Difference inequalities of fractional order (Q2862064)

The paper presents six new difference inequalities, too involved to be described here, whose proof is based on the following fractional difference inequality of Gronwall-Bellman type: If \(u(n), a(n), b(n)\) are real valued nonnegative functions defined on \(n\in\{0,1,2,\dots\}\) and NEWLINE\[NEWLINE\nabla^{\alpha}u(n+1)\leq a(n)u(n) +b(n)NEWLINE\]NEWLINE [here \(\alpha\in (0,1)\) and \(\nabla^{\alpha}u(n)=\sum_{j=0}^{n-1} { j-\alpha \choose j} \nabla u(n-j),\) with \(\nabla u(k)=u(k)-u(k-1)\)], then NEWLINE\[NEWLINE u(n)\leq u(0)\prod_{j=0}^{n-1}\big[1+ B(n-1,\alpha;j)a(j)\big] + \sum_{j=0}^{n-1}[B(n-1,\alpha;j)b(j)\prod_{k=j+1}^{n-1}[1+B(n-1,\alpha;k)a(k)]], NEWLINE\]NEWLINE with \(B(s,\alpha;j)={ s- j+\alpha -1 \choose s-j}\). This inequality was obtained by the authors in [Communications in Computer and Information Science 283, 403--412 (2012)].NEWLINENEWLINEThe paper can be viewed as a natural continuation of the article mentioned before and [the authors, Math. Sci., Springer 6, Article ID 69, 9 p. (2012; Zbl 1277.39003)].