On the regularization of linear and nonlinear Abel-type integral equations of the first kind

We consider the linear integral equation 1/Gamma(alpha) integral (t)(0) (t - s)(alpha -1) K(t, s)y(s)ds = f(t), 0 less than or equal to t less than or equal to T, alpha is an element of (0, 1), and the nonlinear integral equation 1/Gamma(alpha) integral (t)(0) (t - s)(alpha -1) K(t, s)F(s, u(s))ds = f(t), 0 less than or equal to t less than or equal to T, alpha is an element of (0, 1), with a continuous kernel K(t,s) satisfying a Holder condition in t and with a continuous function F(s, u) which as an operator superposition is continuous and bounded from L-p(0, T) to L-q(0, T), 1 less than or equal to p less than or equal to infinity, 1 less than or equal to q less than or equal to infinity. We derive stability estimates and discuss their consequences for the problem of regularization. Stability estimates for the solution of the linear equation are in exponentially weighted L-q norms (1 less than or equal to q less than or equal to infinity). For the nonlinear equation these estimates are established via the inverse modulus of continuity of the function F(t, u) with respect to u.