The goal of this article is to study the box dimension of the mixed Katugampola fractional integral of two-dimensional continuous functions on \([0,1]\times [0,1]\). We prove that the box dimension of the mixed Katugampola fractional integral having fractional order \((\alpha =(\alpha _1,\alpha _2);~ \alpha _1>0, \alpha _2>0)\) of two-dimensional continuous functions on \([0,1]\times [0,1]\) is still two. Moreover, the results are also established for the mixed Hadamard fractional integral. Our new results are to improve the existing studies. We pose also some open problems for further research.

本文的目的是研究\([0,1]\times [0,1]\)上二维连续函数的混合 Katugampola 分数积分的盒维数。我们证明了混合 Katugampola 分数阶积分的盒维数具有分数阶\ ((\alpha =(\alpha _1,\alpha _2);~ \alpha _1>0, \alpha _2>0)\) \([0,1]\times [0,1]\)上的连续函数仍然是两个。此外，还建立了混合Hadamard分数积分的结果。我们的新结果是为了改进现有的研究。我们还提出了一些未解决的问题以供进一步研究。