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Linear integral equations of the third kind usually lead to ill-posed inverse problems if the normed solution space X and the normed data space Y are required to be equal. In the present paper we develop a two-step regularization method—called RPMO method—to regularize such inverse problems. This method does not require Hilbert space properties. Convergence results are presented indicating that there is no general theoretical upper bound less than one for the convergence rates if the corresponding exact data solutions are sufficiently smooth. Moreover, we illustrate the RPMO method by applying its discretized version to an implicit linear boundary value problem which can be transformed into an equivalent third-kind integral equation. [ABSTRACT FROM AUTHOR]