We consider a nonlocal functional equation that is a generalization of the mathematical model used in behavioral sciences. The equation is built upon an operator that introduces a convex combination and a nonlinear mixing of the function arguments. We show that, provided some growth conditions of the coefficients, there exists a unique solution in the natural Lipschitz space. Furthermore, we prove that the regularity of the solution is inherited from the smoothness properties of the coefficients. As a natural numerical method to solve the general case, we consider the collocation scheme of piecewise linear functions. We prove that the method converges with the error bounded by the error of projecting the Lipschitz function onto the piecewise linear polynomial space. Moreover, provided sufficient regularity of the coefficients, the scheme is of the second order measured in the supremum norm. A series of numerical experiments verify the proved claims and show that the implementation is computationally cheap and exceeds the frequently used Picard iteration by orders of magnitude in the calculation time.

翻译：我们考虑一类非局部泛函方程，该方程推广了行为科学中使用的数学模型。该方程基于一个引入凸组合及函数自变量非线性混合的算子构建。我们证明，在系数满足特定增长条件下，该方程在自然的Lipschitz空间中存在唯一解。进一步，我们证明解的正则性继承自系数的光滑性质。作为求解一般情形的自然数值方法，我们考虑分段线性函数的配置格式。我们证明该方法的收敛误差受限于Lipschitz函数到分段线性多项式空间投影的误差。此外，在系数具有充分正则性的条件下，该格式在无穷范数度量下具有二阶精度。一系列数值实验验证了所证明的结论，并表明该实现计算成本低廉，在计算时间上较常用的Picard迭代法有数量级的提升。