Differential equations are frequently used to mathematically describe many problems in real life, but they are always subject to intrinsic phenomena that are neglected and could influence how the model behaves. In some cases like ecosystems, electrical circuits, or even economic models, the model may suddenly change due to outside influences. Occasionally, such changes start off impulsively and continue to exist for specific amounts of time. Non-instantaneous impulses are used in the creation of the models for this kind of scenario. In this paper, a new class of non-instantaneous impulsive $\psi$-Caputo fractional stochastic differential equations under integral boundary conditions driven by the Rosenblatt process was examined. Semigroup theory, stochastic theory, the Banach fixed-point theorem, and fractional calculus were applied to investigating the existence of piecewise continuous mild solutions for the systems under consideration. The impulsive Gronwall’s inequality was employed to establish the unique stability conditions for the system under consideration. Furthermore, we examined the controllability results of the proposed system. Finally, some examples were provided to demonstrate the validity of the presented work.

In this article, we investigate the existence and approximate controllability of a class of Sobolev-type fractional stochastic differential equations of order $1<\delta <2$ with infinite delay. The analysis is carried out in an abstract Hilbert space framework, incorporating fractional dynamics together with stochastic perturbations. By employing techniques from fractional calculus, semigroup theory, and fixed point theory, particularly the Banach contraction principle along with compactness arguments, we establish the existence of mild solutions for the proposed system. Subsequently, sufficient conditions for approximate controllability are derived by combining operator-theoretic methods with stochastic analysis. The novelty of this work lies in extending controllability results to Sobolev-type fractional stochastic systems of order $1<\delta <2$, where both the higher-order fractional structure and stochastic effects are treated simultaneously within a unified framework. This generalizes and complements several existing results in the literature that mainly address deterministic systems or fractional differential equations of order $0<\delta \le 1$. Finally, an illustrative example is presented to demonstrate the applicability and effectiveness of the theoretical findings.

In this manuscript, we investigate fixed points for a subfamily of a non-expansive evolution(NeE) family of bounded linear operators on a Hilbert space $H$. Additionally, we utilize the concept of net and progression algorithm to establish several theorems regarding the strong convergence to a shared fixed point of the aforementioned subfamily. Finally, we furnish an example to bolster our findings.

This work investigates the controllability and stability properties of nonlinear piecewise dynamical systems subject to integral boundary conditions studied in arbitrary time domains. The study employs powerful mathematical ideas like fixed point theorems, Gramian-type matrices and the unified approach given by the theory of time scales. This unified approach provides results applicable seamlessly in various settings of continuous-time systems, discrete-time systems, and systems with hybrid behavior of both. This paper expands existing findings in the literature, providing a more complete view. To show the application of the theoretical results, a detailed example is presented and supported by numerical simulations to confirm the efficiency of the proposed methods.