摘要：
J. Y. Park and T. G. Ha [Nonlinear Anal., 2008, 68: 747-767; 2009, 71: 3203-3217] investigated the existence of anti-periodic solutions for hemivariational inequalities with a pseudomonotone operator. In this note, we point out that the methods used there are not suitable for the proof of the existence of anti-periodic solutions for hemivariational inequalities and we shall give a straightforward approach to handle these problems. The main tools in our study are the maximal monotone property of the derivative operator with antiperiodic conditions and the surjectivity result for L-pseudomonotone operators.
期刊：
Fractional Calculus and Applied Analysis,2016年19(1):94-115 ISSN：1311-0454
通讯作者：
Liu, Xiaoyou;Xu, Youjun
作者机构：
[Liu, XY; Xu, YJ; Liu, Xiaoyou; Xu, Youjun] Univ South China, Sch Math & Phys, Hengyang 421001, Hunan, Peoples R China.
通讯机构：
[Liu, XY; Xu, YJ] U;Univ South China, Sch Math & Phys, Hengyang 421001, Hunan, Peoples R China.
关键词：
fractional differential equation;optimal control;Bogolyubov type theorem;relaxation property;nonconvex constraint
摘要：
We first study the existence results and properties of the solution set of a control system described by fractional differential equations with nonconvex control constraint. Then a problem of minimizing an integral functional over the solution set of the control system is considered. Along with the original minimizing problem, we also consider the problem of minimizing the integral functional whose integrand is the bipolar (with respect to the control variable) of the original integrand over the solution set of the same system but with the convexified control constraint. We prove that the relaxed problem has an optimal solution and obtain some relationships between these two minimizing problems. Finally, an example is given to illustrate the results.
摘要：
In this paper, we research the existence of solutions for a class of Riemann-Liouville fractional evolution inclusions with nonconvex right hand side. Our main results obtain the existence of the extreme solution and the relationship of the solution sets between the original problem and the convexified problem. In the end, we give an example to illustrate the applications of the abstract results.
期刊：
Journal of Function Spaces,2016年2016:1-6 ISSN：2314-8896
通讯作者：
Liu, Xiaoyou
作者机构：
[Fu, Xi] Shaoxing Univ, Dept Math, Shaoxing 312000, Zhejiang, Peoples R China.;[Xu, Zhiyao] Jiangxi Vocat Coll Ind & Engn, Dept Math, Pingxiang 337055, Jiangxi, Peoples R China.;[Liu, Xiaoyou] Univ South China, Sch Math & Phys, Hengyang 421001, Hunan, Peoples R China.
通讯机构：
[Liu, Xiaoyou] U;Univ South China, Sch Math & Phys, Hengyang 421001, Hunan, Peoples R China.
摘要：
Let B be the real unit ball in R-n and f c l(N) (B). Given a multi-index,m = (m(1),...,m(n)) of nonnegative integers with vertical bar m vertical bar = N, we set the quantity sup(x epsilon B,y epsilon E(x,r),x not equal y) (1-vertical bar x vertical bar(2))(alpha) (1-vertical bar y vertical bar(2))(beta) (vertical bar partial derivative(m) f(x)-partial derivative(m) f(y)vertical bar/vertical bar x-y vertical bar(gamma) [x,y](1-gamma)), x+y) where 0 <= gamma <= 1 and alpha + beta - N + 1. In terms of it, we characterize harmonic Bloch and Besov spaces on the real unit ball. This generalizes the main results of Ydneda, 2002, into real harmonic setting.
期刊：
Journal of Computational Analysis and Applications,2016年20(4):734-749 ISSN：1521-1398
通讯作者：
Liu, Xianghu
作者机构：
[Liu, Xianghu; Liu, Yanmin] Zunyi Normal Coll, Sch Math & Comp Sci, Zunyi 563002, Guizhou, Peoples R China.;[Liu, Xiaoyou] Univ South China, Sch Math & Phys, Hengyang 421001, Hunan, Peoples R China.
通讯机构：
[Liu, Xianghu] Z;Zunyi Normal Coll, Sch Math & Comp Sci, Zunyi 563002, Guizhou, Peoples R China.
关键词：
Fractional differential inclusions;boundary value problems;existence results;multivalued maps
摘要：
In this paper, a class of fractional differential inclusions with fractional non-separated (integral) boundary conditions is investigated under both convexity and non-convexity conditions on the multivalued term. Some new existence results are obtained by using standard fixed point theorems. Examples are given to illustrate the results.
期刊：
JOURNAL OF INEQUALITIES AND APPLICATIONS,2015年2015(1):1-10 ISSN：1029-242X
通讯作者：
Liu, Xiaoyou
作者机构：
[Fu, Xi] Shaoxing Univ, Dept Math, Shaoxing 312000, Zhejiang, Peoples R China.;[Liu, Xiaoyou] Univ South China, Sch Math & Phys, Hengyang 421001, Hunan, Peoples R China.
通讯机构：
[Liu, Xiaoyou] U;Univ South China, Sch Math & Phys, Hengyang 421001, Hunan, Peoples R China.
关键词：
pluriharmonic mapping;Bloch space;Besov space
摘要：
We characterize the Bloch spaces and Besov spaces of pluriharmonic mappings on the unit ball of
${\mathbb{C}}^{n}$
by using the following quantity:
$\sup_{\rho(z,w)< r,z\neq w}\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat{D}^{(m)}f(z)-\hat {D}^{(m)}f(w)|}{|z-w|}$
, where
$\alpha+\beta=n+1$
,
$\hat{D}^{(m)}=\frac{\partial ^{m}}{\partial z^{m}}+\frac{\partial^{m}}{\partial\bar{z}^{m}}$
,
$|m|=n$
. This generalizes the main results of (Yoneda in Proc. Edinb. Math. Soc. 45:229-239, 2002) in the higher dimensional case.
摘要：
This paper is concerned with the existence of solutions for nonlinear impulsive fractional differential equations with families of mixed and closed boundary conditions. In both cases, the fractional derivative of lower order is involved in the formulation of impulsive conditions. By means of the Banach fixed point theorem, Schaefer fixed point theorem and Nonlinear alternative of Leray-Sthauder type, some existence results are obtained. Examples are given to illustrate the results.
期刊：
Abstract and Applied Analysis,2014年2014(SI40):1-11 ISSN：1085-3375
通讯作者：
Liu, Xiaoyou
作者机构：
[Fu, Xi] Shaoxing Univ, Dept Math, Shaoxing 312000, Zhejiang, Peoples R China.;[Liu, Xiaoyou] Univ South China, Sch Math & Phys, Hengyang 421001, Hunan, Peoples R China.
通讯机构：
[Liu, Xiaoyou] U;Univ South China, Sch Math & Phys, Hengyang 421001, Hunan, Peoples R China.
关键词：
This paper studies the existence results for nonseparated boundary value problems of fractional differential equations with fractional impulsive conditions. By means of Schaefer fixed point theorem;Banach fixed point theorem;and nonlinear alternative of Leray-Schauder type;some existence results are obtained. Examples are given to illustrate the results. Published: 2014 First available in Project Euclid: 6 October 2014 zbMATH: 07022232 MathSciNet: MR3240533 Digital Object Identifier: 10.1155/2014/364348
关键词：
Fractional differential equation;Optimal control;Relaxation property;Nonconvex constraint;Feedback control
摘要：
We consider the minimization problem of an integral functional with integrand that is not convex in the control on solutions of a control system described by fractional differential equation with mixed nonconvex constraints on the control. A relaxation problem is treated along with the original problem. It is proved that, under general assumptions, the relaxation problem has an optimal solution, and that for each optimal solution there is a minimizing sequence of the original problem that converges to the optimal solution with respect to the trajectory, the control, and the functional in appropriate topologies simultaneously.
