For Hilbert spaces \( Z \) and \( U \) let \( A:Z \to U \) be a bounded linear operator with a non-closed range \( R(A) \) in general. The authors consider the equation \( Az=u (u \in U) \) which is supposed to have a minimum norm solution \( \overline{z} \in Z \) that is sourcewise representable, i.e., \( \overline{z} \in R((A^*A)^{p/2}) \). Here \( A^*: U \to Z \) denotes the adjoint operator of \( A \), and \( p > 0 \) is supposed to be known. It is moreover supposed that \( u_\delta \in U \) and the bounded linear operator \( A_h:Z \to U \) are perturbations satisfying \( \| A_h - A \| \leq h, \| u_\delta - u \| \leq \delta. \) In this situation the following approximations \( z_\eta \in Z \big( \eta = (h,\delta) \big)\) to \( \overline{z} \) are considered, \[ z_\eta = \big(A_h^*A_h\big)^{p/2} v_\eta, \qquad \text{with} \quad v_\eta \in V_\eta, \qquad \| v_\eta \| = \inf\big\{ \| v \| : v \in V_\eta \big\}, \quad \] where the set \( V_\eta \subset Z \) is defined as follows, \[ V_\eta = \Big\{v \in Z : \| A_h\big(A_h^*A_h\big)^{p/2} v - u_\delta \| \leq \mu_\eta + \delta + C h \| v \| \Big\}, \] where \( \mu_\eta = \inf_{z \in Z} \{\| A_h z - u_\delta \| + \delta + h \| z \| \} \), and \( C > 1 \) is a constant. The authors consider a second class of approximations \( z_\eta = \big(A_h^*A_h\big)^{p/2} v_\eta^{\alpha(\eta)} \) where \( v_\eta^{\alpha(\eta)} \in Z \) is obtained by the a priori parameter choice strategy \( \alpha(\eta) = (\delta + h)^2 \) for the following Tikhonov type method, \[ v_\eta^\alpha \in Z, \quad \big((A_h^*A_h)^{p+1} + \alpha I\big) v_\eta^\alpha = (A_h^*A_h)^{p/2} A_h^* u_\delta, \] and finally a generalized quasisolution method is presented. For these methods order optimal error estimates of the kind \( \| z_\eta - \overline{z} \| = O\big((h+\delta)^{p/(p+1)}) \) are obtained, respectively.