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Fixed point theory is a significant area of mathematical analysis with applications across various fields such as differential equations, optimization, and dynamical systems. Recently, multivalued mappings have gained attention due to their ability to model more complex and realistic problems. ln this work, novel classes of nonlinear mappings called k-strictly asymptotically demicontractive-type and asymptotically hemicontractive-type multivalued mappings are introduced in real Hilbert spaces that are symmetric spaces. In addition, we discuss the weak and strong convergence results by considered modified algorithms, and a demiclosedness property, for these classes of mappings are proved. Several non-trivial examples are demonstrated to validate the newly defined mappings. Consequently, the results and iterative methods obtained in this study improve and extend several known outcomes in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a modified adaptive step-size algorithm by basing a generalized Popov's method to solve variational inequality problems in a real Hilbert space. The algorithm only needs to act on the operator once time in each iteration, which greatly reduces the calculation time and calculation difficulty. Of particular interesting, by using our new same generalized Popov's method, we also weaken the conditions of the operator about weak continuity and quasi-monotone property. Of particular interesting, by using our new same generalized Popov's method, we can obtain a weak convergence result under the condition that the operator A is quasi-monotone and a R-linear convergence result under the condition that the operator A is strongly pseudo-monotone in the variational inequality, respectively. Finally, the feasibility and effectiveness of the algorithm are verified by some numerical experiments. [ABSTRACT FROM AUTHOR]

This work proposes a novel Noor iterative scheme, called CT‐iteration, to approximate the fixed points in the new context of generalized nonexpansive mappings with property (E)$$ (E) $$. We establish the strong and weak convergence results in a uniformly convex Banach space. Additionally, numerical examples of the iterative technique are demonstrated using a signal recovery application in a compressed sensing situation. Furthermore, we show the use of the proposed method to generate polynomiographs. The proposed algorithm has been implemented and tested via numerical simulation in MATLAB. The simulation results show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper aims to propose a double inertial Mann-type extragradient algorithm using non-monotonic step size and establish a weak convergence result to find a common solution to the variational inequality problem involving pseudo-monotone mapping and fixed point problem for quasi-nonexpansive mapping. Moreover, we establish a strong convergence for the variational inequality involving strongly pseudo-monotone mapping. To illustrate our algorithm’s potential, we present several numerical experiments, including the Nash–Cournot oligopolistic market equilibrium model and fractional programming problem. We also employ performance profiles to discuss the efficiency of our algorithm over earlier established algorithms based on two performance metrics. [ABSTRACT FROM AUTHOR]

In this paper, we discuss various consequences of Hahn-Banach theorem for bounded b-linear functional in linear n-normed space and describe the notion of reexivity of linear n-normed space with respect to bounded b-linear functional. The concepts of strong convergence and weak convergence of a sequence of vectors with respect to bounded b-linear functionals in linear n-normed space have been introduced and some of their properties are being discussed. [ABSTRACT FROM AUTHOR]

In this paper, we propose a reflected forward-backward splitting algorithm with two different inertial extrapolations to find a zero of the sum of three monotone operators consisting of the maximal monotone operator, Lipschitz continuous monotone operator, and a cocoercive operator in real Hilbert spaces. One of the interesting features of the proposed algorithm is that both the Lipschitz continuous monotone operator and the cocoercive operator are computed explicitly each with one evaluation per iteration. We then obtain weak and strong convergence results under some mild conditions. We finally give a numerical illustration to show that our proposed algorithm is effective and competitive with other related algorithms in the literature. [ABSTRACT FROM AUTHOR]

This paper presents a projection and contraction method with a double inertial extrapolation step and self-adaptive step sizes to solve variational inequalities with quasi-monotonicity in real Hilbert spaces. Weak and strong convergence results are obtained under some mild conditions. We also give linear convergence results under a special case of our proposed method. Preliminary numerical results show that our proposed method is competitive with other related methods in the literature. [ABSTRACT FROM AUTHOR]

Fixed point theory is a significant area of mathematical analysis with applications across various fields such as differential equations, optimization, and dynamical systems. Recently, multivalued mappings have gained attention due to their ability to model more complex and realistic problems. ln this work, novel classes of nonlinear mappings called k-strictly asymptotically demicontractive-type and asymptotically hemicontractive-type multivalued mappings are introduced in real Hilbert spaces that are symmetric spaces. In addition, we discuss the weak and strong convergence results by considered modified algorithms, and a demiclosedness property, for these classes of mappings are proved. Several non-trivial examples are demonstrated to validate the newly defined mappings. Consequently, the results and iterative methods obtained in this study improve and extend several known outcomes in the literature.

This manuscript is devoted to presenting some convergence results of a three-step iterative scheme under the Chatterjea–Suzuki–C ((CSC), for short) condition in the setting of a Banach space. Also, an example of mappings satisfying the (CSC) condition with a unique fixed point is provided. This example proves that the proposed scheme converges to a fixed point of a weak contraction faster than some known and leading schemes. Finally, our main results will be applied to find a solution to functional and fractional differential equations (FDEs) as an application.

This paper proposes a modified D-iteration to approximate the solutions of three quasi-nonexpansive multivalued mappings in a real Hilbert space. Due to the incorporation of an inertial step in the iteration, the sequence generated by the modified method converges faster to the common fixed point of the mappings. Furthermore, the generated sequence strongly converges to the required solution using a shrinking technique. Numerical results obtained indicate that the proposed iteration is computationally efficient and outperforms the standard forward-backward with inertial step.

This manuscript is devoted to presenting some convergence results of a three-step iterative scheme under the Chatterjea–Suzuki–C ((CSC), for short) condition in the setting of a Banach space. Also, an example of mappings satisfying the (CSC) condition with a unique fixed point is provided. This example proves that the proposed scheme converges to a fixed point of a weak contraction faster than some known and leading schemes. Finally, our main results will be applied to find a solution to functional and fractional differential equations (FDEs) as an application. [ABSTRACT FROM AUTHOR]

The subgradient extragradient method with inertial extrapolation step xn + θn(xn − xn−1) (also known as inertial subgradient extragradient method) has been studied extensively in the literature for solving variational inequalities and equilibrium problems. Most of the inertial subgradient extragradient methods in the literature for both variational inequalities and equilibrium problems have not considered the special case when the inertial factor θn = 1. The convergence results have always been obtained when the inertial factor θn is assumed 0 ≤ θn < 1. This paper considers the relaxed inertial version of subgradient extragradient method for equilibrium problems with 0 ≤ θn ≤ 1. We give both weak and strong convergence results using this inertial subgradient extragradient method and also give some numerical illustrations. [ABSTRACT FROM AUTHOR]

In this paper, three G-nonexpansive mappings are implemented and analyzed using an efficient modified three-step iteration algorithm. Assuming coordinate-convexity in a uniformly convex Banach space endowed with a directed graph, conditions for the weak and strong convergence of the scheme are determined. We give numerical comparisons to back up our main theorem, and compare our algorithm's convergence behavior to that of the three-step Noor iteration and SP-iteration. We use our proposed algorithm to solve image deblurring problems as an application. In addition, we discuss a novel approach to signal recovery in situations where the type of noise is unknown.

The inertial projection and contraction method and the inertial forward-backward-forward method have been studied for solving variational inequality problems due to the ability of these methods to work well with only one projection per iteration during computations. The inertial factor in these methods is chosen to be less than 1 in most of the papers that studied these methods. It is known from a computational point of view that the efficiency of these methods improves as the inertial factor approaches 1. In this paper, we modify both the inertial projection and contraction method and the inertial forward-backward-forward method with the possibility of inertial factor taken as 1. Our proposed methods also involve a step-size rule which does not depend on the Lipschitz constant of the cost function and without any line search rule. We analyse the weak convergence of the methods under appropriate conditions. We also modify the proposed methods so that strong convergence is obtained. Preliminary computational results show that our proposed methods are promising and show better performance than some variants of inertial projection and contraction methods and inertial forward-backward-forward methods where the inertial factor is assumed to be less than 1. [ABSTRACT FROM AUTHOR]

In this article, we develop a faster iteration method, called the A ∗ ∗ iteration method, for approximating the fixed points of almost contraction mappings and generalized α -nonexpansive mappings. We establish some weak and strong convergence results of the A ∗ ∗ iteration method for fixed points of generalized α -nonexpansive mappings in uniformly convex Banach spaces. We provide a numerical example to illustrate the efficiency of our new iteration method. The weak w 2 -stability result of the new iteration method is also studied. As an application of our main results, we approximate the solution of a fractional Volterra–Fredholm integro-differential equation. Our results improve and generalize several well-known results in the current literature. [ABSTRACT FROM AUTHOR]

The motive of this paper is to study the convergence analysis of a modified iteration procedure for total asymptotically nonexpansive mapping under some suitable conditions in the setting of CAT(0) spaces. By using MATLAB R2018a, we also illustrate numerical experiment to compare the rate of convergence of the new iteration process with some existing iteration processes.

Abstract In this paper, we study a modified extragradient method for computing a common solution to the split equilibrium problem and fixed point problem of a nonexpansive semigroup in real Hilbert spaces. The weak and strong convergence characteristics of the proposed algorithm are investigated by employing suitable control conditions in such a setting of spaces. As a consequence, we provide a simplified analysis of various existing results concerning the extragradient method in the current literature. We also provide a numerical example to strengthen the theoretical results and the applicability of the proposed algorithm.

In this paper, we present a modified SP iteration with the inertial technical term for three quasi-nonexpansive multivalued mappings in a Hilbert space. We then obtain weak convergence theorem under suitable conditions. The strong convergence theorems are given using CQ and shrinking projection methods with our modified iteration. Finally, we test some numerical experiments to illustrate that our inertial forward–backward method with the inertial technique term has a more effective convergence than that of the standard forward–backward method and Halpern algorithm. [ABSTRACT FROM AUTHOR]

Abstract In this work, the three-dimensional model for the compressible micropolar fluid flow is considered, whereby it is assumed that the fluid is viscous, perfect, and heat conducting. The flow between two coaxial thermoinsulated cylinders, which leads to a cylindrically symmetric model with homogeneous boundary data for velocity, microrotation, and heat flux, is analyzed. The corresponding PDE system is formulated in the Lagrangian setting, and it is proven that this system has a generalized solution locally in time.

In this article, we develop a faster iteration method, called the A∗∗ iteration method, for approximating the fixed points of almost contraction mappings and generalized α-nonexpansive mappings. We establish some weak and strong convergence results of the A∗∗ iteration method for fixed points of generalized α-nonexpansive mappings in uniformly convex Banach spaces. We provide a numerical example to illustrate the efficiency of our new iteration method. The weak w2-stability result of the new iteration method is also studied. As an application of our main results, we approximate the solution of a fractional Volterra–Fredholm integro-differential equation. Our results improve and generalize several well-known results in the current literature.

In this paper, we introduce a new mixed type iterative process, which approximates the common fixed points of single-valued nonexpansive mappings and two multi-valued nonexpansive mappings in a uniformly convex Banach space. We establish strong and weak convergence theorems for the new iterative process in Banach space and give their corresponding applications. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to modify Abbas iteration for two G-nonexpansive mappings. We perform the convergence analysis of newly defined iteration process in uniformly convex Banach space. To compare the rate of convergence of the proposed method with well known Ishikawa iteration, modified S-iteration and Thianwan new iteration processes, a numerical example is also constructed. [ABSTRACT FROM AUTHOR]

We develop versions of the subgradient extragradient method for variational inequalities in Hilbert spaces and establish sufficient conditions for their convergence. First we prove a sufficient condition for a weak convergence of a recent existing algorithm under relaxed assumptions. Then, we propose two other algorithms. Both weak and strong convergence of the considered algorithms are studied. Under additional strong pseudomonotonicity and Lipschitz continuity assumptions, we obtain also a Q -linear convergence rate of these algorithms. Our results improve some recent contributions in the literature. Illustrative numerical experiments are also provided by the end of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new kind of a relaxed CQ algorithm to find the solution of the multiple-set split feasibility problem and the equilibrium problem in a Hilbert space. We prove weak and strong convergence theorems to the proposed algorithm under some mild conditions. Finally, we provide some numerical experiments to show the efficiency and the implementation of our method. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ ℝn be a bounded Lipschitz domain. Let $L: {\mathbb R}^n\rightarrow \bar {\mathbb R}= {\mathbb R}\cup \{+\infty \}$ be a continuous function with superlinear growth at infinity, and consider the functional $\mathcal {I}(u)=\int \nolimits _\Omega L(Du)$ , u ∈ W1,1(Ω). We provide necessary and sufficient conditions on L under which, for all f ∈ W1,1(Ω) such that $\mathcal {I}(f) , the problem of minimizing $\mathcal {I}(u)$ with the boundary condition u|∂Ω = f has a solution which is stable, or – alternatively – is such that all of its solutions are stable. By stability of $\mathcal {I}$ at u we mean that $u_k\rightharpoonup u$ weakly in W1,1(Ω) together with $\mathcal {I}(u_k)\to \mathcal {I}(u)$ imply uk → u strongly in W1,1(Ω). This extends to general boundary data some results obtained by Cellina and Cellina and Zagatti. Furthermore, with respect to the preceding literature on existence results for scalar variational problems, we drop the assumption that the relaxed functional admits a continuous minimizer. [ABSTRACT FROM AUTHOR]

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.

In this paper, we deal with the approximation of fixed point for multivalued nonexpansive mappings via a new three-step algorithm which is independent and faster than the iterative process discussed by Agarwal et al. [3] and establish some strong convergence theorems and a weak convergence theorem in the setting of Banach spaces. Our results extend and generalize the previous works from the existing literature. [ABSTRACT FROM AUTHOR]

In this work, we suggest modifications of the self-adaptive method for solving the split feasibility problem and the fixed point problem of nonexpansive mappings in the framework of Banach spaces. Without the assumption on the norm of the operator, we prove that the sequences generated by our algorithms weakly and strongly converge to a solution of the problems. The numerical experiments are demonstrated to show the efficiency and the implementation of our algorithms. [ABSTRACT FROM AUTHOR]

In this article, we introduce a faster iteration for finding a fixed point of G-monotone nonexpansive mapping in a uniformly convex Banach space with a directed graph. We establish weak and strong convergence theorems of fixed point for G-monotone nonexpansive mapping with a more convenient G-convex interval instead of the previous fixed point dominated conditions in convergence analysis. Moreover, we provide two numerical examples to illustrate the convergence behavior and advantages of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we study multivalued version of Thakur et al. [26] threestep iteration process and establish some strong and a weak convergence theorem in the setting of Banach spaces. Our results extend and generalize some corresponding recent results in the existing literature. [ABSTRACT FROM AUTHOR]

We investigate H-distributions for sequences in the dual pairs of Bessel spaces, (Hsq,H-sp),s∈R,q>1 and q=p/(p-1), by the use of unbounded multipliers, with the finite regularity, as test functions. The results relating weak convergence, H-distributions and strong convergence are applied in the analysis of strong convergence for a sequence of approximated solutions to a class of differential equations P(x,D)un=fn, where P(x, D) is a differential operator of order k with coefficients in the Schwartz class and (fn) is a strongly convergent sequence in an appropriate Bessel potential space. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an extension of multiple set split variational inequality problem (Censor et al. Numer. Algor. 59, 301-323 2012) to multiple set split equilibrium problem (MSSEP) and propose two new parallel extragradient algorithms for solving MSSEP when the equilibrium bifunctions are Lipschitz-type continuous and pseudo-monotone with respect to their solution sets. By using extragradient method combining with cutting techniques, we obtain algorithms for these problems without using any product space. Under certain conditions on parameters, the iteration sequences generated by the proposed algorithms are proved to be weakly and strongly convergent to a solution of MSSEP. An application to multiple set split variational inequality problems and a numerical example and preliminary computational results are also provided. [ABSTRACT FROM AUTHOR]

In this paper, we construct two iteration schemes for approximating a common element of the set of solutions of equilibrium problems (GMEP and GEP) and the set of common fixed points of a finite family of k-strictly asymptotically pseudo-contractions in Hilbert spaces. Fixed point theorems are established in Hilbert spaces. Numerical examples and applications are provided. The main results of this paper modify and improve many important recent results in the literature. [ABSTRACT FROM AUTHOR]

In this paper we have defined a multistep iterative scheme with errors involving a family of asymptotically nonexpansive nonself mappings in Banach spaces. A retraction has been used in the construction of theiteration. We prove here weak and strong convergences of the iteration to common fixed points of the family of asymptotically nonexpansive nonself mappings. We have used several concepts of Banach space geometry. Our results improve and extend some recent results.En este artículo definimos un esquema de multi paso iterativo con errores que involucran una familia de aplicaciones no expansivas y no auto asintóticamente en espacios deBanach. Una retracción se ha usado en la construcción de la iteración. Probamos convergencias débiles y fuertes de las iteraciones a puntos fijos clásicos de la familia de aplicaciones no expansivas no auto asintóticamente. Hemos usado varios conceptos de geometría en espacios de Banach. Nuestro resultado mejora y extiende algunos resultados recientes.

In this paper, we introduce an iteration process for approximating common fixed points of two nonself asymptotically nonexpansive map- pings in Banach spaces. Our process contains Mann iteration process and some other processes for nonself mappings but is independent of Ishikawa iteration process. We prove some weak and strong convergence theorems for this iteration process. Our results generalize and improve some results in contemporary literature.

The purpose of this paper is to prove convergence of a one-step iterative algorithm to approximate a common fixed point of finite family of asymptotically quasi-nonexpansive mappings in a uniformly convex Banach space by assuming some control conditions on the parameters. Our results extend and improve the corresponding results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we first propose a weak convergence algorithm, called the linesearch algorithm, for solving a split equilibrium problem and nonexpansive mapping (SEPNM) in real Hilbert spaces, in which the first bifunction is pseudomonotone with respect to its solution set, the second bifunction is monotone, and fixed point mappings are nonexpansive. In this algorithm, we combine the extragradient method incorporated with the Armijo linesearch rule for solving equilibrium problems and the Mann method for finding a fixed point of an nonexpansive mapping. We then combine the proposed algorithm with hybrid cutting technique to get a strong convergence algorithm for SEPNM. Special cases of these algorithms are also given. [ABSTRACT FROM AUTHOR]

A new class of contractive mappings called pointwise asymptotically ϕ-strict pseudo-contractions in Hilbert spaces is introduced and weak convergence of the sequence generated by Mann's iterative scheme to a fixed point of a uniformly Lipschitzian and pointwise asymptotically ϕ-strict pseudo-contractive mappingTin a Hilbert space is established. Also, a new kind of monotone hybrid method which is a modification of Mann's iterative scheme for finding a common fixed point of an infinitely countable family of uniformly Lipschitzian and pointwise asymptotically ϕ-strict pseudo-contractive mappings is proposed. Strong convergence of the sequence generated by the proposedmonotone hybrid method for an infinitely countable family of uniformly Lipschitzian and pointwise asymptotically ϕ-strict pseudo-contractive mappings in a Hilbert space is also shown. The results presented in this article extend and improve some known results in the literature. [ABSTRACT FROM PUBLISHER]

The theory of Ky Fan minimax inequalities provides a powerful general framework for the study of convex programming, variational inequalities and economic equilibrium problems. One of the fundamental methods for finding a solution of Ky Fan minimax inequalities is the proximal point algorithm, where a lot of papers have been dedicated to this subject. In this paper, a general class of two-level hierarchical Ky Fan minimax inequalities is introduced in real Hilbert spaces. For a wide class of Bregman functions, an association of inexact implicit Bregman-penalization proximal and Bregman-splitting proximal algorithms are suggested and analysed. Weak and strong convergences are proved under essentially weaker conditions. We conclude this paper with a hierarchical minimization problem and a numerical example. [ABSTRACT FROM PUBLISHER]

We consider the multivariate point process determined by the crossing times of the components of a multivariate jump process through a multivariate boundary, assuming to reset each component to an initial value after its boundary crossing. We prove that this point process converges weakly to the point process determined by the crossing times of the limit process. This holds for both diffusion and deterministic limit processes. The almost sure convergence of the first passage times under the almost sure convergence of the processes is also proved. The particular case of a multivariate Stein process converging to a multivariate Ornstein–Uhlenbeck process is discussed as a guideline for applying diffusion limits for jump processes. We apply our theoretical findings to neural network modeling. The proposed model gives a mathematical foundation to the generalization of the class of Leaky Integrate-and-Fire models for single neural dynamics to the case of a firing network of neurons. This will help future study of dependent spike trains. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new type iterative scheme for approximating common fixed points of two asymptotically non- expansive mappings is defined, and weak and strong convergence theorem are proved for the new iterative scheme in a uniformly convex Banach space. The results obtained in this article represent an extension as well as refinement of previous known results. [ABSTRACT FROM AUTHOR]

In this article, we prove some strong and weak convergence theorems for quasi-nonexpansive multivalued mappings in Banach spaces. The iterative process used is independent of Ishikawa iterative process and converges faster. Some examples are provided to validate our results. Our results extend and unify some results in the contemporary literature. [ABSTRACT FROM AUTHOR]