# zbMATH — the first resource for mathematics

Comparison theorems for nonlinear ODE’s. (English) Zbl 0760.34030
The nonoscillatory solutions of the equation $$(Lu(t)\equiv)L_ nu(t)+p(t)f[u(g(t))]=0$$ have similar properties as those of the inequality $$Lu(t)\text{sgn} u[g(t)]\leq 0$$ or $$Lu(t)\text{sgn} u[g(t)]\geq 0$$. By means of that the equation $$L_ nu(t)+p(t)f[u(g(t))]=0$$ is compared with the equation $$M_ nz(t)+q(t)h[z(r(t))]=0$$. The linear equation is compared with the Euler equation.

##### MSC:
 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34C11 Growth and boundedness of solutions to ordinary differential equations 34D05 Asymptotic properties of solutions to ordinary differential equations 34A40 Differential inequalities involving functions of a single real variable
Full Text:
##### References:
 [1] CHANTURIJA T. A.: On the oscillation of solutions of higher order linear differential equations. (Russian), Sem. I. Vekua Institute of Applied Math. (1982). [2] KITAMURA, Y, KUSANO T.: Nonlinear oscillation of higher order functional differential equations with deviating arguments. J. Math. Anal. Appl. 77 (1981), 100-119. · Zbl 0465.34044 · doi:10.1016/0022-247X(80)90263-2 [3] KUSANO T., NAITO M.: Comparison theorems for functional differential equations with deviating arguments. J. Math. Soc. Japan 33 (1981), 509-532. · Zbl 0494.34049 · doi:10.2969/jmsj/03330509 [4] MAHFOUD W. E.: Comparison theorems for delay differential equations. Pacific J. Math. 83 (1979), 187-197. · Zbl 0441.34053 · doi:10.2140/pjm.1979.83.187 [5] NAITO M.: On strong oscillation of retarded differential equations. Hiroshima Math. J. 11 (1981), 553-560. · Zbl 0512.34056 [6] OHRISKA J.: Oscillatory and asymptotic properties of third and four order linear differential equations. Czechoslovak Math. J. 39(114) (1989), 215-224. · Zbl 0688.34018 · eudml:13771 [7] ŠEDA W.: Nonoscilatory solutions of differential equations with deviating argument. Czechoslovak Math. J. 36(111) (1986), 93-107. · Zbl 0603.34064
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.