Nonlinear integral equations with potential-type kernels in the nonperiodic case

We find conditions under which a generalized potential-type operator acts continuously from a Lebesgue space with a general weight to its dual space and possesses the positivity property. Using these conditions, the global existence and uniqueness theorems for various classes of nonlinear integral equations of convolution type in real weighted Lebesgue spaces are proved by the method of monotonic (in the Browder–Minty sense) operators. Also we obtain estimates of the norms of solutions which imply that the corresponding homogeneous equations have only a trivial solution.