## Some new finite difference inequalities.(English)Zbl 0809.26009

Let $$N= \{0,1,2,\dots\}$$. For real-valued, nonnegative functions $$u$$ defined on $$N^ m$$, the author studies (nonlinear, discrete Gronwall- Wendroff-Bihari type) inequalities of the form $u(n)\leq \{c^ 2+ T(n,k,r; F(i,u(i)))\}^{1/2},$ where (i) in the case $$m=1$$ (Theorem 1) $T(n,u)= T(n,k,r; F(i,u(i)))= \sum^{n-1}_{i_ 1= 0} r_ 1(i_ 1) \sum^{i_ 1- 1}_{i_ 2= 0} r_ 2(i_ 2)\cdots \sum^{i_{k-1}- 1}_{i_ k= 0} F(i_ k, u(i_ k)),\quad n\in N,$ $$r_ j$$ are assumed to be positive functions; (ii) in the case $$m= 2$$ (Theorem 2) $$n= (n_ 1,n_ 2)$$, $$i = (i_ 1,i_ 2)$$, $$u= (x_ 1,x_ 2)$$; $T(n,u)= T(n_ 1,k_ 1, 1; T(n_ 2,k_ 2,1; F((i_ 1,i_ 2),u(i_ 1,i_ 2))));$ (iii) in the case of arbitrary fixed $$m$$ (Theorem 3) $$n= (n_ 1,\dots, n_ m)$$, $$i= (i_ 1,\dots i_ m)$$; $T(n,u)= T(n_ 1, 1, 1; T(n_ 2, 1,1;\dots T(n_ m, 1,1; F(i,u(i)))\dots)).$ The function $$F$$ is taken (for example) $F(i,u(i)) =f(i) u^ 2(i)+ g(i) u(i)\quad\text{ or }\quad F(i,u(i))= f(i) u(i) W(u(i))+ g(i) u(i).$ See also the review below.
Remark: The initial inequalities are given in the form $$(u(n))^ 2\leq \dots$$, however regarding proofs these inequalities are like described above. Without any difficulties the obtained results can be extended to the inequalities $$(u(n))^ p\leq\dots\;$$.

### MSC:

 26D15 Inequalities for sums, series and integrals 39A12 Discrete version of topics in analysis

Zbl 0809.26010
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### References:

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