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In this paper, we study the variational inequality over the solution set of the split common solution problem with multiple output sets for monotone operator equations in real Hilbert spaces. We propose a new approach for solving this problem without depending on the norm of the transfer mappings. Some of its applications for solving the variational inequality over the solution set of the split common fixed point problem with multiple output sets and related problems are also mentioned. Finally, we give numerical examples to illustrate the performance of the proposed algorithm and compare it with several existing previous algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we propose a method for efficiently obtaining an approximate solution for constrained nonlinear monotone operator equations. The search direction of the proposed method closely aligns with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) direction, known for its low storage requirement. Notably, the search direction is shown to be sufficiently descent and bounded without using the line search condition. Furthermore, under some standard assumptions, the proposed method converges globally. As an application, the proposed method is applied to solve image restoration problems. The efficiency and robustness of the method in comparison to other methods are tested by numerical experiments using some test problems. [ABSTRACT FROM AUTHOR]

A basic fact, having fundamental significance in quantum mechanics, is that hermitian (or self-adjoint) operators have only real eigenvalues. However, in certain applications in molecular physics, one deals with non-hermitian operators. We discuss a condition for non-hermitian operators to have real eigenvalues, proving that it is the case if and only if it can be decomposed as a product of two, generally non-commuting hermitian operators, one of which is positive definite. The theorem is illustrated on the example of non-hermitian effective Hamiltonians occurring in the non-perturbative form of the Bloch equation. [ABSTRACT FROM AUTHOR]

We are devoted in this work to dealing with a class of time-fractional nonstationary incompressible Navier–Stokes–Voigt equation involving the Caputo fractional derivative. By exploiting the properties of the operators in the equation, we use the Rothe method to show the existence of weak solutions to the equation by verifying all the conditions of the surjectivity theorem for nonlinear weakly continuous operators. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of variational inequality problems the constraint set of which is the set of common solutions of a finite family of operator equations, involving hemi-continuous accretive operators on a reflexive and strictly convex Banach space with a Gâteaux differentiable norm. We present a sequential regularization method of Lavrentiev type and an iterative regularization one in combination with an inertial term to speed up convergence. The strong convergence of the methods is proved without the co-coercivity imposed on any operator in the family. An application of our results to solving the split common fixed point problem with pseudocontractive and nonexpansive operators is given with computational experiments for illustration. [ABSTRACT FROM AUTHOR]

This paper is a study of a generalization of the quantum Riemannian Hamiltonian evolution, previously analyzed by us, in the geometrization of quantum mechanical evolution in a Finsler geometry. We find results with dynamical equations governing the evolution of the trajectories defined by the expectation values of the position. The analysis appears to provide an underlying geometry described by a geodesic equation, with a connection form with a second term which is an essentially quantum effect. These dynamical equations provide a new geometric approach to the quantum evolution where we suggest a definition for "local instability" in the quantum theory. [ABSTRACT FROM AUTHOR]

Certain vertical motions associated with meso-microscale systems are favorable for convection development and maintenance. Correct initialization of updraft motions is thus significant in convective precipitation forecasts. A three-dimensional variational-based vertical velocity (w) assimilation scheme has been developed within the high-resolution (3 km) CMA-MESO (the Mesoscale Weather Numerical Forecast System of the China Meteorological Administration) model. This scheme utilizes the adiabatic Richardson equation as the observation operator for w , enabling the update of horizontal winds and mass fields of the model's background. The tangent linear and adjoint operators are subsequently developed and undergo an accuracy check. A single-point w observation assimilation experiment reveals that the observational information is effectively spread both horizontally and vertically. Specifically, the assimilation of w contributes to the generation of horizontal wind convergence at lower model levels and divergence at higher model levels, thereby adjusting the locations of convection occurrence. The impact of assimilating w on the forecast is then examined through a series of continuous 10 d runs. Further assimilation of w , in addition to the assimilation of conventional and radial wind data, significantly improves the forecast accuracy of precipitation, resulting in higher FSS (fractions skill score) values and higher ETS (equitable threat score) skills at higher thresholds (5 and 20 mmh-1). However, it should be noted that further assimilation of w can potentially lead to some false precipitation, resulting in slightly lower ETS values at lower thresholds (1 mmh-1) and a neutral impact on BIAS (bias score) skills. An individual case study conducted within the batch experiments reveals that assimilating w has a beneficial impact on the enhancement of vertical motion across different layers of the model, facilitating the transport of moisture from lower to middle–high model levels, thereby leading to an improvement in forecast skills. [ABSTRACT FROM AUTHOR]

An extension of slant Hankel operator, namely, the k-th-order λ-slant Hankel operator on the Lebesgue space L 2 (T n) , where T is the unit circle and n ≥ 1, a natural number, is described in terms of the solution of a system of operator equations, which is subsequently expressed in terms of the product of a slant Hankel operator and a unitary operator. The study is further lifted in Calkin algebra in terms of essentially k-th-order λ-slant Hankel operators on L 2 (T n) . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of reconstructing the past distribution for diffusion equations in multidimensional space with the presence of a poly-fractional operator which takes the following form$ {\mathbb{P}_\alpha }\left({{{\left({ - \Delta } \right)}^\gamma }} \right)u: = \sum\limits_{i = 1}^m {{\alpha _i}} {\left({ - \Delta } \right)^{{\gamma _i}}}u, $for $ m\in {\mathbb N} $, $ 0 < {\gamma _1} \le \cdots \le {\gamma _m} \le 1 $ and $ {\alpha _i} > 0 $ for $ i = \overline{1, m} $. The problem essentially arises from the superposition of multiple stochastic processes with different scales, including classical random walks and Lévy flights. We prove that the investigated problem is extremely ill-posed in the sense of Hadamard. Using an appropriate filtering method, we construct a regularized solution which converges to the original solution in $ L^2 $-sense. If we further impose a Gevrey-type a-priori condition, a Hölder convergence rate can be obtained. Three numerical examples using Matlab are also performed to demonstrate the ill-posedness of the problem and effectiveness of the regularization method. [ABSTRACT FROM AUTHOR]

We obtain necessary and sufficient conditions for the existence of a unique solution bounded on the entire axis of a second-order differential equation with piecewise constant operator coefficients. [ABSTRACT FROM AUTHOR]

We consider evolutionary equations with fractional differentiation operators whose restrictions to certain spaces of type S coincide with pseudodifferential operators constructed on the basis of smooth symbols, which are multipliers in these spaces. We establish the well-posedness of the time-nonlocal multipoint problem for these equations with initial function, which is an element of the space of generalized functions of the ultradistribution type. It is shown that the solutions of these problems stabilize to zero in the spaces of generalized functions of the type S′ (weak stabilization) and also stabilize to zero uniformly on R in the case where the initial generalized function has a bounded support. [ABSTRACT FROM AUTHOR]

Nonlinearity presents a significant challenge in developing quantum algorithms involving differential equations, prompting the exploration of various linearization techniques, including the well-known Carleman Linearization. Instead, this paper introduces the Koopman Spectral Linearization method tailored for nonlinear autonomous ordinary differential equations. This innovative linearization approach harnesses the interpolation methods and the Koopman Operator Theory to yield a lifted linear system. It promises to serve as an alternative approach that can be employed in scenarios where Carleman Linearization is traditionally applied. Numerical experiments demonstrate the effectiveness of this linearization approach for several commonly used nonlinear ordinary differential equations. [ABSTRACT FROM AUTHOR]

In this paper, the Cauchy problem for convolution-type nonlocal linear and nonlinear Schrödinger equations (NSEs) is studied. The equations include the general differential operators. The existence, uniqueness, Lp-regularity properties of linear problem and the existence, uniqueness, and blow-up at finite time of the nonlinear problem are obtained. By choosing differential operators including in equations, the regularity properties of a different type of nonlocal Schrödinger equation are studied. [ABSTRACT FROM AUTHOR]

We study nonlocal elliptic and parabolic equations on C1,τ open sets in weighted Sobolev spaces, where τ ∈ (0, 1). The operators we consider are infinitesimal generators of symmetric stable Lévy processes, whose Lévy measures are allowed to be very singular. Additionally, for parabolic equations, the measures are assumed to be merely measurable in the time variable. [ABSTRACT FROM AUTHOR]

Koplienko [Sib. Mat. Zh. 25 (1984), 62–71; English transl. in Siberian Math. J. 25 (1984), 735–743] found a trace formula for perturbations of self‐adjoint operators by operators of Hilbert–Schmidt class B2(H)$\mathcal {B}_2(\mathcal {H})$. Later, Neidhardt introduced a similar formula in the case of pairs of unitaries (U,U0)$(U,U_0)$ via multiplicative path in [Math. Nachr. 138 (1988), 7–25]. In 2012, Potapov and Sukochev [Comm. Math. Phys. 309 (2012), no. 3, 693–702] obtained a trace formula like the Koplienko trace formula for pairs of contractions by answering an open question posed by Gesztesy, Pushnitski, and Simon [Zh. Mat. Fiz. Anal. Geom. 4 (2008), no. 1, 63–107, 202; Open Question 11.2]. In this paper, we supply a new proof of the Koplienko trace formula in the case of pairs of contractions (T,T0)$(T,T_0)$, where the initial operator T0$T_0$ is normal, via linear path by reducing the problem to a finite‐dimensional one as in the proof of Krein's trace formula by Voiculescu [Oper. Theory Adv. Appl. 24 (1987) 329–332] and Sinha and Mohapatra [Proc. Indian Acad. Sci. Math. Sci. 104 (1994), no. 4, 819–853] and [Integral Equations Operator Theory 24 (1996), no. 3, 285–297]. Consequently, we obtain the Koplienko trace formula for a class of pairs of contractions using the Schäffer matrix unitary dilation. Moreover, we also obtain the Koplienko trace formula for a pair of self‐adjoint operators and maximal dissipative operators using the Cayley transform. At the end, we extend the Koplienko–Neidhardt trace formula for a class of pairs of contractions (T,T0)$(T,T_0)$ via multiplicative path using the finite‐dimensional approximation method. [ABSTRACT FROM AUTHOR]

In this paper, we give the necessary and sufficient conditions for the existence of general solutions, self-adjoint solutions, and positive solutions to the system of A i X B i + C i Y D i + E i Z F i = G i (i = 1 , 2) under additional conditions. In addition, we derive the representation of general solutions to the system of A i X B i + C i Y D i + E i Z F i = G i (i = 1 , 2) , and provide the matrix representation of the self-adjoint solutions and the positive solutions in the sense of the star order. [ABSTRACT FROM AUTHOR]

In this paper, we establish the improved uniform error bounds for the Lawson-type exponential wave integrator Fourier pseudo-spectral (LEWIFP) method for the nonlinear Schrödinger equation with wave operator (NLSW) and weak nonlinearity characterized by a small constant ε ∈ (0 , 1 ]. We first convert the NLSW to a coupled system and then consider the LEWIFP method for the coupled system. The LEWIFP method is proved to be time symmetric and mass-conservative which is very important from the point of view of numerical geometric integration. Through careful and rigorous convergence analysis, we establish improved uniform error bounds for the full-discretization at O (h m + ε 2 p τ 2) in the long-time domain up to O (1 / ε 2 p) where m is determined by the regularity conditions. These error bounds are much better than the classical error bounds O (h m + τ 2) provided by the traditional error analysis. In error analysis, in addition to the classical tools such as mathematical induction and energy method, we adopt the regularity compensation oscillation (RCO) technique to analyze the accumulation of errors carefully. The numerical experiments support our error estimates and demonstrate the discrete mass-conservation. In addition, the numerical results show that the discrete energy is stable in the time domain up to O (1 / ε 2 p). [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes a mixed variational formulation for the Navier-Stokes equations with variable viscosity that depends nonlinearly on the velocity gradient. Differently from previous works in which augmented terms are added to the formulation, here we employ a technique that had been previously applied to the stationary Boussinesq problem and the Navier-Stokes equations with constant viscosity. Firstly, a modified pseudostress tensor is introduced involving the diffusive and convective terms and the pressure. Secondly, by using the incompressibility condition, the pressure is eliminated, and the gradient of velocity is incorporated as an auxiliary unknown to handle the aforementioned nonlinearity. As a consequence, a Banach spaces-based formulation is obtained, which can be written as a perturbed twofold saddle point operator equation. We address the continuous and discrete solvabilities of this problem by linearizing the perturbation and employing a fixed-point approach along with a particular case of a known abstract theory. Given an integer ℓ ≥ 0 , feasible choices of finite element subspaces include discontinuous piecewise polynomials of degree ≤ ℓ for each entry of the velocity gradient, Raviart-Thomas spaces of order ℓ for the pseudostress, and discontinuous piecewise polynomials of degree ≤ ℓ for the velocity as well. Finally, optimal a priori error estimates are derived, and several numerical results confirming in general the theoretical rates of convergence, and illustrating the good performance of the scheme, are reported. [ABSTRACT FROM AUTHOR]

We introduce a regularization method with two differential operators for solving a linear ill-posed operator equation system. The existence and uniqueness of regularized solutions to the problem are derived. With an a priori as well as an a posteriori parameter choice strategy, convergence and convergence rates of the regularized solution are also obtained. As an application, we apply the regularization to a simultaneous inversion of the source term and the initial value problem for a heat conduction equation, numerical experiments are presented to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We consider solutions of some two-dimensional fourth-order equation with a biharmonic operator and exponential nonlinearity of a counterpart of the classical Gauss–Bieberbach–Rademacher second-order equation, which was previously inspected by many authors in connection with problems of the geometry of surfaces with negative Gaussian curvature, rarefied gas dynamics, and the theory of automorphic functions. We obtain some conditions for the absence of a solution in a disk of sufficiently large radius and show that global solutions on the plane can exist only if the coefficient of nonlinearity decays at infinity at the rate at least . Otherwise the mean value of the solution on a circle of radius would tend to with exponential rate as . The Pokhozhaev–Mitidieri nonlinear capacity method, based on the choice of appropriate cutoff test functions, proves the impossibility of the existence of such global solution. Also, for the solutions in , periodic in all but variables, the absence of global solutions is obtained by similar methods when the nonlinearity coefficient decays at rate slower than . [ABSTRACT FROM AUTHOR]

Green function of a 2n-th order differential equation with normal coefficients on the half-axis is studied. We first consider the Green function of our equation with “frozen” coefficients. Using Levi’s method, we obtain a Fredholm-type integral equation for the Green function of our problem, whose kernel is a Green function of a problem with constant coefficients. We prove an existence and uniqueness theorem for this integral equation in some Banach spaces of operator-valued functions. The main result of this paper is a theorem stating that the solution of the obtained integral equation is a Green function of our problem. [ABSTRACT FROM AUTHOR]

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We show the existence of unbounded connected components of 2π-periodic positive solutions for the equations with one-dimensional Minkowski-curvature operator − u ′ 1 − u ′ 2 ′ = λ a (x) f (u , u ′ ) , x ∈ R , where λ > 0 is a parameter, a ∈ C (R , R) is a 2π-periodic sign-changing function with ∫ 0 2 π a (x) d x < 0 , f ∈ C (R × R , R) satisfies a generalized regular-oscillation condition. Moreover, for the special case that f does not contain derivative term, we also investigate the global structure of 2π-periodic odd/even sign-changing solutions set under some parity conditions. The proof of our main results are based upon bifurcation techniques. [ABSTRACT FROM AUTHOR]

The present paper aims to study a quarter symmetric metric connection in the tangent bundle and investigate an induced metric and connection on a submanifold of co-dimension 2 and hypersurface by means of mathematical operators concerning the QSM connection in the tangent bundle TM. Totally geodesic (TG) and totally umbilical (TU) concerning the QSM connection on the submanifold of co-dimension 2 and hypersurface in TM are obtained. [ABSTRACT FROM AUTHOR]

This study investigates the existence of triple weak solutions for a system of nonlinear elliptic equations with a fourth-order operator. The problem arises in the mathematical modeling of complex physical phenomena. [ABSTRACT FROM AUTHOR]

In the view of existing works on operators (namely, λ- Hankel and λ- Slant Hankel operators etc.) done by Avendano, Datt and Aggrawal, we take up an operator equation ... and solve it for λ, μ ∈ C. Its solution yields an important class of slant Hankel operators, (namely, (λ, μ)-slant Hankel operators). After that, we provide certain basic properties of (λ, μ)- slant Hankel operator on L²(T) of unit circle T. Finally, we come up with a results for (λ, μ)- slant Hankel operator, which is analogous to classical Kronecker theorem for Hankel matrix. [ABSTRACT FROM AUTHOR]

We consider operators of the form $ L u(x) = \sum_{y \in \mathbb{Z}} k(x-y) \big(u(y) - u(x)\big) $ on the one-dimensional lattice with symmetric, integrable kernel $ k $. We prove several results stating that under certain conditions on the kernel the operator $ L $ satisfies the curvature-dimension condition $ CD_\Upsilon (0, F) $ (recently introduced by the last two authors) with some $ CD $-function $ F $, where attention is also paid to the asymptotic properties of $ F $ (exponential growth at infinity and power-type behaviour near zero). We show that $ CD_\Upsilon (0, F) $ implies a Li-Yau inequality for positive solutions of the heat equation associated with the operator $ L $. The Li-Yau estimate in turn leads to a Harnack inequality, from which we also derive heat kernel bounds. Our results apply to a wide class of operators including the fractional discrete Laplacian. [ABSTRACT FROM AUTHOR]

In this paper, we develop a novel class of linearly implicit and energy-preserving integrating factor methods for the 2D nonlinear Schrödinger equation with wave operator (NLSW), combining the scalar auxiliary variable approach and the integrating factor methods. To begin, a second-order scheme is proposed, which is rigorously proved to be energy-preserving. By using the energy methods, we analyze its optimal convergence without any restrictions on the grid ratio, where a novel technique and an improved induction argument are proposed to circumvent the difficulty arising from the unavailability of a priori |$L^{\infty }$| estimates of numerical solutions. Based on the integrating factor Runge–Kutta methods, we extend the proposed scheme to arbitrarily high order, which is also linearly implicit and conservative. Numerical experiments are presented to confirm the theoretical analysis and demonstrate the advantages of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we study a complete second-order integro-differential operator equation in a Hilbert space. The difference-type kernel of an integral perturbation is a holomorphic semigroup bordered by unbounded operators. The asymptotic behavior of solutions of this equation is studied. Asymptotic formulas for solutions are proved in the case when the right-hand side is close to an almost periodic function. The obtained formulas are applied to the study of the problem of forced longitudinal vibrations of a viscoelastic rod with Kelvin–Voigt friction. [ABSTRACT FROM AUTHOR]

In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form F ⁢ ( u ) = y {F(u)=y} , where F : 풟 ⁢ ( F ) ⊂ 햴 → 햸 {F:\mathcal{D}(F)\subset\mathsf{U}\to\mathsf{Y}} is a nonlinear operator between Hilbert spaces 햴 {\mathsf{U}} and 햸 {\mathsf{Y}} . The method is defined as follows: u n + 1 δ = u n δ - ( T 0 ∗ ⁢ T 0 + α n ⁢ I ) - 1 ⁢ T 0 ∗ ⁢ ( F ⁢ ( u n δ ) - y δ ) , u_{n+1}^{\delta}=u_{n}^{\delta}-(T_{0}^{\ast}T_{0}+\alpha_{n}I)^{-1}T_{0}^{% \ast}(F(u_{n}^{\delta})-y^{\delta}), where T 0 = F ′ ⁢ ( u 0 ) {T_{0}=F^{\prime}(u_{0})} and T 0 ∗ = F ′ ⁢ ( u 0 ) ∗ {T_{0}^{\ast}=F^{\prime}(u_{0})^{\ast}} . Here F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} denotes the Frèchet derivative of F at an initial guess u 0 ∈ 풟 ⁢ ( F ) {u_{0}\in\mathcal{D}(F)} for the exact solution u † {u^{\dagger}} , F ′ ⁢ ( u 0 ) ∗ {F^{\prime}(u_{0})^{\ast}} is the adjoint of F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} and { α n } {\{\alpha_{n}\}} is an a priori chosen sequence of non-negative real numbers satisfying suitable properties. We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearity conditions on operator F, we show convergence of the method and also obtain a convergence rate result under a Hölder-type source condition on the element u 0 - u † {u_{0}-u^{\dagger}} . Furthermore, we derive convergence of the method for the case when no source conditions are used and the study concludes with numerical examples which validate the theoretical conclusions. [ABSTRACT FROM AUTHOR]

In the half-plane, a solution to a two-dimensional hyperbolic equation with a translation operator in the lowest derivative with respect to a spatial variable varying along the entire real axis is constructed in an explicit form. It is proven that the solutions obtained are classical if the real part of the symbol of a differential-difference operator in the equation is positive. [ABSTRACT FROM AUTHOR]

In the case of one spatial independent variable, we study hyperbolic differential-difference equations with potentials represented as linear combinations of translations of the desired function along the spatial variable. The qualitative novelty of this investigation is that, unlike previous research, it is not assumed that the real part of the symbol of the differential-difference operator contained in the equation has a constant sign. Previously, it was possible to remove that substantial restriction (i.e., the specified sign constancy) only for the case where the nonlocal term (i.e., the translated potential) is unique. In the present paper, we consider the case of the general-kind one-variable nonlocal potential, i.e., equations with an arbitrary amount of translated terms. No commensurability assumptions are imposed on the translation lengths. The following results are presented: We find a condition relating the coefficients at the nonlocal terms of the investigated equation and the length of the translations, providing the global solvability of the investigated equation. Under this condition, we explicitly construct a three-parametric family of smooth global solutions of the investigated equation. [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]

In this paper, we initiate the study of asymptotic and oscillatory properties of solutions to second-order functional differential equations with noncanonical operators and unbounded neutral coefficients, using a recent method of iteratively improved monotonicity properties of nonoscillatory solutions. Our results rely on ideas that essentially improve standard techniques for the investigation of differential equations with unbounded neutral terms with delay or advanced argument. The core of the method is presented in a form that suggests further generalizations for higher-order differential equations with unbounded neutral coefficients. [ABSTRACT FROM AUTHOR]

In this article, we analyse the Dyson equation for the density–density response function (DDRF) that plays a central role in linear response time-dependent density functional theory (LR-TDDFT). First, we present a functional analytic setting that allows for a unified treatment of the Dyson equation with general adiabatic approximations for discrete (finite and infinite) and continuum systems. In this setting, we derive a representation formula for the solution of the Dyson equation in terms of an operator version of the Casida matrix. While the Casida matrix is well-known in the physics literature, its general formulation as an (unbounded) operator in the N -body wavefunction space appears to be new. Moreover, we derive several consequences of the solution formula obtained here; in particular, we discuss the stability of the solution and characterise the maximal meromorphic extension of its Fourier transform. We then show that for adiabatic approximations satisfying a suitable compactness condition, the maximal domains of meromorphic continuation of the initial DDRF and the solution of the Dyson equation are the same. The results derived here apply to widely used adiabatic approximations such as (but not limited to) the random phase approximation and the adiabatic local density approximation. In particular, these results show that neither of these approximations can shift the ionisation threshold of the Kohn–Sham system. [ABSTRACT FROM AUTHOR]

Inspired by the ROF model and the L¹/TV image denoising model, we propose a combined model to remove Gaussian noise and salt-and-pepper noise simultaneously. This model combines the L¹ -data fidelity term, L² -data fidelity term and a fractional-order total variation regularization term, and is termed the L¹L²/TVα model. We have used the proximity algorithm to solve the proposed model. Through this method, the non-differentiable term is solved by using the fixed-point equations of the proximity operator. The numerical experiments show that the proposed model can effectively remove Gaussian noise and salt and pepper noise through implementation of the proximity algorithm. As we varied the fractional order α from 0.8 to 1.9 in increments of 0.1, we observed that different images correspond to different optimal values of a. [ABSTRACT FROM AUTHOR]

In this paper, we study some connections between solutions A and B satisfying the operator equation AXB = X. We also investigate several properties between such solutions A and B. In particular, we show that if A has the single valued extension property, then so does B when X is injective. Moreover, we consider the (weak) Fuglede-Putnam type property (defined below) and investigate the local spectral properties between the solutions A and B under the Fuglede-Putnam type property. [ABSTRACT FROM AUTHOR]

In this paper, under appropriate hypotheses, we have the existence of a solution semigroup of partial differential equations with delay operator. These equations are used to describe time–age-structured cell cycle model. We also prove that the solution semigroup is a frequently hypercyclic semigroup. [ABSTRACT FROM AUTHOR]

We present the representation of solution sets of the operator equation XAX = BX on Hilbert spaces by taking advantage of the invariant subspace of B, including solution sets of nonzero solutions, normal solutions and self-adjoint solutions. Moreover, necessary conditions and sufficient conditions are established for existence of nonzero solutions of the operator equation XAX = BX which commute with B. [ABSTRACT FROM AUTHOR]

This paper is devoted to develop a new energy dissipation scheme by finite element method (FEM) for a nonlinear Schrödinger equation with wave operator (NLSW) and investigate its superconvergence behavior through introducing a new second order discrete approximation with respect to time. The new scheme not only guarantees the mass and energy stable but also is efficient in achieving the numerical solution at the last point. Also, an application to other general nonlinear equations with wave operator is discussed. On the one hand, the boundedness of the solution about the numerical equation is deduced and as a result, the existence and uniqueness of the solution are obtained. On the other hand, the superconvergence error estimate is derived and some numerical results are given to confirm the theoretical predictions. [ABSTRACT FROM AUTHOR]

• The symplectic method is applied to derive the analytical solution of micropolar plane annular sector in polar coordinate. • The solution to an eigenproblem of the Hamiltonian operator matrix is obtained in form of the Bessel and power functions. • The analytical solutions of the micropolar plane annular sector under the different boundary conditions are obtained. • Size-dependent behaviors in the micropolar plane annular sector are captured. The symplectic approach is utilized to derive the solutions of the plane annular sector in micropolar elasticity. According to the Hellinger-Reissner variational principle, the Hamiltonian canonical equations are obtained for the plane problem in an annular sectorial region. Through the method of separation of variables, we obtain an eigenvalue equation of the Hamiltonian operator matrix which is a system of linear differential equations with variable coefficients. By the method of series solutions, its general solutions can be expressed as the linear combination of Bessel functions and power functions. The corresponding eigensolutions for three types of homogeneous boundary conditions are derived. According to the adjoint symplectic orthogonality of the eigensolutions, the solution of the micropolar plane annular sector is expressed as a series expansion of eigensolutions. Numerical analyses are presented to demonstrate the validity of the proposed method and confirm its ability to accurately capture the size effect. [ABSTRACT FROM AUTHOR]

This paper is concerned with a regularity analysis of parametric operator equations with a perspective on uncertainty quantification. We study the regularity of mappings between Banach spaces near branches of isolated solutions that are implicitly defined by a residual equation. Under s -Gevrey assumptions on the residual equation, we establish s -Gevrey bounds on the Fréchet derivatives of the locally defined data-to-solution mapping. This abstract framework is illustrated in a proof of regularity bounds for a semilinear elliptic partial differential equation with parametric and random field input. [ABSTRACT FROM AUTHOR]

An optimal control problem is investigated for an abstract semilinear differential equation of the first order in time in a Hilbert space with an unbounded operator and control involved linearly in the right-hand side. The cost functional is assumed to be additively separated with respect to state and control, with a rather general dependence on the state. For this problem, the existence of an optimal control is proved and the properties of the set of optimal controls are established. The author's previous results on the total preservation of unique global solvability (totally global solvability) and on solution estimation for such equations are developed in the context of the nonlinearity of the equation under study. The indicated estimate is found important for the present study. A nonlinear heat equation and a nonlinear wave equation are considered as examples. [ABSTRACT FROM AUTHOR]

A key observation underlying this paper is the fact that the range invariance condition for convergence of regularization methods for nonlinear ill-posed operator equations—such as coefficient identification in partial differential equations (PDEs) from boundary observations—can often be achieved by extending the searched for parameter in the sense of allowing it to depend on additional variables. This clearly counteracts unique identifiability of the parameter, though. The second key idea of this paper is now to restore the original restricted dependency of the parameter by penalization. This is shown to lead to convergence of variational (Tikhonov type) and iterative (Newton-type) regularization methods. We concretize the abstract convergence analysis in a framework typical of parameter identification in PDEs in a reduced and an all-at-once setting. This is further illustrated by three examples of coefficient identification from boundary observations in elliptic and time-dependent PDEs. [ABSTRACT FROM AUTHOR]

In this work we introduce discrete convolution operators and study their most basic properties. We then solve linear difference equations depending on such operators. The theory herein developed generalizes, in particular, the theory of discrete fractional calculus and fractional difference equations. To that matter we make use of the so-called Sonine pairs of kernels. [ABSTRACT FROM AUTHOR]

The problem of determining a term in the right-hand side of elliptic equations from an observation on a part of the boundary is investigated. The inverse problem is formulated as an operator equation and then stabilized by Tikhonov regularization method. The regularized problem is discretized based on Hinze's variational discretization concept and the regularization parameter is chosen guaranteeing that when noise level and the discretization mesh size tend to zero, the solution of the discretized regularized problem converges to the $ f^* $ f ∗ -minimum norm solution of the continuous inverse problem. Some numerical examples are presented for illustrating the performance of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a parabolic–elliptic Keller–Segel system with a logistic term on ℝN$$ {\mathrm{\mathbb{R}}}&#x0005E;N $$, involving mixed local and nonlocal operators −Δ+(−Δ)s$$ -\Delta &#x0002B;{\left(-\Delta \right)}&#x0005E;s $$ with s∈(0,1)$$ s\in \left(0,1\right) $$. At first, we prove that the semigroup {T(x)}t>0$$ {\left\{T(x)\right\}}_{t>0} $$ generated by Δ−(−Δ)s−I$$ \Delta -{\left(-\Delta \right)}&#x0005E;s-I $$ is an analytical semigroup and uses blow‐up arguments in combination with the classical Liouville‐type theorem to demonstrate the regularity of weak solutions of the parabolic equation with the mixed operators. Next, the local existence and uniqueness of classical solutions are established by applying the semigroup theory and regularity results in case of s∈N2N+4,1$$ \mathbf{s}\in \left(\frac{N}{2N&#x0002B;4},1\right) $$. Moreover, we obtain the global existence and boundedness of classical solutions under some conditions on given initial values and the asymptotic behavior of the global solutions with strictly positive initial conditions. [ABSTRACT FROM AUTHOR]

In this paper, we have considered an inverse scattering problem for discrete Dirac operator L with eigenparameter dependent boundary condition. At the outset, the Jost solution and the Jost function of L are given. Then the zeros of the Jost function and some properties of the scattering function are examined. After finding the scattering dataset and the main equation for the operator L, the uniqueness of the kernel, continuity of the scattering function and the appropriate Levinson type formula are obtained. [ABSTRACT FROM AUTHOR]