On the most general form of Bellman-type linear inequalities involving multiple-fold integral functionals
In the paper four inequalities of the Gronwall type are presented. The main initial inequality is of the form \[ (*)\quad x(t)\leq p(t)+\sum^{n}_{j=1}\sum^{r_ j}_{i=1}I_ i^{(j)}(t)(x), \] where the multiple-fold integral functionals \(I_ i^{(j)}\) are defined by \[ I_ i^{(j)}(t)(x)= \] \[ \int^{t}_{0}f^{(j)}_{i,1}(t,t_ 1)\int^{t_ 1}_{0}f^{(j)}_{i,2}(t_ 1,t_ 2)...\int^{t_{j- 1}}_{0}f^{(j)}_{i,j}(t_{j-1},t_ j)x(t_ j)dt_ j...dt_ 1, \] j\(=1,2,...,n\), \(i=1,2,...,r_ j\), \(r_ j\geq 0\). All the functions are assumed to be real-valued, continuous and positive, furthermore p is nondecreasing and \(f^{(j)}_{i,k}(t,s)\) nondecreasing in t for \(s\in [0,h)\) fixed. The special cases of (*) are considered in the first three theorems, i.e.: \(r_ n=1,\) \(r_ j=0\) for \(j<n\) (Th. 1); \(r_ n=m,\) \(r_ j=0\) for \(j<n\) (Th. 2); \(r_ j=1,\) \(j=1,...,n\) (Th. 3). The presented theorems generalize many recently obtained results by \textit{B. G. Pachpatte} [Indian J. Pure Appl. Math. 8, 1093-1107 (1977; Zbl 0402.26008), ibid. 1157-1175 (1977; Zbl 0414.26009), J. Math. Phys. Sci. 10, 101-116 (1976; Zbl 0345.45013)], \textit{R. P. Agarwal} and \textit{E. Thandapani} [An. Ştiinţ. Univ. Al. I. Cuza Iaşi, N. Ser., Secţ. Ia 28, No.2, 71-75 (1982; Zbl 0553.26004)]. The application to obtain boundedness of all solutions of an n-th order linear differential equation is given.
scientific article; zbMATH DE number 3914618