EXISTENCE RESULTS FOR DEGENERATE ELLIPTIC EQUATIONS WITH CRITICAL CONE SOBOLEV EXPONENTS

In this paper, we study the existence result for degenerate elliptic equations with singular potential and critical cone sobolev exponents on singular manifolds. With the help of the variational method and the theory of genus, we obtain several results under different conditions.

Existence Result for Discrete Problems with Dependence on the First Order Difference

In this paper, we prove the existence of a positive solution and a negative solution for a class of second order difference equations with dependence on the first order difference. Our proofs are based on the Mountain Pass Lemma and iterative methods.

On Nonlinear Conformable Fractional Order Dynamical System via Differential Transform Method

This article studies a nonlinear fractional order Lotka-Volterra prey-predator type dynamical system.For the proposed study,we consider the model under the conformable fractional order derivative(CFOD).We investigate the mentioned dynamical system for the existence and uniqueness of at least one solution.Indeed,Schauder and Banach fixed point theorems are utilized to prove our claim.Further,an algorithm for the approximate analytical solution to the proposed problem has been established.In this regard,the conformable fractional differential transform(CFDT)technique is used to compute the required results in the form of a series.Using Matlab-16,we simulate the series solution to illustrate our results graphically.Finally,a comparison of our solution to that obtained for the Caputo fractional order derivative via the perturbation method is given.

MULTIVALUED DIFFERENTIAL EQUATIONS IN BANACH SPACES AND THEIR APPLICATIONS

This paper studies the existence of solutions to a class of multivalued differential equations by using a surjectivity result for multivalued (S+) type mappings. The authors then apply their results to evolution hemivariational inequalities and parabolic equations with nonmonotone discontinuities, which generalize and extend previously known theorems.

ON QUASILINEAR ELLIPTIC HEMIVARIATIONAL INEQUALITIES

lit the present paper, quasilinear elliptic hemivariational inequalities as a generalization to nonconvex functionals of the elliptic variational inequalities are studied. This extension is strongly motivated by various problems in mechanics. By using the notion of the generalized gradient of Clarke and the theory of pseudomonotone operators, the existence of solutions is proved.

A CLASS OF PARABOLIC HEMIVARIATIONAL INEQUALITIES

Quasilinear parabolic hemivariational inequalities as a generalization to nonconvex functions of the parabolic variational inequalities are discussed. This extension is strongly motivated by various problems in mechanics. By use of the notion of the generalized gradient of Clarke and the theory of pseudomonotone operators, it is proved there exists at least one solution.

On Henig Regularization of Material Design Problems for Quasi-Linear p-Biharmonic Equation

We study a Dirichlet optimal design problem for a quasi-linear monotone p-biharmonic equation with control and state constraints. We take the coefficient of the p-biharmonic operator as a design variable in . In this article, we discuss the relaxation of such problem.

THREE POINT BOUNDARY VALUE PROBLEMS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

In this paper, we study existence and uniqueness of solutions to nonlinear three point boundary value problems for fractional differential equation of the type c D δ 0＋ u（t） = f （t, u（t）, c D σ 0＋ u（t））, t ∈ [0, T ], u（0） = αu（η）, u（T ） = βu（η）, where 1 〈 δ 〈 2, 0 〈 σ 〈 1, α, β∈ R, η∈ （0, T ）, αη（1 -β） ＋ （1-α）（T βη） = 0 and c D δ 0＋ , c D σ 0＋ are the Caputo fractional derivatives. We use Schauder fixed point theorem and contraction mapping principle to obtain existence and uniqueness results. Examples are also included to show the applicability of our results.

Differential Evolution Hemivariational Inequalities with Anti-periodic Conditions

The goal of this paper is to deal with a new dynamic system called a differential evolution hemivariational inequality(DEHVI)which couples an abstract parabolic evolution hemivariational inequality and a nonlinear differential equation in a Banach space.First,by apply ing surjectivity result for pseudomonotone multivalued mappins and the properties of Clarke's subgradient,we show the nonempty of the solution set for the parabolic hemivariational inequality.Then,some topological properties of the solution set are established such as boundedness,closedness and convexity.Furthermore,we explore the upper semicontinuity of the solution mapping.Finally,we prove the solution set of the system(DEHVI)is nonempty and the set of all trajectories of(DEHVI)is weakly compact in C(I,X).

Existence of Solutions for Degenerate Quasilinear Parabolic Equationsof Higher Order

In the paper, existence results for degenerate parabolic boundary value problems of higher order are proved. The weak solution is sought in a suitable weighted Sobolev space by using the generalized degree theory.

On the Solvability of Degenerate Quasilinear Parabolic Equations of Second Order

In this paper,we consider nonlinear degenerate quasilinear parabolic initial boundary value problems of second order.Using results from the theory of pseudomonotone operators,we show that there exists at least one weak solution in a suitable weighted Sobolev space.

On Double Degenerate Quasilinear Parabolic Variational Inequalities

We deal with the existence of weak solutions of double degenerate quasilinear parabolic inequalities with a Signorini-Dirichlet-Neumann type mixed boundary condition, which may degenerate in certain subset of the boundary or on a segment in the interior of the domain and in time. The main tools in our study are the maximM monotone property of the derivative operator with zero-initial valued conditions and the theory of pseudomonotone perturbations of maximal monotone mappings.

Solvability of the Dirichlet Problem in W^(2,p )for a Class of Elliptic Equations with Singular Data

We prove an existence and uniqueness result for the Dirichlet problem for a class of elliptic equations with singular data in weighted Sobolev spaces.

A CLASS OF QUASILINEAR ELLIPTIC HEMIVARIATIONAL INEQUALITIES

In this paper, we shall deal with quasilinear elliptic hemivariational inequalities. By the use of the theory of multivalued pseudomonotone mappings, we will prove the existence of solutions.

PROPERTIES OF SOLUTIONS OF n-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Consider the n-dimensional incompressible Navier-Stokes equations δ/(δt)u-α△u +(u ·△↓)u + △↓p = f(x, t), △↓· u = 0,△↓· f = 0,u(x, 0) = u0(x), △↓·u0=0.There exists a global weak solution under some assumptions on the initial function and the external force. It is well known that the global weak solutions become sufficiently small and smooth after a long time. Here are several very interesting questions about the global weak solutions of the Cauchy problems for the n-dimensional incompressible Navier-Stokes equations.· Can we establish better decay estimates with sharp rates not only for the global weak solutions but also for all order derivatives of the global weak solutions?· Can we accomplish the exact limits of all order derivatives of the global weak solutions in terms of the given information?· Can we use the global smooth solution of the linear heat equation, with the same initial function and the external force, to approximate the global weak solutions of the Navier-Stokes equations?· If we drop the nonlinear terms in the Navier-Stokes equations, will the exact limits reduce to the exact limits of the solutions of the linear heat equation?· Will the exact limits of the derivatives of the global weak solutions of the Navier-Stokes equations and the exact limits of the derivatives of the global smooth solution of the heat equation increase at the same rate as the order m of the derivative increases? In another word, will the ratio of the exact limits for the derivatives of the global weak solutions of the Navier-Stokes equations be the same as the ratio of the exact limits for the derivatives of the global smooth solutions for the linear heat equation?The positive solutions to these questions obtained in this paper will definitely help us to better understand the properties of the global weak solutions of the incompressible Navier-Stokes equations and hopefully to discover new special structures of the Navier-Stokes equations.