This paper introduces some numerical algorithms for finding solutions of nonlinear problems like functional equations, split feasibility problems (SFPs) and variational inequality problems (VIPs) in the setting of Hilbert and Banach spaces. Our approach is based on the Thakur-Thakur-Postolache (TTP) iterative algorithm and the class of mean nonexpansive mappings. First we provide some convergence results (including weak and strong convergence) in the setting of Banach space. To support these results, we provide a numerical example and prove that our TTP algorithm in this case converges faster to fixed point compared to other iterative algorithms of the literature. After that, we consider two new TTP type projection iterative algorithms to solve SFPs and VIPs on the Hilbert space setting. Our result are new in analysis and suggest new type effective numerical algorithms for finding approximate solutions of some nonlinear problems.

This manuscript is devoted to presenting some convergence results of a three-step iterative scheme under the Chatterjea–Suzuki–C ((CSC), for short) condition in the setting of a Banach space. Also, an example of mappings satisfying the (CSC) condition with a unique fixed point is provided. This example proves that the proposed scheme converges to a fixed point of a weak contraction faster than some known and leading schemes. Finally, our main results will be applied to find a solution to functional and fractional differential equations (FDEs) as an application.

In this paper, we studied the AA-iterative algorithm for finding fixed points of the class of nonlinear generalized $(\alpha ,\beta )$-nonexpansive mappings. First, we proved weak convergence and then proved several strong convergence results of the scheme in a ground setting of uniformly convex Banach spaces. We gave a few numerical examples of generalized $(\alpha ,\beta )$-nonexpansive mappings to illustrate the major outcomes. One example was constructed over a subset of a real line while the other one was on the two dimensional space with a taxicab norm. We considered both these examples in our numerical computations to show that our iterative algorithm was more effective in the rate of convergence corresponding to other fixed point algorithms of the literature. Some 2D and 3D graphs were obtained that supported graphically our results and claims. As applications of our major results, we solved a class of fractional differential equations, 2D Voltera differential equation, and a convex minimization problem. Our findings improved and extended the corresponding results of the current literature.

In this paper, we introduce the concept of elliptic-valued b-metric spaces, extending the notions of elliptic-valued metric spaces and complex-valued metric spaces. We present several fixed-point results that involve rational and product terms within this novel space framework. To support our main findings, we offer numerical examples. Additionally, we demonstrate an application of Urysohn integral equations.

The purpose of this manuscript was to introduce a new iterative approach based on Green's function for approximating numerical solutions of the Troesch's problem. A Banach space of continuous functions was considered for establishing the main outcome. First, we set an integral operator using a Green's function and embedded this new operator into a three-step iterative scheme. We proved the main convergence result with the help of some mild assumptions on the parameters involved in our scheme and in the problem. Moreover, we proved that the new iterative approach was weak $w^2$-stable. The high accuracy and stability of the scheme was confirmed by several numerical simulations. As an application of the main result, we solved a class of fractional boundary value problems (BVPs). The main results improved and unified several known results of the literature.