Let $ {Yn} $ be a sequence of dependent random variables and $ {Phi_n (cdot, cdot) } $ be a sequence of Borel functions. Let $ \theta_n $ be a solution of the equation $ M_n(x) = 0 $ for each $ n geqq 1 $, where $ M_n(x) = mathrm{E} Phi_n(x, Y_n) $. A Robbins-Monro type stochastic approximation procedure $ X_{n+1} = X_n - a_n Phi_n(X_n, Y_n) $ is considered for estimating $ \theta_n $ for $ n $ sufficiently large. Under some assumptions about $ {a_n},{\theta_n},{Y_n} $ and $ {Phi_n(cdot, cdot)} $ which may not include the fundamental condition $ mathrm{E}[Phi_n(X_n, Y_n) mid X_1, cdots, X_n] = M_n(X_n) $ a.s., the a.s. convergence and in mean-square convergence of $ mid X_n - \theta_n mid $ to zero are studied.