We consider a linear nonautonomous higher order ordinary differential equation and establish the positivity conditions and two-sided bounds for Green’s function for the two-point boundary value problem. Applications of the obtained results to nonlinear equations are also discussed.
We consider a second order regular differential operator whose coefficients are nonselfadjoint bounded operators acting in a Hilbert space. An estimate for the resolvent and a bound for the spectrum are established. An operator is said to be stable if its spectrum lies in the right half-plane. By the obtained bounds, stability and instability conditions are established.
We consider a second order regular differential operator whose coefficients are nonselfadjoint bounded operators acting in a Hilbert space. An estimate for the resolvent and a bound for the spectrum are established. An operator is said to be stable if its spectrum lies in the right half-plane. By the obtained bounds, stability and instability conditions are established.
Let be a bounded linear operator in a complex separable Hilbert space , and be a selfadjoint operator in . Assuming that belongs to the Schatten-von Neumann ideal we derive a bound for , where are the eigenvalues of . Our results are formulated in terms of the “extended” eigenvalue sets in the sense introduced by T. Kato. In addition, in the case we refine the Weyl inequality between the real parts of the eigenvalues of and the eigenvalues of...
We consider the equation , where is the generator of an analytic semigroup on a Banach space , is a variable bounded operator in . It is assumed that the commutator has the following property: there is a linear operator having a bounded left-inverse operator such that is integrable and the operator is bounded. Under these conditions an exponential stability test is derived. As an example we consider a coupled system of parabolic equations.
The “freezing” method for ordinary differential equations is extended to multivariable retarded systems with distributed delays and slowly varying coefficients. Explicit stability conditions are derived. The main tool of the paper is a combined usage of the generalized Bohl-Perron principle and norm estimates for the fundamental solutions of the considered equations.
Let A, B and C be matrices. We consider the matrix equations Y-AYB=C and AX-XB=C. Sharp norm estimates for solutions of these equations are derived. By these estimates a bound for the distance between invariant subspaces of matrices is obtained.
We consider nonlinear non-autonomous multivariable systems governed by differential equations with differentiable linear parts. Explicit conditions for the exponential stability are established. These conditions are formulated in terms of the norms of the derivatives and eigenvalues of the variable matrices, and certain scalar functions characterizing the nonlinearity. Moreover, an estimate for the solutions is derived. It gives us a bound for the region of attraction of the steady state....
We consider the system (ẋ(t) ≡ dx(t)/dt), where x(t) is the state, u(t) is the input, R(τ),R̃(τ) are matrix-valued functions, and F is a causal (Volterra) mapping. Such equations enable us to consider various classes of systems from the unified point of view. Explicit input-to-state stability conditions in terms of the L²-norm are derived. Our main tool is the norm estimates for the matrix resolvents, as well as estimates for fundamental solutions of the linear parts of the considered systems,...
The “freezing” method for ordinary differential equations is extended to multivariable retarded systems with distributed delays and slowly varying coefficients. Explicit stability conditions are derived. The main tool of the paper is a combined usage of the generalized Bohl-Perron principle and norm estimates for the fundamental solutions of the considered equations.
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