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Two qualitative inequalities. (English) Zbl 0598.34007
The following differential inequalities, on the entire real axis, are investigated, and estimates given for their solutions subject to some qualitative restrictions, such as being bounded or belonging to a certain function space: $(1)\quad x'(t)\geq \lambda (x(t))-f(t)\mu (x(t)),\quad (2)\quad x''(t)\geq \lambda (x(t))-f(t)\mu (x(t)),$ where the functions $$\lambda$$ and $$\mu$$ satisfy some easily verifiable conditions (monotonicity type), while f(t) is nonnegative, locally integrable, and belongs to the space M defined by the condition $$\sup \int^{t+1}_{t}f(s)ds<\infty,$$ $$t\in R$$. Under assumed conditions, for both inequalities, the estimate $$\sup x(t)\leq \nu^{-1}(K| f|_ M)$$ is shown to be true for any nonnegative bounded solution. The function $$\nu$$ is defined, roughly speaking, by $$\nu (r)=\lambda (r)/\mu (r),$$ while K stands for a constant. Inequalities on a half-axis are also discussed.

##### MSC:
 34A40 Differential inequalities involving functions of a single real variable 34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
##### Keywords:
qualitative inequality; autonomous inequalities
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##### References:
  Corduneanu, C, Bounded and almost periodic solutions of certain nonlinear elliptic equations, Tôhoku math. J., 32, 265-278, (1980) · Zbl 0443.35027  Corduneanu, C, Bounded and almost periodic solutions of certain nonlinear parabolic equations, Libertas math., II, 131-139, (1982) · Zbl 0504.35007  Corduneanu, C, Almost periodicity criteria for ordinary differential equations, Libertas math., III, 21-43, (1983) · Zbl 0527.34042  Corduneanu, C, Almost periodic solutions to nonlinear elliptic and parabolic equations, Nonlinear anal., TME 7, 357-363, (1983) · Zbl 0512.35008  Corduneanu, C, Almost periodic solutions to some nonlinear parabolic equations, (), 139-141 · Zbl 0504.35007  Corduneanu, C, Application of differential inequalities to stability theory, An. stiint. univ. “al. I. cuza” iaşi, seçt. ia mat., VI, 47-58, (1960), [Russian] · Zbl 0108.27802  Corduneanu, C; Goldstein, J.A, Almost periodicity of bounded solutions to abstract differential equations, (), 115-121 · Zbl 0561.34047  Haraux, A, Nonlinear evolution equations—global behavior of solutions, () · Zbl 0583.35007  Krasovskii, N.N, Stability of motion, (1963), Stanford Univ. Press Stanford, Calif · Zbl 0109.06001  Massera, J.L; Schaffer, J.J, Linear differential equations and function spaces, (1966), Academic Press New York · Zbl 0202.14701  Perov, A.I; Trubnikov, Yu.V; Perov, A.I; Trubnikov, Yu.V; Perov, A.I; Trubnikov, Yu.V, Monotonic differential equations III, Differential equations, Differential equations, Differential equations, 14, 161-170, (1978) · Zbl 0398.34053  Sansone, G, Equazioni differenziali nel campo reale, (1948), Edit. Zanichelli Bologna · JFM 67.0306.01
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