In this article, we establish a new Mann-type method combining both inertial terms and errors to find a fixed point of a nonexpansive mapping in a Hilbert space. We show strong convergence of the iterate under some appropriate assumptions in order to find a solution to an investigative fixed point problem. For the virtue of the main theorem, it can be applied to an approximately zero point of the sum of three monotone operators. We compare the convergent performance of our proposed method, the Mann-type algorithm without both inertial terms and errors, and the Halpern-type algorithm in convex minimization problem with the constraint of a non-zero asymmetric linear transformation. Finally, we illustrate the functionality of the algorithm through numerical experiments addressing image restoration problems.

The purposes of this paper are to establish an alternative forward-backward method with penalization terms called new forward-backward penalty method (NFBP) and to investigate the convergence behavior of the new method via numerical experiment. It was proved that the proposed method (NFBP) converges in norm to a zero point of the monotone inclusion problem involving the sum of a maximally monotone operator and the normal cone of the set of zeros of another maximally monotone operator. Under the observation of some appropriate choices for the available properties of the considered functions and scalars, we can generate a suitable method that weakly ergodic converges to a solution of the monotone inclusion problem. Further, we also provide a numerical example to compare the new forward-backward penalty method with the algorithm introduced by Attouch [Attouch, H., Czarnecki, M.-O. and Peypouquet, J., Coupling forward-backward with penalty schemes and parallel splitting for constrained variational inequalities, SIAM J. Optim., 21 (2011), 1251-1274].