The purpose of this research is to provide sufficient conditions for the local and global existence of solutions for two-dimensional nonlinear fractional Volterra and Fredholm integral equations, based on the Schauder’s and Tychonoff’s fixed-point theorems. Also, we provide sufficient conditions for the uniqueness of the solutions. Moreover, we use operational matrices of hybrid of two-dimensional block-pulse functions and two-variable shifted Legendre polynomials via collocation method to find approximate solutions of the mentioned equations. In addition, a discussion on error bound and convergence analysis of the proposed method is presented. Finally, the accuracy and efficiency of the presented method are confirmed by solving three illustrative examples and comparing the results of the proposed method with other existing numerical methods in the literature.

中文翻译：

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Banach空间中二维非线性分数阶Volterra和Fredholm积分方程的存在性，唯一性和数值解

本研究的目的是为基于Schauder和Tychonoff定点定理的二维非线性分数阶Volterra和Fredholm积分方程解的局部和全局存在提供充分的条件。此外，我们为解决方案的独特性提供了充分的条件。此外，我们通过搭配方法使用二维块脉冲函数和两变量移位的勒让德多项式的混合运算矩阵，找到上述方程的近似解。此外，还讨论了该方法的误差范围和收敛性分析。最后，