A FRACTIONAL NONLINEAR EVOLUTIONARY DELAY SYSTEM DRIVEN BY A HEMI-VARIATIONAL INEQUALITY IN BANACH SPACES

This article deals with a new fractional nonlinear delay evolution system driven by a hemi-variational inequality in a Banach space.Utilizing the KKM theorem,a result concerned with the upper semicontinuity and measurability of the solution set of a hemivariational inequality is established.By using a fixed point theorem for a condensing setvalued map,the nonemptiness and compactness of the set of mild solutions are also obtained for such a system under mild conditions.Finally,an example is presented to illustrate our main results.

MIXED VARIATIONAL INEQUALITIES DRIVEN BY FRACTIONAL EVOLUTIONARY EQUATIONS

The goal of the present paper is to investigate an abstract system, called fractional differential variational inequality, which consists of a mixed variational inequality combined with a fractional evolution equation in the framework of Banach spaces. Using discrete approximation approach, an existence theorem of solutions for the inequality is established under some suitable assumptions.

FRACTIONAL INTEGRAL INEQUALITIES AND THEIR APPLICATIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS

In this paper, first we obtain some new fractional integral inequalities. Then using these inequalities and fixed point theorems, we prove the existence of solutions for two different classes of functional fractional differential equations.

The Brezis–Nirenberg Problem for the Fractional p-Laplacian in Unbounded Domains

In this paper we study the existence of nontrivial solutions to the well-known Brezis–Nirenberg problem involving the fractional p-Laplace operator in unbounded cylinder type domains.By means of the fractional Poincaréinequality in unbounded cylindrical domains,we first study the asymptotic property of the first eigenvalueλp,s(ωδ)with respect to the domainωδ.Then,by applying the concentration-compactness principle for fractional Sobolev spaces in unbounded domains,we prove the existence results.The present work complements the results of Mosconi–Perera–Squassina–Yang[The Brezis–Nirenberg problem for the fractional p-Laplacian.C alc.Var.Partial Differential Equations,55(4),25 pp.2016]to unbounded domains and extends the classical Brezis–Nirenberg type results of Ramos–Wang–Willem[Positive solutions for elliptic equations with critical growth in unbounded domains.In:Chapman Hall/CRC Press,Boca Raton,2000,192–199]to the fractional p-Laplacian setting.

Linearized Transformed L1 Galerkin FEMs with Unconditional Convergence for Nonlinear Time Fractional Schr¨odinger Equations

A linearized transformed L1 Galerkin finite element method(FEM)is presented for numerically solving the multi-dimensional time fractional Schr¨odinger equations.Unconditionally optimal error estimates of the fully-discrete scheme are proved.Such error estimates are obtained by combining a new discrete fractional Gr¨onwall inequality,the corresponding Sobolev embedding theorems and some inverse inequalities.While the previous unconditional convergence results are usually obtained by using the temporal-spatial error spitting approaches.Numerical examples are presented to confirm the theoretical results.

ALIKHANOV LINEARIZED GALERKIN FINITE ELEMENT METHODS FOR NONLINEAR TIME-FRACTIONAL SCHRODINGER EQUATIONS

We present Alikhanov linearized Galerkin methods for solving the nonlinear time fractional Schrödinger equations.Unconditionally optimal estimates of the fully-discrete scheme are obtained by using the fractional time-spatial splitting argument.The convergence results indicate that the error estimates hold without any spatial-temporal stepsize restrictions.Numerical experiments are done to verify the theoretical results.

Analysis of L1-Galerkin FEMs for Time-Fractional Nonlinear Parabolic Problems

This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods.The analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality.In this paper,we establish such a fundamental inequality for the L1 approximation to the Caputo fractional derivative.In terms of the Gronwall type inequality,we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems.The theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.

Fractional Sobolev-Poincar Inequalities in Irregular Domains

This paper is devoted to the study of fractional(q, p)-Sobolev-Poincar′e inequalities in irregular domains. In particular, the author establishes(essentially) sharp fractional(q, p)-Sobolev-Poincar′e inequalities in s-John domains and in domains satisfying the quasihyperbolic boundary conditions. When the order of the fractional derivative tends to 1, our results tend to the results for the usual derivatives. Furthermore, the author verifies that those domains which support the fractional(q, p)-Sobolev-Poincar′e inequalities together with a separation property are s-diam John domains for certain s, depending only on the associated data. An inaccurate statement in [Buckley, S. and Koskela, P.,Sobolev-Poincar′e implies John, Math. Res. Lett., 2(5), 1995, 577–593] is also pointed out.

A Compact Difference Scheme on Graded Meshes for the Nonlinear Fractional Integro-differential Equation with Non-smooth Solutions

In this paper,a compact finite difference scheme for the nonlinear fractional integro-differential equation with weak singularity at the initial time is developed,with O(N^(-(2-α))+M^(-4))accuracy order,where N;M denote the numbers of grids in temporal and spatial direction,α ∈(0,1)is the fractional order.To recover the full accuracy based on the regularity requirement of the solution,we adopt the L1 method and the trapezoidal product integration(PI)rule with graded meshes to discretize the Caputo derivative and the Riemann-Liouville integral,respectively,further handle the nonlinear term carefully by the Newton linearized method.Based on the discrete fractional Gr¨onwall inequality and preserved discrete coefficients of Riemann-Liouville fractional integral,the stability and convergence of the proposed scheme are analyzed by the energy method.Theoretical results are also confirmed by a numerical example.

A Derivation of the Sharp Moser-Trudinger-Onofri Inequalities from the Fractional Sobolev Inequalities

We derive the sharp Moser-Trudinger-Onofri inequalities on the standard n-sphere and CR(2n+1)-sphere as the limit of the sharp fractional Sobolev inequalities for all n≥1.On the 2-sphere and 4-sphere,this was established recently by Chang and Wang.Our proof uses an alternative and elementary argument.