【作者】 于杰；

【导师】 杨秀忠；

【作者基本信息】 河北师范大学 ， 数学， 2023， 硕士

【摘要】 数学家Ulam在1940年首次提出著名的Ulam稳定性问题.一年以后,Hyers用直接法证明了可加函数方程在Banach空间上的稳定性.Aoki和Rassias利用Hyers的方法将控制常量变为无界函数,从而推广了Hyers的结果.随后,函数方程的稳定性问题得到了广泛的研究.本论文在随机(n,β)–范空间、直觉模糊n–范空间、矩阵随机赋范空间、(n,β)–范空间和模糊–范空间上研究函数方程的稳定性.主要结果如下:1.我们定义了随机(n,ββ)–范空间,并给出r–维二次函数方程的解,而后分别利用直接法和不动点法证明了r–维二次函数方程在该空间上的稳定性.2.我们给出了一般八次函数方程的解,并利用直接法和不动点法证明了一般八次函数方程在直觉模糊n–范空间上的稳定性.3.我们得到了m–维可加函数方程的解,并利用直接法和不动点法证明了m–维可加函数方程在矩阵随机赋范空间上的稳定性.4.我们采用直接法证明了η–型混合可加四次函数方程在(n,β)–范空间上的稳定性,并利用直接法和不动点法证明了其在模糊–范空间上的稳定性.更多还原

【Abstract】 In 1940,the mathematician Ulam first presented the famous Ulam stability problem.In the next year,Hyers proved that stability of additive equation in Banach spaces by using the direct method.Aoki and Rassias extended the result of Hyers by turning the control constant in Hyers method into an unbounded function.Subsequently,the stability of problems for the functional equations have been extensively studied.In this paper,we investigate the stability of functional equations in random(n,β)–normed spaces,intuitionistic fuzzy n–normed spaces,matrix random normed spaces,(n,β)–normed spaces and fuzzy –normed spaces.The main results are as follows:1.We define random(n,β)–normed spaces and give the general solution of the r–variable quadratic functional equation,βthen prove the stability of r–variable quadratic functional equation in random(n,β)–normed spaces by using the direct method and fixed point method,respectively.2.We give the solution of the generalized octic functional equation,βand use the direct and fixed point methods to prove the stability of the generalized octic functional equation in intuitionistic fuzzy n–normed spaces.3.We get the general solution of the m–dimensional additive functional equation,βand prove the stability of the m–dimensional additive functional equation in matrix random normed spaces by using the direct and fixed point methods.4.We adopt the direct method to prove the stability of the η–type mixed additive and quartic functional equation in(n,β)–normed spaces,and the direct and fixed point methods to prove its stability in fuzzy –normed spaces.更多还原