Zhenhai Liu

• Professor
• Professor (Full) at Guangxi Minzu University

We consider a degenerate nonlinear elliptic equation driven by the weighted p-Laplacian differential operator and a reaction consisting of a concave term and of a resonant perturbation. Using variational tools and critical groups, we show the existence of a nontrivial bounded solution.

The primary objective of this paper is to investigate an abstract differential system comprising an evolution equation with history-dependent operator and a doubly nonlinear inclusion with nonmonotone perturbation. The latter arises from enthalpy formulation of heat conduction problems involving phase change and nonmonotone source. We firstly intro...

In this paper, we investigate the generic regularity of conservation solutions to the N − abc family of Camassa-Holm type equation with (N + 1)-order nonlinearities. This quasi-linear equation is nonlocal with higher order nonlinearities, compared to the Camassa-Holm equation (N = 1) and Novikov eqution (N = 2). For an open dense set of C 3 initial...

We consider an eigenvalue problem for the nonautonomous double phase differential operator. We show that there exist two positive numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt}...

The aim of this paper is to examine an inverse problem of parameter identification in an evolutionary quasi-variational hemivariational inequality in infinite dimensional reflexive Banach spaces. First, the solvability and compactness of the solution set to the inequality are established by employing a fixed point argument and tools of non-linear a...

The primary objective of this paper is to study a new class of parabolic quasi variational–hemivariational inequalities. First, we prove a unique solvability result for such class under some mild conditions. Second, we show the existence of an optimal solution for an associated control problem. Finally, these results are applied to a model of quasi...

The aim of this paper is to examine an inverse problem of parameter identification in an evolutionary quasi-variational hemivariational inequality in in nite dimensional re exive Banach spaces. First, the solvability and compactness of the solution set to the inequality are established by employing a fixed point argument and tools of nonlinear anal...

We consider an eigenvalue problem driven by the anisotropic ( p , q )-Laplacian and with a Carathéodory reaction which is ( p ( z ) − 1)-sublinear as x → + ∞. We look for positive solutions. We prove an existence, nonexistence and multiplicity theorem which is global in the parameter λ > 0, that is, we prove a bifurcation-type theorem which describ...

In this paper, we are concerned with a generalized evolution dynamical system, called fractional differential variational-hemivariational inequality (FDVHVI, for short), which is composed of a nonlinear fractional evolution inclusion and a time-dependent mixed variational-hemivariational inequality in the framework of Banach spaces. The objective o...

We consider a nonlinear Dirichlet problem driven by the double phase operator. The reaction has the combined effects of parametric concave term and of an indefinite convex one (“concave-convex” problem with indefinite weight). Using the Nehari method we prove the existence of two bounded, positive ground state solutions.

The main purpose of this paper is to study an abstract system which consists of a non-linear differential inclusion with $C_0$-semigroups and history-dependent operators combined with an evolutionary non-linear inclusion involving pseudomonotone operators, which contains several interesting problems as special cases. We first introduce a hybrid ite...

The main purpose of this paper is to study an abstract system which consists of a non-linear differential inclusion with $C_0$-semigroups and history-dependent operators combined with an evolutionary non-linear inclusion involving pseudomonotone operators, which contains several interesting problems as special cases. We first introduce a hybrid ite...

We consider a first order periodic system in ℝN, involving a time dependent maximal monotone operator which need not have a full domain and a multivalued perturbation. We prove the existence theorems for both the convex and nonconvex problems. We also show the existence of extremal periodic solutions and provide a strong relaxation theorem. Finally...

In this paper, we investigate the − family of Camassa-Holm type equation with (+ 1)-order nonlinearities. This quasi-linear equation is nonlocal with higher order nonlinearities, compared to the Camassa-Holm equation (= 1) and Novikov equation (= 2). Using both the lower order and the higher order energy conservation laws, as well as the characteri...

We consider a Dirichlet problem driven by the anisotropic (p, q)-Laplacian (double phase problem) and with a reaction term which exhibits asymmetric behavior as x→±∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlen...

This paper is concerned with an evolutionary quasi-variational hemivariational inequality in which both the convex and nonconvex energy functionals depend on the unknown solution. The inequality serves as a direct problem of the inverse problem of parameters identification. Employing a fixed point argument and tools from nonlinear analysis, we esta...

The main purpose of this paper is to study an abstract system which consists of a parabolic hemivariational inequality with a Hilfer fractional evolution equation involving history-dependent operators, which is called a Hilfer fractional differential hemivariational inequality. We first show existence and a priori estimates for the parabolic hemiva...

In this paper, we study elliptic equations in which the reaction (right hand side) exhibits an asymmetric behavior as \(x\rightarrow \pm \infty \). More precisely, we assume that we have resonance as \(x\rightarrow -\infty \), while as \(x\rightarrow +\infty \) the equation is superlinear. Using variational tools combined with the theory of critica...

We consider a Neumann problem driven by a (p(z), q(z))-Laplacian (anisotropic problem) plus a parametric potential term with λ > 0 being the parameter. The reaction is superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter λ m...

We consider a Dirichlet nonlinear equation driven by the (p, 2)-Laplacian and with a reaction having the competing effects of a parametric asymmetric superlinear term and a resonant perturbation. We show that for all small values of the parameter the problem has at least five nontrivial smooth solutions all with sign information.

We consider an anisotropic Dirichlet problem driven by the variable (p, q)-Laplacian (double phase problem). In the reaction, we have the competing effects of a singular term and of a superlinear perturbation. Contrary to most of the previous papers, we assume that the perturbation changes sign. We prove a multiplicity result producing two positive...

The goal of the paper is to investigate a Kirchhoff-type elliptic problem driven by a generalized nonlocal fractional p-Laplacian whose nonlocal term vanishes at finitely many points. Multiple nontrivial solutions are obtained by applying a variational method combined with truncation techniques.

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 applying surjectivity result for pseudomonotone multivalued mappins and the propert...

We consider a nonlinear eigenvalue problem driven by the anisotropic (p, q)-Laplacian. Using variational tools and truncation and comparison techniques, we show the existence of a continuous spectrum (a bifurcation-type theorem). We also show the existence of a minimal positive solution and determine the properties of the minimal solution map.

We consider a nonlinear eigenvalue problem for the Dirichlet (p,q)$(p,q)$‐Laplacian with a sign‐changing Carathé$\acute{\rm e}$odory reaction. Using variational tools, truncation and comparison techniques, and critical groups, we prove an existence and multiplicity result which is global in the parameter λ>0$\lambda >0$ (bifurcation‐type theorem)....

We consider a Dirichlet problem driven by a weighted (p,2)-Laplacian with a reaction which is resonant both at \(\pm\infty\) and at zero (double resonance). We prove a multiplicity theorem producing three nontrivial smooth solutions with sign information and ordered. In the appendix we develop the spectral properties of the weighted r-Laplace diffe...

We consider a parametric double phase equation with unbalanced growth and a logistic-type reaction. We assume that in the reaction the perturbation term may be sign-changing. We show that for all large values of the parameter λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepa...

We consider a Dirichlet problem driven by the anisotropic p‐Laplacian, with a reaction having the competing effects of a singular term and a parametric superlinear perturbation. We prove a bifurcation‐type theorem describing the changes of the set of positive solutions as the parameter varies. We also prove the existence of minimal positive solutio...

In this paper, we consider an abstract system which consists of a nonlinear differential inclusion and a parabolic hemivariational inequality (DPHVI) in Banach spaces. The objective of this paper is four fold. The first target is to deal with the existence of solutions and the properties which involve the boundedness and continuous dependence resul...

The aim of this paper is to investigate a nonlinear optimal control problem governed by a complicated dynamic variational-hemivariational inequality (DVHVI, for short) with history-dependent operators in the framework of an evolution triple of spaces. First, we prove a generalized existence and uniqueness theorem to a class of dynamic variational-h...

We consider a (p, q)-equation with unbalanced growth (double phase problem) and a logistic reaction of the equidiffusive type. We show the existence and uniqueness of a positive solution.

The purpose of this paper is to establish some new results on the Levitin–Polyak well-posedness to a class of split hemivariational inequality problems on Hadamard manifolds. We first consider a new class of split hemivariational inequality problems (for short, SHIP) on Hadamard manifolds and introduce the regularized gap functions for these proble...

In the present paper, we are concerned with the study of a nonlinear complementarity problem (NCP, for short) with a nonlinear and nonhomogeneous partial differential operator (called double phase differential operator), a convection term (i.e., a reaction depending on the gradient), a generalized multivalued boundary condition, and two nonlocal te...

We consider a double phase Dirichlet problem with a gradient dependent reaction term (convection). Using the theory of nonlinear operators of monotone type, we show the existence of a bounded strictly positive solution. Moreover, we show that the set of these solutions is compact in the corresponding generalized Sobolev–Orlicz space.

In this paper, we investigate the N − abc family of Camassa-Holm type equation with (N + 1)-order nonlinearities. Using both the lower order and the higher order energy conservation laws, as well as the characteristic method, we establish the global existence and uniqueness of the Hölder continuous energy weak solution to the N − abc family of Cama...

We consider a Dirichlet problem having a double phase differential operator with unbalanced growth and reaction involving the combined effects of a concave (sublinear) and of a convex (superlinear) terms. We allow the coefficient \(\mathcal E\in L^\infty(\Omega)\) of the concave term to be sign changing. We show that when \(\|\mathcal E\|_\infty \)...

In this paper, we establish several kinds of integral inequalities in two independent variables, which improve well-known versions of Gronwall-Bellman inequalities and extend them to fractional integral form. By using these inequalities, we can provide explicit bounds on unknown functions. The integral inequalities play an important role in the qua...

In this paper, we will study optimal feedback control problems derived by a class of Riemann-Liouville fractional evolution equations with history-dependent operators in separable reflexive Banach spaces. We firstly introduce suitable hypotheses to prove the existence and uniqueness of mild solutions for this kind of Riemann-Liouville fractional ev...

We consider a double phase Dirichlet problem with both convex and nonconvex unilateral constraints (variational-hemivariational inequality). Using variational techniques and tools from nonsmooth analysis (convex and nonconvex), we establish the existence of a nontrivial bounded solution.

The goal of this paper is to consider fractional differential hemivariational inequalities (FDHVIs, for short) in the framework of Banach spaces. Our first aim is to investigate the existence of mild solutions to FDHVIs by means of a fixed point technique avoiding the hypothesis of compactness on the semigroup. The second step of the paper is to st...

We consider a double phase Dirichlet problem with a reaction exhibiting the competing effects of a concave (sublinear) term and a parametric convex (superlinear) term. Using the Nehari method, we show that for all small values of the parameter, the problem has at least two positive bounded solutions.

We consider a double phase problem with a gradient dependent reaction (convection). Using the theory of nonlinear operators of monotone type, we show the existence of a nontrivial, positive, bounded solution.

We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian and with a Carathéodory reaction f(z, x) which is (p-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{docu...

The primary objective is to investigate a class of noncoercive variational–hemivariational inequalities on a Banach space. We start with several new existence results for the abstract inequalities in which our approach is based on arguments of recession analysis and the theory of pseudomonotone operators. A nonsmooth elastic contact problem is cons...

We consider a nonlinear Robin problem driven by the anisotropic (p, q)-Laplacian and with a reaction exhibiting the competing effects of a parametric sublinear (concave) term and of a superlinear (convex) term. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter varies. We also prove the exis...

In this paper we consider a class of feedback control systems described by an evolution hemivariational inequality involving history-dependent operators. Under the mild conditions, first, we prove a priori estimates of the solutions to the feedback control system. Then, an existence theorem for the feedback control system is obtained by using the w...

We consider a nonlinear Dirichlet problem driven by the variable exponent (anisotropic) $p$-Laplacian and a reaction that has the competing effects of a singular term and of a superlinear perturbation. There is no parameter in the equation (nonparametric problem). Using variational tools together with truncation and comparison techniques, we show t...

The goal of this paper is to deal with approximate controllability of control systems described by non-autonomous second-order evolution hemivariational inequalities with nonlocal conditions in Hilbert spaces. First we define the concept of mild solution relying on the existence of an evolution operator for the corresponding linear equation and the...

We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator. The reaction has a parametric concave term and negative sublinear perturbation. In contrast to the case of a positive perturbation, we show that now for all big values of the parameter λ> 0 , we have at least two positive solutions which do not vanish in the...

In this paper, we investigate the initial value problem (IVP henceforth) associated with the perturbed nonlinear Schrödinger equations with the Kerr law nonlinearity. First, by using Fourier restriction norm method and Tao’s [k, Z]-multiplier method, we establish a trilinear estimate on the Bourgain space Xs,b.\documentclass[12pt]{minimal} \usepack...

In this paper, we first recall a class of parametric variational–hemivariational inequalities (PVHIs) introduced in Jiang et al. (2020). Then, based on the properties of the Clarke generalized gradient, we establish the Hölder continuity of the solution mapping for PVHIs in terms of regularized gap functions under some assumptions imposed on the da...

In this paper, we investigate the long time behavior of the damped forced generalized Ostrovsky equation below the energy space. First, by using Fourier restriction norm method and Tao’s [ k , Z ] [k,Z] - multiplier method, we establish the multi-linear estimates, including the bilinear and trilinear estimates on the Bourgain space X s , b . X_{s,b...

In this paper, we investigate the perturbed nonlinear Schrödinger equation with Kerr law nonlinearity and localized damping. First, by using the semigroup method, we prove that the global existence and uniqueness of the solution to the linear problem. To overcome some difficulties, such as the presence of the perturbed effect and smooth effect, ben...

We consider a Dirichlet double phase problem with unbalanced growth. In the reaction we have the combined effects of a critical term and of a locally defined Carathéodory perturbation. Using cut-off functions and truncation techniques we bypass the critical term and deal with a coercive problem. Using this auxillary problem, we show that the origin...

In this paper, we study the existence of solutions for non-coercive variational-hemivariational inequalities involving nonlocal fractional p-Laplacian. Our approach is based on the theory of pseudomonotone operators in the sense of Brézis, recession analysis and the properties of nonlocal fractional p-Laplace operators recently established.The inno...

In this work, we deal with the quasilinearization technique for a class of nonlinear Riemann-Liouville fractional-order two-point boundary value problems. Using quasilinearization technique, we construct a monotone sequence of approximate solutions which has quadratic convergence to the unique solution of the original problem, and establish the cor...

The aim the paper is to study a large class of variational-hemivariational inequalities involving constraints in a Banach space. First, we establish a general existence theorem for this class. Second, we introduce a sequence of penalized problems without constraints. Under the suitable assumptions, we prove that the Kuratowski upper limit with resp...

We consider a nonlinear parametric Dirichlet problem driven by the (p, q)-Laplacian (double phase problem) with a reaction exhibiting the competing effects of three different terms. A parametric one consisting of the sum of a singular term and of a drift term (convection) and of a nonparametric perturbation which is resonant. Using the frozen varia...

In this paper, we consider the Cauchy problem for the N − abc family of the Camassa–Holm type equation with both dissipation and dispersion. First, we establish the global well-posedness of the strong solutions under certain conditions on the initial datum. Then, we investigate the propagation speed with compactly supported initial data. This resul...

In this paper, we consider the regularization of a class of elliptic variational‐hemivariational inequalities driven by the fractional Laplace operator. First, we demonstrate characterizations of nonlocal elliptic variational‐hemivariational inequalities. Next, we provide coercivity conditions that guarantee the existence and uniqueness of solution...

In the paper we study a class of semilinear differential variational systems with nonlocal boundary conditions, which are obtained by mixing evolution equations and generalized variational inequalities. Firstly, we show the properties of the solution set for generalized variational inequalities. Then, the existence results are established and prove...

The paper sets forth a new type of variational problem without any ellipticity or monotonicity condition. A prototype is a differential inclusion whose driving operator is the competing weighted $(p,q)$-Laplacian $-\Delta_p u+\mu\Delta_q u$ with $\mu\in \mathbb{R}$. Local and nonlocal boundary value problems fitting into this nonstandard setting ar...

The purpose of this paper is to consider a control system of nonlinear evolution hemivariational inequalities with mixed nonconvex constraints on the control. First, we investigate the relaxed problem for the hemivariational inequalities systems with feedback control and the integrand convexicated with respect to the control. Second, under some gen...

We consider a frictionless contact problem, Problem [Formula: see text], for elastic materials. The process is assumed to be static and the contact is modelled with unilateral constraints. We list the assumptions on the data and derive a variational formulation of the problem, Problem [Formula: see text]. Then we consider a perturbation of Problem...

The purpose of this paper is to investigate a class of differential variational inequalities involving a constraint set in Banach spaces. A well-posedness result for the inequality is obtained, including the existence, uniqueness, and stability of the solution in mild sense. Further, we introduce a penalized problem without constraints and prove th...

The goal of this paper is to provide systematic approaches to study the feedback control systems governed by impulsive evolution equations in separable reflexive Banach spaces. We firstly give some existence results of mild solutions for the equations by applying the Banach’s fixed point theorem and the Leray–Schauder alternative fixed point theore...

In this paper a class of generalized differential variational inequalities with constraints involving history-dependent operators in Banach spaces is investigated. The unique solvability and regularity results are obtained via surjectivity of multivalued pseudomonotone operators combined with a fixed point principle. From abstract results, a theore...

The purpose of this paper is to discuss the control systems of nonlinear evolution hemivariational inequalities and their “bang-bang” principle in Banach space. At first, we show that extremal trajectories are in fact dense in the trajectories of the original system with convexified feedback control. Second, by using the density results, the nonlin...

This article analyzes nonlinear, second-order difference equations subject to "left-focal" two-point boundary conditions. Our research questions are: RQ1: What are new, sufficient conditions under which solutions to our "discrete" problem will exist?; RQ2: What, if any, is the relationship between solutions to the discrete problem and solutions o...

This paper deals with a new class of second order nonlinear differential variational inequalities. Firstly, we study the properties of a solution set for a variational inequality by using the KKM technique and a monotonicity property. Then, we prove the existence of solutions to a second order nonlinear differential variational inequality with anti...

In this paper, by introducing a new concept of the (f, g, h)-quasimonotonicity and applying the maximal monotonicity of bifunctions and KKM technique, we show the existence results of solutions for quasi mixed equilibrium problems when the constraint set is compact, bounded and unbounded, respectively, which extends and improves several well-known...

In this paper, we investigate the initial value problem(IVP henceforth) associated with the higher order nonlinear dispersive equation given in Jones et al. (Int J Math Math Sci 24:371–377, 2000): $$\begin{aligned} \left\{ \begin{array}{ll} \partial _tu+\alpha \partial _x^7u+\beta \partial _x^5u+\gamma \partial _x^3u+\mu \partial _xu+\lambda u\part...

This paper is devoted to the existence of solutions for space-fractional parabolic hemivariational inequalities by means of the well-known surjectivity result for multivalued (S + ) type mappings. © 2019 American Institute of Mathematical Sciences. All Rights Reserved.

The purpose of this paper is to study a class of semilinear differential variational systems with nonlocal boundary conditions, which are obtained by mixing semilinear evolution equations and generalized variational inequalities. First we prove essential properties of the solution set for generalized variational inequalities. Then without requiring...

We study an abstract second order inclusion involving two nonlinear single-valued operators and a nonlinear multivalued term. Our goal is to establish the existence of solutions to the problem by applying numerical scheme based on time discretization. We show that the sequence of approximate solution converges weakly to a solution of the exact prob...

In this paper, we study a differential hemivariational inequality (DHVI, for short) in the framework of reflexive Banach spaces. Our aim is three fold. The first one is to investigate the existence and the uniqueness of mild solution, by applying a general fixed-point principle. The second one is to study its exponential stability, by employing the...

In this paper, we are concerned with the existence of mild solution and controllability for a class of nonlinear fractional control systems with damping in Hilbert spaces. Our first step is to give the representation of mild solution for this control system by utilizing the general method of Laplace transform and the theory of (�,)-regularized fami...

In this paper, we study a nonlinear Dirichlet problem of p-Laplacian type with combined effects of nonlinear singular and convection terms. An existence theorem for positive solutions is established as well as the compactness of solution set. Our approach is based on Leray–Schauder alternative principle, method of sub-supersolution, nonlinear regul...

In this paper, we suggest a new iterative scheme for finding a common element of the set of solutions of a split equilibrium problem and the set of fixed points of 2-generalized hybrid mappings in Hilbert spaces. We show that the iteration converges strongly to a common solution of the considered problems. A numerical example is illustrated to veri...

In this paper, a sensitivity analysis of optimal control problem for a class of systems described by nonlinear fractional evolution inclusions (NFEIs, for short) on Banach spaces is investigated. Firstly, the nonemptiness as well as the compactness of the mild solutions set S ( ζ ) ( ζ being the initial condition) for the NFEIs are obtained, and we...

In this paper, we deal with the control systems described by a large class of fractional semilinear parabolic equations. Firstly, we reformulate the fractional parabolic equations into abstract fractional differential equations associated with a semigroup on an appropriate Banach space. Secondly, we introduce a suitable concept on a mild solution f...