A note on certain integral inequalities with delay
The author establishes explicit bounds for solutions of three delay integral inequalities. The typical inequality considered in the paper is of the form \[ x^2(t) \leq c^2 + 2 \int^t_0 [f(s) x(\sigma(s)) W(x(\sigma(s))) + h(s) x(\sigma(s))]ds, \tag{A} \] for \(t \in [0,\infty)\), with the condition \(x(t) = \psi(t)\), \(t \in [a,0]\), \(\psi(\sigma(t)) \leq c\) for every \(t \geq 0\) and \(\sigma(t) \leq 0\), where \(c \geq 0\) is a constant, \(f, h \in C(R_+, R_+)\), \(W \in C(R_+,R_+)\) is nondecreasing, with \(W(0) = 0\) and \(W (x)>0\) for \(x > 0\), \(\sigma \in C(R_+, R_+)\) with \(\sigma(t) \leq t\) and \(-\infty < a:= \inf\{\sigma(t) : t\in R_+\} \leq 0\), \(x \in C([a,\infty), R_+)\). The key step of the argument used in all proofs of the paper is to show that, under the assumed conditions, all functions \(x(\sigma(s))\) involved in the right member of the delay inequality satisfy the relation \(x(\sigma(x)) \leq x(s)\) for all \(s \in R_+\). Hence, after replacing all \(x(\sigma(s))\) by \(x(s)\), the delay integral inequality under consideration yields a corresponding integral inequality without delay, and it can be treated easily by a known method. Some generalizations of Ou-Iang's inequality given by the same author [J. Math. Anal. Appl. 182, No. 1, 143-157 (1994; Zbl 0806.26009)] can be obtained from the Theorem of the paper as consequences for the case when \(\sigma (t) = t\) for \(t \in R_+\). The argument in the proofs of the paper had been used by \textit{P. Ch. Tsamatos} and \textit{S.K. Ntouyas} in Period. Math. Hung. 23, No. 1, 91-94 (1991; Zbl 0746.26011)].
scientific article; zbMATH DE number 892563