10 research outputs found
New gaps between zeros of fourth-order differential equations via Opial inequalities
In this paper, for a fourth-order differential equation, we will establish some lower bounds for the distance between zeros of a nontrivial solution and also lower bounds for the distance between zeros of a solution and/or its derivatives. We also give some new results related to some boundary value problems in the theory of bending of beams. The main results will be proved by making use of some generalizations of Opial and Wirtinger-type inequalities. Some examples are considered to illustrate the main results
On Lyapunov-type inequality for a class of quasilinear systems
In this paper, we establish a new Lyapunov-type inequality for quasilinear systems. Our result in special case reduces to the result of Watanabe et al. [J. Inequal. Appl. 242(2012), 1-8]. As an application, we also obtain lower bounds for the eigenvalues of corresponding systems
Lyapunov-type inequality for a class of Dirichlet quasilinear systems involving the (p1,p2,…,pn)-Laplacian
AbstractWe state and prove a generalized Lyapunov-type inequality for one-dimensional Dirichlet quasilinear systems involving the (p1,p2,…,pn)-Laplacian. Our result generalize the Lyapunov-type inequality given in Napoli and Pinasco (2006) [12]
An applied mathematical excursion through Lyapunov inequalities, classical analysis and differential equations
Several different problems make the study of the so called Lyapunov type
inequalities of great interest, both in pure and applied mathematics. Although
the original historical motivation was the study of the stability properties of
the Hill equation (which applies to many problems in physics and engineering),
other questions that arise in systems at resonance, crystallography,
isoperimetric problems, Rayleigh type quotients, etc. lead to the study of
Lyapunov inequalities () for differential equations.
In this work we review some recent results on these kinds of questions which
can be formulated as optimal control problems. In the case of Ordinary
Differential Equations, we consider periodic and antiperiodic boundary
conditions at higher eigenvalues and by using a more accurate version of the
Sturm separation theory, an explicit optimal result is obtained. Then, we
establish Lyapunov inequalities for systems of equations. To this respect, a
key point is the characterization of the best Lyapunov constant for the
scalar given problem, as a minimum of some especial (constrained or
unconstrained) variational problems defined in appropriate subsets of the usual
Sobolev spaces. For Partial Differential Equations on a domain , it is proved that the relation between the quantities and
plays a crucial role in order to obtain nontrivial Lyapunov type
inequalities (which are called Sobolev inequalities by many authors). This fact
shows a deep difference with respect to the ordinary case. Combining the linear
results with Schauder fixed point theorem, we can obtain some new results about
the existence and uniqueness of solutions for resonant nonlinear problems for
ODE or PDE, both in the scalar case and in the case of systems of equationsComment: 36 page
Lyapunov-type inequality and eigenvalue estimates for fractional problems
In this work, we establish the Lyapunov-type inequalities for the fractional boundary value problems with Hilfer derivative for different boundary conditions. We apply this inequality to fractional eigenvalue problems and prove one of the important results of real zeros of certain Mittag-Leffler functions and improve the bound of the eigenvalue using the Cauchy-Schwarz inequality and Semi-maximum norm. We extend it for higher order cases