In recent years, many researchers have studied the fixed points of generalized $\alpha$-nonexpansive (GAN) mappings, yet there has been limited research on multi-valued GAN mappings. In this paper, we focused on the class of multi-valued GAN mappings in the context of Banach spaces. We introduced a novel iterative process to find fixed points of these mappings and established weak and strong convergence theorems under mild conditions. To demonstrate the numerical efficiency of our new approach, we presented a supportive example and compare the accuracy of our method with existing iteration processes. To show practical usability of the research, we considered a larger class of boundary value problems (BVPs) and proved a convergence result with a supportive example. Our results are new and unify several results of the current literature.

In this paper, we will introduce the idea of grand variable weighted Herz spaces ${\dot{K}}_{q(\cdot )}^{\alpha (\cdot ),ϵ),\theta }(\tau )$ in which $\alpha$ is also a variable. Our main purpose in this paper is to prove the boundedness of Hardy operators on grand variable weighted Herz spaces.

In this article, by utilizing the idea of controlled functions, we present a novel notion of triple controlled quasi rectangular metric like spaces and prove Banach fixed point principal in such spaces. A topology in such spaces and its topological properties have been discussed. The result, presented here is a new contribution to the field of fixed point theory. Examples of this new structure are given.

The manuscript introduces the concept of a double-composed metric-like space, which is an extension of the notion of a double-composed metric space. In this new space, the self-distance is not necessarily zero, but if the distance metric equals zero, it must be for identical points of distance. Furthermore, this paper presents several results related to this novel concept in the literature, with a particular focus on Hardy–Rogers type contractions. It establishes fixed point theorems accompanied by some illustrative examples that elucidate the findings. Finally, this research provides an application to nonlinear integral equation to substantiate our theorems.