This paper addresses the existence of mild solutions and the approximate controllability of a class of higher-order Hilfer fractional semi-linear neutral stochastic differential equations with non-instantaneous impulses in Hilbert spaces. The system is driven by both fractional Brownian motion and Poisson jumps, thereby capturing long-range dependence as well as random discontinuities. By combining techniques from fractional calculus, stochastic analysis, and operator theory, we establish sufficient conditions for the existence of mild solutions. The analysis is carried out through the construction of suitable solution operator families and the application of Sadovskii’s fixed point theorem in an appropriate phase space framework. In addition, we investigate the controllability properties of the system and derive criteria ensuring approximate controllability of the underlying fractional neutral dynamics. The proposed approach relies on the structural properties of the higher-order Hilfer fractional derivative, estimates for stochastic integrals with respect to fractional Brownian motion, and compactness arguments adapted to non-instantaneous impulsive effects. The inclusion of Poisson jumps and neutral terms introduces significant analytical difficulties, which are overcome using refined resolvent operator techniques and fractional power estimates. An illustrative example is presented to demonstrate the applicability of the theoretical results. The results obtained generalize and unify several recent developments in the theory of fractional stochastic systems and provide a flexible framework for analyzing controlled dynamical models with memory, randomness, and impulsive behavior.

The paper examines the Hyers-Ulam and the generalized Hyers-Ulam-Rassias stability of a group of feedback-controlled neutral delay differential equations. Using Fourier transforms and resolvent kernels, we can prove strong stability results and obtain a quantitative approximation of error in approximations and exact solutions. The parameters of stability that are sufficient are defined in terms of system parameters, delay, and controller gain. Moreover, the effect of state-feedback control on the stability improvement is analytically considered, showing that with the increase of feedback gain, the Hyers-Ulam stability constant decreases, and consequently, the system robustness is enhanced. A number of theoretical findings are formulated, such as generalized stability theorems and controller-dependent estimates, to point out stabilizing effects of feedback mechanisms. To confirm the theoretical results as well as demonstrate the efficiency of feedback control in narrowing stability limits, a numerical case is provided. The findings are valuable to the development of the stability theory of neutral delay systems and provide useful data on the design of robust control strategies in the real world.

Moment stability of McKean-Vlasov stochastic recurrent neural networks with mixed delays.

This paper investigates the Hyers–Ulam and generalized Hyers–Ulam–Rassias stability of a class of feedback-controlled fractional differential equations involving the Caputo derivative. By employing Laplace transform techniques and resolvent kernel representations, we derive explicit stability bounds that quantify the de viation between approximate and exact solutions. Sufficient conditions ensuring stability are obtained in terms of the fractional order, system parameters, time de lay, and feedback gain. Furthermore, the influence of the controller gain on stability performance is rigorously analyzed, and it is shown that increasing the feedback gain leads to a strict reduction in the Hyers–Ulam stability constant, thereby en hancing system robustness. The generalized Hyers–Ulam–Rassias stability is also established under time-varying perturbations. To validate the theoretical findings, a numerical simulation based on a predictor–corrector scheme is presented, demon strating the effectiveness of the proposed control strategy in improving stability and convergence. The obtained results significantly extend existing Hyers–Ulam stability theory to controlled fractional-order dynamical systems and provide new insights into robust control design for systems with memory effects.

We present a semi-analytic investigation of the resolvent operator, and its associated forcing and response modes for quasi-one-dimensional shock-laden flows. Using a Green’s function approach, we derive resolvent solutions for isentropic (subsonic and supersonic) and transonic flows with shocks in converging–diverging nozzles of arbitrary geometry. Our analysis demonstrates that shock-induced heightened sensitivity in the resolvent across flow discontinuities leads to significant discrepancies between numerically computed and the analytical input and output modes if shock effects are not properly accounted for. In particular, we find that the resolvent operator exhibits singular behaviour at the shock location. Specifically, the inviscid (where the shock is treated purely as a flow discontinuity) and viscous analytical leading resolvent modes do not converge as the viscosity parameter $\mu \rightarrow 0$ , which affects the accuracy of flow control and stability analyses that rely on resolvent-based methods. Furthermore, the derived solutions serve as benchmarks for verifying numerical schemes designed to compute adjoint and resolvent modes in shock-laden flows, ensuring that they capture the correct physical behaviour in the presence of shocks.

Based on Krasnosel’skii fixed point theorem, k-set contractions fixed point theorem, Banach’s fixed point theorem, the existence and uniqueness of solutions is investigated for Mittag-Leffler kernel-type fractional differential equations under nonlocal delay and impulsive boundary value conditions, considering both compact and non-compact resolvent operators. However, a challenge arises due to the order of the derivative ($$0<\vartheta <1$$), which complicates the proof of equicontinuity; one of the research objectives of this study is to solve this issue. In addition, another contribution in this process is that the constant L is generalized as an unbounded Lebesgue integrable function in case of non-compact measure conditions. Moreover, the Lipschitz conditions of nonlinear terms and $$\hbar,\rho $$ are expanded from non-negative constants $$L,\hbar _{1},\rho _{1}$$ to unbounded Lebesgue integrable functions. Finally, three examples are provided to demonstrate the validity of the present work.

ABSTRACT This study deals with the controllability of a class of non‐densely defined abstract partial impulsive integro‐differential systems having finite delay in a nonlocal domain. We derive sufficient criteria to ensure the controllability of the system. Our analysis assumes the existence of a resolvent operator for the homogeneous part and applies Schauder's fixed‐point theorem. To conclude, the results are illustrated with an example.

The purpose of this work is to investigate the controllability of Langevin-type stochastic neutral impulsive integro-differential equations governed by the Caputo fractional derivative and driven by fractional Brownian motion, which arise naturally in systems exhibiting memory, impulsive effects, and stochastic disturbances. Using resolvent operators and fixed-point techniques, necessary and sufficient controllability conditions are established for the associated linear system, while the controllability of the nonlinear system is demonstrated via the Banach contraction principle. The theoretical results confirm that appropriate control functions can steer the system to a desired state within a finite time interval. Finally, illustrative numerical examples are provided to demonstrate the applicability and effectiveness of the obtained results, highlighting their relevance to practical stochastic control problems.

We study the spatial decay behaviour of resolvent kernels for a large class of non-local Lévy operators and bound states of the corresponding Schrödinger operators. Our findings naturally lead us to proving results for Lévy measures, which have subexponential or exponential decay, respectively. This leads to sharp transitions in the decay rates of the resolvent kernels. We obtain estimates that allow us to describe and understand the intricate decay behaviour of the resolvent kernels and the bound states in either regime, extending findings by Carmona, Masters and Simon for fractional Laplacians (the subexponential regime) and relativistic operators (the exponential regime). Our proofs are mainly based on methods from the theory of operator semigroups.

Abstract In this paper we consider a two–dimensional Hamiltonian integral system on an interval $$[ a,b )$$ [ a , b ) . We investigate the definiteness and surjectivity properties and define associated with this system the maximal $$S_{\max }$$ S max and minimal $$S_{\min }$$ S min linear relations. For an improper gauge $${\mathfrak L}$$ L , we calculate the $${\mathfrak L}$$ L -resolvent matrix and describe the set of $${\mathfrak L}$$ L -resolvents of the linear relation $$S_{\min }$$ S min in the cases when the system is either quasiregular, limit circle, or limit point at b .

Abstract The foundation of spectral theory on the S -spectrum can be traced back to the quaternionic framework of quantum mechanics. The concept of S -spectrum for quaternionic operators emerged as the natural spectrum in slice hyperholomorphic functional calculi, known as the S -functional calculus and also utilized in the quaternionic spectral theorem. This spectral theory extends to Clifford operators. A key distinction from classical complex spectral theory lies in the definition of the S -spectrum, which is second order in the operator T , and in the S -resolvent operators that turns out to be the product of two different operators. This study delves into the analyticity of the S -resolvent operators under specified boundary conditions for the S -spectral problem. The spectral theory on the S -spectrum also provides deeper insights into classical spectral theory.

This paper investigates the existence, uniqueness, stability, and controllability of mild solutions for a class of nonlinear fractional differential equations involving the ρ-Caputo derivative. By employing resolvent operator theory within Hilbert spaces, we first establish explicit representation formulas for the mild solutions. Sufficient conditions for their existence and uniqueness are then derived via the Banach fixed-point theorem. Furthermore, we analyse Ulam-Hyers, Ulam-Hyers-Rassias stability, and semi Ulam-Hyers-Rassias stability providing quantitative estimates on the robustness of solutions to perturbations. Controllability is addressed for both linear and nonlinear cases using controllability operators and Gramian techniques, establishing criteria for exact and approximate controllability. The applicability of the theoretical framework is demonstrated through an illustrative example. These findings extend existing results in fractional evolution equations and underscore the efficacy of the ρ-Caputo operator for modelling systems with memory and hereditary properties.

In this study, we present a problem which consists quantum difference equation with matrix value impulsive conditions and eigenparameter dependent boundary condition. Our examination is based on the concept of main spectral and scattering properties of this problem. We introduce Jost solution, Jost function and scattering function of the problem. Then, we find resolvent operator, Green function, eigenvalues and continuous spectrum of the operator generated by same problem.

ABSTRACT In this paper, we study a splitting proximal method for minimizing the sum of convex functions defined on metric spaces with negative curvature. Our approach utilizes the resolvent operator and is tailored to the geometry of such spaces. We establish convergence rate theorems for the proposed splitting method by imposing additional conditions on the objective function. Finally, we apply our results to convex optimization problems arising in convex feasibility problems, the centroid problem, and, in particular, the computation of Karcher means.

This research establishes an innovative analytical framework for examining C0-solutions of pseudo-almost automorphic type in evolution systems governed by nonlocal initial conditions within Banach spaces. Our methodological approach is founded upon the compactness properties of the resolvent operator (I−A)−1 and the semigroup generated by A. By integrating semigroup theory with fixed-point methodologies, we develop a unified strategy that successfully overcomes challenges arising from nonlocal initial specifications. The practical applicability of our theoretical framework is verified through its implementation on a transport equation with nonlocal initial history, thereby demonstrating both the existence of solutions and their distinctive character.

Nonlocal light-matter interactions with giant atoms in high-dimensional environments are not only fundamentally intriguing for testing quantum electrodynamics beyond the dipole approximation but also crucial for building high-dimensional quantum networks and engineering multipartite entangled states. Given the enigmatic and largely uncharted collective signatures exhibited by multiple giant atoms within two-dimensional optical baths, we delve into their nonperturbative collective dynamics within the single-excitation subspace, focusing on the case where they are coupled to a common two-dimensional photonic reservoir and employing a resolvent operator approach. We demonstrate that precisely engineered atomic arrangements lead to unconventional quantum dynamics, featuring non-Markovianity-induced beats and long-lived bound states in the continuum, thereby providing a versatile platform for implementing two-dimensional quantum memory. Phenomenologically, we observe the emergence of exotic photon emission patterns in both two- and three-dimensional (3D) baths. The emission directions are shown to be precisely controllable on demand through exact phase engineering of the coupling parameters, enabling a highly efficient chiral light-matter interface. Moreover, our generalization to a 3D bath reveals that coherent dipole-dipole interactions can survive despite the coupling to a continuum of modes, a finding that challenges conventional wisdom regarding decoherence.

In this paper, we investigate a class of impulsive fractional neutral hemivariational inequality with infinite delay in Banach spaces. First, we give a proper definition of the resolvent operator in Banach space, and define the concept of a mild solution by using fractional calculus, the semigroup of operators theory, and properties of generalized Clarke subdifferential. Second, an optimal control problem is constructed for the corresponding linear control system, and the optimal control expression is derived based on the resolvent operator consisting of duality mapping. Then, under the assumption that the impulsive function is only continuous, we prove the existence of mild solutions and approximate controllability results by using an approximating technique, the fractional power operator, and the fixed point theorem of condensing multivalued maps. Finally, an example is given to verify the validity of the main results.

This paper addresses the issue of approximate controllability for a class of control system which is represented by nonlinear factional order stochastic differential equations with infinite delay. By using fractional calculus, semigroup theory, stochastic analysis, and the technique of stochastic control theory, a new set of sufficient conditions for the approximate controllability of a fractional stochastic differential system are formulated and proved. More precisely, the results are established under the assumption that the corresponding deterministic fractional linear system is approximately controllable with using resolvent condition and techniques on fractional power operators. As an illustration of the application of the obtained results, an example is also provided.

Computational cost precludes direct numerical simulation or wall-resolved large-eddy simulations of non-equilibrium, wall-bounded turbulent flows in realistic conditions. Wall-modelled large-eddy simulations (WMLES) and hybrid RANS/LES methods can be used to analyse these flows at much decreased cost, but require modelling of the near-wall layer and, in particular, a means to address the deficit of turbulent activity, or eddies, in the vicinity of the interface between the outer flow and the wall model. We report a computational framework to populate the wall region with synthetic but realistic eddies and reflect their integrated effect on the flow in the inner layer. Two means of generating spatio-temporal representations for the synthetic eddies are investigated: low-order, resolvent-based representations of the wall layer and a coarse-grained, data-driven spectral proper orthogonal decomposition (SPOD) model, both generated in turbulent channel flow at a friction Reynolds number, R e τ = 1000 . The eddy-augmented WMLES models are then tested in R e τ = 5000 and 20,000 channels and compared with experimental and numerical data. The inherent scaling of the resolvent operator can be used to scale the resolvent model to higher Reynolds numbers (and potentially populate new, self-similar eddies as the wall layer grows in inner units), while the SPOD model is energetically optimal for reconstruction of the flow at Reynolds numbers close to that where it is obtained, but degrades as the Reynolds number is increased. The results show that the effect of the introduction of synthetic eddies is twofold: first, a direct contribution to the stress due to the presence of the synthetic eddies and, second, an improved prediction of the normal Reynolds stresses in the inner layer due to an accompanying, coupled reduction in the time- and length-scales of the variation of the URANS-like velocity in the inner layer. Implications and extensions of the method for more complex flows, for example external boundary layers with pressure gradient and separation, are briefly discussed. • A method to enrich the near-wall region in WMLES is introduced. • Data-driven and user-driven approaches are used to generate synthetic eddies. • Appropriate choice of the eddy wavenumber and y-support yields improved results. • Future developments of the technique are proposed.

This paper presents dual new algorithms for answering the Equilibrium Problem and Fixed Point Problems in Banach spaces. By operating the Bregman distance, we make a sweeping statement projection-based ways to unimpressed boundaries in non-Euclidean spaces, mainly in instances where conventional ideas of Lipschitz continuity or monotonicity are restrictive. The firstly algorithm services the generalized resolvent operator and Bregman projections in reflexive Banach spaces, founding strong coming together below relaxed settings. The next algorithm, planned for Hilbert spaces, integrates mistake open-mindedness machineries to confirm constancy in the attendance of computational perturbations.