分数阶差分方程解的振动性

Oscillation results for certain fractional difference equations

分数阶微积分是研究任意阶微分和积分性质及应用的一种理论,它可以更加精确的描述一些系统的物理特性,更加适应系统的变化,可以应用于描述生物医学中的肿瘤生长(生长刺激与生长抑制)过程。为了研究2类分数阶差分方程解的振动性,主要利用反证法,即假设方程有非振动解,对于第1类方程首先确定函数符号,通过构造Riccati函数,对其求差分,利用函数满足的条件得到矛盾,即假设不成立,验证了解的振动性。对于第2类带有初值条件的方程,首先证明了与该分数阶差分方程等价的和分形式,然后分别考虑0<α≤1和α>1两种情况,运用Stirling公式及阶乘函数的性质,放大处理得到与已知条件相矛盾,假设不成立,获得分数阶差分方程有界解振动的充分条件。以上结果优化了相关结论,丰富了相关成果,并把结果应用到具体方程之中,验证了方程解的振动性质。

Fractional calculus is a theory that studies the properties and application of arbitrary order differentiation and integration.It can describe the physical properties of some systems more accurately,and better adapt to changes in the system,playing an important role in many fields.For example,it can describe the process of tumor growth（growth stimulation and growth inhibition）in biomedical science.The oscillation of solutions of two kinds of fractional difference equations is studied,mainly using the proof by contradiction,that is,assuming the equation has a nonstationary solution.For the first kind of equation,the function symbol is firstly determined,and by constructing the Riccati function,the difference is calculated.Then the condition of the function is used to satisfy the contradiction,that is,the assumption is false,which verifies the oscillation of the solution.For the second kind of equation with initial condition,the equivalent fractional sum form of the fractional difference equation are firstly proved.With considering 0α≤1 and α1,respectively,by using the properties of Stirling formula and factorial function,the contradictory is got through enhanced processing,namely the assuming is not established,and the sufficient condition for the bounded solutions of the fractional difference equation is obtained.The above results will optimize the relevant conclusions and enrich the relevant results.The results are applied to the specific equations,and the oscillation of the solutions of equations is proved.