Approximate solution of nonlinear equations with weighted potential type operators

Global theorems on existence, uniqueness and ways of finding solutions are proved in a real space $L_2(-\infty,\infty)$ for different classes of nonlinear integral equations with weighted potential type operators
\begin{gather*} F(x,u(x))+\int_{-\infty}^\infty\frac{[a(x)-a(t)]\,u(t)}{|x-t|^{1-\alpha}}\,dt=f(x),\\ u(x)+\int_{-\infty}^\infty\frac{[a(x)-a(t)]\,F(t,u(t))}{|x-t|^{1-\alpha}}\,dt=f(x),\\ u(x)+F\left(x,\int_{-\infty}^\infty\frac{[a(x)-a(t)]\,u(t)}{|x-t|^{1-\alpha}}\,dt\right)=f(x) \end{gather*}
by means of combining the basic principle of monotone operators theory by Browder–Minty with the Banach contraction mapping principle. It is shown that the solutions can be found by using the Picard successive approximations method and speed estimates of their convergence are proved. The obtained results cover, in particular, the linear integral equations case with potential type kernels of a special form.