Operator Equations Research Articles

On the solution of the Yang–Baxter–like operator equation on Hilbert spaces

This paper deals with solutions of the Yang–Baxter–like operator equation AXA = XAX on the infinite dimensional Hilbert space H , where A is a rank-two bounded linear operator and X is the unknown bounded linear operator. For an operator X, necessary and sufficient conditions such that X is a solution of the equation AXA = XAX are given. With these criteria, it is in fact shown that all solutions of the nonlinear equation AXA = XAX under the corresponding conditions can be derived merely through solving some systems of linear equations.

Inverse Problems of Recovering Lower-Order Coefficients from Boundary Integral Data

We study inverse problems of identification of lower-order coefficients in a second-order parabolic equation. The coefficients are sought in the form of a finite series segment with unknown coefficients, depending on time. The linear case is also considered. Overdetermination conditions are the integrals over the boundary of a solution’s domain with weights. We focus on existence and uniqueness theorems and stability estimates for solutions to these inverse problems. An operator equation to which the problem is reduced is studied with the use of the contraction mapping principle. A solution belongs to some Sobolev space and has all generalized derivatives occurring into the equation summable to some power. The method of the proof is constructive, and it can be used for developing new numerical algorithms for solving the problem.

A Unified Concept of the Degree of Ill-posedness for Compact and Non-Compact Linear Operator Equations in Hilbert Spaces Under the Auspices of the Spectral Theorem

Covering ill-posed problems with compact and non-compact operators regarding the degree of ill-posedness is a never ending story written by many authors in the inverse problems literature. This paper tries to add a new narrative and some new facets with respect to this story under the auspices of the spectral theorem. The latter states that any self-adjoint and bounded operator is unitarily equivalent to a multiplication operator on some (semi-finite) measure space. We will exploit this fact and derive a distribution function from the corresponding multiplier, the growth behavior of which at zero allows us to characterize the degree of ill-posedness. We prove that this new concept coincides with the well-known one for equations with compact operators (by means of their singular values), and illustrate the implications along examples with non-compact operators, including the Hausdorff moment operator and convolutions.

Applications of KKAT Transform Technique in Convolution Kind Linear Volterra Operator Equations

In this paper, Karry-Kalim Adnan transform (KKAT) is utilized to find the solution of convolution kind linear Volterra equations of the first and second kind. Examples are offered to illustrate the Karry-Kalim Adnan technique for solving convolution kind Volterra operator equations. It was noted that this technique is much easier in terms of mathematical operations and simplicity in arriving at the exact solution and this is explained through the applications in the paper

Characterizations of second-order differential operators

Let N, k be positive integers with k≥2, and Ω⊂RN be a domain. By the well-known properties of the Laplacian and the gradient, we have ∇2(f·g)=g∇2f+f∇2g+2⟨∇f,∇g⟩for all f,g∈Ck(Ω,R). H. König and V. Milman showed that the converse is also true, i.e. this operator equation characterizes the Laplacian and the gradient under some assumptions. Thus the main aim of this paper is to provide an extension of this result and to study the corresponding equation T(f·g)=fT(g)+T(f)g+2B(A(f),A(g))f,g∈P,where Q and R are commutative rings, P is a subring of Q and T:P→Q and A:P→R are additive, while B:R×R→Q is a symmetric and bi-additive. Related identities with one function will also be considered.

A quadratic operator equation and a non-commutative uncertainty relation

In this paper, we explore equivalent conditions for the positivity of a quadratic equation P ( x ) = A x 2 + Bx + C with operator coefficients. Specifically, we show that A x 2 + 2 Bx + C ≥ 0 for all x ∈ R if and only if A , C ≥ 0 , B = B ∗ , and there exists a self-adjoint operator H such that [ A + C + H B + i ( A − C ) B + i ( A − C ) A + C − H ] ≥ 0. Also, we show that if A and B are two unital C ∗ -algebras, Φ : A → B is a unital tracial positive linear map, and ϕ : A → A is a linear map satisfying ϕ ( X ) ∗ ϕ ( X ) ≤ ϕ ( X ∗ X ) for all X ∈ A , then V Ψ ( A ) ♯ V Ψ ( B ) ≥ ± 1 2 Ψ ( [ A , B ] ) , where A and B are self-adjoint elements in A , Ψ = Φ ∘ ϕ , and V Ψ ( X ) = Ψ ( X ∗ X ) − Ψ ( X ∗ ) Ψ ( X ) denotes the generalized variance of an observable X.

A Self-Adaptive Explicit Iterative Algorithm for Solving the Split Common Solution Problem

This paper aims to introduce a novel self-adaptive explicit iterative algorithm for solving the split common solution problem with monotone operator equations in real Hilbert spaces. This new approach does not employ resolvent operators nor does it use the norms of the bounded linear operators (transfer mappings) from the source space to the image spaces.

Periodic solutions to integro-differential equations: variational formulation, symmetry, and regularity

. We consider nonconstant periodic constrained minimizers of semilinear elliptic equations for integro-differential operators in ℝ. We prove that, after an appropriate translation, each of them is necessarily an even function which is decreasing in half its period. In particular, it has only two critical points in half its period, the absolute maximum and minimum. If these statements hold for all nonconstant periodic solutions, and not only for constrained minimizers, remains as an open problem. Our results apply to operators with kernels in two different classes: kernels K which are convex and kernels for which K(τ 1∕2) is a completely monotonic function of τ. This last new class arose in our previous work on nonlocal Delaunay surfaces in ℝ n . Due to their symmetry of revolution, it gave rise to a 1d problem involving an operator with a nonconvex kernel. Our proofs are based on a not so well-known Riesz rearrangement inequality on the circle 𝕊 1 established in 1976. We also put in evidence a new regularity fact which is a truly nonlocal-semilinear effect and also occurs in the nonperiodic setting. Namely, for nonlinearities in C β and when 2 s + β < 1 (2s being the order of the operator), the solution is not always C 2 s + β − ϵ for all ϵ > 0.

A New Facet of a Decreasing Rearrangement Approach to Solving a Class of Urysohn Integral Equations

This article deals in an L 2 -setting with ill-posed nonlinear operator equations [ F ( x , κ ) ] ( s ) : = ∫ 0 1 x ( t ) κ − s dt = y ( s ) ( 0 ≤ s ≤ 1 ) . Here, a family of parameter-dependent specific integral equations of Urysohn type is considered, equipped with some exponent κ > 0 as parameter. As inverse problem, the decreasing rearrangement x ∗ of the uniformly bounded positive function x and the parameter κ are to be determined simultaneously from observations of the right-hand side y. Under some additional conditions, it can be shown that the pair ( x ∗ , κ ) of solutions is uniquely determined whenever y belongs to the range of the forward operator F. Exploiting the analytical and numerical results from [1] for the special exponent κ = 2 , one is able to get some solutions for general exponents in steps. Alternatively, by use of a classical regularization approach and a suitable discretization technique, computational results are presented in two example situations.

A general simulation with trajectories in Heisenberg picture for quantum non-Markovian dynamics

Abstract We present a general numerical simulation method to solve non-Markovian dynamics of an open quantum system influenced by quantum Brownian motion. Based on the determined memory kernel function, this method enabling the resolution of non-Markovian dynamics for a wide range of system Hamiltonians and spectral densities. The system dynamics are described by exact integro-differential operator equations without any common approximations and they are simulated in this work by definite-number equations with stochastic initial conditions. This approach ensures the applicability of mature numerical methods and maintains computational complexity that remains largely invariant, even when dealing with more complex models. The high accuracy of our simulation is evident from a comparison with the results obtained from corresponding exact master equations, underscoring the reliability and precision of our method.

Divergent contribution of environmental factors to soil organic and inorganic carbon in different land use types in a forest-grassland ecotone of Inner Mongolia, China.

Understanding the comprehensive impacts of environmental factors on soil organic carbon (SOC) and soil inorganic carbon (SIC) in different land use types is of great significance for sustainable soil management. Least absolute shrinkage and selection operator (LASSO) and structural equation modelling were applied to reveal the driving mechanism of SOC, SIC and the ratio between SOC to SIC (SOC/SIC) in three major land use types (forest, grassland and farmland) in a forest-grassland ecotone (FGE) of Inner Mongolia, Northeast China. Mean annual temperature (MAT), mean annual temperature (MAP) and Enhanced Vegetation Index (EVI) were selected by LASSO as the three most important environmental factors affecting SOC, SIC and SOC/SIC in all land use types. MAT had the strongest direct effects on SOC in forest and grassland among all the environmental factors, suggesting temperature was the most important control factor of SOC in forest and grassland. EVI had relatively strong indirect effects on SOC by influencing total nitrogen and total phosphorus in grassland and farmland, respectively. MAT and MAP had significantly direct effects on SIC in all land use types, demonstrating the balance between precipitation and evapotranspiration affected by temperature were the major control factors of SIC. In addition, MAT had significantly positive direct effects (>42%) on SOC/SIC in all land use types, suggesting climate warming can have positive feedbacks on the proportion of SOC in the soil carbon pool in FGE. This study provides a comprehensive understanding of the driving mechanism of SOC, SIC and SOC/SIC in FGE of Northeast China.

Estimates of solutions for ambiguously solvable linear matrix equations

The article examines the problem of estimating solutions of operator equations under conditions of uncertainty. We obtain the expressions for guaranteed errors of solutions of indefinite linear equations in the spaces of rectangular matrices in the presence of additional data with deterministic errors belonging to special sets. In a particular case, explicit formulas are obtained for guaranteed linear vector estimates and guaranteed vector estimation errors and for guaranteed posterior estimates and measurement errors. These estimation results are illustrated by a test example in the case of operators that act in the space of 2×2 matrices with a non-zero kernel.

Preservation of global solvability and estimation of solutions of some controlled nonlinear partial differential equations of the second order

Let $U$ be the set of admissible controls, $T&gt;0$, and let $W[0;\tau]$, $\tau\in(0;T]$, be a scale of Banach spaces such that the set of restrictions of functions from $W=W[0;T]$ to $[0;\tau]$ coincides with $W[0;\tau]$; let $F[.;u]\colon W\to W$ be a controlled Volterra operator, $u\in U$. Earlier, for the operator equation $x=F[x;u]$, $x\in W$, the author introduced a comparison system in the form of a functional integral equation in the space $\mathbf{C}[0;T]$. It was established that to preserve (under small perturbations of the right-hand side) the global solvability of the operator equation, it is sufficient to preserve the global solvability of the specified comparison system, and the corresponding sufficient conditions were established. In this paper, further examples of application of this theory are considered: nonlinear wave equation, strongly nonlinear wave equation, nonlinear heat equation, strongly nonlinear parabolic equation.

A new preconditioned Richardson iterative method

In this paper, we propose a new iterative technique for solving an operator equation $Ax=y$ based on the Richardson iterative method.Then, by using the Chebyshev polynomials, we modify the proposed method to accelerate the convergence rate.Also, we present the results of some numerical experiments that demonstrate the efficiency and effectivenessof the proposed methods compared to the existing, state-of-the-art methods.

The Existence and Uniqueness Results for a Nonlocal Bounbary Value Problem of Caputo-type Hadamard Hybrid Fractional Integro-differential Equations

This article is dedicated to study the existence and uniqueness of solutions for a non local bounbary value problem of Caputo-type Hadamard hybrid fractional integro-differential equations in Banach space, the recent researches considered the study of differential equations of Caputo-type Hadamard hybrid fractional integro-differential equations with classical order and the study of existence and uniqueness of solutions using approched numerical methodes, the objective of this paper is the study of the existence and uniqueness of fractional order of integro-differential equations involving the Caputo-type Hadamard derivative using fixed point theory. This work have two important results, the first result was the discussion of a new results owing to the fixed point theorem. Before the prove of results the problem was trandformed to Hadamard type problem. The first result based on Dhage fixed point theorem, after transforming our nonlocal boundary value problem into integral equation we defined operator equation, then we applied the fixed point theorem to get the existence resutl. The second result was the existence and uniqueness of solution for our nonlocal boundary value problem, we get this result using the Banach fixed point theorem. We illustrate our results by example to ending our theorical study.

Generalized Riccati Operator Equations, Invariant Subspaces and Toeplitz Corona Problem

We consider generalized Riccati operator equations and study their solutions in terms of Berezin transforms. We also investigate invariant subspaces of some Toeplitz operators on the Bergman space L a 2 over the unit disc D of the complex plane C . Some other related questions are also discussed.

Extended Kurchatov-type methods for solving nonlinear equations

A plethora of applications from diverse disciplines can be solved if reduced to nonlinear equations in suitable abstract spaces. Such equations are solved mostly iteratively. That is why, three-step iterative methods of the Kurchatov-type for solving nonlinear operator equations are investigated using approximation by the Frechet derivative of an operator of a nonlinear equation by divided di erences. We study the local and the semi-local convergence using conditions only on the operators on the methods. The conditions and speed of convergence of these methods are determined. Moreover, the domain of uniqueness is found for the solution. The results of numerical experiments validate the theoretical results. The new idea can be used on other iterative methods utilizing inverses of divided di erences of order one.

SOLVABILITY OF NONLINEAR BOUNDARY VALUE PROBLEMS FOR DIFFERENTIAL EQUILIBRIUM EQUATIONS OF NON-FLAT TIMOSHENKO TYPE SHELLS OF NON-ZERO GAUSSIAN CURVATURE IN ISOMETRIC COORDINATES

We study the solvability of a boundary value problem for a system of five nonlinear second-order partial differential equations under given nonlinear boundary conditions, which describes the equilibrium state of elastic non-flat inhomogeneous isotropic shells with loose edges in the framework of the Timoshenko shear model, referred to isometric coordinates.The boundary value problem is reduced to a nonlinear operator equation with respect to generalized displacements in Sobolev space, the solvability of which is established using the principle of compressed mappings.

Squashed 7-spheres, octonions and the swampland

Abstract We give a brief account of how to derive the entire eigenvalue spectrum of the operators on the squashed S 7 that appear in the compactification of eleven-dimensional supergravity. These spectra determine the mass spectrum of the fields in AdS 4 and are important for the corresponding 𝒩 = 1 supermultiplet structure. By an orientation-flip on the squashed S 7 we can also determine the spectrum of the corresponding non-supersymmetric theory, and, e.g., its spectrum of marginal operators on the boundary of AdS 4 which has some relevance for the AdS stability conjecture in the swampland program. Here we review recent work in [1, 2, 3] which is a continuation of the work in [4] where the complete spectrum of irreducible isometry representations of the fields in AdS 4 was derived for this compactification. Details are here given primarily for 2-forms while comments are also made on the key role of G 2 and octonions for the structure of the operator equations and mode functions on the squashed S 7. Key features of these improved methods were obtained in Joel Karlsson’s 2021 MSc thesis [5]. This is a write-up of a talk given at ISQS28, Prague, Czech Republic, July 4, 2024.

Proper splittings of Hilbert space operators

Proper splittings of Hilbert space operators