation. From the point of view of the fixed point problem, this means that any convergence result for Krasnoselskij-Mann iterative algorithms in the class of k-demicontractive mappings can be derived from its corresponding counterpart from quasi-nonexpansive mappings. [ABSTRACT FROM AUTHOR]

It is known that a strict contraction on a complete metric space with a graph possesses a fixed point. In the present paper, we show that this property holds for single valued and set-valued self-mappings of metric spaces with graphs that are of the contractive type. We also show the convergence of iterates of these mappings to fixed points. In particular, our results are true for metric spaces with symmetric graphs. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a novel approach for estimating the fixed points of multi-valued non-expansive mappings using the AR-iteration scheme in the context of uniformly convex Banach spaces. We establish strong and weak convergence results, providing rigorous analytical proofs and illustrating the results with a detailed example. Our approach showcases the potential of the AR-iteration scheme in solving real-world problems, particularly in addressing two-point boundary value problems. We demonstrate the applicability and effectiveness of the AR-iteration scheme in numerical analysis and computational mathematics. Our results contribute significantly to the advancement of numerical methods in solving boundary value problems, offering new insights and directions for future research in this area. Furthermore, we provide a detailed explanation of Green’s function approach and its implications for various scientific and engineering applications, paving the way for future research in this area. [ABSTRACT FROM AUTHOR]

The aim of this note is threefold: first, to present a few relevant facts about the way in which the technique of enriching contractive mappings was introduced; secondly, to expose the main contributions in the area of enriched mappings established by the authors and their collaborators by using this technique; and third, to survey some related developments in the very recent literature which were authored by other researchers. [ABSTRACT FROM AUTHOR]

The study of the zero existence problem for a nonnegative set-valued functional on a metric space is continued. The zero existence problem for a functional related by a certain -continuity condition to a parametric family of -search functionals on an open subset of a metric space is examined. A theorem containing several sufficient conditions for this functional to have zeros is proved. As corollaries of this result, theorems on the existence of coincidence and fixed points are also proved for set-valued mappings related by the -continuity condition to families of set-valued mappings with the property that the existence of coincidence and fixed points in an open subset of a metric space is preserved under parameter variation. For uniformly convex metric spaces, analogs of M. Edelstein's 1972 asymptotic center theorem and M. Frigon's 1996 fixed point theorem for nonexpansive mappings of Banach spaces are obtained and compared with the main results of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we establish strong and weak convergence theorems for Suzuki's generalized nonexpansive mapping in uniformly convex Banach spaces using the iterative scheme introduced by Panigrahy et al [9]. Next, we see an example of Suzuki's generalized nonexpansive mapping, which is not a nonexpansive mapping. Using this example and some numerical tests, we infer empirically that the Panigrahy iteration process converges faster than the Krasnoselskij, Thakur, and M-iteration processes. [ABSTRACT FROM AUTHOR]

We study simple, properly ordered nonexpansive Bratteli-Vershik ($ BV $) systems. Correcting a mistake in an earlier paper, we redefine the classes standard nonexpansive ($ SNE $) and strong standard nonexpansive ($ SSNE $). We define also the classes of very well timed and well timed systems, their opposing classes of untimed and very untimed systems (which feature, as subclasses of 'Case (2)', in the work of Downarowicz and Maass as well as Hoynes on expansiveness of $ BV $ systems of finite topological rank), and several related classes according to the existence of indistinguishable pairs (of some 'depth') and their synchronization ('common cuts'). We establish some properties of these types of systems and some relations among them. We provide several relevant examples, including a problematic one that is conjugate to a well timed system while also (vacuously) in the classes 'Case (2)'. We prove that the class of all simple, properly ordered nonexpansive $ BV $ systems is the disjoint union of the ones conjugate to well timed systems and those conjugate to untimed systems, thereby showing that nonexpansiveness in $ BV $ systems arises in one of two mutually exclusive ways. [ABSTRACT FROM AUTHOR]

This paper focuses on investigating the problem of finding a common element of the set of solutions of a generalized nonlinear implicit variational-like inclusion problem involving an $ (A,\eta) $ (A , η) -maximal m-relaxed monotone mapping in the sense of L.-G.-s.i.p. and the set of fixed points of a total asymptotically nonexpansive mapping. To achieve such a purpose, a new iterative algorithm is constructed. Applying the concepts of graph convergence and generalized resolvent operator associated with an $ (A,\eta) $ (A , η) -maximal m-relaxed monotone mapping in the sense of L.-G.-s.i.p. As an application of the obtained equivalence relationship, the strong convergence of the sequence generated by our proposed iterative algorithm to a point belonging to the intersection of the two sets mentioned above is proved. [ABSTRACT FROM AUTHOR]

We provide a concrete isometric description of all the preduals of ⃵1 for which the standard basis in ⃵1 has a finite number of w*-limit points. Then, we apply this result to give an example of an ⃵1-predual X such that its dual X* lacks the weak* fixed point property for nonexpansive mappings (briey, wα-FPP), but X does not contain an isometric copy of any hyperplane Wα of the space c of convergent sequences such that Wα is a predual of ⃵1 and W*α ff lacks the w*-FPP. This answers a question left open in the 2017 paper of the present authors. [ABSTRACT FROM AUTHOR]