Article
Ufa Mathematical Journal
Volume 3, Number 4, pp. 7-12
Approximate solution of nonlinear equations with weighted potential type operators
Askhabov S.N.
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In real space $L_2(-\infty, \infty)$, by combining the basic principle of monotone operators theory by Browder-Minty and Banach contraction mapping principle, for different classes of nonlinear integral equations with weighted potential type operators $$F(x, u(x))+\int\limits_{-\infty}^{\infty}\frac{[a(x)-a(t)]\,u(t)}{|x-t|^{1-\alpha}}dt=f(x)\,,$$ $$u(x)+\int\limits_{-\infty}^{\infty}\frac{[a(x)-a(t)]\,F(t, u(t))}{|x-t|^{1-\alpha}}dt=f(x)\,,$$ $$u(x)+F\left(x, \int\limits_{-\infty}^{\infty}\frac{[a(x)-a(t)]\,u(t)}{|x-t|^{1-\alpha}}dt\right)=f(x)\,,$$ there are proved global theorems on existence, uniqueness and ways of finding solutions. It is shown that the solutions can be found by using the Picard's type successive approximations method and proved speed estimates of their convergence. The obtained results cover, in particular, the linear integral equations case with potential type kernels of special form.