Approximate solutions of nonlinear convolution type equations on segment

For various classes of integral convolution type equations with a monotone nonlinearity, we prove global solvability and uniqueness theorems as well as theorems on the ways for finding the solutions in real Lebesgue spaces. It is shown that the solutions can be found in space $L_2(0, 1)$ by a Picard's type successive approximations method and we prove the estimates for the rate of convergence. The obtained results cover, in particular, linear integral convolution type equations. In the case of a power nonlinearity, it is shown that the solutions can be found by the gradient method in the space $L_p(0, 1)$ and weighted spaces $L_p(\varrho)$.