This work focused on the investigation of a generalized variation inclusion problem. The resolvent operator for generalized $\eta$-co-monotone mapping was structured, the Lipschitz constant was estimated and its relationship with the graph convergence was accomplished. An Ishikawa type iterative algorithm was designed by incorporating the resolvent operator and total asymptotically non-expansive mapping. By employing the novel implication of graph convergence and analyzing the convergence of the considered iterative method, the common solution of the generalized variational inclusion and the set of fixed points of a total asymptotically non-expansive mapping was obtained. Moreover, a generalized resolvent dynamical system was investigated. Some of its attributes were discussed and implemented to examine the considered generalized variation inclusion problem.

In this article, our goal is to propose and design a hybrid Picard $S$-type semi-implicit mid-point method to approximate the fixed point of a contractive-like mapping. The convergence result and stability of the proposed method are established under suitable assumptions. We implemented the newly constructed method to approximate the common element, which is the fixed point of a contractive-like mapping and simultaneously solves a general variational inequality. Finally, the significance of the proposed scheme is illustrated through the study of a fractional diffusion equation.

Herein, we aimed to develop two hybrid inertial viscosity-type forward-backward splitting algorithms for estimating the common solution of the variational inclusion problem and fixed point problem in Hilbert spaces. At the beginning of each iteration, the first algorithm estimated viscosity, fixed point, and inertial extrapolation. On the other hand, the second method estimated viscosity and inertial extrapolation alone. We demonstrated that the sequence induced by the proposed hybrid algorithms has strong convergence. We discussed a few special cases of the proposed algorithms and also presented theoretical applications of our findings. We furnished suitable numerical examples to validate the effectiveness of the recommended approaches.

In this paper, we construct a new four-step iterative method and show that our newly designed scheme converges faster than a number of iterative methods. We corroborate our claims by performing numerical experiments. We analyze the strong convergence result to approximate the fixed point of a contractive-like mapping and establish the weak ${\omega }^2$-stability of our scheme. Further, strong and weak convergence results are incorporated for a generalized $\alpha$-Reich-Suzuki nonexpansive mapping under some mild assumptions. We also illustrate numerical examples to validate our theoretical claims. Finally, we set forth our scheme to explore a Caputo-type nonlinear fractional Volterra-Fredholm integro-differential equation and a fractional diffusion equation.

This manuscript introduces two Mann-type hybrid inertial Yosida approximation iterative schemes for exploring a split variational inclusion problem and a fixed point of a nonexpansive mapping. Unlike existing methods, our schemes initiate the process by computing a Mann-type iteration that incorporates both an inertial extrapolation and a fixed-point iteration. The Yosida approximation operators associated with the corresponding monotone mappings are employed. We establish strong convergence theorems for the proposed schemes under suitable assumptions without estimating the norm of a bounded linear operator. Numerical examples are presented to validate the theoretical results, and a comparison of the proposed iterative schemes with existing methods is provided. Finally, an application of our schemes for solving the split common fixed point problem (SCFPP) is also discussed.