THE STUDY of many mathematical physics problems leads to solving an operator of the first kind. There are many examples, e.g. determination of the initial temperature distribution given the temperature distribution at a later time, location of tumors by tomography, determination of atmospheric temperature profiles from telemetry data, seismic prospecting, system identification, etc. In any practical situation there are small errors in measuring. Therefore, given quantities in the equation are not known precisely, this type of problem is ill-posed. In particular, if the solution exists it generally depends discontinuously upon the data. Many authors have studied methods of solving ill-posed problems; we mentioned the famous Picard theorem and Smidt theorem [l]. Tikhonov and Arsenin [2], Lavrent’ev et al. [3], Morozov [4], Groetsch [S] have also studied the problem of finding stabilizing approximate solutions and discussing the convergence of approximate solutions. A particularly useful method is the method of Tikhonov regularization. This method determines the regularization operator and regularization parameter. The stabilizing approximate solution obtained by this method is called the Tikhonov regular solution. In this paper, the method of Tikhonov regularization is used to solve an operator equation of the first kind with approximate operator and the right-hand side. We choose the regularization parameter by using the general Arcangeli’s criterion to give the convergence of the Tikhonov regular solution, and give the uniform convergence of the regular solution of an operator equation with a Fredholm operator. Let X and Y be real Hilbert spaces, T: X * Y be a compact linear operator, and consider the operator equation of the first kind TX = y, (1-l)