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The inertial forward–backward splitting algorithm can be considered as a modified form of the forward–backward algorithm for variational inequality problems with monotone and Lipschitz continuous cost mappings. By using parallel and inertial techniques and the forward–backward splitting algorithm, in this paper, we propose a new parallel inertial forward–backward splitting algorithm for solving variational inequality problems, where the constraints are the intersection of common solution sets of a finite family of variational inclusion problems. Then, strong convergence of proposed iteration sequences is showed under standard assumptions imposed on cost mappings in a real Hilbert space. Finally, some numerical experiments demonstrate the reliability and benefits of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We consider the stabilization of a class of linear evolution systems z ′ = A z + B v under the observation y = C z by means of a finite dimensional control v. The control is based on the design of a Luenberger observer which can be infinite or finite dimensional (of dimension large enough). In the infinite dimensional case, the operator A is supposed to generate an analytical semigroup with compact resolvent and the operators B and C are unbounded operators whereas in the finite dimensional case, A is assumed to be a self-adjoint operator with compact resolvent, B and C are supposed to be bounded operators. In both cases, we show that if (A, B) and (A, C) verify the Fattorini-Hautus Criterion, then we can construct an observer-based control v of finite dimension (greater or equal than largest geometric multiplicity of the unstable eigenvalues of A) such that the evolution problem is exponentially stable. As an application, we study the stabilization of the diffusion system. [ABSTRACT FROM AUTHOR]

In real Hilbert spaces, given a single-valued Lipschitz continuous and monotone operator, we study generalized split feasibility problem (GSFP) over solution set of monotone variational inclusion problem. An inertia iterative method is proposed to solve this problem, by showing that the sequence generated by the iteration converges strongly to solution of GSFP. As against previous methods, our step size is chosen to be simple and not depending on norm of associated bounded linear map as well as Lipschitz constant of the single-valued operator. The obtained result was applied to study split linear inverse problem, precisely, the LASSO problem. Lastly, with the aid of numerical examples, we exhibited efficiency of our algorithm and its dominance over other existing schemes. [ABSTRACT FROM AUTHOR]

In the framework of a second‐order semilinear integro‐differential control system in Hilbert spaces, the paper provides sufficient conditions for proving the existence of optimal control. The Banach fixed point theorem is used to investigate the existence and uniqueness of mild solutions for the proposed problem. Additionally, it is shown that, under specific assumptions, there exists at least one optimal control pair for the Lagrange's problem as presented in the article. An example for validation is included in the paper to further support the theoretical findings. [ABSTRACT FROM AUTHOR]

We prove an extended Granger–Johansen representation theorem (GJRT) for finite‐ or infinite‐order integrated autoregressive time series on Banach space. We assume only that the resolvent of the autoregressive polynomial for the series is analytic on and inside the unit circle except for an isolated singularity at unity. If the singularity is a pole of finite order the time series is integrated of the same order. If the singularity is an essential singularity the time series is integrated of order infinity. When there is no deterministic forcing the value of the series at each time is the sum of an almost surely convergent stochastic trend, a deterministic term depending on the initial conditions and a finite sum of embedded white noise terms in the prior observations. This is the extended GJRT. In each case the original series is the sum of two separate autoregressive time series on complementary subspaces – a singular component which is integrated of the same order as the original series and a regular component which is not integrated. The extended GJRT applies to all integrated autoregressive processes irrespective of the spatial dimension, the number of stochastic trends and cointegrating relations in the system and the order of integration. [ABSTRACT FROM AUTHOR]

This paper studies the stability of an abstract thermoelastic system with Cattaneo’s law, which describes finite heat propagation speed in a medium. We introduce a region of parameters containing coupling, thermal dissipation, and possible inertial characteristics. The region is partitioned into distinct subregions based on the spectral properties of the infinitesimal generator of the corresponding semigroup. By a careful estimation of the resolvent operator on the imaginary axis, we obtain distinct polynomial decay rates for systems with parameters located in different subregions. Furthermore, the optimality of these decay rates is proved. Finally, we apply our results to several coupled systems of partial differential equations. [ABSTRACT FROM AUTHOR]

This article focuses on the existence and Ulam–Hyers–Rassias stability outcomes pertaining to a specific category of impulsive integro‐differential inclusions (with instantaneous and non‐instantaneous impulses). These problems are examined using resolvent operators, drawing from the Grimmer perspective. Our analysis is based on Bohnenblust–Karlin's and Darbo's fixed point theorems for multivalued mappings in Banach spaces. Additionally, we provide an illustrative example to reinforce and demonstrate the validity of our findings. [ABSTRACT FROM AUTHOR]

The biharmonic equation is one of the important equations of mathematical physics, describing the behaviour of harmonic functions in higher-dimensional spaces. The main purpose of this study was to construct a finite-dimensional perturbation for the Dirichlet boundary value problem associated with the biharmonic equation. The methodological basis for this study was an integrated approach that includes mathematical analysis, algebraic methods, operator theory, and the theorem on the existence and uniqueness of a solution for a boundary value. The main tool is a finite-dimensional perturbation, which allows for examining the properties and behaviour of boundary value problems in as much detail as possible. In the study, descriptions of correctly solvable internal boundary value problems for a biharmonic equation in non-simply connected domains were considered in detail. The study is also devoted to the search for solutions and the analytical representation of resolvents of boundary value problems for a biharmonic equation in multi-connected domains. Within the framework of the study, theorems and their consequences were proved, and a finite-dimensional perturbation was constructed for the Dirichlet boundary value problem. Analytical representations of resolvents of boundary value problems for a biharmonic equation in multi-connected domains were also obtained. The examination of a finite-dimensional perturbation of the Dirichlet boundary value problem for a biharmonic equation has expanded the understanding of the properties of this equation in various contexts. [ABSTRACT FROM AUTHOR]

This research investigates the existence of piecewise asymptotically almost automorphic mild solutions for integrodifferential equations with infinite delay. The existence results are proved by using the Mönch's fixed point theorem, the concept of measures of non-compactness theorem and resolvent operator. Finally, an example is presented to illustrate our obtained results. [ABSTRACT FROM AUTHOR]

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This paper proves the existence of R$$ \mathcal{R} $$‐bounded solution operator families of the resolvent problem of Navier–Stokes–Korteweg model in half‐space (R+N)$$ \left({\mathbf{R}}_{+}^N\right) $$ with slip boundary condition. Especially we investigate the model for arbitrary constant viscosity and capillarity. We employ the R$$ \mathcal{R} $$‐bounded solution operators of the model obtained from the whole space cases and partial Fourier transform techniques to analyze the model. [ABSTRACT FROM AUTHOR]

We study families of analytic semigroups, acting on a Banach space, and depending on a parameter, and give sufficient conditions for existence of uniform with respect to the parameter norm bounds using spectral properties of the respective semigroup generators. In particular, we use estimates of the resolvent operators of the generators along vertical segments to estimate the growth/decay rate of the norm for the family of analytic semigroups. These results are applied to prove the Lyapunov linear stability of planar traveling waves of systems of reaction–diffusion equations, and the bidomain equation, important in electrophysiology. [ABSTRACT FROM AUTHOR]

The goal of this work is to introduce the notion of topological degree via the principle of the degree of nondensifiability (DND for short). We establish some new fixed point theorems, concerning, Schaefer's fixed point theorem and the nonlinear alternative of Leray–Schauder type. As applications, we study the existence of mild solution of functional semilinear integro-differential equations. [ABSTRACT FROM AUTHOR]

The partial neutral functional fractional differential equation described by the fractional operator is considered in the present investigation. The used fractional operator is the Caputo derivative. In the present paper, the fractional resolvent operators have been defined and used to prove the existence of the unique solution of the fractional neutral differential equations. The fixed point theorem has been used in existence investigations. For an illustration of our results in this paper, an example has been provided as well. [ABSTRACT FROM AUTHOR]

Iterative diagonalization of large matrices to search for a subset of eigenvalues that may be of interest has become routine throughout the field of quantum chemistry. Lanczos and Davidson algorithms hold a monopoly, in particular, owing to their excellent performance on diagonally dominant matrices. However, if the eigenvalues happen to be clustered inside overlapping Gershgorin disks, the convergence rate of both strategies can be noticeably degraded. In this work, we show how Davidson, Jacobi–Davidson, Lanczos, and preconditioned Lanczos correction vectors can be formulated using the reduced partitioning procedure, which takes advantage of the inherent flexibility promoted by Brillouin–Wigner perturbation (BW-PT) theory's resolvent operator. In doing so, we establish a connection between various preconditioning definitions and the BW-PT resolvent operator. Using Natural Localized Molecular Orbitals (NLMOs) to construct Configuration Interaction Singles (CIS) matrices, we study the impact the preconditioner choice has on the convergence rate for these comparatively dense matrices. We find that an attractive by-product of preconditioning the Lanczos algorithm is that the preconditioned variant only needs 21%–35% and 54%–61% of matrix-vector operations to extract the lowest energy solution of several Hartree–Fock- and NLMO-based CIS matrices, respectively. On the other hand, the standard Davidson preconditioning definition seems to be generally optimal in terms of requisite matrix-vector operations. [ABSTRACT FROM AUTHOR]

We regard anisotropic Maxwell's equations as a boundary control and observation system on a bounded Lipschitz domain. The boundary is split into two parts: one part with perfect conductor boundary conditions and the other where the control and observation takes place. We apply a feedback control law that stabilizes the system in a semi-uniform manner without any further geometric assumption on the domain. This will be achieved by separating the equilibriums from the system and show that the remaining system is described by an operator with compact resolvent. Furthermore, we will apply a unique continuation principle on the resolvent equation to show that there are no eigenvalues on the imaginary axis. [ABSTRACT FROM AUTHOR]

We primarily investigate the existence of solutions for fractional neutral integro-differential equations with nonlocal initial conditions, which are crucial for understanding natural phenomena. Taking into account factors such as neutral type, fractional-order integrals, and fractional-order derivatives, we employ probability density functions, Laplace transforms, and resolvent operators to formulate a well-defined concept of a mild solution for the specified equation. Following this, by using fixed-point theorems, we establish the existence of mild solutions under more relaxed conditions. [ABSTRACT FROM AUTHOR]

We present an extension of the RSVD- Δ t algorithm initially developed for resolvent analysis of statistically stationary flows to handle harmonic resolvent analysis of time-periodic flows. The harmonic resolvent operator, as proposed by Padovan et al. (J Fluid Mech 900, 2020), characterizes the linearized dynamics of time-periodic flows in the frequency domain, and its singular value decomposition reveals forcing and response modes with optimal energetic gain. However, computing harmonic resolvent modes poses challenges due to (i) the coupling of all N ω retained frequencies into a single harmonic resolvent operator and (ii) the singularity or near-singularity of the operator, making harmonic resolvent analysis considerably more computationally expensive than a standard resolvent analysis. To overcome these challenges, the RSVD- Δ t algorithm leverages time stepping of the underlying time-periodic linearized Navier–Stokes operator, which is N ω times smaller than the harmonic resolvent operator, to compute the action of the harmonic resolvent operator. We develop strategies to minimize the algorithm's CPU and memory consumption, and our results demonstrate that these costs scale linearly with the problem dimension. We validate the RSVD- Δ t algorithm by computing modes for a periodically varying Ginzburg–Landau equation and demonstrate its performance using the flow over an airfoil. [ABSTRACT FROM AUTHOR]

In this paper, we study the convergence of generalized pseudo-spectrum associated with bounded operators in a Hilbert space. We prove that the approximate generalized pseudo-spectrum converges to the exact set under norm convergence. To prove this result, we use the Hausdorff distance and the assumption that the generalized resolvent operator is not constant on any open subset of the generalized resolvent set. [ABSTRACT FROM AUTHOR]

Sufficient conditions are given for the unique solvability of nonlocal problems for abstract singular equations that are formulated in terms of the zeros of the modified Bessel function and the resolvent of the operator coefficient of the equations under consideration. Examples are presented. [ABSTRACT FROM AUTHOR]

Resolvent degree is an invariant measuring the complexity of algebraic and geometric phenomena, including the complexity of finite groups. To date, the resolvent degree of a finite simple group G has only been investigated when G is a cyclic group; an alternating group; a simple factor of a Weyl group of type E 6 , E 7 , or E 8 ; or PSL (2 , F 7). In this paper, we establish upper bounds on the resolvent degrees of the sporadic groups by using the invariant theory of their projective representations. To do so, we introduce the notion of (weak) RD k ≤ d -versality, which we connect to the existence of "special points" on varieties. [ABSTRACT FROM AUTHOR]

We consider an inverse scattering problems for the biharmonic Schrödinger operator Δ2 + A · ∇ + V in three dimensions. By the Helmholtz decomposition, we take A = ∇p + ∇ ×ψ. The main contributions of this work are twofold. First, we derive a stability estimate of determining the divergence-free part ∇ ×ψ of A by far-field data at multiple wavenumbers. As a consequence, we further derive a quantitative stability estimate of determining − 1 2 ∇ ⋅ A + V. Both the stability estimates improve as the upper bound of the wavenumber increases, which exhibit the phenomenon of increased stability. Second, we obtain the uniqueness of recovering both A and V by partial far-field data. The analysis employs scattering theory to obtain an analytic domain and an upper bound for the resolvent of the fourth order elliptic operator. Notice that due to an obstruction to uniqueness, the corresponding results do not hold in general for the Laplacian, i.e., Δ + A · ∇ + V. This can be explained by the fact that the resolvent of the biharmonic operator enjoys a faster decay estimate with respect to the wavenumber compared with the Laplacian. [ABSTRACT FROM AUTHOR]

In this work, we mainly focus on the local existence and regularity of integral solutions for a class of nondensely defined partial neutral functional integrodifferential equations with unbounded delay. We use the theory of integrated resolvent operators introduced by Oka [H. Oka, Integrated resolvent operators, J. Integral Equations Appl. 7 1995, 2, 193–232]. Finally, we provide an example to demonstrate the basic findings of our work. [ABSTRACT FROM AUTHOR]

This work primarily delves into three key areas: the presence of mild solutions, exploration of the topological and geometrical makeup of solution sets, and the continuous dependency of solutions on a second-order semilinear integro-differential inclusion. The Bohnenblust-Karlin fixedpoint method has been integrated with Grimmer's theory of resolvent operators. Ultimately, the study delves into a mild solution for a partial integro-differential inclusion to showcase the achieved outcomes. [ABSTRACT FROM AUTHOR]

In this paper, our main purpose is to establish the controllability results for nonlocal neutral Hilfer fractional differential equations of Sobolev-type in a Hilbert Space as well as to generalize the results that existed in the literature on this topic. We present three types of conditions on the nonlocal initial condition's function to prove the existence of a mild solution for nonlocal neutral Hilfer fractional differential equations of Sobolev-type, and we then derive the approximate controllability results for the system. With help of an approximate technique, we establish the existence and controllability results under the weaker hypothesis (continuous only) on the nonlocal initial condition's function. The main tools applied in our analysis are semigroup theory, fractional calculus, resolvent operator theory, the theory of fractional powers of operators, Krasnoselskii's fixed point theorem, Schauder's fixed point theorem, and approximating technique. Finally, we provide two examples as applications to illustrate our main results. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the scattering function and discrete spectrum of the impulsive Sturm‐Liouville type differential operator with a turning point and nonlinear eigenparameter‐dependent boundary condition. Using hyperbolic type representations of the fundamental solutions, the operator's discrete spectrum was constructed. We presented asymptotic equations for the fundamental solutions and the Jost function. Finally, we stated an example to demonstrate the paper's major points. [ABSTRACT FROM AUTHOR]

We revise Krein's extension theory of semi-bounded Hermitian operators by reducing the problem to finding all positive and contractive extensions of the "resolvent operator" (I + T)-1 of T. Our treatment is somewhat simpler and more natural than Krein's original method which was based on the Krein transform (I-T)(I+T)-1. Apart from being positive and symmetric, we do not impose any further constraints on the operator T: neither its closedness nor the density of its domain is assumed. Moreover, our arguments remain valid in both real or complex Hilbert spaces. [ABSTRACT FROM AUTHOR]

This paper is concerned with the principal spectral theory of time-periodic cooperative systems with nonlocal dispersal and Neumann boundary condition. First we present a sufficient condition for the existence of principal eigenvalues by using the theory of resolvent positive operators with their perturbations. Then we establish the monotonicity of principal eigenvalues with respect to the frequency and investigate the limiting properties of principal eigenvalues as the frequency tends to zero or infinity. We also study the effects of dispersal rates and dispersal ranges on the principal eigenvalues, and the difficulty is that principal eigenvalues of time-periodic cooperative systems with Neumann boundary conditions are not monotone with respect to the domain. Finally, we apply our theory to a man-environment-man epidemic model and consider the impacts of dispersal rates, frequency, and dispersal ranges on the basic reproduction number and positive time-periodic solutions. [ABSTRACT FROM AUTHOR]

We study a class of two-sided optimal control problems of general linear diffusions under a so-called Poisson constraint: the controlling is only allowed at the arrival times of an independent Poisson signal processes. We give a weak and easily verifiable set of sufficient conditions under which we derive a quasi-explicit unique solution to the problem in terms of the minimal r-excessive mappings of the diffusion. We also investigate limiting properties of the solutions with respect to the signal intensity of the Poisson process. Lastly, we illustrate our results with an explicit example. [ABSTRACT FROM AUTHOR]

This paper uses the resolvent operator technique to investigate second-order non-autonomous neutral integrodifferential equations with impulsive conditions in a Banach space. We study the existence of a mild solution and the system's approximate controllability. The semigroup and resolvent operator theory, graph norm, and Krasnoselskii's fixed point theorem are used to demonstrate the results. Finally, we present our findings with an example. [ABSTRACT FROM AUTHOR]

The purpose of this work is to study new systems given by evolution equations via resolvent operators for solving Yosida inclusion problems, equilibrium problems and fixed point problems on Hadamard manifolds. Then we show that the systems have a unique solution under some suitable assumptions. Furthermore, the exponential stability and invariance properties of the systems are also investigated. Our main results in this paper are new and extend the existing ones in the literature. [ABSTRACT FROM AUTHOR]

We develop two "Nesterov's accelerated" variants of the well-known extragradient method to approximate a solution of a co-hypomonotone inclusion constituted by the sum of two operators, where one is Lipschitz continuous and the other is possibly multivalued. The first scheme can be viewed as an accelerated variant of Tseng's forward-backward-forward splitting (FBFS) method, while the second one is a Nesterov's accelerated variant of the "past" FBFS scheme, which requires only one evaluation of the Lipschitz operator and one resolvent of the multivalued mapping. Under appropriate conditions on the parameters, we theoretically prove that both algorithms achieve O 1 / k last-iterate convergence rates on the residual norm, where k is the iteration counter. Our results can be viewed as alternatives of a recent class of Halpern-type methods for root-finding problems. For comparison, we also provide a new convergence analysis of the two recent extra-anchored gradient-type methods for solving co-hypomonotone inclusions. [ABSTRACT FROM AUTHOR]

This paper aims to establish the existence and uniqueness of mild solutions to abstract sequential fractional differential equations. The approach employed involves the utilization of resolvent operators and the fixed-point theorem. Additionally, we investigate a specific example concerning a partial differential equation incorporating the Caputo fractional derivative. [ABSTRACT FROM AUTHOR]

We extend the resolvent framework to two-phase flows with low-inertia particles. The particle velocities are modelled using the equilibrium Eulerian model. We analyse the turbulent flow in a vertical pipe with Reynolds number of $5300$ (based on diameter and bulk velocity), for Stokes numbers $St^+=0-1$ , Froude numbers $Fr_z=-4,-0.4,0.4,4$ and $1/Fr_z = 0$ (gravity omitted). The governing equations are written in input–output form and a singular value decomposition is performed on the resolvent operator. As for single-phase flows, the operator is low rank around the critical layer, and the true response can be approximated using one singular vector. Even with a crude forcing model, the formulation can predict physical phenomena observed in Lagrangian simulations, such as particle clustering and gravitational effects. Increasing the Stokes number shifts the predicted concentration spectra to lower wavelengths; this shift also appears in the direct numerical simulation spectra and is due to particle clustering. When gravity is present, there are two critical layers, one for the concentration field, and one for the velocity field. For upward flow, the peak of concentration fluctuations shifts closer to the wall, in agreement with the literature. We explain this with the aid of the different locations of the two critical layers. Finally, the model correctly predicts the interaction of near-wall vortices with particle clusters. Overall, the resolvent operator provides a useful framework to explain and interpret many features observed in Lagrangian simulations. The application of the resolvent framework to higher $St^+$ flows in combination with Lagrangian simulations is also discussed. [ABSTRACT FROM AUTHOR]

This paper attempts to prove the Lipschitz continuity of the resolvent operator associated with a (P , η) -accretive mapping and compute an estimate of its Lipschitz constant. This is done under some new appropriate conditions that are imposed on the parameter and mappings involved in it; with the goal of approximating a common element of the solution set of a system of generalized variational-like inclusions and the fixed point set of a total asymptotically nonexpansive mapping in the framework of real Banach spaces. A new iterative algorithm based on the resolvent operator technique is proposed. Under suitable conditions, we prove the strong convergence of the sequence generated by our proposed iterative algorithm to a common element of the two sets mentioned above. The final section is dedicated to investigating and analyzing the notion of a generalized H(.,.)-accretive mapping introduced and studied by Kazmi et al. (Appl Math Comput 217:9679–9688, 2011). In this section, we provide some comments based on the relevant results presented in their work. [ABSTRACT FROM AUTHOR]

We study the final value problem involving a class of semilinear fractional pseudo‐parabolic equations, where the nonlinearity probably takes values in fractional Sobolev spaces. By establishing some estimates for resolvent operators and employing the embedding related to Hilbert scales and fractional Sobolev spaces, we are able to obtain the existence and uniqueness result to the mentioned problem. In addition, the behavior of solutions at initial time is analyzed with respect to the final data. It will be shown that various cases of the nonlinearity functions meet our setting, including functions with gradient term. [ABSTRACT FROM AUTHOR]

The linearized Poisson–Nernst–Planck (PNP) equation under closed ends around a neutral state is studied. It is reduced to a damped heat equation under non‐local boundary conditions, which leads to a stochastic interpretation of the linearized equation as a Brownian particle which jump and is reflected, at Poisson distributed time, to one of the end points of the channel, with a probability which is proportional to its distance from this end point. An explicit expansion of the heat kernel reveals the eigenvalues and eigenstates of both the PNP equation and its adjoint. For this, we take advantage of the representation of the resolvent operator and recover the heat kernel by applying the inverse Laplace transform. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the (recently introduced by Alavrez et al.) concept of the so-called asymptotically (ω,c) periodic functions. We investigate further properties of this class of functions which complement the currently known ones. These properties allowed us to prove the existence and uniqueness of asymptotically (ω,c) - periodic mild solution of an integro-differential equation in a Banach space, a result which generalizes some recent works in the class of S-asymptotically - periodic functions by Brindle and N'Guérékata. [ABSTRACT FROM AUTHOR]

Recently, structured inverse variational inequality (SIVI) problems have attracted much attention. In this paper, we propose new splitting methods to solve SIVI problems by employing the idea of the classical Douglas-Rachford splitting method (DRSM). In particular, the proposed methods can be regarded as a novel application of the DRSM to SIVI problems by decoupling the linear equality constraint, leading to smaller and easier subproblems. The main computational tasks per iteration are the evaluations of certain resolvent operators, which are much cheaper than those methods without taking advantage of the problem structures. To make the methods more implementable in the general cases where the resolvent operator is evaluated in an iterative scheme, we further propose to solve the subproblems in an approximate manner. Under quite mild conditions, global convergence, sublinear rate of convergence, and linear rate of convergence results are established for both the exact and the inexact methods. Finally, we present preliminary numerical results to illustrate the performance of the proposed methods. [ABSTRACT FROM AUTHOR]

We study a relaxed inertial forward–backward–half-forward splitting approach with variable step size to solve a monotone inclusion problem involving a maximal monotone operator, a cocoercive operator, and a monotone Lipschitz operator. The convergence of the sequence of iterations generated by the discretisations of a continuous-time dynamical system is established under suitable conditions. Given the challenges associated with computing the resolvent of the composite operator, the proposed method is employed to tackle the composite monotone inclusion problem. Additionally, a convergence analysis is conducted under certain conditions. To demonstrate the effectiveness of the algorithm, numerical experiments are performed on the image deblurring problem. [ABSTRACT FROM AUTHOR]

This article considers the optimal and time optimal feedback control problems for a semi-linear neutral retarded integro-differential system. The existence of mild solutions and feasible pairs for the considered system is studied by applying theory of resolvent operators for linear neutral integro-differential evolution systems, fractional powers of operators and Schauder fixed point theorem. Then the Lagrange optimal feedback control problem for the system is investigated via limit arguments under some suitable assumptions. The time optimal feedback control problem is proposed and discussed deliberately here as well. An example is presented in the end to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the weak convergence of the Rosenbrock semi-implicit method for semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. We are interested in SPDEs where the nonlinear part is stronger than the linear part, also called stochastic reaction dominated transport equations. For such SPDEs, many standard numerical schemes lose their stability properties. Exponential Rosenbrock and Rosenbrock-type methods were proved to be efficient for such SPDEs, but only their strong convergence were recently analyzed. Here, we investigate the weak convergence of the Rosenbrock semi-implicit method. We obtain a weak convergence rate which is twice the rate of the strong convergence. Our error analysis does not rely on Malliavin calculus, but rather only uses the Kolmogorov equation and the smoothing properties of the resolvent operator resulting from the Rosenbrock semi-implicit approximation. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the stability analysis of one-dimensional fully dynamic piezoelectric beam systems with internal fractional delay under different feedback controls (boundary and internal). By introducing two new equations to deal with fractional delay, equivalent new systems are obtained. Based on classical semigroup theory, we prove the well-posedness of the related systems. In the analysis of their stability, different research methods are adopted in this paper, aiming to reach conclusions in a more concise way. One of the most critical is some estimations which are caused by the existence of fractional delay terms. When the feedback control acts on the boundary, we give the non-trivial conditions of the delay parameter and the control parameter, then the exponential decay of the system with boundary feedback is obtained by using Lyapunov functional theory. When the feedback control operates in the internal domain, the delay parameter is less than the control parameter, the exponential stability of system with internal feedback is obtained by estimating of the resolvent operator norm. This is the first study of piezoelectric beam system with fractional delay and magnetic effect, which has certain significance in its development. [ABSTRACT FROM AUTHOR]

In this paper, we shall establish sufficient conditions for the existence, approximate controllability, and Ulam–Hyers–Rassias stability of solutions for impulsive integrodifferential equations of second order with state-dependent delay using the resolvent operator theory, the approximating technique, Picard operators, and the theory of fixed point with measures of noncompactness. An example is presented to illustrate the efficiency of the result obtained. [ABSTRACT FROM AUTHOR]

The main objective of this work is to investigate the existence of mild solutions and approximate controllability for some integrodifferential evolution equations with nonlocal conditions. Assuming that the linear part is exactly null and approximately controllable, using resolvent operator theory, we provide our main results. An example is given to illustrate the basic results of this work. [ABSTRACT FROM AUTHOR]

This paper focuses on investigating the problem of finding a common element of the set of solutions of a generalized nonlinear implicit variational-like inclusion problem involving an $ (A,\eta) $ (A , η) -maximal m-relaxed monotone mapping in the sense of L.-G.-s.i.p. and the set of fixed points of a total asymptotically nonexpansive mapping. To achieve such a purpose, a new iterative algorithm is constructed. Applying the concepts of graph convergence and generalized resolvent operator associated with an $ (A,\eta) $ (A , η) -maximal m-relaxed monotone mapping in the sense of L.-G.-s.i.p. As an application of the obtained equivalence relationship, the strong convergence of the sequence generated by our proposed iterative algorithm to a point belonging to the intersection of the two sets mentioned above is proved. [ABSTRACT FROM AUTHOR]

In this work we introduce a general formulation of resolvent family, also named resolvent operator, to describe explicit formulas for the solutions of dynamic equations on time scales of order 0 < α ≤ 1 . The treatment developed here is based on a formulation of Laplace Transform on time scales that includes continuous and discrete cases, to obtain concise expressions for such explicit formulas. Moreover, this formulation of Laplace Transform allows to obtain discrete counterparts of some important properties in the context of fractional calculus. As main results in this work, we study the relationship between an abstract resolvent family and its infinitesimal generator, along with the main properties of resolvent families, and the existence of solutions for abstract dynamic equations on time scales. In addition, we introduce formulas for resolvent families as solutions of several types of dynamic equations of order 0 < α ≤ 1 on continuous, discrete and quantum time scales. [ABSTRACT FROM AUTHOR]

This paper focuses on studying the uniqueness of the mild solution for an abstract fractional differential equation. We use Banach's fixed point theorem to prove this uniqueness. Additionally, we examine the stability properties of the equation using Ulam's stability. To analyze these properties, we consider the involvement of Hadamard fractional derivatives. Throughout this study, we put significant emphasis on the role and properties of resolvent operators. Furthermore, we investigate Ulam-type stability by providing examples of partial fractional differential equations that incorporate Hadamard derivatives. [ABSTRACT FROM AUTHOR]

Inertial methods play a vital role in accelerating the convergence speed of optimization algorithms. We present an inertial forward-backward-half forward splitting algorithm, which mainly finds a zero of the sum of three operators, where two of them are cocoercive operator and monotone-Lipschitz continuous respectively. Meanwhile, the convergence analysis of the proposed algorithm is established under mild conditions. To overcome the difficulty in the calculation for the resolvent of the composite operator, relying on a primal-dual idea, we expand the proposed algorithm to solve the composite inclusion problem involving a linearly composed monotone operator. As an application, we make use of the obtained inertial algorithm to deal with a composite convex optimization problem. We also show extensive numerical experiments on the total variation-based image deblurring problem to demonstrate the efficiency of the proposed algorithm. Specifically, the proposed algorithm not only has a better quality of the deblurring image but also converges more rapidly than the original one. [ABSTRACT FROM AUTHOR]

This paper provides a first attempt to incorporate the massive discontinuous changes in the spatio-temporal dynamics of epidemics. Namely, we propose an extended class of epidemic models, governed by coupled stochastic semilinear partial differential equations, driven by pure-jump Lévy noise. Based on the considered type of incidence functions, by virtue of semigroup theory, a truncation technique and Banach fixed point theorem, we prove the existence and pathwise uniqueness of mild solutions, depending continuously on the initial datum. Moreover, by means of a regularization technique, based on the resolvent operator, we acquire that mild solutions can be approximated by a suitable converging sequence of strong solutions. With this result at hand, for positive initial states, we derive the almost-sure positiveness of the obtained solutions. Finally, we present the outcome of several numerical simulations, in order to exhibit the effect of the considered type of stochastic noise, in comparison to Gaussian noise, which has been used in the previous literature. Our established results lay the ground-work for investigating other problems associated with the new proposed class of epidemic models, such as asymptotic behavior analyses, optimal control as well as identification problems, which primarily rely on the existence and uniqueness of biologically feasible solutions. • A new class of epidemic models incorporating random discontinuous changes in the spatio-temporal dynamics is proposed. • The existence and pathwise uniqueness of mild solutions, for different types of incidence functions, is established. • The biological feasibility of mild solutions, for different types of incidence functions, is derived. • Numerical simulations showing the effect of discontinuous random changes on the spatio-temporal dynamics are presented. [ABSTRACT FROM AUTHOR]