摘要
为了研究一类非线性分数阶q型差分方程边值问题非平凡解的存在唯一性。首先,在一个新的集合上定义一个新概念,再利用正规锥的定义,建立了2个混合单调算子唯一不动点的存在性,获得了线性分数阶q型边值问题的Green函数,并且对Green函数的上下界进行了估计,由此可得到特解的表达形式。其次,运用抽象定理,讨论了符合定理条件的非线性项,建立了上述问题的唯一解的存在性,并获得逼近唯一解的迭代序列,进而证明了分数阶q型差分方程边值问题非平凡解的存在唯一性。最后,通过列举一个例子来说明主要定理和结果的有效性。研究结果表明,定理条件得证且方程组边值问题非平凡解满足存在唯一性。研究方法在理论证明和边值问题方面都得到了良好的结果,对探究其他边值问题具有一定的借鉴意义。
In order to study the existence and uniqueness of nontrivial solutions for boundary value problems of a class of nonlinear fractional difference equations.Firstly,a new concept is defined on a new set.Then,by using the definition of normal cone,the existence of unique fixed points for two mixed monotone operators is established.Green functions for linear fractionalorder boundary value problems are obtained,and the upper and lower bounds of Green functions are estimated.Thus,the expression of the particular solution can be obtained.Secondly,by using the abstract theorem,we discuss the non-linear terms which meet the conditions of the theorem,establish the existence of the unique solution of the above problem,and obtain the iteration sequence approaching the unique solution.Furthermore,the existence and uniqueness of nontrivial solutions for boundary value problems of fractional difference equations are proved.Finally,we give an example to illustrate the validity of the main theorems and results.The results show that the conditions of the theorem are proved and the nontrivial solution of the boundary value problem satisfies the existence and uniqueness.The research method has obtained good results in theoretical proof and boundary value problem,and has certain reference significance for other boundary value problems.
作者
韩伟
孟晓宇
桑彦彬
HAN Wei;MENG Xiaoyu;SANG Yanbin(School of Science,North University of China,Taiyuan,Shanxi 030051,China)
出处
《河北科技大学学报》
CAS
2019年第4期307-316,共10页
Journal of Hebei University of Science and Technology
基金
国家自然科学基金(11571324)
山西省高等学校科技创新项目(201802085)
山西省高等学校优秀青年学术带头人支持计划项目
关键词
非线性偏微分方程
分数阶q型差分方程
混合单调算子
存在唯一性
非平凡解
nonlinear partial differential equations
fractional q-difference equation
mixed monotone operator
existence and uniqueness
nontrivial solution
