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In this paper, we propose a new method for solving variational inclusion problems in Hilbert spaces. This algorithm uses two inertial terms to speed up the convergence. In order to avoid computing the Lipschitz stepsize, we use an updated stepsize which is not necessary to know the Lipschitz constant of the operator. The weak convergence is established under some mild conditions. We present numerical performance of the proposed algorithm and compare our algorithm with other algorithms in literature. Finally, we deduce our algorithm for solving the convex minimization problem and give an application to the data classification problem of heart failure dataset. [ABSTRACT FROM AUTHOR]

The symbol TD(X) denotes the hyperspace of all nonempty totally disconnected compact subsets of a Hausdorff space X. This hyperspace is endowed with the Vietoris topology. For a mapping between Hausdorff spaces f : X → Y, define the induced mapping TD(f) : TD(X) → TD(Y) by TD(f)(A) = f(A) (the image of A under f). In the current paper, we study the relationships between the condition f belongs to a class of mappings between Hausdorff spaces 필 and the condition TD(f) belongs to 필. [ABSTRACT FROM AUTHOR]

In this paper, we consider the tensor complementarity problem (TCP). From the perspective of non-coupled equality constraint minimization problem for the symmetric TCP, we propose a generalized alternating direction method of multipliers (G-ADMM) to solve the general TCP in which the tensor may not be symmetric. The global convergence and the O (1 k) convergence rate of the proposed method are proved under the assumption which is much weaker than the monotone assumption. Numerical results show that the method is efficient for solving the TCP. [ABSTRACT FROM AUTHOR]

We study the existence and uniqueness of solutions to the inverse quasi-variational inequality problem. Motivated by the dynamical approach to solving optimization problems such as variational inequality, monotone inclusion, and inverse variational problems, we consider a dynamical system associated with the inverse quasi-variational inequality problem, and establish the existence and uniqueness of a solution to the proposed system. We prove that every trajectory of the proposed dynamical system converges to the unique solution of the inverse quasi-variational inequality problem and that the system is globally asymptotically stable at its equilibrium point. We also prove that if the function which governs the inverse quasi-variational inequality problem is strongly monotone and Lipschitz continuous, then the dynamical system is globally exponentially stable at its equilibrium point. We discretize the dynamical system and show that the sequence generated by the discretization of the system converges strongly to the unique solution of the inverse quasi-variational inequality problem under certain assumptions on the parameters involved. Finally, we provide numerical examples to support and illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new inertial type algorithm with self -- adaptive step -- size technique for approximating a common element of the set of solutions of pseudomonotone variational inequality problem and the set of common fixed point of a finite family of generic generalized nonspreading mappings in uniformly smooth and 2 -- uniformly convex real Banach space. Furthermore, we prove a strong convergence theorem of our algorithm without prior knowledge of the Lipschitz constant of the operator under some mild assumptions. We also give numerical examples to illustrate the performance of our algorithm. Our result generalize and improve many existing results in the literature. [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]

Abstract This work is devoted to the analysis for a new class of set-valued inverse mixed variational inequalities (SIMVIs) in reflexive Banach spaces, when both the mapping and the constraint set are perturbed simultaneously by two parameters. Several equivalence characterizations are given for SIMVIs to have nonempty and bounded solution sets. Based on the equivalence conditions, under the premise of monotone mappings, the stability result for the SIMVIs is obtained in the reflexive Banach space. Furthermore, to illustrate the results, an example of the traffic network equilibrium control problem is provided at the end of this paper. The results presented in this paper generalize and extend some known results in this area.

This work is devoted to the analysis for a new class of set-valued inverse mixed variational inequalities (SIMVIs) in reflexive Banach spaces, when both the mapping and the constraint set are perturbed simultaneously by two parameters. Several equivalence characterizations are given for SIMVIs to have nonempty and bounded solution sets. Based on the equivalence conditions, under the premise of monotone mappings, the stability result for the SIMVIs is obtained in the reflexive Banach space. Furthermore, to illustrate the results, an example of the traffic network equilibrium control problem is provided at the end of this paper. The results presented in this paper generalize and extend some known results in this area. [ABSTRACT FROM AUTHOR]

In this paper, we propose a self‐adaptive inertial‐like algorithm with Bregman distance for approximating a common solution of systems of variational inequalities for a class of monotone and Lipschitz continuous mappings in real reflexive Banach spaces. Our algorithm is constructed without using hybrid projection method and shrinking projection method, and its strong convergence is proved without the prior information of the Lipschitz constant of the mapping. Finally, we provide some numerical experiments to illustrate the performance of the newly proposed method including a comparison with related works in solving signal restoration problems. [ABSTRACT FROM AUTHOR]

In this work, we give necessary and sufficient conditions for the existence of solutions to the variational inequality problem: find x 0 ∈ K such that ⟨ F (x 0) , y - x 0 ⟩ ≥ 0 , for every y ∈ K , where K is a nonempty closed convex subset of a real Hilbert space H and F : K → H is a monotone and continuous operator. These characterizations are given in terms of approximate fixed points sequences, as well as by Leray–Schauder condition. We apply our obtained results in a constrained convex minimization problem. [ABSTRACT FROM AUTHOR]

In this article, we propose a Halpern-type subgradient extragradient algorithm for solving a common element of the set of solutions of variational inequality problems for continuous monotone mappings and the set of f-fixed points of continuous f-pseudocontractive mappings in reflexive real Banach spaces. In addition, we prove a strong convergence theorem for the sequence generated by the algorithm. As a consequence, we obtain a scheme that converges strongly to a common f-fixed point of continuous f-pseudocontractive mappings and a scheme that converges strongly to a common zero of continuous monotone mappings in Banach spaces. Furthermore, we provide a numerical example to illustrate the implementability of our algorithm.

Let G : H → H and K : H → H be monotone mappings that are either sequentially weakly continuous or continuous, where H is a real Hilbert space. In this work, we introduce two new iterative methods for approximating solutions of the Hammerstein equation u + G K u = 0 , if they exist. The first iterative method is shown to always converge weakly to an element in the solution set of the Hammerstein equation if this solution set is nonempty. The second iterative method is a modification of the first method to upgrade weak convergence to strong convergence. Convergence results are obtained without requiring the maps to be bounded. Numerical examples are provided to demonstrate the convergence of one of these methods. Comparisons with some existing methods show that the method is cost effective in terms of the number of iterations required to obtain a solution and the computational time. [ABSTRACT FROM AUTHOR]

In this paper, we construct a Halpern-type subgradient extragradient iterative algorithm which converges strongly to a common point of the f-fixed point set of a continuous f-pseudocontractive mapping and the solutions set of a variational inequality problem for Lipschitz monotone mapping in reflexive real Banach spaces. In addition, a numerical example is given to demonstrate the behaviour of the convergence of the algorithm. Our results improve, generalize and unify some related recent results in the literature. [ABSTRACT FROM AUTHOR]

The authors consider the existence, multiplicity, and uniqueness for polyharmonic problem with Navier boundary conditions. One of the interesting features in our proof is that we give a new attempt to consider the uniqueness of nontrivial solution for the above polyharmonic problem by using the theory of monotone mappings. This is probably the first time this theory is used to solve polyharmonic problems. Then we apply the fixed point theorems on cones to analyze the existence and multiplicity of positive solutions for the above polyharmonic problem. This is very difficult for partial differential equations, especially for polyharmonic equations. The main reason is that the Green's function for the above polyharmonic problem is unbounded. We overcome the difficulties by using some new techniques. The uniqueness of nontrivial solution and the existence of positive solutions for polyharmonic equations with Dirichlet boundary conditions are also considered. [ABSTRACT FROM AUTHOR]

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Convex analysis and the theory of differential inclusions with maximal monotone right-hand sides suggests casting consensus algorithms as systems involving switching between such differential inclusions. Convergence of solutions to such switching systems is shown and applications to consensus are presented. Robustness of pointwise asymptotic stability for a single differential inclusion which has some monotonicity-related properties, but needs not be monotone, is shown.

Convex analysis and the theory of differential inclusions with maximal monotone right-hand sides suggests casting consensus algorithms as systems involving switching between such differential inclusions. Convergence of solutions to such switching systems is shown and applications to consensus are presented. Robustness of pointwise asymptotic stability for a single differential inclusion which has some monotonicity-related properties, but needs not be monotone, is shown.

In this paper, we introduce a new self-adaptive algorithm for approximating a common element of the set of solutions of generalized mixed equilibrium problems, split common null point and common fixed point of a finite family of quasi-ɸ-nonexpansive mappings in 2-uniformly smooth and uniformly convex real Banach space. Our proposed algorithm is constructed in such a way that its convergence analysis does not require a prior estimate of the operator norm. Furthermore, we establish a strong convergence result under some mild conditions. Our result improves and generalizes many existing results in the literature. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to construct an algorithm for approximating a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of f-fixed points of a finite family of f-pseudocontractive mappings and the set of solutions of a finite family of variational inequality problems for Lipschitz monotone mappings in real reflexive Banach spaces. [ABSTRACT FROM AUTHOR]

In this article, we propose hybrid projection iterative algorithm for finding a common element of solution set of a finite family of generalized mixed equilibrium problem, semi-fixed point set of a finite family of continuous semi-pseudocontractive mappings and solution set of a finite family of variational inequality for a finite family of monotone and L-Lipschitz mappings in Banach spaces and proved strong convergence theorem. The main result of the article generalizes some of known results in the literature. Furthermore, we give a numerical example to demonstrate the behavior of the convergence of the algorithm. [ABSTRACT FROM AUTHOR]

We show the nonlinear ergodic theorem for monotone 1-Lipschitz mappings in uniformly convex spaces: if C is a bounded closed convex subset of an ordered uniformly convex space (X, ∣·∣, ⪯), T:C → C a monotone 1-Lipschitz mapping and x ⪯ T(x), then the sequence of averages 1n∑i=0n−1Ti(x)$ \frac{1}{n}\sum\nolimits_{i=0}^{n-1}T^{i}(x) $ converges weakly to a fixed point of T. As a consequence, it is shown that the sequence of Picard’s iteration {Tn(x)} also converges weakly to a fixed point of T. The results are new even in a Hilbert space. The Krasnosel’skiĭ-Mann and the Halpern iteration schemes are studied as well.

We are concerned in this article with the existence of solutions to inclusions containing generalized pseudomonotone perturbations of maximal monotone mappings in general Banach spaces. Our approach is based on a truncation–regularization technique and an extension of the Moreau–Yosida–Brezis–Crandall–Pazy regularization for maximal monotone mappings in general Banach spaces. We also consider some applications to multivalued variational inequalities containing elliptic operators with rapidly growing coefficients in Orlicz–Sobolev spaces. [ABSTRACT FROM AUTHOR]

For q > 1, let E be a q-uniformly smooth real Banach space with dual space E Let A : E → E be a Lipschitz and strongly monotone mapping such that A-1(0) = For given x1 E, let {xn} be generated iteratively by the algorithm ... where J is the normalized duality mapping from E into E and λ is a positive real number choosen in a suitable interval. Then it is proved that the sequence {xn} converges strongly to x, the unique point of A-1(0). Our theorems are applied to the convex minimization problem. Futhermore, our technique of proof is of independent interest. [ABSTRACT FROM AUTHOR]

Let E be a real reflexive Banach space with its dual E ∗ and f be a proper, convex and lower-semi-continuous function on E. The purpose of this paper is to introduce and study a new class of mappings from E into E ∗ called f-pseudocontractive mappings with the notion of f-fixed points. In the case that E is a real reflexive Banach space and f is a strongly coercive, bounded and uniformly Fréchet differentiable Legendre function which is strongly convex on bounded subsets of E, a sequence is constructed which converges strongly to a common f-fixed point of two f-pseudocontractive mappings. As a consequence, we obtain a scheme which converges strongly to a common zero of monotone mappings. Furthermore, this analog is applied to approximate solutions to convex optimization problems. Our results improve and generalize many of the results in the literature. [ABSTRACT FROM AUTHOR]

Ran and Reurings (Proc Am Math Soc 132(5):1435–1443, 2003) extended the Banach contraction principle to the setting of partially ordered metric spaces and recently Proinov (J Fixed Point Theory Appl 22:21, 2020) extended and unified many earlier fixed point theorems. In this paper we will present analogous results for the significantly wider class of mappings on preordered metric spaces. We give non-trivial examples of Kannan-type mappings. [ABSTRACT FROM AUTHOR]

In this article, we introduced an iterative scheme for finding a common element of the set of fixed points of a multi-valued hemicontractive-type mapping, the set of common solutions of a finite family of split equilibrium problems and the set of common solutions of a finite family of variational inequality problems in real Hilbert spaces. Moreover, the sequence generated by the proposed algorithm is proved to be strongly convergent to a common solution of these three problems under mild conditions on parameters. Our results improve and generalize many well-known recent results existing in the literature in this field of research.

In this paper, we first introduce a monotone mapping and its resolvent in general metric spaces.Then, we give two new iterative methods by combining the resolvent method with Halpern's iterative method and viscosity approximation method for finding a fixed point of monotone mappings and a solution of variational inequalities. We prove convergence theorems of the proposed iterations in CAT(0) metric spaces.

Abstract In this paper, we introduce an algorithm for solving classical variational inequalities problem with Lipschitz continuous and monotone mapping in Banach space. We modify the subgradient extragradient methods with a new and simple iterative step size, the strong convergence of algorithm is established without the knowledge of the Lipschitz constant of the mapping. Finally, a numerical experiment is presented to show the efficiency and advantage of the proposed algorithm. Our results generalize some of the work in Hilbert spaces to Banach spaces.

Abstract For a variational inequality problem, the inertial projection and contraction method have been studied. It has a weak convergence result. In this paper, we propose a strong convergence iterative method for finding a solution of a variational inequality problem with a monotone mapping by projection and contraction method and inertial hybrid algorithm. Our result can be used to solve other related problems in Hilbert spaces.

In this paper, a new class of set-valued inverse variational inequalities (SIVIs) are introduced and investigated in reflexive Banach spaces. Several equivalent characterizations are given for the set-valued inverse variational inequality to have a nonempty and bounded solution set. Based on the equivalent condition, we propose the stability result for the set-valued inverse variational inequality with both the mapping and the constraint set that are perturbed in a reflexive Banach space, provided that the mapping is monotone. Furthermore, some examples are shown to support the main results. The results in this paper generalize and extend some known results in this area. [ABSTRACT FROM AUTHOR]

The purpose of this article is to introduce a general inertial projected gradient method with a self-adaptive stepsize for solving variational inequality problems. The proposed method incorporates two different extrapolations with respect to the previous iterates into the projected gradient method. The weak convergence for our method is proved under standard assumptions without any requirement of the knowledge of the Lipschitz constant of the mapping. Furthermore, R-linear convergence rate is established under the strong monotonicity assumption of the mapping. Finally preliminary results from numerical experiments and applications to optimal control problems are performed which show the advantage of the proposed method. [ABSTRACT FROM AUTHOR]

The tensor complementarity problem (TCP) is a special instance of nonlinear complementarity problems, which has many applications in multi-person noncooperative games, hypergraph clustering problems, and traffic equilibrium problems. How to solve the TCP, via analyzing the structure of the related tensor, is one of important research issues. In this paper, we propose an alternating direction method of multipliers (ADMM) to solve the TCP. We show that the solution set of the TCP, where the involved multilinear mapping is monotone, is nonempty and compact if the involved tensor is an S-tensor. Moreover, the ADMM for the TCP with a monotone involved multilinear mapping is proven to be globally convergent with a linear convergence rate. Some preliminary numerical results show that the proposed ADMM method is promising and effective. [ABSTRACT FROM AUTHOR]

Two new iterative algorithms are introduced for solving the split common null point problem (SCNPP). The first algorithm is shown to converge weakly and the second is shown to converge strongly to a solution of SCNPP in a Hilbert space. The algorithms are forward methods (i.e, involving no computation of the resolvent of a monotone operator). Another feature of the algorithms is that the selection of the step-sizes does not need prior knowledge of operator norms. Numerical experiments are presented to illustrate the performance of the algorithms. [ABSTRACT FROM AUTHOR]

We consider a neural network approach to an inverse variational inequality which is assumed to have a non-empty set of solutions. In the case of gradient mappings, we prove that every trajectory of the network converges to the solution set in the case of convex potentials and if the solution set is singleton, the network is globally asymptotically stable at the equilibrium point. We also prove that if the network has a strongly convex potential, then the network is globally exponentially stable at the equilibrium point. Another purpose of this paper is to point out certain fatal mistakes in the paper by Zou et al. [A novel method to solve inverse variational inequality problems based on neural networks. Neurocomputing. 2016;173:1163–1168]. [ABSTRACT FROM AUTHOR]

In this manuscript we introduce a new class of monotone generalized nonexpansive mappings and establish some weak and strong convergence theorems for Krasnoselskii iteration in the setting of a Banach space with partial order. We consider also an application to the space L1([0,1]). Our results generalize and unify the several related results in the literature.

The article introduces a new algorithm for solving a class of variational inequality problems for monotone operators and system of nonlinear variational inequalities problems for two inverse strongly monotone operators. We describe how to incorporate the extragradient like technique based on altering points technique with inertial effects. A weak convergence theorem is established for the proposed algorithm. Numerical examples are performed to illustrate the numerical efficiency of the algorithm and compare with other algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we propose a generalized explicit algorithm for approximating the common solution of generalized split feasibility problem and the fixed point of demigeneralized mapping in uniformly smooth and 2-uniformly convex real Banach spaces. The generalized split feasibility problem is a general mathematical problem in the sense that it unifies several mathematical models arising in (symmetry and non-symmetry) optimization theory and also finds many applications in applied science. We designed the algorithm in such a way that the convergence analysis does not need a prior estimate of the operator norm. More so, we establish the strong convergence of our algorithm and present some computational examples to illustrate the performance of the proposed method. In addition, we give an application of our result for solving the image restoration problem and compare with other algorithms in the literature. This result improves and generalizes many important related results in the contemporary literature.

Abstract The purpose of this article is to propose a modified viscosity implicit-type proximal point algorithm for approximating a common solution of a monotone inclusion problem and a fixed point problem for an asymptotically nonexpansive mapping in Hadamard spaces. Under suitable conditions, some strong convergence theorems of the proposed algorithms to such a common solution are proved. Our results extend and complement some recent results in this direction.

Abstract Inspired by the work of Zegeye (J. Math. Anal. Appl. 343:663–671, 2008) and the recent papers of Chidume et al. (Fixed Point Theory Appl. 2016:97, 2016; Br. J. Math. Comput. Sci. 18:1–14, 2016), we devise a viscosity iterative algorithm without involving the resolvent operator for approximating the zero of a monotone mapping in the setting of uniformly convex Banach spaces. Under concise parameter conditions we establish strong convergence of the proposed algorithm. Moreover, applications to constrained convex minimization problems and solution of Hammerstein integral equations are included. Finally, the performances and computational examples and a comparison with related algorithms are presented to illustrate the efficiency and applicability of our new algorithm.

The purpose of this note is to discuss the recent paper of Espínola and Wiśnicki about the fixed point theory of monotone nonexpansive mappings. In their work, it is claimed that most of the fixed point results of this class of mappings boil down to the classical Knaster-Tarski fixed point theorem. We will show that their approach is very restrictive and fail to have any meaningful usefulness in applications.

The purpose of this paper is to study a forward–backward algorithm for approximating a zero of the sum of maximally monotone mappings in the setting of Banach spaces. Under some mild conditions, we prove a new strong convergence theorem for the algorithm produced by the method in real reflexive Banach spaces. In addition, we give some applications to the minimization problems. Finally, we provide a numerical example, which supports our main result. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a hybrid extra-gradient iterative method to approximate the common solution of a system of unrelated generalized mixed equilibrium problems for monotone and Lipschitz continuous mappings and system of unrelated hierarchical fixed point problems for nonexpansive mappings in Hilbert space. We prove a strong convergence theorem for the sequences generated by the proposed iterative algorithm. Further, we give some consequences and applications of our main result. Finally, we discuss a numerical example to demonstrate the applicability of the iterative algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of finding a common solution to split generalized mixed equilibrium problem and fixed point problem for quasi-φ-nonexpansive mappings in 2-uniformly convex and uniformly smooth Banach space E1 and a smooth, strictly convex, and reflexive Banach space E2 . An iterative algorithm with Armijo linesearch rule for solving the problem is presented and its strong convergence theorem is established. The convergence result is obtained without using the hybrid method which is mostly used when strong convergence is desired. Finally, numerical experiments are presented to demonstrate the practicability, efficiency, and performance of our algorithm in comparison with other existing algorithms in the literature. Our results extend and improve many recent results in this directio [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iterative algorithm for solving classical variational inequalities problem with Lipschitz continuous and monotone mapping in real Hilbert space. We modify the subgradient extragradient methods with a step size; an advantage of the algorithm is the computation of only one value of the mapping and one projection onto the admissible set per one iteration. The convergence of the algorithm is established without the knowledge of the Lipschitz constant of the mapping. Meanwhile, R-linear convergence rate is obtained under strong monotonicity assumption of the mapping. Also, we generalize the method with Bregman projection. Finally, some numerical experiments are presented to show the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

An algorithm is constructed to approximate a zero of a maximal monotone operator in a uniformly convex and uniformly smooth real Banach space. The sequence of the algorithm is proved to converge strongly to a zero of the maximal monotone map. In the case where the Banach space is a real Hilbert space, our theorem complements the celebrated proximal point algorithm of Martinet and Rockafellar. Furthermore, our convergence theorem is applied to approximate a solution of a Hammerstein integral equation in our general setting. Finally, numerical experiments are presented to illustrate the convergence of our algorithm. [ABSTRACT FROM AUTHOR]

Recently, researchers are showing more interest on both modular vector spaces and modular function spaces. Looking at the number of results it is pertinent to say that, exploration in this direction especially in the area of fixed point theory and applications is still ongoing, many good results can still be unveiled. As a contribution from our part, we study some fixed point results in modular vector spaces associated with order relation. As an application, we were able to study the existence of fixed point(s) of both depolarizing quantum operation and Markov operators through modular functions/modular spaces. The awareness on the importance of quantum theory and Economics globally were the sole motivations of the application choices in our work. Our work complement the existing results. In fact, it adds to the number of application areas that modular vector/function spaces covered. [ABSTRACT FROM AUTHOR]

We present some new coincidence fixed point theorems for generalized multi-valued weak L-contraction mappings. Our outcomes extend several recent results in the framework of complete metric spaces endowed with a graph. Two illustrative examples are included and some consequences are derived. [ABSTRACT FROM AUTHOR]

A coincidence point of a pair of mappings is an element, at which these mappings take on the same value. Coincidence points of mappings of partially ordered spaces were studied by A.V. Arutyunov, E.S. Zhukovskiy, and S.E. Zhukovskiy (see Topology and its Applications, 2015, V. 179, No. 1, p. 13–33); they proved, in particular, that an orderly covering mapping and a monotone one, both acting from a partially ordered space to a partially ordered one, possess a coincidence point. In this paper, we study the existence of a coincidence point of a pair of mappings that act from a partially ordered space to a set, where no binary relation is defined, and, consequently, it is impossible to define covering and monotonicity properties of mappings. In order to study the mentioned problem, we define the notion of a "quasi-coincidence" point. We understand it as an element, for which there exists another element, which does not exceed the initial one and is such that the value of the first mapping at it equals the value of the second mapping at the initial element. It turns out that the following condition is sufficient for the existence of a coincidence point: any chain of "quasi-coincidence" points is bounded and has a lower bound, which also is a "quasi-coincidence" point. We give an example of mappings that satisfy the above requirements and do not allow the application of results obtained for coincidence points of orderly covering and monotone mappings. In addition, we give an interpretation of the stability notion for coincidence points of mappings that act in partially ordered spaces with respect to their small perturbations and establish the corresponding stability conditions. [ABSTRACT FROM AUTHOR]