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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 $$(*)\quad x(n)\le p(n)+\sum\sp{q}\sb{j=1}\sum\sp{r\sb j}\sb{i=1}J\sb i\sp{(j)}(n,x)\quad (:=p(n)+A(x)),\quad n\in N,$$ where $$J\sb i\sp{(j)}(n,x)=\sum\sp{n- 1}\sb{s\sb 1=n\sb 0}f\sb{i1}\sp{(j)}(n,s\sb 1)...\sum\sp{s\sb{j-1}- 1}\sb{s\sb j=n\sb 0}f\sb{ij}\sp{(j)}(s\sb{j-1},s\sb j)x(s\sb j),$$ all the functions x, p, $f\sb{ik}\sp{(j)}$ are real-valued and nonnegative, p - nondecreasing, $f\sb{ik}\sp{(j)} - nondecrea\sin g$ in n for every $s\in N$ fixed. In the first two theorems some special cases of (*) are considered. Theorems 3, 4 concern nonlinear inequalities $x(n)\le p(n)+g(n)H\sp{-1}(A(H(x)))$ with H nonnegative, strictly increasing, subadditive, $H(0)=0$, and furthermore $g\equiv 1$ (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 {\it B. G. Pachpatte} [e.g. Indian J. Pure Appl. Math. 8, 1093-1107 (1977; Zbl 0402.26008)]. See also {\it R. P. Agarwal} and {\it 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:
 26D10 Inequalities involving derivatives, differential and integral operators 39A12 Discrete version of topics in analysis
Full Text:
##### References:
 [1] Agarwal, R. P.; Thandapani, E.: On discrete generalizations of Gronwall’s inequality. Bull. inst. Math. acad. Sinica 9, No. 2, 235-248 (1981) · Zbl 0474.26009 [2] Bellman, R.; Cooke, K. L.: Differential-difference equations. (1963) · Zbl 0105.06402 [3] Bykov, Ja.V.; Linenko, V. G.: The stability of the solutions of summary difference equations. Differentsial’nye uravneniya 9, 349-354 (1973) [4] Bopaev, K. B.: On some discrete inequalities. Differentsial’nye uravneniya (Alma-ata), 35-44 (1981) [5] Coffman, C. V.: Asymptotic behavior of solutions of ordinary difference equations. Trans. amer. Math. soc. 110, 22-51 (1964) · Zbl 0122.09703 [6] Jones, G. S.: Fundamental inequalities for discrete and discontinuous functional equations. SIAM J. Appl. math. 12, 43-47 (1964) · Zbl 0154.05702 [7] Demidovič, V. B.: A certain criterion for the stability of difference equations. Differentsial’nye uravneniya 5, 1247-1255 (1969) [8] Masolockaja, L. V.: Stability of difference inequalities. Differentsial’nye uravneniya 34, 147-156 (1967) [9] Mckee, S.: Generalized discrete Gronwall lemmas. Z. angew. Math. mech. 62, No. 9, 429-434 (1982) · Zbl 0524.26013 [10] Pachpatte, B. G.: Finite difference inequalities and their applications. Proc. nat. Acad. sci. India, sect. A 43, 348-356 (1973) · Zbl 0302.39001 [11] Pachpatte, B. G.: On discrete inequalities related to Gronwall inequality. Proc. indian acad. Sci. sect. A 85, 26-40 (1977) · Zbl 0349.39002 [12] Pachpatte, B. G.: One some new integral inequalities and their discrete analogues. Indian J. Pure appl. Math. 8, 1093-1107 (1977) · Zbl 0402.26008 [13] Pachpatte, B. G.: On some new discrete inequalities and their applications to a class of sum-difference equations. An. ştiinţ. Univ. ”al. I cuza” laşi secţ. I a mat. (N.S.) 24, 315-326 (1978) [14] Popenda, J.; Werborwki, J.: On the discrete analogy of Gronwall lemma. Fasc. math. 11, 143-154 (1979) · Zbl 0458.26008 [15] Redheffer, R.; Walter, W.: A comparison theorem for difference inequalities. J. differential equations 44, 111-117 (1982) · Zbl 0455.35009 [16] Sugiyama, S.: Difference inequalities and their applications to stability problems. Lecture notes in mathematics 243 (1971) · Zbl 0236.39002 [17] Sugiyama, S.: Stability problems on difference and functional difference equations. Proc. Japan acad. 45, 526-529 (1969) · Zbl 0202.43102 [18] Willett, D.; Wong, J. S. W.: On the discrete analogues of some generalizations of Gronwall’s inequality. Monatsh. math. 69, 362-367 (1964) · Zbl 0145.06003 [19] Yang, En Hao: On some new discrete inequalities of the Bellman-bihari type. Nonlinear anal. 7, No. 11, 1237-1246 (1983) · Zbl 0526.26009 [20] Yang, En Hao: On the most general form of Bellman type linear inequalities involving multiple-fold integral functional. J. math. Anal. appl. 103, 184-197 (1984) · Zbl 0573.26008 [21] Yang, En Hao: On some new integral inequalities in N independent variables. J. math. Anal. appl. 109, 171-181 (1985) · Zbl 0577.26010 [22] Zamkovaja, L.; Krjukov, B. I.: The stability of nonlinear systems of differential and difference equations. Differentsial’nye uravneniya 13, 756-757 (1977)