Multivalued Monotone Operators Research Articles

Complementarity problems for multivalued monotone operator in Banach spaces

We utilize S. Park's maximal element theorem in this paper to prove the existence theorems of solutions of the complementarity problems for multivalued monotone operator in Banach spaces.

An iterative algorithm for maximal monotone multivalued operator equations

A proximal iterative algorithm for the mulitivalue operator equation 0 υ T(x) is presented, where T is a maximal monotone operator. It is an improvement of the proximal point algorithm as well know. The convergence of the algorithm is discussed and an example is given.

The iterative solution for a class of monotone operator equations

The iterative solution for a class of multivalued monotone operator equations just likeA(u)∈−B(u) is discussed, whereA is a positive definite linear single-valued operator,B is a bounded and monotone multivalued operator. The existence and convergence of approximate solutions are proved. The method of numerical realization is demonstrated in some examples.

Variational principles for a class of finite step elastoplastic problems with non-linear mixed hardening

A class of elastoplastic relations with non-linear mixed hardening is addressed in the framework of an internal variable theory. The relevant finite step structural problem, resulting from the time integration of the constitutive laws according to a backward difference scheme, is then formulated in a geometrically linear range. Convex analysis and potential theory for monotone multi-valued operators are shown to provide the rationale for the development of a consistent variational theory. As a special result, a convex minimum principle in the final values of displacements, plastic strains and plastic parameters is formulated and a critical comparison with an analogous result proposed in the literature is presented.

Variational formulations of non-linear and non-smooth structural problems

The inverse problem of variational calculus is addressed with reference to structural models governed by non-linear field equations and monotone multi-valued constitutive operators. For such a class of models a non-smooth analysis must be necessarily carried out. The concept of consistency of non-linear strain operators is first recalled in view of a Lagrangian formulation of equilibrium. The structural problem is then recast in terms of a single structural operator which encompasses the field and constitutive equations by means of two sub-operators. The first one, which accounts for equilibrium and compatibility, is proved to be conservative and its potential explicitly derived. The second one is assumed to be conservative since it embodies multi-valued constitutive relations which are expressed as Subdifferentials of convex functionals. The original problem is then amenable to a weak formulation and, recalling recent results on the potential theory of monotone multi-valued operators, a constructive method for the variational formulation of problems expressed in terms of conservative multi-valued operators is presented. The structural operator is accordingly integrated in the product space of all the state variables to get the expression of the associated potential. Further, by enforcing constraint relations and kinematic compatibility, a family of non-smooth func̀tionals is derived and the related stationarity conditions are suitably defined starting from the concept of local subdifferential. Finally, it is shown that the stationarity of each of these functionals yields back an operator form of the structural problem.

A variational theory for finite-step elasto-plastic problems

An extended version ofgeneralized standard elasto-plastic material is considered in the framework of an internal variable theory of associated plasticity. According to a backward difference scheme for time integration of the flow rule, a finite-step structural problem is formulated in a geometrically linear range. Convex analysis and a brand new potential theory for monotone multivalued operators are shown to provide the natural mathematical setting for the derivation of the related variational formulation. A general stationarity principle is obtained and then specialized to obtain a minimum principle in terms of displacements, plastic strains and internal variables. A critical comparison with an analogous minimum principle recently proposed in literature is performed, showing the inadequacy of classical procedures in deriving non-smooth variational formulations.

A potential theory for monotone multivalued operators

The concept of cyclic monotonicity of a multivalued map has been introduced by R. T. Rockafellar with reference to the subdifferential operator of a convex functional. He observed that cyclic monotonicity could be viewed heuristically as a discrete substitute for the classical condition of conservativity, i.e., the vanishing of all the circuital integrals of a vector field. In the present paper a potential theory for monotone multivalued operators is developed, and, in this context, an answer to Rockafellar’s conjecture is provided. It is first proved that the integral of a monotone multivalued map along lines and polylines can be properly defined. This result allows us to introduce the concept of conservativity of a monotone multivalued map and to state its relation with cyclic monotonicity. Further, as a generalization of a classical result of integral calculus, it is proved that the potential of the subdifferential of a convex functional coincides, to within an additive constant, with the restriction of the functional on the domain of its subdifferential map. It is then shown that any conservative monotone graph admits a pair of proper convex potentials which meet a complementarity relation. Finally, sufficient conditions are given under which the complementary and the Fenchel’s conjugate of the potential associated with a conservative maximal monotone graph do coincide.

Browder-Hartman-Stampacchia variational inequalities for multi-valued monotone operators

An existence theorem of Browder-Hartman-Stampacchia variational inequalities is extended to multi-valued monotone operators. Surjectivity for multi-valued monotone operators and properties of solution sets are discussed.

The iterative solution of the equation f ∈ x + Tx for a monotone operator T in L p spaces

Suppose T is a multivalued monotone operator (not necessarily continuous) with open domain D ( T ) in L p (2⩽ p -∞), f ∈ R ( I + T ) and the equation f ∈ x + Tx has a solution q ∈ D ( T ). Then there exists a neighbourhood B ∋ D ( T ) of q and a real number r 1 > 0 such that for any r ⩾ r 1 , for any initial guess x 1 ∈ B , and any singlevalued section T 0 of T , the sequence { x n } n ∞ = 1 generated from x 1 by x n + 1 = (1 - C n ) x n + C n ( f − T 0 x n ) remains in D ( T ) and converges strongly to q with ‖ x n − q ‖ = 0( n −1/2 ). Furthermore, for X = L p ( E ), μ( E ) < ∞, μ = Lebesgue measure and 1 < p < 2, suppose T is a singe-valued locally Lipschitzian monotone operator with open domain D ( T ) in X . For f ∈ R ( I + T ), a solution of the equation x + Tx = f is obtained as the limit of an iteratively constructed sequence with an explicit error estimate.

On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space

Throughout, H denotes a real Hilbert space with inner product (., .), C a nonempty closed convex subset of H, and A a multivalued monotone operator with domain D(A) 3 C. (That is, A is a subset of H x H with the property that whenever [xi , zui] E A for i = 1,2, then (wr wa , x1 XJ > 0. By definition A(x) = {w E H: [.x, w] E A} and D(A) = {x: A(x) # a).) We denote by E the set of solutions u E C of the variational inequality

Nonlinear perturbations of uncoupled systems of elliptic operators

Here f = (fi 1)~ in a continuous function from ~ x A into IR M and fl = (fli)~ in a maximal monotone (multivalued) operator in the euclidean space ~M. In § 2 we develop abstract results which are applied to Eqs. (1.1) and (i,2) in § 3. The results and techniques used here are somewhat analogous to those in Brezis [2], Brezis et al. [3], Browder [-4], Da Prato [5], Hess [7, 8], Martin [10], Ton [ i2] , as well as several references cited in these papers.

A note on multivalued monotone operators.

A note on multivalued monotone operators.

Multivaluedness In Structural Mechanics With Applications To Plasticity

A general theory for structural models governed by multivalued relations is presented. It is based on the definition of a generalized elastic model and provides an unified framework for many different models usually adopted in the literature. The main properties of this model are investigated, existence and uniqueness conditions are discussed and the relevant variational principles are derived. An application to elasto-plasticity enlights the unifying features of the theory. INTRODUCTION An accurate analysis of most of the results and solution procedures developed in the literature for different structural models which, at a first glance, are characteristic of a particular context, reveals that they are in fact derived by more general properties which allow a treatment in an unified framework. As an example, elasto-static problems can be addressed by means of the classical potential theory [1] since they are governed by linear operators and by univocal relations between the state variables. When the relations fail to be univocal different approaches have usually been exploited to establish the basic results of existence and uniqueness, to state variational principles and to perform numerical computations. This occurs for most of the models introduced to analyse elasto-plastic and viscous behaviour of materials or to study damage and fracture problems. Actually these issues are commonly treated in the literature by means of specific formulations. The aim of this work is to present a more general approach which is based on a new potential theory [2] for monotone multivalued operators. Transactions on Engineering Sciences vol 7, © 1995 WIT Press, www.witpress.com, ISSN 1743-3533 20 Contact Mechanics II A generalized elastic model is defined as a reference model. Its analysis provides general results which can be applied to most of the structural problems quoted above. It is shown that the basic properties of maximality, cyclic monotonicity and conservativity of the graph of the constitutive laws and the convexity of the admissible domains allow to define complementary constitutive potentials and to exploit the tools of the Convex Analysis [3]. For generalized elastic structures an existence theorem is provided, uniqueness conditions are discussed and a general procedure is presented to derive the main family of variational principles. Finally the finite step elasto-plastic problem is treated as a particular case of the generalized elastic one and most of the variational principles which can be found in the literature [4 6] are derived as special forms of the ones of the main family. Applications of the generalized elastic model to visco-elasticity and visco-plasticity can be performed in an analogous way [7]. 1. GENERALIZED ELASTICITY Theory of elasticity is a fundamental issue of Mathematics Applied to Physics and its results can be exploited to solve other problems which are addressed by a formally analogous mathematical approach. In this section we introduce the generalized elastic constitutive law and study its main properties. 1.1. Constitutive properties The fundamental assumption of elasticity is that there is a relation between the static and kinematic internal state variables which involves only their actual values and not their history. In this sense we deal with a material which has no memory. We shall analyse the essential properties of this material starting from a more general formulation than the classical one, considering the internal constitutive relation. The study of the external constraints can be addressed by a perfectly analogous strategy. Let T> and S be the vectorial spaces of the deformations and stresses. A generalized elastic constitutive law £ is a relation between the elements e £ T> and a G S defined by the multivalued mappings: x S belonging to the graph, it results: