Maximal Monotone Mappings Research Articles

APPROXIMATION METHOD FOR COMMON FIXED POINT OF A COUNTABLE FAMILY OF MULTI-VALUED QUASI-ϕ-NONEXPANSIVE MAPPINGS IN BANACH SPACES

In this paper, we study a new iterative algorithm to approximate a common fixed point of acountable family of multi-valued quasi-φ nonexpansive mappings in a uniformly smooth and uniformlyconvex real Banach space. We prove strong convergence theorem which improve and generalize recentlyannounced results in the literature. We also give some example to illustrate our theorem and we alsogive application to zero point problem of maximal monotone mappings

Metric regularity properties of monotone mappings

The theory of metric regularity deals with properties of set-valued mappings that provide estimates useful in solving inverse problems and generalized equations. Maximal monotone mappings, which dominate applications related to convex optimization, have valuable special features in this respect that have not previously been recorded. Here it is shown that the property of strong metric subregularity is generic in an almost everywhere sense. Metric regularity not only coincides with strong metric regularity but also implies local single-valuedness of the inverse, rather than just of a graphical localization of the inverse. Consequences are given for the solution mapping associated with a monotone generalized equation.

On a multivalued prescribed mean curvature problem and inclusions defined on dual spaces

This article addresses two main objectives. First, it establishes a functional analytic framework and presents existence results for a quasilinear inclusion describing a prescribed mean curvature problem with homogeneous Dirichlet boundary conditions, involving a multivalued lower order term. The formulation of the problem is done in the space of functions with bounded variation. The second objective is to introduce a general existence theory for inclusions defined on nonreflexive Banach spaces, which is specifically applicable to the aforementioned prescribed mean curvature problem. This problem can be formulated as a multivalued variational inequality in the space of functions with bounded variation, which, under suitable conditions, is equivalent to an inclusion involving a maximal monotone mapping of type (D) and a generalized pseudomonotone mapping. We prove an abstract existence theorem for inclusions of this form, under some coercivity conditions involving both the maximal monotone and the generalized pseudomonotone mappings.

Variational Inequalities Over the Intersection of Fixed Point Sets of Generalized Demimetric Mappings and Zero Point Sets of Maximal Monotone Mappings

In this paper, we consider a variational inequality problem which is defined over the intersection of the set of common fixed points of a finite family of generalized demimetric mappings and the set of common zero points of a finite family of maximal monotone mappings. We propose an iterative algorithm which combines the hybrid steepest descent method with the inertial technique to solve this variational inequality problem. Strong convergence theorem of the proposed method is established under standard and mild conditions. Moreover, we do not require any prior information regarding the Lipschitz and strongly monotone constants of the mapping. The results of this paper improve and extend several known results in the literature.

Existence and uniqueness of weak periodic solutions for a coupled parabolic-elliptic system

ased on the maximal monotone mapping theory and applying the Schauder fixed point the- orem, we prove the existence and the uniqueness of weak periodic solution for nonlinear parabolic-elliptic equations in Orlicz-Sobolev spaces, with growth nonlinearity in gradient associated with some appropriate N-functions.

Revised algorithm for finding a common solution of variational inclusion and fixed point problems

Recent research has uncovered an algorithm for locating the common solution to variational inclusion problems with multivalued maximal monotone mapping and ?-inverse strongly monotone mapping, as well as the points that are invariant under non-expansive mapping. In their algorithm, Zhang et al. [S. Zhang, J. H. W. Lee, C. K. Chan, Algorithms of common solutions to quasi-variational inclusion and fixed point problems, Appl. Math. Mech. 29(5) (2008), 571-581.], ? must satisfy a very strict condition, namely ? ? [0, 2?]; thus, it cannot be used for all Lipschitz continuous mappings, despite the fact that inverse strongly monotone implies Lipschitz continuous. This manuscript aims to define a new algorithm that addresses the flaws of the previously described algorithm. Our algorithm is used to solve minimization problems involving the fixed point set of a non-expansive mapping. In addition, we support all of our claims with numerical examples derived from computer simulation.

Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem

<p style='text-indent:20px;'>In this paper, we first establish a surjectivity result for the sum of a maximal monotone mapping and a generalized pseudomontone mapping by using the generalized Yosida approximation technique. Then, we study a double obstacle quasilinear parabolic variational inequality problem <inline-formula><tex-math id="M1">\begin{document}$ {{\rm{(VIP)}}} $\end{document}</tex-math></inline-formula> by using the surjectivity result and penalty approximation method. In order to deal with the double obstacle constraints, we construct a quasilinear parabolic partial differential penalty equation, then we obtain the solvability of the quasilinear parabolic partial differential penalty equation. Moreover, we show that the set of the solutions to the penalty equation is bounded and every weak cluster point of this set is a solution of the problem <inline-formula><tex-math id="M2">\begin{document}$ {{\rm{(VIP)}}} $\end{document}</tex-math></inline-formula>. At last, as an application, we obtain numerical solutions of two double obstacle parabolic variational inequality problems by using the power penalty approximation method. Through the figures in the examples, we can intuitively see the different numerical solutions of the problems at different times.</p>

Total asymptotically nonexpansive mappings and generalized variational-like inclusion problems in semi-inner product spaces

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

A strong convergence theorem under a new shrinking projection method for nonlinear mappings in reflexive Banach spaces

In this paper, using a new shrinking projection method, we study the strong convergence theorem for finding a common point of the set of common fixed points of a finite family of Bregman demigeneralized mappings, common zero points of a finite family of Bregman inverse strongly monotone mappings and zero points of maximal monotone mapping in reflexive Banach space. Using this result, we get a new strong convergence theorem in Banach space. Our result extends and improves important recent results presented by many authors.

Optimal Design Versus Maximal Monge–Kantorovich Metrics

The connection between the minimal elastic compliance problem and Monge transport involving the euclidean metric cost has been evidenced in the year 1997. The aim of this paper is to renew this connection and adapt it to some variants in optimal design, focusing in particular on the optimal pre-stressed membrane problem. We show that the underlying metric cost is associated with an unknown maximal monotone map which maximizes the Monge-Kantorovich distance between two measures. In parallel with the classical duality theory leading to existence and PDE optimality conditions, we present a geometrical approach arising from a two-point scheme in which geodesics with respect to the optimal metric play a central role. It turns out from examples that optimal structures are very often truss-like, i.e. supported by piecewise affine geodesics. In case of a discrete load, we are able to relate the existence of such truss-like solutions to an extension property of maximal monotone maps that is of independent interest.

A Strong Convergence Theorem for a Finite Family of Bregman Demimetric Mappings in a Banach Space under a New Shrinking Projection Method

In this paper, using a new shrinking projection method and new generalizedk-demimetric mappings, we consider the strong convergence for finding a common point of the sets of zero points of maximal monotone mappings, common fixed points of a finite family of Bregmank-demimetric mappings, and common zero points of a finite family of Bregman inverse strongly monotone mappings in a reflexive Banach space. To the best of our knowledge, such a theorem for Bregmank-demimetric mapping is the first of its kind in a Banach space. This manuscript is online on arXiv by the link http://arxiv.org/abs/2107.13254.

A New Approximation Method for Finding Zeros of Maximal Monotone Operators

One of the major problems in the theory of maximal monotone operators is to find a point in the solution set Zer( ), set of zeros of maximal monotone mapping . The problem of finding a zero of a maximal monotone in real Hilbert space has been investigated by many researchers. Rockafellar considered the proximal point algorithm and proved the weak convergence of this algorithm with the maximal monotone operator. Güler gave an example showing that Rockafellar’s proximal point algorithm does not converge strongly in an infinite-dimensional Hilbert space. In this paper, we consider an explicit method that is strong convergence in an infinite-dimensional Hilbert space and a simple variant of the hybrid steepest-descent method, introduced by Yamada. The strong convergence of this method is proved under some mild conditions. Finally, we give an application for the optimization problem and present some numerical experiments to illustrate the effectiveness of the proposed algorithm.

A new algorithm for finding fixed points of Bregman asymptotically regular quasi-nonexpansive mapping and solutions of equilibrium problems

In this paper, we propose a new algorithm for finding a common element of the set of fixed points of Bregman asymptotically regular quasi-nonexpansive mappings and the set of zeros of maximal monotone mappings and the set of solutions of equilibrium problems for pseudomonotone and Bregman Lipschitz-type continuous bifunctions in reflexive Banach spaces. Moreover, the strong convergence of the sequence generated by this algorithm is established under some suitable conditions.

Existence of weak solutions for a nonlinear parabolic equations by Topological degree

We study the nonlinear parabolic initial boundary value problem associated to the equation ut − diva(x, t, u, grad u) = f(x, t), where the terme − diva(x, t, u, grad u) is a Leray-Lions operator, The right-hand side f is assumed to belong to L^q(Q). We prove the existence of a weak solution for this problem by using the Topological degree theory for operators of the form L + S, where L is a linear densely defined maximal monotone map and S is a bounded demicontinuous map of class (S+) with respect to the domain of L.

Convergence results for a zero of the sum of a finite family of maximal monotone mappings in Banach spaces

ABSTRACT The purpose of this paper is to study the method of approximation for zeros of the sum of a finite family of maximal monotone mappings in the setting of Banach spaces. Under some mild conditions, we establish strong convergence results of the proposed approximation method. The assumptions that one of the mappings is single valued and α-inverse strongly monotone are dispensed with. 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.

A Self-Adaptive Algorithm for Split Null Point Problems and Fixed Point Problems for Demicontractive Multivalued Mappings

In this work, we study the split null point problem and the fixed point problem in Hilbert spaces. We introduce a self-adaptive algorithm based on the viscosity approximation method without prior knowledge of the operator norm for finding a common solution of the considered problem for maximal monotone mappings and demicontractive multivalued mappings. A strong convergence result of our proposed algorithm is established under some suitable conditions. Some convergence results for the split feasibility problem and the split minimization problem are consequences of our main result. Finally, we also give numerical examples for supporting our main result.

Compositions and convex combinations of Bregman weakly relatively nonexpansive operators in reflexive Banach spaces

In this paper, we investigate the right and left Bregman weakly relatively nonexpansive operators in reflexive Banach spaces. We first introduce the notions of convex hull demi-closedness principle, strictly strongly convex hull demi-closedness principle, strongly demi-closedness principle, and composition demi-closedness principle of a family of nonlinear mappings using another new notions of convex asymptotic fixed points, and composition asymptotic fixed points for such mappings in a Banach space E. We analyze, in particular, compositions and convex combinations of Bregman weakly relatively nonexpansive operators, and prove the convergence of the Mann iterative method for operators of these types. Finally, we use our results to approximate common zeros of maximal monotone mappings and solutions to convex feasibility problems. To support our results, we include nontrivial examples in the paper. Therefore, our results improve and generalize many known results in the current literature.

A strong convergence theorem for approximation of a zero of the sum of two maximal monotone mappings in Banach spaces

The purpose of this paper is to study the method of approximation for a zero of the sum of two maximal monotone mappings in Banach spaces and prove strong convergence of the proposed method under suitable conditions. The method of proof is of independent interest. In addition, we give some applications to the minimization problems. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

A Strong Convergence Theorem under a New Shrinking Projection Method for Finite Families of Nonlinear Mappings in a Hilbert Space

In this paper, using a new shrinking projection method, we deal with the strong convergence for finding a common point of the sets of zero points of a maximal monotone mapping, common fixed points of a finite family of demimetric mappings and common zero points of a finite family of inverse strongly monotone mappings in a Hilbert space. Using this result, we get well-known and new strong convergence theorems in a Hilbert space.

Relaxation methods for optimal control problems

We consider a nonlinear optimal control problem with dynamics described by a differential inclusion involving a maximal monotone map [Formula: see text]. We do not assume that [Formula: see text], incorporating in this way systems with unilateral constraints in our framework. We present two relaxation methods. The first one is an outgrowth of the reduction method from the existence theory, while the second method uses Young measures. We show that the two relaxation methods are equivalent and admissible.