Mohammad Dilshad

In this manuscript, we design and present a Jungck-type iterative method to find the common fixed point of a pair of mappings with weak compatibility in hyperbolic metric spaces. Strong convergence is perfomed to estimate the common fixed point, and stability of the designed iterative scheme is established under suitable assumptions. Further, $ \Delta $-convergence is established, and a theoretical result is outlined to exhibit the coherence and effectiveness of our scheme with some of its counterparts. A numerical case study is set forth to evidence the convergence and effectiveness of the proposed method. Finally, the efficacy and applicability of our proposed method is illustrated by implementing it to explore a nonlinear delay integral equation (NDIE).

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.

In this paper, we propose three variants of Mann-type inertial forward–backward iterative methods for approximating the minimum-norm solution of the monotone inclusion problem and the fixed points of nonexpansive mappings. In the first two methods, we compute the Mann-type iteration together with the inertial extrapolation and fixed-point iteration in the initiation of the process, while the last method computes only the Mann-type iteration with inertial extrapolation at the start of the process. We establish the strong convergence results for each method with appropriate assumptions and discuss some applications of the presented methods. Finally, we present numerical examples in both finite- and infinite-dimensional Hilbert spaces to demonstrate their efficiency. A comparative analysis with existing methods is also provided.

This paper studies an efficient method for solving stochastic optimization problems formulated as stochastic variational inequalities with a quasi-monotone operator, where the cost function extends the classical monotone and pseudomonotone operators. Our proposed method iterates an adaptive stepsize that adjusts automatically without linesearch and includes a momentum term to accelerate the convergence. Each iteration requires only a single projection onto the feasible set, ensuring low computational complexity. Under standard assumptions, the algorithm achieves almost sure convergence and a proven convergence rate. Furthermore, numerical experiments demonstrate its superior performance, accuracy, stability, and efficiency compared with existing stochastic approximation schemes. We also apply the method to problems such as stochastic network bandwidth allocation, stochastic complementarity problems, and the networked stochastic Nash–Cournot game, showing its strength and practical usefulness. The obtained result is an extension of existing works in the literature.

In this paper, we design some generalized split problems which can be seen as an extended form of the split variational inequality problems. We present several iterative algorithms for solving generalized split problems and demonstrate the weak convergence results under some appropriate assumptions within the context of real Hilbert spaces. Finally, we support these results with the help of numerical examples in both the finite and infinite dimensional spaces. As a result of this work, a new direction will be opened in studying split problems.

Herein, we present two hybrid inertial self-adaptive iterative methods for determining the combined solution of the split variational inclusions and fixed-point problems. Our methods include viscosity approximation, fixed-point iteration, and inertial extrapolation in the initial step of each iteration. We employ two self-adaptive step sizes to compute the iterative sequence, which do not require the pre-calculated norm of a bounded linear operator. We prove strong convergence theorems to approximate the common solution of the split variational inclusions and fixed-point problems. Further, we implement our methods and results to examine split variational inequality and split common fixed-point problems. Finally, we illustrate our methods and compare them with some known methods existing in the literature.

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.