Comparison and oscillation theorems for functional differential equations with deviating arguments. (English) Zbl 0714.34106

Grace, S. R.; Lalli, B. S.

Math. Nachr. 144, 65-79 (1989).

The authors are interested in comparing the oscillatory behaviour of the two equations (I) \(L_ nx(t)+f(t,x(r(t)))=0,\) n even, \(L_ 0x(t)=x(t)\), \(L_ kx(t)=(1/a_ k(t))(L_{k-1}x(t))',\) \(k=1,...,n-1\) and (II) \(M_ nx(t)+q(t)h(x(s(t)))=0,\) n even \(M_ 0x(t)=x(t)\), \(M_ kx(t)=1/b_ k(t)(M_{k-1}x(t))',\) \(k=1,...,n-1\). This problem includes the earlier studied equations \(x^{(n)}(t)+P(t)F(x(r(t)))=0\) (Mahfoud) and also \(M_ nx(t)+q(t)x(t)=0\) (Kusano, Kitamura). The results connected with (I) and (II) are of high degree of generality and includes a lot of former theorems. Also a comparison technique between (II) and a set of first order delay differential equations is studied.

Reviewer: T.Dłotko

MSC:

34K99	Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Keywords:

comparing of solution; first order delay differential equations

References:

[1]	Ccar, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 pp 749– (1977)
[2]	Grace, J. Math. Anal. Appl. 104 pp 79– (1984)
[3]	Grace, Houston J. Math. 10 pp 173– (1984)
[4]	Grace, Houston J. Math. 10 pp 453– (1984)
[5]	Grace, J. Math. Anal. Appl. 120 pp 39– (1986)
[6]	Grace, Siam J. Math. Anal. 15 pp 1082– (1984)
[7]	Hille, Trans. Amer. Math. Soc. 64 pp 234– (1948)
[8]	Kartsatos, J. Math. Anal. Appl. 52 pp 1– (1975)
[9]	Kartsatos, J. Math. Anal. Appl. 66 pp 297– (1978)
[10]	Koplatadze, Differential’nye Uravnenija 18 pp 1463– (1982)
[11]	Kusano, Pacific J. Math. 92 pp 345– (1981) · Zbl 0475.34019
[12]	Lovelady, Rocky Mountains J. Math. 6 pp 299– (1976)
[13]	Mahfoud, J. Differential Equations 28 pp 437– (1978) · Zbl 0373.34036
[14]	Philos, Utilitos Math. 17 pp 259– (1980)
[15]	Philos, Arch. Math. 36 pp 168– (1981)
[16]	Trench, Proc. Amer. Math. Soc. 52 pp 147– (1975)