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November  2016, 15(6): 2281-2300. doi: 10.3934/cpaa.2016037

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

1. 

Department of Mathematics, Texas A\&M University-Kingsville, 700 University Blvd., Kingsville, TX 78363-8202 , United States

2. 

Department of Mathematics, Atilim University 06836, Incek, Ankara

Received  February 2016 Revised  June 2016 Published  September 2016

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.
Citation: Ravi P. Agarwal, Abdullah Özbekler. Lyapunov type inequalities for $n$th order forced differential equations with mixed nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2281-2300. doi: 10.3934/cpaa.2016037
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show all references

References:
[1]

Nonlinear Anal., 7 (1983), 259-270. doi: 10.1016/0362-546X(83)90070-6.  Google Scholar

[2]

Internat. Schriftenreihe Numer. Math., 64, Birkhäuser, Basel, 1983.  Google Scholar

[3]

Singapore: World Scientific, 1986. doi: 10.1142/0266.  Google Scholar

[4]

Computers Math. Applic., 17 (1989), 1397-1421. doi: 10.1016/0898-1221(89)90023-0.  Google Scholar

[5]

Dordrecht, Boston, London: Kluwer Academic Publishers, 1993. doi: 10.1007/978-94-011-2026-5.  Google Scholar

[6]

Computers Math. Applic., 46 (2003), 1799-1826. doi: 10.1016/S0898-1221(03)90238-0.  Google Scholar

[7]

Bound. Value Probl., 2012 (2012), 1-23. doi: 10.1186/1687-2770-2012-49.  Google Scholar

[8]

Dynam. Systems Appl., 24 (2015), 211-220.  Google Scholar

[9]

J. Inequal. Appl., 2015 (2015), 142, 10 pp. doi: 10.1186/s13660-015-0633-4.  Google Scholar

[10]

Appl. Math. Comput., 265 (2015), 456-468. doi: 10.1016/j.amc.2015.05.038.  Google Scholar

[11]

Pasific J. Math., 12 (1962), 801-812.  Google Scholar

[12]

Acta Math., 77 (1945), 127-136.  Google Scholar

[13]

Amer. J. Math., 71 (1949), 67-70.  Google Scholar

[14]

Proc. Amer. Math. Soc., 125 (1997), 1123-1129. doi: 10.1090/S0002-9939-97-03907-5.  Google Scholar

[15]

Appl. Math. Comput., 216 (2010), 368-373. doi: 10.1016/j.amc.2010.01.010.  Google Scholar

[16]

Hokkaido Math., 12 (1983), 105-112. doi: 10.14492/hokmj/1381757783.  Google Scholar

[17]

Fasc. Math., 23 (1991), 25-41.  Google Scholar

[18]

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[19]

Trans. Amer. Math. Soc., 182 (1973), 469-480.  Google Scholar

[20]

J. Math. Anal. Appl., 51 (1975), 670-677.  Google Scholar

[21]

Heidelberg: Elsevier Ltd, 2005.  Google Scholar

[22]

Colloq Math Soc János Bolyai, 30 (1979), 158-180. Google Scholar

[23]

J. London Math. Soc., 2 (1970), 461-466.  Google Scholar

[24]

SIAM J. Appl. Math., 27 (1974), 180-199.  Google Scholar

[25]

Canad. Math. Bull., 17 (1974), 499-504.  Google Scholar

[26]

Rocky Mountain J. Math., 6 (1976), 457-492.  Google Scholar

[27]

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[28]

J. Math. Anal. Appl., 35 (2007), 1195-1206. doi: 10.1016/j.jmaa.2007.01.095.  Google Scholar

[29]

New York, 1964 and Birkhäuser, Boston: Wiley, 1982.  Google Scholar

[30]

Commun. Pure. Appl. Anal., 11 (2012), 465-473. doi: 10.3934/cpaa.2012.11.465.  Google Scholar

[31]

Proc. Amer. Math. Soc., 14 (1963), 525-526.  Google Scholar

[32]

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[34]

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[35]

J. Math. Anal. Appl., 83 (1981), 486-494. doi: 10.1016/0022-247X(81)90137-2.  Google Scholar

[36]

Appl. Math. Lett., 17 (2004), 847-853. doi: 10.1016/j.aml.2004.06.016.  Google Scholar

[37]

Ann Fac Sci Univ Toulouse 2 (1907), 27-247, Reprinted as Ann Math Studies, No. 17, Princeton, 1947. Google Scholar

[38]

Dordrecht: 53 Kluwer Academic Publishers Group, 1991. doi: 10.1007/978-94-011-3562-7.  Google Scholar

[39]

J. Differential Equations, 227 (2006), 102-115. doi: 10.1016/j.jde.2006.01.004.  Google Scholar

[40]

J. Anal. Math., 7 (1959), 79-88.  Google Scholar

[41]

Stanford, CA: Stanford University Press, 1962.  Google Scholar

[42]

J. Anal. Math., 195 (1995), 527-536. doi: 10.1006/jmaa.1995.1372.  Google Scholar

[43]

Georgian Math. J., 4 (1997), 139-148. doi: 10.1023/A:1022930116838.  Google Scholar

[44]

Facta. Univ. Ser. Math. Inform., 16 (2001), 35-44.  Google Scholar

[45]

Electron J Differential Equations, 2009 (2009), 1-14.  Google Scholar

[46]

J. Math. Anal. Appl., 233 (1999), 445-460. doi: 10.1006/jmaa.1999.6265.  Google Scholar

[47]

Math. Slovaca, 52 (2002), 31-46.  Google Scholar

[48]

Quart. Appl. Math. Soc., 23 (1965), 83-87.  Google Scholar

[49]

J. Math. Anal. Appl., 32 (1970), 424-434.  Google Scholar

[50]

Tamkang J. Math., 6 (1975), 5-11.  Google Scholar

[51]

J. Math. Anal. Appl., 332 (2007), 497-511. doi: 10.1016/j.jmaa.2006.10.010.  Google Scholar

[52]

Advances in Dynam. Sys. Appl., 5 (2010), 231-248.  Google Scholar

[53]

Comput. Math. Appl., 55 (2008), 2631-2642. doi: 10.1016/j.camwa.2007.10.014.  Google Scholar

[54]

Turkish J. Math., 32 (2008), 255-275.  Google Scholar

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Appl. Math. Comput., 134 (2003), 307-317. doi: 10.1016/S0096-3003(01)00285-5.  Google Scholar

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Amer. J. Math., 73 (1951), 368-380.  Google Scholar

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J. Inequal. Appl., 2012 (2012), 1-7. doi: 10.1186/1029-242X-2012-5.  Google Scholar

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