In this paper, we explain the approximate controllability of $\mathsf{\Psi }$-Hilfer fractional neutral differential equations with infinite delay. The outcome is demonstrated using the infinitesimal operator, fractional calculus, semigroup theory, and the Krasnoselskii’s fixed point theorem. To begin, we emphasise the presence of the mild solution and show that the $\mathsf{\Psi }$-Hilfer fractional system is approximately controllable. Additionally, we present theoretical and practical examples.

In this paper, we concentrate on a control system with a non-local condition that is governed by a Hilfer fractional neutral stochastic evolution hemivariational inequality (HFNSEHVI). By using concepts of the generalized Clarke sub-differential and a fixed point theorem for multivalued maps, we first demonstrate adequate requirements for the existence of mild solutions to the concerned control system. Then, using limited Lagrange optimal systems, we demonstrate the existence of optimal state-control pairs that are regulated by an HFNSEHVI with a non-local condition. In order to demonstrate the existence of fixed points, the symmetric structure of the spaces and operators that we create is essential. Without considering the uniqueness of the control system’s solutions, the best control results are established. Lastly, an illustration is used to demonstrate the major result.

In this article, we examine the existence of Hilfer fractional $(\mathtt{HF})$ stochastic differential systems with nonlocal conditions and delay via almost sectorial operators. The major methods depend on the semigroup of operators method and the $M\ddot{o}nch$ fixed-point technique via the measure of noncompactness, and the fundamental theory of fractional calculus. Finally, to clarify our key points, we provide an application.

The purpose of this work is to investigate the controllability of non-instantaneous impulsive (NII) Hilfer fractional (HF) neutral stochastic evolution equations with a non-dense domain. We construct a new set of adequate assumptions for the existence of mild solutions using fractional calculus, semigroup theory, stochastic analysis, and the fixed point theorem. Then, the discussion is driven by some suitable assumptions, including the Hille–Yosida condition without the compactness of the semigroup of the linear part. Finally, we provide examples to illustrate our main result.

Mobile ad hoc networks (MANET) have been seen as a related advancement to Group Key Management (GKM) applications. Remembering the true objective to guarantee amass applications and disallow uncertified clients from getting to the correspondence data that cannot be anchored by a remote MANET, including IP multicast, the singular gathered data content must remain encoded by a typical shared gathering key. Key administration is required to anchor the assurance of gathering the key and to safeguard those gathering data. GKM framework is associated with the remote system condition partners with three issues: execution, security, and system versatility. This article focuses on the Unmanned Aerial Vehicle (UAV)-mobile backbone node (MBN) remote system performance. The UAV-MBN Network condition is a military system that includes a proposal to group an important administrative structure. A half-and-half gathering key administration technique, which works on each target of UAV-MBN, is included in an arrangement to start two basic remote gathering key administration difficulties: (1) operational performance and (2) multiple-enrollment development. By working with minimal small-scale key administration, this strategy can diminish the execution cost associated with the key administration along with the increment operational execution of remote GKM. Scaled-down key organization is carried out in the context of these movement units. The key administration approach also restricted the operational procedure and decreased the operations cost in terms of key generation, figuring, and associated correspondence. Overall, the HGKM strategy that has been introduced enhances the operational process and functions effectively in reasonable remote areas.

We define the concept of rough limit set of a triple sequence space of Bernstein-Stancu polynomials of fuzzy numbers and obtain the relation between the set of rough limit and the extreme limit points of a triple sequence space of Bernstein-Stancu polynomials of fuzzy numbers. Finally, we investigate some properties of the rough limit set of Bernstein-Stancu polynomials.