Basis problems for matrix valued functions

Basis problems for self-adjoint matrix valued functions are studied. We suggest a new and nonstandard method to solve basis problems both in finite and infinite dimensional spaces. Although many results in this paper are given for operator functions in infinite dimensional Hilbert spaces, but to demonstrate practicability of this method and to present a full solution of basis problems, in this paper we often restrict ourselves to matrix valued functions which generate Rayleigh systems on the n-dimensional complex space Cn. The suggested method is an improvement of an approach given recently in our paper [M. Hasanov, A class of nonlinear equations in Hilbert space and its applications to completeness problems, J. Math. Anal. Appl. 328 (2007) 1487-1494], which is based on the extension of the resolvent of a self-adjoint operator function to isolated eigenvalues and the properties of quadratic forms of the extended resolvent. This approach is especially useful for nonanalytic and nonsmooth operator functions when a suitable factorization formula fails to exist.     

A class of nonlinear complementarity problems for multifunctions

Given a continuous mapF:R ⁿ R ⁿ and a lower semicontinuous positively homogeneous convex functionh:R ⁿ R, the nonlinear complementarity problem considered here is to findxR ₊ ⁿ andyh(x), the subdifferential ofh atx, such thatF(x)+y0 andx ^T (F(x)+y)=0. Some existence theorems for the above problem are given under certain conditions on the mapF. An application to quasidifferentiable convex programming is also shown.The authors are grateful to Professor O. L. Mangasarian and the referee for their substantive suggestions.     

Conditions for existence of a global strong solution to one class of nonlinear evolution equations in Hilbert space

We obtain a criterion of global strong solvability for one class of nonlinear evolution equations in Hilbert space.     

On the Drazin inverse of the sum of two operators and its application to operator matrices

Given two bounded linear operators F,G on a Banach space X such that G²F=GF²=0, we derive an explicit expression for the Drazin inverse of F+G. For this purpose, firstly, we obtain a formula for the resolvent of an auxiliary operator matrix in the form . From the provided representation of D(F+G) several special cases are considered. In particular, we recover the case GF=0 studied by Hartwig et al. [R.E. Hartwig, G. Wang, Y. Wei, Some additive results on Drazin inverse, Linear Algebra Appl. 322 (2001) 207-217] for matrices and by Djordjevi? and Wei [D.S. Djordjevi?, Y. Wei, Additive results for the generalized Drazin inverse, J. Aust. Math. Soc. 73 (1) (2002) 115-126] for operators. Finally, we apply our results to obtain representations for the Drazin inverse of operator matrices in the form which are extensions of some cases given in the literature.     

On the Riesz basis of a family of analytic operators in the sense of Kato and application to the problem of radiation of a vibrating structure in a light fluid

In this paper, we prove that the system formed by some of generalized eigenvectors of the operator T₀+εT₁+ε²T₂+? which are analytic on ε, forms a Riesz basis of the separable Hilbert space H, where ε∈C and T₀,T₁,T₂,… some linear transformations on H which have the same domain D⊆H. After that, we give an application for a problem concerning the radiation of a vibrating structure in a light fluid.     

On a class of Riccati equations in a Hilbert space

We establish existence and uniqueness of solutions of a class of Riccati equations in Hilbert space ocurring in filtering problems for distributed parameter systems using point sensors.     

On bounds for solutions to nonlinear wave equations in Hilbert space with applications to nonlinear elastodynamics

For the nonlinear wave equationu _tt -Nu +G(t,u, u _t ) = ? in Hilbert space, with associated homogeneous initial data, we show how ana priori bound of the form ∫ ₀ ^T ∥G(τ,u, u _τ)∥² dτ ≤ κ ∫ ₀ ^T ∥?(τ)∥² dτ leads to upper and lower bounds for ∥u∥ in terms of ∥?∥. An application to nonlinear elastodynamics is presented.     

A class of function spaces and its application in solving second-order nonlinear differential equations

In this paper, we prove an existence theorem for time global monotone positive solutions of nonlinear second-order ordinary differential equations by applying the Schauder-Tikhonov fixed point theorem. This result generalizes the result of existence on a half-line given in Yin (2003) [8].     

Conditions for existence of a global strong solution to one class of nonlinear evolution equations in Hilbert space. II

We obtain a criterion of global strong solvability for one class of nonlinear evolution equations in Hilbert space.     

Conditions for the existence of a global strong solution to a class of nonlinear evolution equations in a Hilbert space

We study a nonlinear operator differential equation in a Hilbert space. This equation represents an abstract model for the system of Navier-Stokes equations. The main result consists in proving the existence of a strong solution to this equation under the condition that a certain other system of equations (related to the original equation) has only the zero solution.     

Approximation of solutions of nonlinear equations of Hammerstein type in Hilbert space

Let be a real Hilbert space. Let , be bounded monotone mappings with , where and are closed convex subsets of satisfying certain conditions. Suppose the equation has a solution in . Then explicit iterative methods are constructed that converge strongly to such a solution. No invertibility assumption is imposed on , and the operators and need not be defined on compact subsets of .

     

EXP-function method and its application to nonlinear equations

Exp-function method is used to find a unified solution of a nonlinear wave equation. Variant Boussinesq equations are selected to illustrate the effectiveness and simplicity of the method. A generalized solitary solution with free parameters is obtained.     

Stable sequential convex programming in a Hilbert space and its application for solving unstable problems

A parametric convex programming problem with an operator equality constraint and a finite set of functional inequality constraints is considered in a Hilbert space. The instability of this problem and, as a consequence, the instability of the classical Lagrange principle for it is closely related to its regularity and the subdifferentiability properties of the value function in the optimization problem. A sequential Lagrange principle in nondifferential form is proved for the indicated convex programming problem. The principle is stable with respect to errors in the initial data and covers the normal, regular, and abnormal cases of the problem and the case where the classical Lagrange principle does not hold. It is shown that the classical Lagrange principle in this problem can be naturally treated as a limiting variant of its stable sequential counterpart. The possibility of using the stable sequential Lagrange principle for directly solving unstable optimal control problems and inverse problems is discussed. For two illustrative problems of these kinds, the corresponding stable Lagrange principles are formulated in sequential form.