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DEMONSTRATIO MATHEMATICAVol. XXVIINo 21994Baotong CuiOSCILLATION C R I T E R I A F O R S E C O N D O R D E RNONLINEAR DIFFERENTIAL EQUATIONS1. IntroductionWe shall study the oscillatory behaviour of the solutions of second ordernonlinear differential equation(1)(a(i)V(x(0)x'(0)' + pCWOdtmt) + 9(<)/(x(0) = 0,where a, p, q are continuous on [io> oo), to > 0, a(t) > 0;:RR arecontinuous, ip(u) > 0 and uf(u) > 0 for u ^ 0.We restrict our attention to those solutions of equation (1) which existon [ij,oo) where t\ > to, and which are nontrivial in any neighborhood ofinfinity. Such a solution is called oscillatory if it has arbitrarily large zeros.Otherwise the solution is called nonoscillatory.Y. Jurang [1]—[4] considered the second order differential equation withdamping(2)(a(t)x'(t))' + p(t)X'(t)+ q(t)X(t) = 0 ,where a, p, q are continuous on [£0, oo), a(t) > 0, and established sufficientconditions so that all solutions of equation (2) are oscillatory. Philos [2]considered the equation of the form(3)x"(0 + K0x'(*) + i(*)/(x(0) = 0,where p, q : [io>°o) — R , / : RR are continuous, x f ( x ) > 0, f'(x) > 0for x ^ 0 and / is strongly sublinear, i.e. J
Demonstratio Mathematica – de Gruyter
Published: Apr 1, 1994
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