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On some new discrete generalizations of Gronwall’s inequality. (English) Zbl 0643.26013

The main result of the paper (Theorem 3) concerns a linear discrete inequality of the type

(*)x(n)p(n)+ j=1 q i=1 r j J i (j) (n,x)(:=p(n)+A(x)),nN,

where

J i (j) (n,x)= s 1 =n 0 n-1 f i1 (j) (n,s 1 )··· s j =n 0 s j-1 -1 f ij (j) (s j-1 ,s j )x(s j ),

all the functions x, p, f ik (j) are real-valued and nonnegative, p - nondecreasing, f ik (j) -nondecreasing in n for every sN fixed. In the first two theorems some special cases of (*) are considered. Theorems 3, 4 concern nonlinear inequalities x(n)p(n)+g(n)H -1 (A(H(x))) with H nonnegative, strictly increasing, subadditive, H(0)=0, and furthermore g1 (Theorem 3); H - submultiplicative, g - nonnegative (Theorem 4). Linear inequalities are discrete analogies of those proved by the author in J. Math. Anal. Appl. 103, 184-197 (1984; Zbl 0573.26008) and extend many results proved by B. G. Pachpatte [e.g. Indian J. Pure Appl. Math. 8, 1093-1107 (1977; Zbl 0402.26008)]. See also R. P. Agarwal and E. Thandapani [Bull. Inst. Math., Acad. Sin. 9, 235-248 (1981; Zbl 0474.26009); An. Ştiinţ. Univ. Al. I. Cuza Iaşi, N. Ser., Secţ. Ia 28, 71-75 (1982; Zbl 0553.26004)].

Reviewer: J.Popenda
MSC:
26D10Inequalities involving derivatives, differential and integral operators
39A12Discrete version of topics in analysis