In this paper, we present self-adaptive inertial iterative algorithms involving Yosida approximation to investigate a split variational inclusion problem (SVIP) and common solutions of a fixed point problem (FPP) and SVIP in Hilbert spaces. We analyze the weak convergence of the proposed iterative algorithm to explore the approximate solution of the SVIP and strong convergence to estimate the common solution of the SVIP and FPP under some mild suppositions. A numerical example is demonstrated to validate the theoretical findings, and comparison of our iterative methods with some known schemes is outlined.

In this paper, we have introduced some viscosity-type inertial iterative methods for solving fixed point and variational inclusion problems in Hilbert spaces. Our methods calculated the viscosity approximation, fixed point iteration, and inertial extrapolation jointly in the starting of every iteration. Assuming some suitable assumptions, we demonstrated the strong convergence theorems without computing the resolvent of the associated monotone operators. We used some numerical examples to illustrate the efficiency of our iterative approaches and compared them with the related work.

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.

This article contains some outcomes on fixed points for a graph preserving nonlinear contraction in a metric space endued with a transitive directed graph. Our results improved, enriched, and subsumed various known fixed point theorems. To argue for reliability of our results, we presented two examples. We concluded the manuscript to investigate a unique solution of a certain first-order boundary value problem by means of our results.