Existence of solutions of parabolic variational inequalities with one-sided restrictions

We prove sufficient conditions for the existence of a solution of a strong nonlinear variational inequality of parabolic type. The theory can be used for solving parabolic equations with one-sided boundary conditions. As an example, we prove the existence of a solution of a strong parabolic variational inequality with p-Laplacian in the Sobolev space L _p (0, T, W _p ¹ ()), p [2, ).Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 460–476.Original Russian Text Copyright © 2005 by O. V. Solonukha.This revised version was published online in April 2005 with a corrected issue number.     

Existence theorems for solutions of variational inequalities

Summary The existence of nonzero solutions for a class of generalized variational inequalities is studied by fixed point index approach for multi-valued mappings in finite dimensional spaces and reflexive Banach spaces.     

A Class of differential inverse variational inequalities in finite dimensional spaces

In this paper, we study a class of differential inverse variational inequality (for short, DIVI) in finite dimensional Euclidean spaces. Firstly, under some suitable assumptions, we obtain linear growth of the solution set for the inverse variational inequalities. Secondly, we prove existence theorems for weak solutions of the DIVI in the weak sense of Carath\"{e}odory by using measurable selection lemma. Thirdly, by employing the results from differential inclusions we establish a convergence result on Euler time dependent procedure for solving the DIVI. Finally, we give a numerical experiment to verify the validity of the algorithm.     

On the nonemptiness and compactness of the solution sets for vector variational inequalities

In this article, we investigate conditions for nonemptiness and compactness of the sets of solutions of pseudomonotone vector variational inequalities by using the concept of asymptotical cones. We show that a pseudomonotone vector variational inequality has a nonempty and compact solution set provided that it is strictly feasible. We also obtain some necessary conditions for the set of solutions of a pseudomonotone vector variational inequality to be nonempty and compact.     

Existence of multiple solutions for a wide class of differential inclusions†

We give a unified approach to study the existence of multiple positive solutions of nonlinear differential inclusions of the form $$-u^{\prime \prime }(t)\in F(t,u(t)),\text{a.e.}t\in (0,1),$$ subject to various nonlocal boundary conditions. We study these problems via a perturbed integral inclusion of the form $u(t)\in Bu(t)+{\int }_0^{1}k(t,s)F(s,u(s))ds$.     

Positive weak solutions of semilinear second order elliptic inequalities via variational inequalities

Existence, uniqueness and convergence of approximants of positive weak solutions for semilinear second order elliptic inequalities are obtained. The nonlinearities involved in these inequalities satisfy suitable upper or lower bound conditions or monotonicity conditions. The lower bound conditions are allowed to contain the critical Sobolev exponents. The methodology is to establish variational inequality principles for demicontinuous pseudo-contractive maps in Hilbert spaces by considering convergence of approximants and apply them to the corresponding variational inequalities arising from the semilinear second order elliptic inequalities. Examples on the existence, uniqueness and convergence of approximants of positive weak solutions of the semilinear second order elliptic inequalities are given.     

Multiplicity of solutions for semilinear variational inequalities via linking and ∇-theorems

We prove the existence of three distinct nontrivial solutions for a class of semilinear elliptic variational inequalities involving a superlinear nonlinearity. The approach is variational and is based on linking and ∇-theorems. Some nonstandard adaptations are required to overcome the lack of the Palais-Smale condition.     

Existence and compactness of solutions to impulsive differential equations with nonlocal conditions

The new type of nonlinear integral inequalities of Volterra–Fredholm type for discontinuous functions is investigated. Then, by using these inequalities and Schaefer fixed‐point theorem, we present new existence results for impulsive semilinear differential equations with nonlocal conditions. Moreover, the compactness of solution sets can be shown in some certain conditions. Copyright © 2015 John Wiley & Sons, Ltd.     

Approximate solutions and optimality conditions of vector variational inequalities in Banach spaces

In this paper, we introduce and discuss the notion of ε-solutions of vector variational inequalities. Using convex analysis and nonsmooth analysis, we provide some sufficient conditions and necessary conditions for a point to be an ε-solution of vector variational inequalities.      

On the Existence of a Generalized Solution to a Three-Dimensional Elliptic Equation with Radiation Boundary Condition

For a second order elliptic equation with a nonlinear radiation-type boundary condition on the surface of a three-dimensional domain, we prove existence of generalized solutions without explicit conditions (like ) on the trace of solutions. In the boundary condition, we admit polynomial growth of any fixed degree in the unknown solution, and the heat exchange and emissivity coefficients may vary along the radiating surface. Our generalized solution is contained in a Sobolev space with an exponent q which is greater than for the fourth power law.     

Existence results for a class of semilinear nonlocal evolution equations

In this paper, we study a class of semilinear evolution equations with nonlocal initial conditions and give some new results on the existence of mild solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

Penalty method for solving a class of stochastic differential variational inequalities with an application

The aim of this paper is to study the penalty method for solving a class of stochastic differential variational inequalities (SDVIs). The penalty problem for solving SDVIs is first constructed and the convergence of the sequences generated by the penalty problem is proved under some mild conditions. As an application, the convergence of the sequences generated by the penalty problem is obtained for solving a stochastic migration equilibrium problem with movement cost.     

Existence of the mild solution for fractional semilinear initial value problems

In this paper we will study the existence and uniqueness of mild solution for the semilinear initial value problem of non-integer order:

u^(α)(t)=Au(t)+f(t,u(t),Gu(t),Su(t)),$u^{\left(\alpha \right)}\left(t\right)=Au\left(t\right)+f(t,u\left(t\right),Gu\left(t\right),Su\left(t\right))\text{,}$

Existence,uniqueness, and multiplicity results on positive solutions for a class of Hadamard-type fractional boundary value problem on an infinite interval

This paper focuses on a class of Hadamard-type fractional differential equation with nonlocal boundary conditions on an infinite interval. New existence, uniqueness, and multiplicity results of positive solutions are obtained by using Schauder's fixed point theorem, Banach's contraction mapping principle, the monotone iterative method, and the Avery-Peterson fixed point theorem. Examples are included to illustrate our main results.     

Enclosure results for quasilinear systems of variational inequalities

In this paper we consider systems of quasilinear elliptic variational inequalities, and prove the existence of minimal and maximal (in the set theoretical sense) solutions within some ordered interval of an appropriately defined pair of sub- and supersolutions. We show that the notion of sub- and supersolutions of variational inequalities introduced here is consistent with the usual notion of sub-supersolutions for (variational) equations. For weakly coupled quasimonotone systems of variational inequalities the existence of smallest and greatest solutions is proved.     

Semigroups of locally Lipschitz operators associated with semilinear evolution equations

In this paper we introduce the notion of semigroups of locally Lipschitz operators which provide us with mild solutions to the Cauchy problem for semilinear evolution equations, and characterize such semigroups of locally Lipschitz operators. This notion of the semigroups is derived from the well-posedness concept of the initial-boundary value problem for differential equations whose solution operators are not quasi-contractive even in a local sense but locally Lipschitz continuous with respect to their initial data. The result obtained is applied to the initial-boundary value problem for the complex Ginzburg–Landau equation.