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We consider the problem of approximating Nash equilibria of N functions f 1 , ... , f N of N variables. In particular, we show systems of the form u ̇ j (t) = − ∇ x j f j (u (t)) (j = 1 , ... , N) are well-posed and the large time limits of their solutions u (t) = (u 1 (t) , ... , u N (t)) are Nash equilibria for f 1 , ... , f N provided that these functions satisfy an appropriate monotonicity condition. To this end, we will invoke the theory of maximal monotone operators on a Hilbert space. We will also identify an application of these ideas in game theory and show how to approximate equilibria in some game theoretic problems in function spaces. [ABSTRACT FROM AUTHOR]

In real Hilbert spaces, given a single-valued Lipschitz continuous and monotone operator, we study generalized split feasibility problem (GSFP) over solution set of monotone variational inclusion problem. An inertia iterative method is proposed to solve this problem, by showing that the sequence generated by the iteration converges strongly to solution of GSFP. As against previous methods, our step size is chosen to be simple and not depending on norm of associated bounded linear map as well as Lipschitz constant of the single-valued operator. The obtained result was applied to study split linear inverse problem, precisely, the LASSO problem. Lastly, with the aid of numerical examples, we exhibited efficiency of our algorithm and its dominance over other existing schemes. [ABSTRACT FROM AUTHOR]

This paper deals with the research of solutions of bounded variation (BV) to evolution inclusion coupled with a time and state dependent maximal monotone operator. Different problems are studied: existence of solutions, unicity of the solution, existence of periodic and bounded variation right continuous (BVRC) solutions. Second-order evolution inclusions and fractional (Caputo and Riemann–Liouville) differential inclusions are also considered. A result of the Skorohod problem driven by a time- and space-dependent operator under rough signal and a Volterra integral perturbation in the BRC setting is given. The paper finishes with some results for fractional differential inclusions under rough signals and Young integrals. Many of the given results are novel. [ABSTRACT FROM AUTHOR]

We study the split common zero point problem with multiple output sets (SCZPPMOS, for short) in Banach spaces. We introduce two new cyclic projection algorithms for finding a solution to SCZPPMOS and establish the strong convergence of the sequences generated by them. [ABSTRACT FROM AUTHOR]

We leverage path differentiability and a recent result on nonsmooth implicit differentiation calculus to give sufficient conditions ensuring that the solution to a monotone inclusion problem will be path differentiable, with formulas for computing its generalized gradient. A direct consequence of our result is that these solutions happen to be differentiable almost everywhere. Our approach is fully compatible with automatic differentiation and comes with the following assumptions which are easy to check (roughly speaking): semialgebraicity and strong monotonicity. We illustrate the scope of our results by considering the following three fundamental composite problem settings: strongly convex problems, dual solutions to convex minimization problems, and primal-dual solutions to min-max problems. [ABSTRACT FROM AUTHOR]

A differential inclusion with a time-dependent maximal monotone operator and a perturbation is studied in a separable Hilbert space. The perturbation is the sum of a time-dependent single-valued operator and a multivalued mapping with closed nonconvex values. A particular feature of the single-valued operator is that its sum with the identity operator multiplied by a positive square-integrable function is a monotone operator. The multivalued mapping is Lipschitz continuous with respect to the phase variable. We prove the existence of a solution and the density in the corresponding topology of the solution set of the initial inclusion in the solution set of the inclusion with a convexified multivalued mapping. For these purposes, new distances between maximal monotone operators are introduced. [ABSTRACT FROM AUTHOR]

In this paper, we propose two inertial algorithms with a new self-adaptive step size for approximating a solution of the split common null-point problem in the framework of Banach spaces. The step sizes are adaptively updated over each iteration by a simple process without the prior knowledge of the operator norm of the bounded linear operator. Under suitable conditions, we prove the weak-convergence results for the proposed algorithms in p-uniformly convex and uniformly smooth Banach spaces. Finally, we give several numerical results in both finite- and infinite-dimensional spaces to illustrate the efficiency and advantage of the proposed methods over some existing methods. Also, data classifications of heart diseases and diabetes mellitus are presented as the applications of our methods. [ABSTRACT FROM AUTHOR]

In this article, we propose an iterative scheme for solving a generalized split common null point problem. The iterative method uses self‐adaptive stepsizes that do not depend on the Lipschitz constants of the underlying operators. Instead, the stepsizes are updated using a simple updating formula. We prove the strong convergence theorems of the sequence generated by the iterative scheme under some mild assumptions. Furthermore, we derive iterative methods for solving split generalized variational inequality, convex feasibility, and minimization problems. In addition, some numerical illustrations are provided to support the main result. [ABSTRACT FROM AUTHOR]

We prove two key inequalities for metric and generalized projections in a certain Banach space. We then obtain some asymptotic behavior of a sequence generated by the shrinking projection method introduced by Takahashi et al. (J. Math. Anal. Appl. 341:276–286, 2008) where the computation allows some nonsummable errors. We follow the idea proposed by Kimura (Banach and Function Spaces IV (ISBFS 2012), pp. 303–311, 2014). The mappings studied in this paper are more general than the ones in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018). In particular, the results in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018) are both extended and supplemented. Finally, we discuss our results for finding a zero of maximal monotone operator and a minimizer of convex functions on a Banach space. [ABSTRACT FROM AUTHOR]

We investigate a special class of nonlinear infinite dimensional systems. These systems are obtained by modifying the second order differential equation that is part of the description of conservative linear systems "out of thin air" introduced by Tucsnak and Weiss in 2003. The modified differential equation contains a new nonlinear damping term that is maximal monotone and possibly set-valued. We show that this new class of nonlinear infinite dimensional systems is incrementally scattering passive (hence well-posed). Our approach uses the theory of maximal monotone operators and the Crandall--Pazy theorem about nonlinear contraction semigroups, which we apply to a Lax--Phillips type nonlinear semigroup that represents the whole system. We illustrate our result on the n-dimensional wave equation. [ABSTRACT FROM AUTHOR]

We introduce a series of distances between maximal monotone operators and study their properties. As applications, we consider the existence of solutions to evolutionary inclusions with maximal monotone operators. [ABSTRACT FROM AUTHOR]

In this paper we prove, in a separable Hilbert space, the existence of solutions to a couple of differential inclusions, both of them governed by time and state-dependent maximal monotone operators, and with multi-valued perturbations. An application to an optimal control problem is also presented. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iterative forward–backward splitting algorithm with errors for solving the split monotone variational inclusion problem of the sum of two monotone operators in real Hilbert spaces. We suggest and analyze this method under some mild appropriate conditions imposed on the parameters such that another strong convergence theorem for this problem is obtained. We also apply our main result to image-feature extraction with the multiple-image blends problem, the split minimization problem, and the convex minimization problem, and provide numerical experiments to illustrate the convergence behavior and show the effectiveness of the sequence constructed by the inertial technique. [ABSTRACT FROM AUTHOR]

We are concerned in the present work with diverse perturbations of time dependent monotone/accretive evolution. New applications such as integral–differential Volterra perturbation, periodicity, relaxation, asymptotic properties are presented. [ABSTRACT FROM AUTHOR]

In this survey, we review some old and new results initiated with the study of expansive mappings. From a variational perspective, we study the convergence analysis of expansive and almost-expansive curves and sequences governed by an evolution equation of the monotone or non-monotone type. Finally, we propose two well-defined algorithms to remedy the shortcomings concerning the ill-posedness of expansive-type evolution systems. [ABSTRACT FROM AUTHOR]

The aim of this work is to investigate the existence and uniqueness of μ-pseudo almost periodic (resp. μ-pseudo compact almost automorphic) weak solutions for the following partial functional differential inclusion: x ′ (t) + A x (t) ∋ f (t , x t) f o r t ∈ R , where A : H ⟶ 2 H is a strongly maximal monotone operator on a real Hilbert space H , f : R × C ⟶ H is a Stepanov μ-pseudo almost periodic (resp. μ-pseudo compact almost automorphic) function of class r, C = C ([ − r , 0 ] , H) is the Banach space of all continuous functions from [ − r , 0 ] to H and the history function x t is defined by x t (θ) = x (t + θ) for θ ∈ [ − r , 0 ]. Two examples are given for parabolic and subdifferential systems. [ABSTRACT FROM AUTHOR]

In this paper, we present a method for finding the zero of the sum of finite family of maximal monotone operators on real Hilbert spaces. In the case where the number of maximal monotone operators is three, we define a function such that its fixed points are solutions of our problem. Some illustrative examples are given at the end of this paper. [ABSTRACT FROM AUTHOR]

A multidomain macro-hybrid mixed variational subpotential approach of models in continuum mechanics is developed. Variational principles of mass conservation, balance momentum and energy, as well as material constitutive relations, are formulated as subpotential subdifferential evolution systems. Under the basis of variational divergence formulae and subdifferential generalized boundary conditions, variational mixed initial/boundary-value inclusion problems are established in reflexive Banach trajectory real functional frameworks. Multidomain motion velocity, mass conservation density, material constitutive stress, deformation porosity as well as balance internal energy evolution coupled variational mechanical systems are achieved. As a distinctive modeling feature, multidomain synchronizing transmission constraints are implemented in terms of dual Lagrangian internal boundary subpotential subdifferential inclusions. Lastly, evolution internal variational approximations, in general, for nonconforming macro-hybrid mixed finite element spatial discretizations as well as semi-implicit time marching fully discrete schemes are briefly discussed. [ABSTRACT FROM AUTHOR]

The problem of generalized equation 0 ∈ T(z), for maximal monotone operator T : H ⇉ H , covers many important optimization problems. The solution trajectory uλ(t) of the classical Yosida-regularization based differential equation system together with numerical approximation trajectories u λ α (t) , for finding solutions to the generalized equation, is investigated in this paper. The uniform convergence of the approximate trajectories u λ α (t) → u λ (t) on interval [ 0 , + ∞) is proved, as α tends to 0. Importantly, it is proved that the solution trajectory uλ(t) has the exponential convergence rate under the upper Lipschitz continuity of T− 1 at the origin. Moreover, the proposed differential equation approach is applied to the Karush-Kuhn-Tucker system for a smooth convex optimization problem. A numerical algorithm for getting an approximate solution trajectories of Yosida-regularization based differential equation of nonlinear programming is presented and an illustrative example is implemented by the numerical algorithm to verify the theory developed. [ABSTRACT FROM AUTHOR]

In this paper, the problem of the large deviations for the invariant measures of the multivalued stochastic differential equations is considered. Under the assumptions of diffusion coefficient being non-Lipschitz and elliptic, we establish the large deviation principle for the invariant measures of the solutions to the multivalued stochastic differential equations. The proof is based on the work of large deviations and invariant measures for the solutions to the multivalued stochastic differential equations. [ABSTRACT FROM AUTHOR]

In this paper we provide a new approach in the study of a variational-hemivariational inequality in Hilbert space, based on the theory of maximal monotone operators and the Banach fixed point theorem. First, we introduce the inequality problem we are interested in, list the assumptions on the data and show that it is governed by a multivalued maximal monotone operator. Then, we prove that solving the variational-hemivariational inequality is equivalent to finding a fixed point for the resolvent of this operator. Based on this equivalence result, we use the Banach contraction principle to prove the unique solvability of the problem. Moreover, we construct the corresponding Picard, Krasnoselski and Mann iterations and deduce their convergence to the unique solution of the variational-hemivariational inequality. [ABSTRACT FROM AUTHOR]

We consider a control problem in a finite-dimensional setting, which consists in finding a minimizer for a standard functional defined by way of two continuous and bounded below functions and a convex function, where the control functions take values in a closed convex set and the state functions solve a differential system made up of a differential inclusion governed by a maximal monotone operator; and an ordinary differential equation with a Lipschitz mapping in the right-hand side. We first show the existence of a unique absolutely continuous solution of our system, by transforming it to a sole evolution differential inclusion, and then use a result from the literature. Secondly, we prove the existence of an optimal solution to our problem. The main novelties are: the presence of the time-dependent maximal monotone operators, which may depend as well as their domains on the time variable; and the discretization scheme for the approximation of the solution. [ABSTRACT FROM AUTHOR]

The present paper deals with a maximal monotone operator A of type (D) in a Banach space whose dual space is strictly convex. We establish some representations for the value Ax at a given point x via its values at nearby points of x. We show that the faces of Ax are contained in the set of all weak∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$^*$$\end{document} convergent limits of bounded nets of the operator at nearby points of x, then we obtain a representation for Ax by use of this set. In addition, representations for the support function of Ax based on the minimal-norm selection of the operator in certain Banach spaces are given. [ABSTRACT FROM AUTHOR]

Consider in a real Hilbert space H the following problem, denoted (P ɛ μ) , − ɛ u ′ ′ (t) + μ u ′ (t) + A u (t) + B u (t) ∋ f (t) , 0 < t < T , u (0) = u 0 , u ′ (T) = 0 , where T > 0 is a given time instant, ɛ > 0 , μ ≥ 0 are small parameters, A : D (A) ⊂ H → H is a maximal monotone operator (possibly multivalued), and B : H → H is a Lipschitz operator (or monotone and Lipschitz on bounded sets). Consider also the following reduced problem, denoted (P μ) , μ u ′ (t) + A u (t) + B u (t) ∋ f (t) , 0 < t < T , u (0) = u 0 , where μ > 0 , as well as the algebraic equation (inclusion), A u (t) + B u (t) ∋ f (t) , 0 ≤ t ≤ T. (E 00) In this paper we are concerned with the following topics: (a) existence and uniqueness of solutions to the above problems and to equation (E 00) ; (b) continuity of the solution to problem (P ɛ μ) with respect to ɛ > 0 and μ ≥ 0 ; (c) convergence of the solution of problem (P ɛ μ) to the solution of problem P μ 0 , as ɛ → 0 + and μ → μ 0 , where μ 0 is a fixed positive number; (d) convergence of the solution of problem (P ɛ μ) to the solution of the equation A u + B u ∋ f (t) as ɛ → 0 + and μ → 0 + ; (e) applications. [ABSTRACT FROM AUTHOR]

In this paper, we study the so-called generalized monotone quasi-inclusion problem which is a generalization and extension of well-known monotone quasi-inclusion problem. We propose a forward–backward splitting method for solving this problem in the framework of reflexive Banach spaces. Based on Bregman distance function, we prove a strong convergence result of the proposed algorithm to a common zero of the problem. As an application, we apply the main result to the variational inequality problem. Finally, we provide some numerical examples to demonstrate our algorithm performance. The results presented in this paper improve and extend many known results in the literature. [ABSTRACT FROM AUTHOR]

We consider a quasilinear parabolic boundary value problem with a nonlocal boundary condition of Bitsadze–Samarskii type. A theorem on the existence and uniqueness of a periodic solution of this problem is proved. [ABSTRACT FROM AUTHOR]

In this paper. we study the convergence analysis of Generalized Split Feasibility Problem (GSFP) in real Hilbert spaces. We introduce a new modified iterative algorithm which does not require any prior knowledge of operator norm for approximating the solution of GSFP and established the strong convergence of sequence generated by our iterative algorithm. The strong convergence result is proved without the prior knowledge of the operator norm. [ABSTRACT FROM AUTHOR]

This paper deals with a split equality hierarchical fixed point problem in real Hilbert spaces which is an important and natural extension of hierarchical fixed point problem and split equality fixed point problem. An iterative algorithm where the stepsizes do not depend on the operator norms, so called simultaneous Krasnoselski-Mann algorithm is suggested for solving the split equality hierarchical fixed point problem. Further we prove a weak convergence theorem for the sequence generated by this algorithm. This special aspect of the algorithm together with the convergence result makes it an interesting scheme. Furthermore, we give some examples to justify the main result. Finally, we show that our purposed iterative algorithm is more efficient than some other known iterative algorithms. On the other hand, the framework is general and allows us to treat in a unified way several iterative algorithms, recovering, developing and improving some recently known related convergence results in the literature. [ABSTRACT FROM AUTHOR]

The main objective of the present paper is to prove the existence of absolutely continuous solutions for an evolution problem in a Hilbert space H, of the form − (d u / d t) (t) ∈ A (t , u (t)) u (t) where, for each (t , x) ∈ [ 0 , T ] × H , the maximal monotone operator A (t , x) : H → 2 H is of absolutely continuous variation in time and Lipschitz continuous in state, in the sense of the pseudo-distance introduced by A. A. Vladimirov. The perturbed problem is also considered. [ABSTRACT FROM AUTHOR]

A large number of nonlinear and optimization problems can be reduced to altering point problems. This paper aims to introduce the parallel normal S-iteration technique and study its convergence rates for solving such problems in infinite-dimensional Hilbert spaces under practical assumptions. We place particular emphasis on the parallel splitting method for the sum of two maximal monotone operators and that can apply for solving a class of convex composite minimization problems. Moreover, we present applications of our iterative methods to some nonlinear problems, such as a system of variational inequalities and a system of inclusion problems. Finally, to demonstrate the applicability of the altering point technique, the performances of our proposed parallel normal S-iteration methods are presented through numerical experiments in signal recovery problems. [ABSTRACT FROM AUTHOR]

The need to solve l¹ regularized linear problems can be motivated by various compressive sensing and sparsity related techniques for data analysis and signal or image processing. These problems lead to nonsmooth convex optimization in high dimensions. Theoretical works predict a sharp phase transition for the exact recovery of compressive sensing problems. Our numerical experiments show that state-of-the-art algorithms are not effective enough to observe this phase transition accurately. This paper proposes a simple formalism that enables us to produce an algorithm that computes an l¹ minimizer under the constraints Au = b up to the machine precision. In addition, a numerical comparison with standard algorithms available in the literature is exhibited. The comparison shows that our algorithm compares advantageously with other state-of-the-art methods, both in terms of accuracy and efficiency. With our algorithm, the aforementioned phase transition is observed at high precision. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to find a common element in the intersection of the set of zeros of the inclusion problem of sum of two monotone mappings and the set of fixed points of a Bregman quasi nonexpansive mapping in a reflexive Banach space by using Bregman distance and shrinking projection method. Under suitable conditions, some strong convergence theorems are proved. As applications, we utilize our results to study the convex minimization problem, variational inequality problem. [ABSTRACT FROM AUTHOR]

Variational inclusion is an important general problem consisting of many useful problems like variational inequality, minimization problem and nonlinear monotone equations. In this article, a new scheme for solving variational inclusion problem is proposed and the scheme uses inertial and relaxation techniques. Moreover, the scheme is self adaptive, that is, the stepsize does not depend on the factorial constants of the underlying operator, instead it can be computed using a simple updating rule. Weak convergence analysis of the iterates generated by the new scheme is presented under mild conditions. In addition, schemes for solving variational inequality problem and split feasibility problem are derived from the proposed scheme and applied in solving Nash-Cournot equilibrium problem and image restoration. Experiments to illustrate the implementation and potential applicability of the proposed schemes in comparison with some existing schemes in the literature are presented. [ABSTRACT FROM AUTHOR]

In this paper, we first consider the split common null point problem in two Banach spaces. Then, using the Bregman generalized resolvents of maximal monotone operators, we prove strong convergence theorems of Halpern type iteration for finding a solution of the split common null point problem in two Banach spaces. As an application of our result, we study the split equilibrium problem in general Banach spaces and approximate a solution of the problem for the first time. Our new technique is based on basic properties of a Bregman distance induced by a Bregman function without using Bregman projection or the requirement of Mosco convergence of the sequences produced by the method. It is well known that the Bregman distance does not satisfy either the symmetry property or the triangle inequality which are required for standard distances. So, the results of the paper improve and extend many recent results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a regularization method for solving the variational inclusion problem of the sum of two monotone operators in real Hilbert spaces. We suggest and analyze this method under some mild appropriate conditions imposed on the parameters, which allow us to obtain a short proof of another strong convergence theorem for this problem. We also apply our main result to the fixed point problem of the nonexpansive variational inequality problem, the common fixed point problem of nonexpansive strict pseudocontractions, the convex minimization problem, and the split feasibility problem. Finally, we provide numerical experiments to illustrate the convergence behavior and to show the effectiveness of the sequences constructed by the inertial technique. [ABSTRACT FROM AUTHOR]

In this paper, a new parallel algorithm for finding a common solution of a system of quasi-variational inclusion problems and a common fixed point of a finite family of nonexpansive mappings in a q-uniformly Banach space is introduced and analyzed. A strong convergence theorem of the proposed algorithm is established under some control conditions. As a consequence, we apply our main results to solve convex minimization problems, multiple sets variational inequality problems and multiple sets equilibrium problems. Some numerical experiments of image restoration problems are also given for supporting the main results. [ABSTRACT FROM AUTHOR]

The Cahn–Hilliard–Navier–Stokes system describes the evolution of two isothermal, incompressible, immiscible fluids in a bounded domain. In this work, we consider the stationary nonlocal Cahn–Hilliard–Navier–Stokes system in two and three dimensions with singular potential. We prove the existence of a weak solution for the system using pseudo-monotonicity arguments and Browder's theorem. Further, we establish the uniqueness and regularity results for the weak solution of the stationary nonlocal Cahn–Hilliard–Navier–Stokes system for constant mobility parameter and viscosity. Finally, in two dimensions, we establish that the stationary solution is exponentially stable (for convex singular potentials) under suitable conditions on mobility parameter and viscosity. [ABSTRACT FROM AUTHOR]

Based on a hybrid steepest‐descent method and a splitting method, we propose a variable metric iterative algorithm, which is useful in computing the elastic net solution. A solution theorem is established in the framework of Hilbert spaces. Numerical results are conducted on the signal processing to demonstrate the capacity and effectiveness of our algorithm. [ABSTRACT FROM AUTHOR]

In this paper, using a discrete scheme, we prove the existence of solutions of differential inclusions for a new class of perturbed second-order time and state-dependent maximal monotone operators. From our main theorem, we derive related results to perturbed second-order sweeping processes. [ABSTRACT FROM AUTHOR]

This paper studies the convergence rates of two accelerated computational methods without assuming nonexpansivity of the underlying operators with convex and affine domains in infinite-dimensional Hilbert spaces. One method is a noninertial method, and its convergence rate is estimated as R T , { x n } (n) = o 1 n in worst case. The other is an inertial method, and its convergence rate is estimated as R T , { y n } (n) = o 1 n under practical conditions. Then, we apply our results to give new results on convergence rates for solving generalized split common fixed-point problems for the class of demimetric operators. We also apply our results to variational inclusion problems and convex optimization problems. Our results significantly improve and/or develop previously discussed fixed-point problems and splitting problems and related algorithms. To demonstrate the applicability of our methods, we provide numerical examples for comparisons and numerical experiments on regression problems for publicly available high-dimensional real datasets taken from different application domains. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new algorithm by incorporating inertial terms in a subgradient extragradient algorithm for solving variational inequality problems involving strongly pseudomonotone and Lipschitz continuous operators in Hilbert spaces. The strong convergence of the algorithm is obtained under mild assumptions. We also provide some numerical examples to illustrate that the acceleration of our algorithm is effective. [ABSTRACT FROM AUTHOR]

Representation formulas for faces and support functions of the values of maximal monotone operators are established in two cases: either the operators are defined on reflexive and locally uniformly convex real Banach spaces with locally uniformly convex duals, or their domains have nonempty interiors on real Banach spaces. Faces and support functions are characterized by the limit values of the minimal-norm selections of maximal monotone operators in the first case while in the second case they are represented by the limit values of any selection of maximal monotone operators. These obtained formulas are applied to study the structure of maximal monotone operators: the local unique determination from their minimal-norm selections, the local and global decompositions, and the unique determination on dense subsets of their domains. [ABSTRACT FROM AUTHOR]

By providing a new iterative method our aim is finding a common element of the set of fixed points of two nonexpansive mappings, the set of solutions to a variational inclusion and the set of solutions of a generalized equilibrium problem in a real Hilbert space. We review the strong convergence of the new iterative method in the framework of Hilbert spaces. Finally, we show that our main result is a generalization for some known theorems in this field. [ABSTRACT FROM AUTHOR]

By using our own approach, we study the strong convergence of an inexact proximal point algorithm with possible unbounded errors for a maximal monotone operator in a Banach space. We give a necessary and sufficient condition for the zero set of the operator to be nonempty and show that, in this case, this iterative sequence converges strongly to a zero of the operator. We present also some applications of our results to equilibrium problems and optimization. Our proximal point algorithm contains, as a special case, the one considered in Hilbert space by Djafari Rouhani and Moradi in (J Optim Theory Appl 172:222–235, 2017) and solves the open problem of extending it to a Banach space, which was stated in that paper and in Djafari Rouhani and Moradi in (J Optim Theory Appl 181:864–882, 2019). Since the nonexpansiveness of the resolvent operator, which holds in Hilbert space, is not valid anymore in Banach space, our results require new methods of proofs, and significantly improve upon the previous results, both in theory and in applications. [ABSTRACT FROM AUTHOR]

In this paper, we prove a moderate deviation principle for the multivalued stochastic differential equations whose proof are based on recently well-developed weak convergence approach. As an application, we obtain the moderate deviation principle for reflected Brownian motion. [ABSTRACT FROM AUTHOR]

The theory of differential inclusions has played a central role in many areas as biological systems, physical problems and population dynamics. The principle aim of our work is to compute explicitly the discrete approximate solution of a differential inclusion including a maximal monotone operator. Also we present a numerical application of our results for showing how to compute the discrete approximate solution of its corresponding differential inclusion. [ABSTRACT FROM AUTHOR]

In a real Hilbert space H, let us consider the boundary-value problem -εu"(t) + βu'(t) + Au(t) + Bu(t) ∋ f(t), t ∈ [0; T]; u(0) = u0, u'(T) = 0, where T > 0 is a given time instant, ε,μ are positive parameters, A: D(A) ⊂ H → H is a (possibly set-valued) maximal monotone operator, and B: H → H is a Lipschitz operator. In this paper, we investigate the behavior of the solutions to this problem in two cases: (i) μ > 0 fixed, 0 < ε → 0, and (ii) ε > 0 fixed and 0 < μ → 0. Notice that if μ = 1 and ε is a positive small parameter, the above problem is a Lions-type regularization of the Cauchy problem u0(t) + Au(t) + Bu(t) ∋ f(t); t 2 [0; T]; u(0) = u0, which was recently studied by L. Barbu and G. Moroşanu [Commun. Contemp. Math. 19 (2017)]. Our abstract results are illustrated with examples related to the heat equation and the telegraph differential system. [ABSTRACT FROM AUTHOR]