Ranges of nonlinear asymptotically linear operators

This paper deals with the solvability of the equation A(u) ? S(u) = f, where A is a continuous self-adjoint operator defined on a real Hilbert space H with values in H, the null-space of A is nontrivial, and N is a nonlinear completely continuous perturbation. Sufficient, and necessary-sufficient conditions are given for the equation to be solvable. Abstract theorems are applied to solving boundary value problems for partial differential equations.     

Eigenfunction methods and nonlinear hyperbolic boundary value problems at resonance

The abstract boundary value problem Lu + Gu = f, u ϵ dom(L) ⊂ H, is considered. Here H is used to denote a real separable Hilbert space, L a closed symmetric linear operator, and G a nonlinear operator assumed to be Lipschitz continuous and strongly monotone. In addition L is assumed to have a complete set of eigenfunctions in H, and is allowed to have an infinite dimensional null space. The existence of unique solutions, depending continuously on f, is established by a constructive approach. Galerkin approximations are considered and error estimates are given. As an application of the main result, the existence of time periodic weak solutions of the n-dimensional wave equation is shown.     

A stability result for abstract evolution problems

Consider an abstract evolution problem in a Hilbert space H (1) where A(t) is a linear, closed, densely defined operator in H with domain independent of t ≥ 0 and G(t,u) is a nonlinear operator such that ‖G(t,u)‖a(t) ‖u‖^p, p = const > 1, ‖f(t)‖ ≤ b(t). We allow the spectrum of A(t) to be in the right half‐plane Re(λ) < λ₀(t), λ₀(t) > 0, but assume that lim_{t → ∞}λ₀(t) = 0. Under suitable assumptions on a(t) and b(t), the boundedness of ‖u(t)‖ as t → ∞ is proved. If f(t) = 0, the Lyapunov stability of the zero solution to problem (1) with u₀ = 0 is established. For f ≠ 0, sufficient conditions for the Lyapunov stability are given. The novel point in our study of the stability of the solutions to abstract evolution equations is the possibility for the linear operator A(t) to have spectrum in the half‐plane Re(λ) < λ₀(t) with λ₀(t) > 0 and lim_{t → ∞}λ₀(t) = 0 at a suitable rate. The new technique, proposed in the paper, is based on an application of a novel nonlinear differential inequality. Copyright © 2012 John Wiley & Sons, Ltd.     

Existence and uniqueness results for a nonlinear differential system

We study the existence, uniqueness and asymptotic behavior of the strong and weak solutions of a nonlinear differential system with 2N equations in a real Hilbert space H, subject to a boundary condition and initial data. This problem is a discrete version with respect to spatial variable x of some partial differential problems (with H = ℝⁿ ), which have applications in integrated circuits modelling     

Algorithm and MATLAB package for some nonlinear 2D evolution equations

We will present LiScM2, a MATLAB package for numerical solutions of some partial differential evolution equations of the form u_t + Lu = N (u, ∇u), with boundary and initial conditions, where L is a 2D linear elliptic operator (Laplace operator for this version) and N is a nonlinear part. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear equations satisfying the nonresonance condition

In this article, the equation Au ± B(u) = f is studied in a separable Hilbert space H. Here A: H → H is a linear self-adjoint operator with domain dense in H and B: H → H is nonlinear. It is studied when this equation has a solution for any f ∈ H. The results obtained are applied to the problem of periodic solutions of the nonlinear wave equation with homogeneous boundary conditions of the third kind and to elliptic equations. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 226–248, 2005.     

Hardy Spaces of Harmonic Functions on Homogeneous Isotropic Trees

A variational method for operator equations of the form Pu + δβ(u) ? f has been given in Dinca [1]. Here P is a (generally) nonlinear operator in a Hilbert space, β: H → ? ∞ is a convex, proper (β ≠ + ∞) and lower-semicontinuous functional and δβ(u) stands for the subdifferential of β at the point u. The present paper has two parts. The first part contents the main results of Dinca [1] without proofs. The second part discusses particular cases and applications to mechanics among which “the climatisation problem for non-linear elliptic equations” and its applications.     

Least squares and bounded variation regularization with nondifferentiable functionals

Let A be an operator from a real Banach space into a real Hilbert space. In this paper we study least squares regularization methods for the ill-posed operator equation A(u) = f using nonlinear nondifferentiable penalty functionals. We introduce a notion of distributional approximation, and use constructs of distributional approximations to establish convergence and stability of approximations of bounded variation solutions of the operator equation. We also show that the results provide a framework for a rigorous analysis of numerical methods based on Euler-Lagrange equations to solve the minimization problem. This justifies many of the numerical implementation schemes of bounded variation minimization that have been recently proposed.     

Generalized equations,variational inequalities and a weak Kantorovich theorem

We prove a Kantorovich-type theorem on the existence and uniqueness of the solution of a generalized equation of the form f(u)+g(u) ' 0f(u)+g(u)\owns 0 where f is a Fréchet-differentiable function and g is a maximal monotone operator defined on a Hilbert space. The depth and scope of this theorem is such that when we specialize it to nonlinear operator equations, variational inequalities and nonlinear complementarity problems we obtain novel results for these problems as well. Our approach to the solution of a generalized equation is iterative, and the solution is obtained as the limit of the solutions of partially linearized generalized Newton subproblems of the type Az+g(z) ' bAz+g(z)\owns b where A is a linear operator.     

Spline approximation of a non-linear Riemann–Hilbert problem†

In this article we use linear spline approximation of a non-linear Riemann–Hilbert problem on the unit disk. The boundary condition for the holomorphic function is reformulated as a non-linear singular integral equation A(u) = 0, where A : H ¹(Γ) → H ¹(Γ) is defined via a Nemytski operator. We approximate A by A ⁿ : H ¹(Γ) → H ¹(Γ) using spline collocation and show that this defines a Fredholm quasi-ruled mapping. Following the results of (A.I. ?nirel'man, The degree of quasi-ruled mapping and a nonlinear Hilbert problem, Math. USSR-Sbornik 18 (1972), pp. 373–396; M.A. Efendiev, On a property of the conjugate integral and a nonlinear Hilbert problem, Soviet Math. Dokl. 35 (1987), pp. 535–539; M.A. Efendiev, W.L. Wendland, Nonlinear Riemann–Hilbert problems for multiply connected domains, Nonlinear Anal. 27 (1996), pp. 37–58; Nonlinear Riemann–Hilbert problems without transversality. Math. Nachr. 183 (1997), pp. 73–89; Nonlinear Riemann–Hilbert problems for doubly connected domains and closed boundary data, Topol. Methods Nonlinear Anal. 17 (2001), pp. 111–124; Nonlinear Riemann–Hilbert problems with Lipschitz, continuous boundary data without transversality, Nonlinear Anal. 47 (2001), pp. 457–466; Nonlinear Riemann–Hilbert problems with Lipschitz-continuous boundary data: Doubly connected domains, Proc. Roy. Soc. London Ser. A 459 (2003), pp. 945–955.), we define a degree of mapping and show the existence of the spline solutions of the fully discrete equations A ⁿ (u) = 0, for n large enough. We conclude this article by discussing the solvability of the non-linear collocation method, where we shall need an additional uniform strong ellipticity condition for employing the spline approximation.     

On completely irreducible operators

Let T be a bounded linear operator on Hilbert space H, M an invariant subspace of T. If there exists another invariant subspace N of T such that H = M + N and M ∩ N = 0, then M is said to be a completely reduced subspace of T. If T has a nontrivial completely reduced subspace, then T is said to be completely reducible; otherwise T is said to be completely irreducible. In the present paper we briefly sum up works on completely irreducible operators that have been done by the Functional Analysis Seminar of Jilin University in the past ten years and more. The paper contains four sections. In section 1 the background of completely irreducible operators is given in detail. Section 2 shows which operator in some well-known classes of operators, for example, weighted shifts, Toeplitz operators, etc., is completely irreducible. In section 3 it is proved that every bounded linear operator on the Hilbert space can be approximated by the finite direct sum of completely irreducible operators. It is clear that a completely irreducible operator is a rather suitable analogue of Jordan blocks in L(H), the set of all bounded linear operators on Hilbert space H. In section 4 several questions concerning completely irreducible operators are discussed and it is shown that some properties of completely irreducible operators are different from properties of unicellular operators. __________ Translated from Acta Sci. Nat. Univ. Jilin, 1992, (4): 20–29     

Faedo-galerkin approximations in equations of evolution

We consider the evolution equation u' = Au + Nu u(0) = ? u is a function on a real interval [0, T] with values in a Hilbert space H, A is a linear operator in H generating a continuous semigroup e^At and N is a nonlinear operator in H. We show that

• 1 Existence of the exact solution implies existence of the Faedo-Galerkin Approximations.
• 2 Existence of the Faedo-Galerkin Approximations implies existence of the exact solution.
• 3 Uniform convergence of the Faedo-Galerkin Approximations to the exact solution.

The Paper consists of two parts. In the first five sections we require that A possesses a complete orthonormal system of eigenfunctions, in section 6 we drop this requirement.     

A New Continuation Method for the Study of Nonlinear Equations at Resonance

A new continuation theorem for the existence of solutions to an equation Lu = N(u), where N is a nonlinear continuous operator and L a linear Fredholm noninvertible one, is proved. The continuation which makes N collapse is replaced by a deformation of L to an invertible linear operator. This implies results concerning sublinear N, N having a linear growth at infinity and superlinear N. These generalize the classical theorems on the solvability of semilinear elliptic BVP′s at resonance. The periodic solutions of Liénard equations are studied.     

Ergodic theory and iterative solution of linear operator equations

In this paper we consider the problem of approximating solutions of linear operator equations of the type u-Tu=f. The main tools are Dotson's extension of the Eberlein ergodic theorem to affine mappings and the DeMoivre-Laplace theorem of probability theory. The main results are applied to obtain theorems on the iterative approximation of solutions of linear operator equations in Hilbert space and the approximation in L ^ρ norm of solutions of a certain functional equation in the space L ^∞     

Dynamical Systems Method of gradient type for solving nonlinear equations with monotone operators

A version of the Dynamical Systems Method (DSM) of gradient type for solving equation F(u)=f where F:H→H is a monotone Fréchet differentiable operator in a Hilbert space H is studied in this paper. A discrepancy principle is proposed and the convergence to the minimal-norm solution is justified. Based on the DSM an iterative scheme is formulated and the convergence of this scheme to the minimal-norm solution is proved.     

Least-squares solutions of multi-valued linear operator equations in Hilbert spaces

Let M be a linear manifold in H₁ H₂, where H₁, and H₂ are Hilbert spaces. Two notions of least-squares solutions for the multi-valued linear operator equation (inclusion) y ε M(x) are introduced and investigated. The main results include (i) equivalent conditions for least-squares solvability, (ii) properties of a least-squares solution, (iii) characterizations of the set of all least-squares solutions in terms of algebraic operator parts and generalized inverses of linear manifolds, and (iv) best approximation properties of generalized inverses and operator parts of multi-valued linear operators. The principal tools in this investigation are an abstract adjoint theory, orthogonal operator parts, and orthogonal generalized inverses of linear manifolds in Hilbert spaces.     

Numerical shadows: Measures and densities on the numerical range

For any operator M acting on an N-dimensional Hilbert space H_N we introduce its numerical shadow, which is a probability measure on the complex plane supported by the numerical range of M. The shadow of M at point z is defined as the probability that the inner product (Mu, u) is equal to z, where u stands for a random complex vector from H_N, satisfying ||u||=1. In the case of N=2 the numerical shadow of a non-normal operator can be interpreted as a shadow of a hollow sphere projected on a plane. A similar interpretation is provided also for higher dimensions. For a hermitian M its numerical shadow forms a probability distribution on the real axis which is shown to be a one dimensional B-spline. In the case of a normal M the numerical shadow corresponds to a shadow of a transparent solid simplex in R^N-1 onto the complex plane. Numerical shadow is found explicitly for Jordan matrices J_N, direct sums of matrices and in all cases where the shadow is rotation invariant. Results concerning the moments of shadow measures play an important role. A general technique to study numerical shadow via the Cartesian decomposition is described, and a link of the numerical shadow of an operator to its higher-rank numerical range is emphasized.     

Existence of bounded solutions for nonlinear elliptic equations in unbounded domains

In this paper we study the existence of bounded weak solutions for some nonlinear Dirichlet problems in unbounded domains. The principal part of the operator behaves like the p-laplacian operator, and the lower order terms, which depend on the solution u and its gradient u, have a power growth of order p–1 with respect to these variables, while they are bounded in the x variable. The source term belongs to a Lebesgue space with a prescribed asymptotic behaviour at infinity.     

Approximation of solutions to evolution equations

The convergence of the Galerkin approximations to solutions of abstract evolution equations of the form u′(t)= ? Au(t) + M(u(t)) is shown. Here A is a closed, positive definite, self-adjoint linear operator with domain D(A) dense in a Hilbert space H and M is a non-linear map defined on D(A^½) which satisfies a Lipschitz condition on balls in D(A^½).     

Critical exponents for large time asymptotics for fractal Hamilton-Jacobi-KPZ equations

Nonlinear and nonlocal evolution equations of the form u_t = ℒ︁u ± |∇u |^q, where ℒ︁ is a pseudodifferential operator representing the infinitesimal generator of a Lévy stochastic process, have been derived (see, [6]) as models for growing interfaces in the case when the continuous Brownian diffusion surface transport is augmented by a random hopping mechanism. The goal of this note is to report properties of solutions to this equation resulting from the interplay between the strengths of the “diffusive” linear and “hyperbolic” nonlinear terms, posed in the whole space R ^N , and supplemented with nonnegative, bounded, and sufficiently regular initial conditions. The full text of the paper, including complete proofs and other results, will appear in the Transactions of the American Mathematical Society. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)