Two qualitative inequalities (Q1079717)

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.