Subdifferential Operator Research Articles

Evolutionary variational\u2013hemivariational inequalities with applications to dynamic viscoelastic contact mechanics

The purpose of this work is to introduce and investigate a complicated variational–hemivariational inequality of parabolic type with history-dependent operators. First, we establish an existence and uniqueness theorem for a first-order nonlinear evolution inclusion problem, which is driven by a convex subdifferential operator for a proper convex function and a generalized Clarke subdifferential operator for a locally Lipschitz superpotential. Then, we employ the fixed point principle for history-dependent operators to deliver the unique solvability of the parabolic variational–hemivariational inequality. Finally, a dynamic viscoelastic contact problem with the nonlinear constitutive law involving a convex subdifferential inclusion is considered as an illustrative application, where normal contact and friction are described, respectively, by two nonconvex and nonsmooth multi-valued terms.

Existence for a quasistatic variational-hemivariational inequality

<p style='text-indent:20px;'>This paper deals with an evolution inclusion which is an equivalent form of a variational-hemivariational inequality arising in quasistatic contact problems for viscoelastic materials. Existence of a weak solution is proved in a framework of evolution triple of spaces via the Rothe method and the theory of monotone operators. Comments on applications of the abstract result to frictional contact problems are made. The work extends the known existence result of a quasistatic hemivariational inequality by S. Migórski and A. Ochal [SIAM J. Math. Anal., 41 (2009) 1415-1435]. One of the linear and bounded operators in the inclusion is generalized to be a nonlinear and unbounded subdifferential operator of a convex functional, and a smallness condition of the coefficients is removed. Moreover, the existence of a hemivariational inequality is extended to a variational-hemivariational inequality which has wider applications.

Asymptotic profiles of nonlinear homogeneous evolution equations of gradient flow type

This work is concerned with the gradient flow of absolutely $p$-homogeneous convex functionals on a Hilbert space, which we show to exhibit finite ($p<2$) or infinite extinction time ($p \geq 2$). We give upper bounds for the finite extinction time and establish convergence rates of the flow. Moreover, we study next order asymptotics and prove that asymptotic profiles of the solution are eigenfunctions of the subdifferential operator of the functional. To this end, we compare with solutions of an ordinary differential equation which describes the evolution of eigenfunction under the flow. Our work applies, for instance, to local and nonlocal versions of PDEs like $p$-Laplacian evolution equations, the porous medium equation, and fast diffusion equations, herewith generalizing many results from the literature to an abstract setting. We also demonstrate how our theory extends to general homogeneous evolution equations which are not necessarily a gradient flow. Here we discover an interesting integrability condition which characterizes whether or not asymptotic profiles are eigenfunctions.

Fractional flows driven by subdifferentials in Hilbert spaces

This paper presents an abstract theory on well-posedness for time-fractional evolution equations governed by subdifferential operators in Hilbert spaces. The proof relies on a regularization argument based on maximal monotonicity of time-fractional differential operators as well as energy estimates based on a nonlocal chain-rule formula for subdifferentials. Moreover, it will be extended to a Lipschitz perturbation problem. These abstract results will be also applied to time-fractional nonlinear PDEs such as time-fractional porous medium, fast diffusion, p-Laplace parabolic, Allen-Cahn equations.

Douglas–Rachford Splitting for the Sum of a Lipschitz Continuous and a Strongly Monotone Operator

The Douglas–Rachford method is a popular splitting technique for finding a zero of the sum of two subdifferential operators of proper, closed, and convex functions and, more generally, two maximally monotone operators. Recent results concerned with linear rates of convergence of the method require additional properties of the underlying monotone operators, such as strong monotonicity and cocoercivity. In this paper, we study the case, when one operator is Lipschitz continuous but not necessarily a subdifferential operator and the other operator is strongly monotone. This situation arises in optimization methods based on primal–dual approaches. We provide new linear convergence results in this setting.

Generalized Nonsmooth Exponential-Type Vector Variational-Like Inequalities and Nonsmooth Vector Optimization Problems in Asplund Spaces

The aim of this article is to study new types of generalized nonsmooth exponential type vector variational-like inequality problems involving Mordukhovich limiting subdifferential operator. We establish some relationships between generalized nonsmooth exponential type vector variational-like inequality problems and vector optimization problems under some invexity assumptions. The celebrated Fan-KKM theorem is used to obtain the existence of solution of generalized nonsmooth exponential-type vector variational like inequality problems. In support of our main result, some examples are given. Our results presented in this article improve, extend, and generalize some known results offer in the literature.

On Fractional Differential Inclusions with Nonlocal Boundary Conditions

Abstract The main purpose of this paper is to study a class of boundary value problem governed by a fractional differential inclusion in a separable Banach space E $$\begin{array}{} \displaystyle \left\{ \begin{array}{lll} D ^\alpha u(t) +\lambda D^{\alpha-1 }u(t) \in F(t, u(t), D ^{\alpha-1}u(t)), \hskip 2pt t \in [0, 1] \\ I_{0^+}^{\beta }u(t)\left\vert _{t=0}\right. = 0, \quad u(1)=I_{0^+}^{\gamma }u(1) \end{array} \right. \end{array}$$ in both Bochner and Pettis settings, where α ∈ ]1, 2], β ∈ [0, 2 – α], λ ≥ 0, γ &gt; 0 are given constants, Dα is the standard Riemann-Liouville fractional derivative, and F : [0, 1] × E × E → 2E is a closed valued multifunction. Topological properties of the solution set are presented. Applications to control problems and subdifferential operators are provided.

Viscosity solutions for Hamiltonian equations with Young measures

The paper is concerned with a type value functions which occurs in the control problems subject to evolution inclusions involving time-dependent subdifferential operators where the controls are Young measures. We study their link with the viscosity solution of the associated Hamilton-Jacobi-Bellman equation.

$$\varepsilon $$ ε -Subdifferential as an Enlargement of the Subdifferential

This work introduces a remarkable property of enlargements of maximal monotone operators. The basic tool in our analysis is a family of enlargements, introduced by Svaiter. Using the fact that the $$\varepsilon $$ -subdifferential operator can be regarded as an enlargement of the subdifferential, a sufficient condition for some calculus rules in convex analysis can be provided. We give several corollaries about $$\varepsilon $$ -subdifferential and extend one of them to arbitrary enlargement.

Nonlinear diffusion equations as asymptotic limits of Cahn‐Hilliard systems on unbounded domains via Cauchy's criterion

This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial‐boundary problem for the nonlinear diffusion equation in an unbounded domain ( ), written as urn:x-wiley:mma:media:mma4760:mma4760-math-0003 which represents the porous media, the fast diffusion equations, etc, where β is a single‐valued maximal monotone function on , and T&gt;0. In Kurima and Yokota (J Differential Equations 2017; 263:2024‐2050 and Adv Math Sci Appl 2017; 26:221‐242) existence and uniqueness of solutions for were directly proved under a growth condition for β even though the Stefan problem was excluded from examples of . This paper completely removes the growth condition for β by confirming Cauchy's criterion for solutions of the following approximate problem ε with approximate parameter ε&gt;0: urn:x-wiley:mma:media:mma4760:mma4760-math-0005 which is called the Cahn‐Hilliard system, even if ( ) is an unbounded domain. Moreover, it can be seen that the Stefan problem excluded from Kurima and Yokota (J Differential Equations 2017; 263:2024‐2050 and Adv Math Sci Appl 2017; 26:221‐242) is covered in the framework of this paper.

Intriguing maximally monotone operators derived from nonsunny nonexpansive retractions

Monotone operator theory and fixed point theory for nonexpansive mappings are central areas in modern nonlinear analysis and optimization. Although these areas are fairly well developed, almost all examples published are based on subdifferential operators, linear relations, or combinations thereof. In this paper, we construct an intriguing maximally monotone operator induced by a certain nonexpansive retraction. We analyze this operator, which does not appear to be assembled from subdifferential operators or linear relations, in some detail. Particular emphasis is placed on duality and strong monotonicity.

Anticipated backward stochastic variational inequalities with generalized reflection

We study the existence and uniqueness of the strong solution for the following anticipated backward stochastic variational inequality with oblique subgradients, written under differential form: [Formula: see text] The generalized multivalued product term [Formula: see text] loses the maximal monotony property of its constituent subdifferential operator and, as a consequence, some specific techniques when approaching the above problem are mandatory. The univalued anticipated BSDE addressed in Peng and Yang [19] highlights the duality between these types of equations and stochastic differential delay equations, in order to solve a stochastic control problem. We also provide an example of an anticipated BSDE with time-dependent convex constraints, which can be reduced to our equation.

Optimal Control of Second Order Stochastic Evolution Hemivariational Inequalities with Poisson Jumps

The purpose of this article is to study the optimal control problem of second order stochastic evolution hemivariational inequalities with Poisson jumps by virtue of cosine operator theory in the Hilbert space. Initially, the sufficient conditions for existence of mild solution of the proposed system are verified by applying properties of Clarke's subdifferential operator and fixed point theorem in multivalued maps. Further, we formulated and proved the existence results for optimal control of the proposed system with corresponding cost function by using Balder theorem. Finally an example is provided to illustrate the main results.

Polyhedral sweeping processes with unbounded nonconvex-valued perturbation

A polyhedral sweeping process with a multivalued perturbation whose values are nonconvex unbounded sets is studied in a separable Hilbert space. Polyhedral sweeping processes do not satisfy the traditional assumptions used to prove existence theorems for convex sweeping processes. We consider the polyhedral sweeping process as an evolution inclusion with subdifferential operators depending on time. The widely used assumption of Lipschitz continuity for the multivalued perturbation term is replaced by a weaker notion of (ρ−H) Lipschitzness. The existence of solutions is proved for this sweeping process.

On steady flow of non-Newtonian fluids with frictional boundary conditions in reflexive Orlicz spaces

A stationary viscous incompressible non-Newtonian fluid flow problem is studied with a non-polynomial growth of the extra (viscous) part of the Cauchy stress tensor together with a multivalued nonmonotone frictional boundary condition described by the Clarke subdifferential. We provide an abstract result on existence of solution to a subdifferential operator inclusion and a hemivariational inequality in the reflexive Orlicz–Sobolev space setting modeling the flow phenomenon. We establish the existence result, and under additional conditions, also uniqueness of a weak solution in the Orlicz–Sobolev space to the flow problem.

Backward stochastic dynamics with a subdifferential operator and non-local parabolic variational inequalities

In this article, we provide the first systematic study on the unique existence of the solution of backward stochastic dynamical variational inequalities on a general complete filtered probability space. We also build up a comprehensive analysis of the correspondence between these stochastic variational inequalities (resp. backward stochastic dynamics) and the weak solutions (instead of viscosity ones due to the intrinsic non-local nature of the integral of the gradient involved) of a class of non-local parabolic variational inequalities (resp. parabolic partial differential equations), which is barely touched in the existing literature due to its unconventional setting.

Stability of Equilibria via Regularity of the Diagonal Subdifferential Operator

In this paper we investigate the Aubin property of the solution map of a parametric equilibrium problem, by providing a connection with a suitable behaviour of the diagonal subdifferential operator associated to the equilibrium bifunction. In particular, we shed some light on the relationship between metric regularity and subregularity of the diagonal subdifferential, on one side, and some properties of the bifunction, on the other side.

Subdifferential inclusions for stress formulations of unilateral contact problems

We consider two classes of inclusions involving subdifferential operators, both in the sense of Clarke and in the sense of convex analysis. An inclusion that belongs to the first class is stationary while an inclusion that belongs to the second class is history-dependent. For each class, we prove existence and uniqueness of the solution. The proofs are based on arguments of pseudomonotonicity and fixed points in reflexive Banach spaces. Then we consider two mathematical models that describe the frictionless unilateral contact of a deformable body with a foundation. The constitutive law of the material is expressed in terms of a subdifferential of a nonconvex potential function and, in the second model, involves a memory term. For each model, we list assumptions on the data and derive a variational formulation, expressed in terms of a multivalued variational inequality for the stress tensor. Then we use our abstract existence and uniqueness results on the subdifferential inclusions and prove the unique weak solvability of each contact model. We end this paper with some examples of one-dimensional constitutive laws for which our results can be applied.

Multi-valued backward stochastic differential equations driven byG-Brownian motion and its applications

In this paper, we prove the existence and uniqueness of a solution for a class of backward stochastic differential equations driven by G-Brownian motion with subdifferential operator by means of the Moreau–Yosida approximation method. Moreover, we give a probabilistic interpretation for the viscosity solutions of a kind of nonlinear variational inequalities. Copyright © 2017 John Wiley & Sons, Ltd.

On the Range of the Douglas–Rachford Operator

The problem of finding a minimizer of the sum of two convex functions—or, more generally, that of finding a zero of the sum of two maximally monotone operators—is of central importance in variational analysis. Perhaps the most popular method of solving this problem is the Douglas–Rachford splitting method. Surprisingly, little is known about the range of the Douglas–Rachford operator. In this paper, we set out to study this range systematically. We prove that for 3* monotone operators a very pleasing formula can be found that reveals the range to be nearly equal to a simple set involving the domains and ranges of the underlying operators. A similar formula holds for the range of the corresponding displacement mapping. We discuss applications to subdifferential operators, to the infimal displacement vector, and to firmly nonexpansive mappings. Various examples and counterexamples are presented, including some concerning the celebrated Brezis–Haraux theorem.