Inexact Halpern-type proximal point algorithm

We present several strong convergence results for the modified, Halpern-type, proximal point algorithm $${x_{n+1}=\alpha_{n}u+(1-\alpha_{n})J_{\beta_n}x_n+e_{n}}$$ (n = 0,1, . . .; $${u,\,x_0\in H}$$ given, and $${J_{\beta_n}=(I+\beta_nA)^{-1}}$$, for a maximal monotone operator A) in a real Hilbert space, under new sets of conditions on $${\alpha_n\in(0,1)}$$ and $${\beta_n\in(0,\infty)}$$. These conditions are weaker than those known to us and our results extend and improve some recent results such as those of H. K. Xu. We also show how to apply our results to approximate minimizers of convex functionals. In addition, we give convergence rate estimates for a sequence approximating the minimum value of such a functional.