In this paper, we investigate nonlinear functional Volterra–Urysohn integral equations, a class of nonlinear integral equations of Volterra type. The existence and uniqueness of the solution to the equation is proved by a technique based on the Picard iterative method. For the numerical approximation of the solution, the Euler and trapezoidal discretization methods are utilized which result in a system of nonlinear algebraic equations. Using a Gronwall inequality and its discrete version, first order of convergence to the exact solution for the Euler method and quadratic convergence for the trapezoidal method are proved. To prove the convergence of the trapezoidal method, a new Gronwall inequality is developed. Finally, numerical examples show the functionality of the methods.

在本文中，我们研究了非线性泛函 Volterra-Urysohn 积分方程，这是一类 Volterra 类型的非线性积分方程。该方程解的存在性和唯一性是通过一种基于Picard迭代法的技术证明的。对于解的数值近似，使用了欧拉和梯形离散化方法，这导致了非线性代数方程组。使用 Gronwall 不等式及其离散形式，证明了欧拉方法精确解的一阶收敛性和梯形方法的二次收敛性。为了证明梯形方法的收敛性，开发了一个新的 Gronwall 不等式。最后，数值例子展示了这些方法的功能。