Lyapunov type inequalities for $n$th order forced differential equations with mixed nonlinearities

Ravi P.  Agarwal; Abdullah Özbekler

doi:10.3934/cpaa.2016037

Article Contents

2016, Volume 15, Issue 6: 2281-2300. Doi: 10.3934/cpaa.2016037

This issue Previous Article Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in four and more spatial dimensions Next Article Finite dimensional smooth attractor for the Berger plate with dissipation acting on a portion of the boundary

Lyapunov type inequalities for $n$th order forced differential equations with mixed nonlinearities

Ravi P. Agarwal^1, and
Abdullah Özbekler^2,

1.
Department of Mathematics, Texas A\&M University-Kingsville, 700 University Blvd., Kingsville, TX 78363-8202
2.
Department of Mathematics, Atilim University 06836, Incek, Ankara

Received: January 31, 2016

Revised: May 31, 2016

Published: October 2016

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Abstract

In the case of oscillatory potentials, we present Lyapunov type inequalities for $n$th order forced differential equations of the form \begin{eqnarray} x^{(n)}(t)+\sum_{j=1}^{m}q_j(t)|x(t)|^{\alpha_j-1}x(t)=f(t) \end{eqnarray} satisfying the boundary conditions \begin{eqnarray} x(a_i)=x'(a_i)=x''(a_i)=\cdots=x^{(k_i)}(a_i)=0;\qquad i=1,2,\ldots,r, \end{eqnarray} where $a_1 < a_2 < \cdots < a_r$, $0\leq k_i$ and \begin{eqnarray} \sum_{j=1}^{r}k_j+r=n;\qquad r\geq 2. \end{eqnarray} No sign restriction is imposed on the forcing term and the nonlinearities satisfy \begin{eqnarray} 0 < \alpha_1 < \cdots < \alpha_j < 1 < \alpha_{j+1} < \cdots < \alpha_m < 2. \end{eqnarray} The obtained inequalities generalize and compliment the existing results in the literature.

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Mathematics Subject Classification: Primary: 34C10.

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