The relaxed inertial Tseng-type method for solving the inclusion problem involving a maximally monotone mapping and a monotone mapping is proposed in this article. The study modifies the Tseng forward-backward forward splitting method by using both the relaxation parameter, as well as the inertial extrapolation step. The proposed method follows from time explicit discretization of a dynamical system. A weak convergence of the iterates generated by the method involving monotone operators is given. Moreover, the iterative scheme uses a variable step size, which does not depend on the Lipschitz constant of the underlying operator given by a simple updating rule. Furthermore, the proposed algorithm is modified and used to derive a scheme for solving a split feasibility problem. The proposed schemes are used in solving the image deblurring problem to illustrate the applicability of the proposed methods in comparison with the existing state-of-the-art methods.

Motivated by Gopal et al. (Acta Math. Sci. 36B(3):1-14, 2016). We introduce the notion of α-type F-contraction in the setting of modular metric spaces which is independent from one given in (Hussain et al. in Fixed Point Theory Appl. 2015:158, 2015). Further, we establish some fixed point and periodic point results for such contraction. The obtained results encompass various generalizations of the Banach contraction principle and others.