The Ambartsumian delay equation is used in the theory of surface brightness in the Milky way. The Adomian decomposition method (ADM) is applied in this paper to solve this equation. Two canonical forms are implemented to obtain two types of the approximate solutions. The first solution is provided in the form of a power series which agrees with the solution in the literature, while the second expresses the solution in terms of exponential functions which is viewed as a new solution. A rapid rate of convergence has been achieved and displayed in several graphs. Furthermore, only a few terms of the new approximate solution (expressed in terms of exponential functions) are sufficient to achieve extremely accurate numerical results when compared with a large number of terms of the first solution in the literature. In addition, the residual error using a few terms approaches zero as the delay parameter increases, hence, this confirms the effectiveness of the present approach over the solution in the literature.

In this paper, we develop an accelerated three-step iterative scheme for the approximation of fixed points of contraction mappings in Banach spaces, with a particular focus on applications to fractional models. Strong convergence of the proposed iteration is established under standard contraction assumptions, together with stability and data dependence results. A refined rate of convergence analysis shows that the new scheme achieves a smaller effective contraction factor and converges faster than several classical two- and three-step iterative methods, including the Picard, Mann, Ishikawa, and S-iteration processes. The theoretical results are applied to Caputo-type fractional differential equations by reformulating the associated boundary value problems as fixed-point equations. Existence and uniqueness of solutions follow from the Banach contraction principle, while the accelerated convergence of the proposed iteration leads to improved numerical efficiency. Extensive numerical experiments, including fractional differential equations and nonlinear contraction mappings on the real line, are presented to validate the theoretical findings. The results demonstrate that the proposed three-step iteration provides an effective and reliable computational tool for fractional and non-local models.

This work uses (c)-comparison functions in a metric space endowed with an arbitrary binary relation to establish certain fixed point results under a nonlinear version of almost contractions. The findings addressed here generalize and extend a number of recent findings. A few scenarios are described to demonstrate the validity of our results. The legitimacy of the unique solution of a nonlinear integral problem is assessed utilizing our findings.

This manuscript investigates certain fixed point outcomes for an extended almost weak contraction map in a metric space equipped with a locally transitive binary relation. The insights achieved here improve, broaden, solidify and advance a number of noteworthy findings. We formulate a metrical fixed point theorem for almost Boyd–Wong contractions in the course of the procedure. We provide numerous examples of illustration in favour of our findings. Our results enable us to find a unique solution to a class of nonlinear matrix equations.

The objective of this paper is to propose some fixed-point findings under a relational contraction of Pant type employing a pair of auxiliary functions and through a generalized class of transitive binary relations. Our outcomes extend, sharpen, modify and enrich many existing findings. To facilitate our research, we create a few instances that convey our findings. Through the use of our outcomes, we demonstrate the existence and uniqueness of solutions for a nonlinear integral equation.

This article aims to describe the solvability of certain functional equations related to dynamic programming by fixed-point theorems in relational-metric spaces. To achieve our goal, we exhibit a non-unique fixed-point finding for a map of the Ćirić type in a relational-metric space. In this way, we consolidate and amend innumerable celebrated results in fixed-point theory. A couple of instances are supplied to emphasize the practical value of our findings.