Lyapunov-type Inequalities Research Articles

Lyapunov Type Inequalities for Fractional Impulsive Hamiltonian Systems

Lyapunov Type Inequalities for Fractional Impulsive Hamiltonian Systems

Matrix measure approach to Lyapunov-type inequalities for linear Hamiltonian systems with impulse effect

We present new Lyapunov-type inequalities for Hamiltonian systems, consisting of 2n-first-order linear impulsive differential equations, by making use of matrix measure approach. The matrix measure estimates of fundamental matrices of linear impulsive systems are crucial in obtaining sharp inequalities. To illustrate usefulness of the inequalities we have derived new disconjugacy criteria for Hamiltonian systems under impulse effect and obtained new lower bound estimates for eigenvalues of impulsive eigenvalue problems.

Lyapunov type inequalities for second-order forced dynamic equations with mixed nonlinearities on time scales

In this paper, we present some new Hartman and Lyapunov inequalities for second-order forced dynamic equations on time scales \({\mathbb T}\) with mixed nonlinearities: $$\begin{aligned} x^{\Delta \Delta }(t)+\sum _{k=1}^nq_k(t)|x^{\sigma }(t)|^{\alpha _k-1}x^{\sigma }(t)=f(t);\quad t\in [t_0,\infty )_{\mathbb T}, \end{aligned}$$ where the nonlinearities satisfy $$\begin{aligned} 0<\alpha _1<\cdots <\alpha _m<1<\alpha _{m+1}<\cdots <\alpha _n<2. \end{aligned}$$ No sign restrictions are imposed on the potentials \(q_k\), \(k=1,2,\ldots ,n\), and the forcing term f. The inequalities obtained generalize and compliment the existing results for the special cases of this equation in the literature.

Extremal norms of the potentials recovered from inverse Dirichlet problems

Consider the Sturm–Liouville eigenvalue problem , where , and its spectrum is denoted by . For a real number λ, define and . We will set up a formula for explicitly in terms of λ and specify where the infimum can be attained. As an application, we will give the extremal values of the nth eigenvalue of the Dirichlet problem for potentials on a sphere , . The proofs are based on a new Lyapunov-type inequality for Sturm–Liouville equations with potentials.

Lyapunov-type inequalities for partial differential equations

In this work we present a Lyapunov inequality for linear and quasilinear elliptic differential operators in N -dimensional domains Ω. We also consider singular and degenerate elliptic problems with A p coefficients involving the p -Laplace operator with zero Dirichlet boundary condition. As an application of the inequalities obtained, we derive lower bounds for the first eigenvalue of the p -Laplacian, and compare them with the usual ones in the literature.

Lyapunov type inequalities for second-order differential equations with mixed nonlinearities

AbstractIn this paper, we present some new Lyapunov and Hartman type inequalities for second-order equations with mixed nonlinearities:where

Fractional boundary value problems and Lyapunov-type inequalities with fractional integral boundary conditions

Fractional boundary value problems and Lyapunov-type inequalities with fractional integral boundary conditions

Lyapunov-type inequality for a Riemann-Liouville fractional differential boundary value problem

The aim of this paper is to present a Lyapunov-type inequality for a Riemann-Liouville fractional differential equation of order $2<\alpha\leq3$ subject to mixed boundary conditions.

Lyapunov-type inequalities for third-order linear differential equations

Lyapunov-type inequalities for third-order linear differential equations

Lyapunov-type inequalities for nonlinear impulsive systems with applications

Lyapunov-type inequalities for nonlinear impulsive systems with applications

Lyapunov-type inequalities for higher order half-linear differential equations

The Lyapunov inequality for second order linear differential equations has been extended to higher order linear differential equations in various forms. It has also been extended to second and third order half-linear equations. However, it has not yet been developed for higher order half-linear differential equations due to the nonlinear feature of the equation and the complexity caused by the multiple p-Laplacian operators in the equation. In this paper, by subtle applications of forward and backward inductions, we establish several Lyapunov-type inequalities for even and odd order half-linear differential equations. These inequalities are applied to determine the nonexistence of nontrivial solutions of higher order boundary value problems and to estimate eigenvalues for higher order eigenvalue problems. Our results cover many results in the literature when the equations become linear.

A Lyapunov type inequality for indefinite weights and eigenvalue homogenization

Fil: Salort, Ariel Martin. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Ciudad Universitaria. Instituto de Investigaciones Matematicas ; Argentina

Lyapunov-type inequalities for a class of fractional differential equations

In this paper, we establish new Lyapunov-type inequalities for a class of fractional boundary value problems. As an application, we obtain a lower bound for the eigenvalues of corresponding equations.

New Lyapunov-type inequalities for a class of even-order linear differential equations

In this paper, we obtain some new Lyapunov-type inequalities for a class of even-order linear differential equations, the results are new and generalize and improve some early results in this field.

Matrix Lyapunov inequalities for ordinary and elliptic partial differential equations

This paper is devoted to the study of $L_p$ Lyapunov-type inequalities for linear systems of equations with Neumann boundary conditions and for any constant $p \geq 1$. We consider ordinary and elliptic problems. The results obtained in the linear case are combined with Schauder fixed point theorem to provide new results about the existence and uniqueness of solutions for resonant nonlinear problems. The proof uses in a fundamental way the nontrivial relation between the best Lyapunov constants and the minimum value of some especial minimization problems.

Lyapunov type inequalities for even order differential equations with mixed nonlinearities

In the case of oscillatory potentials, we present Lyapunov and Hartman type inequalities for even order differential equations with mixed nonlinearities: $x^{(2n)}(t)+(-1)^{n-1}\sum_{i=1}^{m}q_{i}(t)|x(t)|^{\alpha_{i}-1}x(t)=0$ , where $n,m\in{\mathbb{N}}$ and the nonlinearities satisfy $0<\alpha_{1}<\cdots<\alpha_{j}<1<\alpha_{j+1}<\cdots<\alpha_{m}<2$ .

Lyapunov-type inequality for a fractional differential equation with fractional boundary conditions

In this paper, a Lyapunov-type inequality is obtained for a fractional differential equation under fractional boundary conditions. We then use this inequality to obtain an interval where a certain Mittag-Leffler function has no real zeros.

On the Lyapunov-type inequalities of a three-point boundary value problem for third order linear differential equations

In this paper, by using Green’s function for second order differential equations with Dirichlet boundary condition, we establish new Lyapunov-type inequalities for third order linear differential equation which improve all existing results in the literature. As an application, we obtain sharp lower bound for the eigenvalues of corresponding equations and for the distance between the end points in the three consecutive zeros of the solution of the equations.

Lyapunov Inequality For Dynamic Equation With Order n +1 On Time Scales

In this paper, we establish a Lyapunov-type inequality for the higher order dynamic equation on an arbitrary time scale 𝕋 where n is a positive integer, α ≥ 1 is the quotient of two odd positive integers and with ak ε Crd(𝕋, (0,∞)) (1 ≤ k≤ n) and p ε Crd(𝕋,ℝ).

Lyapunov-type inequalities for $(m+1)$th order half-linear differential equations with anti-periodic boundary conditions

In this work, we will establish several new Lyapunov-type inequalities for (m + 1)th order half-linear differential equations with anti-periodic boundary condi- tions, the results of this paper are new and generalize and improve some early results in the literature.