## Generalizations of an inequality of Kiguradze.(English)Zbl 0546.34010

For the differential equation $$d^ ny/dx^ n+yF(x,y)=0$$, (with appropriate conditions on F), Kiguradze has shown that if $$y^{(i)}\geq 0$$, $$i=0,1,...,k$$, $$y^{(k+1)}\leq 0$$ on (a,$$\infty)$$, then (1) $$(x- a)y^{(t+1)}\leq (k-t)y^{(t)} t=0,1,...,k$$ on (a,$$\infty)$$. In this note the author generalizes inequality (1) in various directions.
Reviewer: P.N.Bajaj

### MSC:

 34A40 Differential inequalities involving functions of a single real variable

### Keywords:

nonoscillatory solution
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### References:

 [1] Bruckner, A.M; Ostrow, E, Some function classes related to the class of convex functions, Pacific J. math., 12, 1203-1215, (1962) · Zbl 0121.29501 [2] Coppel, W.A, Disconjugacy, () · Zbl 0224.34003 [3] Elias, U, Oscillatory solutions and extremal points for a linear differential equation, Arch. rational mech. anal., 70, 177-198, (1979) · Zbl 0412.34016 [4] Elias, U, Necessary conditions and sufficient conditions for disfocality and disconjugacy of differential equations, Pacific J. math., 81, 379-397, (1979) · Zbl 0378.34027 [5] Foster, K.E; Grimmer, R.C, Nonoscillatory solutions of higher order differential equations, J. math. anal. appl., 71, 1-17, (1979) · Zbl 0428.34029 [6] Foster, K.E; Grimmer, R.C, Nonoscillatory solutions of higher order delay equations, J. math. appl., 77, 150-164, (1980) · Zbl 0455.34053 [7] Grimmer, R.C, Oscillation criteria and growth of nonoscillatory solutions of even order ordinary and delay-differential equations, Trans. amer. math. soc., 198, 215-228, (1974) · Zbl 0292.34026 [8] Grimmer, R.C, Comparison theorems for third-and fourth-order linear equations, J. differential equations, 25, 1-9, (1977) · Zbl 0392.34016 [9] Jones, G.D, An ordering of oscillation types for y(n) + py = 0, SIAM J. math. anal., 12, 72-77, (1981) · Zbl 0458.34019 [10] Karlin, S, Total positivity, (1968), Stanford Univ. Press Stanford, Calif · Zbl 0219.47030 [11] Kiguradze, I.T; Kiguradze, I.T, Oscillation properties of solutions of certain ordinary differential equations, Dokl. akad. nauk SSSR, Soviet math. dokl., 3, 649-652, (1962) · Zbl 0144.11201 [12] Kusano, T; Onose, H, Oscillation of functional differential equations with retarded argument, J. differential equations, 15, 269-277, (1979) · Zbl 0292.34078 [13] Lazer, A.C, The behaviour of solutions of the differential equation y‴ + py′ + qy = 0, Pacific J. math., 17, 435-466, (1966) · Zbl 0143.31501 [14] Polya, G, On the Mean value theorem corresponding to a given linear homogeneous differential equations, Trans. amer. math. soc., 24, 312-324, (1972) · JFM 50.0299.02
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