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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.

In this paper, we introduce the notion of Suzuki type Z-contraction and study the corresponding fixed point property. This kind of contraction generalizes the Banach contraction and unifies several known type of nonlinear contractions. We consider a nonlinear operator satisfying a nonlinear contraction in a metric space and prove fixed point results. As an application, we apply our result to show the solvability of nonlinear Hammerstein integral equations. Theorem 1.1. [8] Let (X, d) be a compact metric space and F : X → X be a mapping. Assume that 1 2 d (x, F x) < d (x, y) ⇒ d (F x, F y) < d (x, y) ,

In this article, we offer two modifications of the modified forward‐backward splitting method based on inertial Tseng method and viscosity method for inclusion problems in real Hilbert spaces. Under standard assumptions, such as Lipschitz continuity and monotonicity (also maximal monotonicity), we establish weak and strong convergence of the proposed algorithms. We give the numerical experiments to show the efficiency and advantage of the proposed methods and we also used our proposed algorithm for solving the image deblurring and image recovery problems. Our result extends some related works in the literature and the primary experiments might also suggest their potential applicability.

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.MSC: 47H09; 47H10; 54H25; 37C25

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