Some new finite difference inequalities
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 \((u(n))^ p\leq\dots\;\).
Gronwall-Wendroff- Bihari type inequalities
scientific article; zbMATH DE number 641449
scientific article