Lyapunov type inequalities and their applications for quasilinear impulsive systems

Abstract

A novel Lyapunov-type inequality for Dirichlet problem associated with the quasilinear impulsive system involving the (p j , q j )-Laplacian operator for j = 1,2 is obtained. Then utility of this new inequality is exemplified in finding disconjugacy criterion, obtaining lower bounds for associated eigenvalue problems and investigating boundedness and asymptotic behaviour of oscillatory solutions. The effectiveness of the obtained disconjugacy criterion is illustrated via an example. Our results not only improve the recent related results but also generalize them to the impulsive case.

A Lyapunov-Type Inequality for Partial Differential Equation Involving the Mixed Caputo Derivative

“…holds, where q is a real-valued continuous function. The Lyapunov inequality (3) was regarded as a very important and useful tool in the study of differential equations, especially in the aspect of stability theory, oscillation theory, intervals of disconjugacy, and eigenvalue problems [2][3][4][5]. Subsequently, there were many improvements and extensions of the inequality (3) related to integer-order derivative, Definition 1 ([14]).…”

Asymptotical stability analysis of conformable fractional systems

“…This derivative seems to be more natural, and it coincides with the classical definition of the first derivative. In 2015, Thabet Abdeljawad proceeded on to develop the definition, some basic concepts about conformable fractional derivative such as chain rule, Grönwall's Inequality, exponential functions and Lyapunov inequality were studied in [19][20][21][22]. In addition, the Laplace transform was introduced to solve the linear differential systems [23].…”