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Nonlinear discrete inequalities with two variables and their applications
Applied Mathematics and Computation 217 (2010) 2217–2225
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Nonlinear discrete inequalities with two variables and their applications Shengfu Deng Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, PR China
a r t i c l e
i n f o
a b s t r a c t An error in the proof of Theorem 1 by Zheng et al. [K. Zheng, Y. Wu, S. Zhong, Discrete nonlinear integral inequalities in two variables and their applications, Appl. Math. Comput. 207 (2009) 140–147] is reported. This paper gives the right proof under some additional condition. Application examples to show boundedness and uniqueness of solutions of a Volterra type difference equation are also given. Ó 2010 Elsevier Inc. All rights reserved.
Keywords: Discrete inequality Nonlinear Volterra type Boundedness Uniqueness
1. Introduction Gronwall–Bellman inequalities and their various linear and nonlinear generalizations play very important roles in the discussion of existence, uniqueness, continuation, boundedness, and stability properties of solutions of differential equations and difference equations. The literature on such inequalities and their applications is vast. For example, see [1–5,8,11– 14,17,22] for continuous cases, and [6,7,9,10,15,16,18–21] for discrete cases. Recently, Zheng et al. [21] considered the following discrete inequality
uðm; nÞ 6 aðm; nÞ þ
k m 1 X n1 X X
fi ðm; n; s; tÞwi ðuðs; tÞÞ;
m; n 2 N0
ð1:1Þ
s¼0 t¼0
i¼1
with k nonlinear terms where N0 = {0, 1, 2, . . . }. In order to prove their Theorem 1, they constructed an auxiliary inequality
~1 Þ þ ~ 1; n uðm; nÞ 6 b1 ðm
k m 1 X n1 X X i¼1
~f ðm ~ 1 ; s; t Þwi ðuðs; tÞÞ; i ~ 1; n
~ 06n6n ~ 0 6 m 6 m;
ð1:2Þ
s¼0 t¼0
~ and n ~ so that the term b1 ðm ~ 1 Þ is a constant with respect to variables m and n. They gave a ~ 1; n for some positive integers m claim for the inequality (1.2). Firstly, they proved that their claim is true for k = 1. Then they assumed that their claim is also true for k. For k + 1, (1.2) was changed into
nðm; nÞ 6 c1 ðm; nÞ þ
k X m1 X n1 X i¼1
~f ðm ~ 1 ; s; t Þ/i ðnðs; tÞÞ: iþ1 ~ 1 ; n
ð1:3Þ
s¼0 t¼0
They said that (1.3) has the same form as (1.2) so they applied their inductive assumption. However, this is not right because ~ 1 Þ in (1.2) is a constant with respect to m and n. In this paper, we require ~ 1; n c1(m, n) in (1.3) is a function of m and n but b1 ðm that a(m, n) satisfies some additional condition, and give not only the right proof but examples to show boundedness and uniqueness of solutions of a Volterra type difference equation. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.07.022
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2. Main results ~ðm; nÞ ¼ max06s6m;06g6n;s;g2N0 aðs; gÞ and define D1u(m, n) = u(m + 1, n) u(m, n). Assume that: Let a ~ðm; nÞ is nondecreasing in n; (C1) a (m, n) is nonnegative for m, n 2 N0 and a(0, 0) > 0; D1 a (C2) all fi(m, n, s, t) (i = 1, . . . , k) are nonnegative for m, n, s, t 2 N0; (C3) all wi (i = 1, . . . , k) are continuous and nondecreasing functions on [0, 1) and are positive on (0, 1). They satisfy the w relationship w1 / w3 / / wm where wi / wi+1 means that wiþ1i is nondecreasing on (0, 1) (see [17]). Let W i ðuÞ ¼
Ru
for u P ui where ui > 0 is a given constant. Then, Wi is strictly increasing so its inverse W 1 is well dei
dz ui wi ðzÞ
fined, continuous and increasing in its corresponding domain. Theorem 2.1. Suppose that (C1)–(C3) hold and u (m, n) is a nonnegative function for m, n 2 N0 satisfying (1.1) . Then
2 uðm; nÞ 6
4W k ða ~ð0; nÞÞ W 1 k
þ
m1 X n1 X
~f ðm; n; s; tÞ þ k
s¼0 t¼0
m 1 X s¼0
D3 rk ðm; n; s; nÞ
3
5; /k W 1 k1 ðr k ð0; 0; s; 0ÞÞ
0 6 m 6 M1 ; 0 6 n 6 N1 ;
ð2:1Þ
where
~f ðm; n; s; tÞ ¼ i
max
06s6m;06g6n;s;g2N0
fi ðs; g; s; tÞ;
ð2:2Þ
rk(m, n, s, t) is determined recursively by
~ðs; tÞ; r 1 ðm; n; s; tÞ ¼ a r iþ1 ðm; n; s; tÞ ¼ W i ðr 1 ðm; n; 0; tÞÞ þ
s1 X t1 X
~f ðm; n; s; gÞ i
s¼0 g¼0
þ
s1 X
s¼0
ð2:3Þ
D3 r i ðm; n; s; tÞ
; /i W 1 i1 ðr i ð0; 0; s; 0ÞÞ
D3 ri ðm; n; s; tÞ ¼ r i ðm; n; s þ 1; tÞ r i ðm; n; s; tÞ; /i ðuÞ ¼
wi ðuÞ , wi1 ðuÞ
i ¼ 1; . . . ; k 1; i ¼ 1; . . . ; k;
/1(u) = w1(u), W0 = I (Identity), and M1 and N1 are positive integers satisfying
~ð0; N1 ÞÞ þ W i ða
M 1 1 N 1 1 X X s¼0
M 1 1 X
~f ðM1 ; N1 ; s; tÞ þ i
t¼0
Remark 2.1. If all wi(i = 1, . . . , k) satisfy not affect our results (see [2]).
s¼0
R1 ui
dz wi ðzÞ
D r ðM ; N ; s; N1 Þ 3 i 1 1 6 /i W 1 i1 ðr i ð0; 0; s; 0ÞÞ
Z
1
ui
dz ; wi ðzÞ
i ¼ 1; . . . ; k:
ð2:4Þ
¼ 1, then we may choose N1 = 1 and M1 = 1. Different choices of ui in Wi do
Lemma 2.1. For i = 1, . . . , k, D3ri(m, n, s, t) is nonnegative and nondecreasing in m, n and t, and ri(m, n, s, t) is nonnegative and nondecreasing in its arguments. ~ðm; nÞ and ~f i ðm; n; s; tÞ, it is easy to check that they are nonnegative and nondecreasing in m and Proof. By the definitions of a ~ðm; nÞ P aðm; nÞ and ~f i ðm; n; s; tÞ P fi ðm; n; s; tÞ where i = 1, . . . , k. a(0, 0) > 0 in (C1) implies that a ~ðm; nÞ > 0 for all n, and a ~ðs; 0Þ > 0, we have m 6 M1 and n 6 N1. Clearly, using the fact r 1 ð0; 0; s; 0Þ ¼ a
D3 r 1 ðm þ 1; n; s; tÞ D3 r1 ðm; n; s; tÞ ¼ 0;
D3 r 2 ðm þ 1; n; s; tÞ D3 r2 ðm; n; s; tÞ ¼
t1 X
~f 1 ðm þ 1; n; s; jÞ
t1 X
j¼0
j¼0
~f 1 ðm; n; s; jÞ þ D3 r 1 ðm þ 1; n; s; tÞ D3 r 1 ðm; n; s; tÞ P 0; w1 ðr 1 ð0; 0; s; 0ÞÞ
which yield that D3r1(m, n, s, t) and D3r2(m, n, s, t) are nondecreasing in m. Assume that D3rl(m, n, s, t) is nondecreasing in m. Then
D3 r lþ1 ðm þ 1; n; s; tÞ D3 r lþ1 ðm; n; s; tÞ ¼
t1 X j¼0
~f ðm þ 1; n; s; jÞ l
t1 X
1; n; s; tÞ D3 r l ðm; n; s; tÞ ~f ðm; n; s; jÞ þ D3 r l ðm þ P 0; l 1 / W j¼0 l l1 ðr l ð0; 0; s; 0ÞÞ
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S. Deng / Applied Mathematics and Computation 217 (2010) 2217–2225
which implies that D3rl+1(m, n, s, j) is nondecreasing in m. By induction, D3ri(m, n, s, j) is nondecreasing in m. Similarly, since ~ðm; nÞ is nondecreasing in n, we can prove that D3ri(m, n, s, j) is also nondecreasing in n and t, and nonnegative by inducD1 a tion again. Then ri(m, n, s, t) is nonnegative and nondecreasing in its arguments. h ~ 6 N 1 . Consider an auxiliary inequality ~ and n ~ where m ~ 6 M1 ; n Proof of Theorem 2.1. Take any arbitrary positive integers m
~ n ~ ; m; nÞ þ uðm; nÞ 6 r 1 ðm;
k m1 n1 X XX
~f ðm; ~ ; s; tÞwi ðuðs; tÞÞ i ~ n
ð2:5Þ
s¼0 t¼0
i¼1
~ 06n6n ~ . Claim that u(m, n) in (2.5) satisfies for 0 6 m 6 m;
2 4W k ðr 1 ðm; ~ n ~ ; 0; nÞÞ þ uðm; nÞ 6 W 1 k
m1 X n1 X
~f ðm; ~ ; s; tÞ þ k ~ n
m1 X
s¼0 t¼0
s¼0
~ n ~; s; nÞ D3 r k ðm;
3
5 /k W 1 k1 ðr k ð0; 0; s; 0ÞÞ
ð2:6Þ
~ M 2 g and 0 6 n 6 minfn ~ ; N 2 g where M2 and N2 are positive integers satisfying for 0 6 m 6 minfm;
~ n ~ ; 0; N 2 ÞÞ þ W i ðr 1 ðm;
M 2 1 N 2 1 X X s¼0
~f ðm; ~; s; tÞ þ i ~ n
t¼0
M 2 1 X
~ n ~ ; s; N2 Þ D r ðm; 3 i 6 1 /i W i1 ðr i ð0; 0; s; 0ÞÞ
s¼0
Z
1 ui
dz ; wi ðzÞ
i ¼ 1; . . . ; k:
ð2:7Þ
~ n ~ ; m; nÞ; D3 r i ðm; ~ n ~ ; m; nÞ and ~f i ðm; ~ n ~ ; m; nÞ Before we prove (2.6), notice that we may choose M1 6 M2 and N1 6 N2. In fact, r i ðm; ~ and n ~ by Lemma 2.1. Thus, M2 and N2 satisfying (2.7) get smaller as m ~ and n ~ are chosen larger. In are nondecreasing in m ~ ¼ M 1 and n ~ ¼ N1 . particular, M2 and N2 satisfy the same (2.4) as M1 and N1 for m We divide the proof of (2.6) into two steps by using a mathematical introduction. Step 1. k = 1. P Pn1 ~ ~ ~ Let zðm; nÞ ¼ m1 s¼0 t¼0 f 1 ðm; n; s; tÞw1 ðuðs; tÞÞ and z(0, n) = 0. It is clear that z(m, n) is nonnegative and nondecreasing in m ~ n ~ ; m; nÞ þ zðm; nÞ for 0 6 m 6 m; ~ 06n6n ~ and by Lemma 2.1 and n. Observe that (2.5) is equivalent to uðm; nÞ 6 r 1 ðm;
D1 zðm; nÞ ¼
n1 X
~f 1 ðm; ~ n ~ ; m; tÞw1 ðuðm; tÞÞ 6
n1 X
t¼0
~f 1 ðm; ~ n ~ ; m; tÞw1 ðr 1 ðm; ~ n ~ ; m; tÞ þ zðm; tÞÞ
t¼0
~ n ~ ; m; nÞ þ zðm; nÞÞ 6 w1 ðr1 ðm;
n1 X
~f 1 ðm; ~ n ~; m; tÞ:
t¼0
~ n ~ ; m; nÞ > 0, we have Since w1 is nondecreasing and r1 ðm; n1 ~ n ~ ; m; nÞ X ~ n ~; m; nÞ D1 zðm; nÞ þ D3 r 1 ðm; D3 r 1 ðm; ~f 1 ðm; ~ n ~ ; m; tÞ þ 6 ~ n ~ ; m; nÞÞ ~ n ~ ; m; nÞÞ w1 ðzðm; nÞ þ r 1 ðm; w ð zðm; nÞ þ r 1 1 ðm; t¼0 n1 X
~ n ~ ; m; nÞ D r ðm; ~f 1 ðm; ~ n ~ ; m; tÞ þ 3 1 : w ðr ð0; 0; m; 0ÞÞ 1 1 t¼0
6
ð2:8Þ
Then
Z
~ n ~ ;mþ1;nÞ zðmþ1;nÞþr 1 ðm;
~ n ~ ;m;nÞ zðm;nÞþr 1 ðm;
ds 6 w1 ðsÞ 6
Z
~ n ~ ;mþ1;nÞ zðmþ1;nÞþr 1 ðm;
~ n ~ ;m;nÞ zðm;nÞþr1 ðm;
~ n ~; m; nÞ ds D1 zðm; nÞ þ D3 r 1 ðm; 6 ~ n ~ ; m; nÞÞ w1 ðzðm; nÞ þ r1 ðm; ~ n ~ ; m; nÞÞ w1 ðzðm; nÞ þ r 1 ðm;
n1 X
~ n ~ ; m; nÞ D r ðm; ~f 1 ðm; ~ n ~ ; m; tÞ þ 3 1 ; w1 ðr 1 ð0; 0; m; 0ÞÞ t¼0
so
Z
~ n ~ ;m;nÞ zðm;nÞþr 1 ðm;
~ n ~ ;0;nÞ zð0;nÞþr 1 ðm;
m1 X ds ¼ w1 ðsÞ s¼0
Z
~ n ~ ;sþ1;nÞ zðsþ1;nÞþr 1 ðm;
~ n ~ ;s;nÞ zðs;nÞþr1 ðm;
m1 n1 m1 X D3 r 1 ðm; XX ~ n ~ ; s; nÞ ds ~f 1 ðm; ~ n ~; s; tÞ þ 6 : w w1 ðsÞ s¼0 t¼0 ðr ð0; 0; s; 0ÞÞ 1 1 s¼0
The definition of W1 in Theorem 2.1 and z(0, n) = 0 show
~ n ~ ; m; nÞÞ 6 W 1 ðr 1 ðm; ~ n ~; 0; nÞÞ þ W 1 ðzðm; nÞ þ r 1 ðm;
m 1 X n1 X
~f 1 ðm; ~ n ~ ; s; tÞ þ
s¼0 t¼0
~ 06n6n ~: 0 6 m 6 m; Eq. (2.7) shows that the right side of (2.9) is in the domain of W 1 1 implies
m 1 X s¼0
~ n ~ ; s; nÞ D3 r 1 ðm; ; w1 ðr 1 ð0; 0; s; 0ÞÞ ð2:9Þ
W 1 1
~ and 0 6 n 6 n ~ . Thus the monotonicity of for all 0 6 m 6 m
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" ~ n ~ ; m; nÞ 6 W 1 ~ n ~; 0; nÞÞ þ uðm; nÞ 6 zðm; nÞ þ r1 ðm; W 1 ðr 1 ðm; 1
m1 X n1 X
~f 1 ðm; ~ n ~ ; s; tÞ þ
s¼0 t¼0
m1 X s¼0
~ n ~ ; s; nÞ D3 r 1 ðm; w1 ðr 1 ð0; 0; s; 0ÞÞ
# ð2:10Þ
~ and 0 6 n 6 n ~ , i.e., (2.6) is true for k = 1. for 0 6 m 6 m Step 2. k + 1. Assume that (2.6) is true for k. Consider
~ n ~ ; m; nÞ þ uðm; nÞ 6 r 1 ðm;
kþ1 m 1 X n1 X X
~f ðm; ~ ; s; tÞwi ðuðs; tÞÞ i ~ n
s¼0 t¼0
i¼1
Pkþ1 Pm1 Pn1 ~ ~ and 0 6 n 6 n ~ . Let zðm; nÞ ¼ i¼1 ~ ~ for 0 6 m 6 m s¼0 t¼0 f i ðm; n; s; tÞwi ðuðs; tÞÞ and z(0, n) = 0. Then z(m, n) is nonnegative ~ n ~ ; m; nÞ þ zðm; nÞ for 0 6 m 6 m ~ and 0 6 n 6 n ~ . Moreover, we and nondecreasing in m and n and satisfies uðm; nÞ 6 r 1 ðm; have
D1 zðm; nÞ ¼
kþ1 X n1 X i¼1
~f i ðm; ~ n ~; m; tÞwi ðuðm; tÞÞ 6
t¼0
kþ1 X n1 X i¼1
~f i ðm; ~ n ~ ; m; tÞwi ðr 1 ðm; ~ n ~; m; tÞ þ zðm; tÞÞ:
t¼0
~ n ~ ; m; nÞ > 0, we have by Lemma 2.1 Since wi is nondecreasing and r1 ðm;
Pkþ1 Pn1 ~ ~ ~ ~ ~ ~ n ~ ; m; nÞ D3 r 1 ðm; i¼1 t¼0 f i ðm; n; m; tÞwi ðzðm; tÞ þ r 1 ðm; n; m; tÞÞ þ ~ ~ ~ ~; m; nÞÞ w1 ðzðm; nÞ þ r 1 ðm; n; m; nÞÞ w1 ðr 1 ðm; n
~ n ~; m; nÞ D1 zðm; nÞ þ D3 r 1 ðm; 6 ~ ~ w1 ðzðm; nÞ þ r1 ðm; n; m; nÞÞ
n1 X
kþ1 X n1 ~ n ~ ; m; nÞ X ~ n ~ ; m; tÞÞ D r ðm; w ðzðm; tÞ þ r1 ðm; ~f 1 ðm; ~f ðm; ~ n ~ ; m; tÞ þ 3 1 ~ ; m; tÞ i þ i ~ n ~ ~; m; tÞÞ w w ðr ð0; 0; m; 0ÞÞ ðzðm; tÞ þ r ð m; n 1 1 1 1 t¼0 i¼2 t¼0
6
k X n1 ~ n ~ ; m; nÞ X D r ðm; ~f 1 ðm; ~f ðm; ~ iþ1 ðzðm; tÞ þ r 1 ðm; ~ n ~ ; m; tÞ þ 3 1 ~ ; m; tÞ/ ~ n ~ ; m; tÞÞ þ iþ1 ~ n w1 ðr 1 ð0; 0; m; 0ÞÞ i¼1 t¼0 t¼0
n1 X
6
~ iþ1 ðuÞ ¼ wiþ1 ðuÞ for i = 1, . . . , k, which gives ~ and 0 6 n 6 n ~ where / for 0 6 m 6 m w1 ðuÞ
Z
~ n ~ ;mþ1;nÞ zðmþ1;nÞþr1 ðm;
~ n ~ ;m;nÞ zðm;nÞþr 1 ðm;
ds 6 w1 ðsÞ 6
Z
~ n ~ ;mþ1;nÞ zðmþ1;nÞþr 1 ðm;
~ n ~ ;m;nÞ zðm;nÞþr1 ðm;
~ n ~ ; m; nÞ ds D1 zðm; nÞ þ D3 r 1 ðm; 6 ~ n ~ ; m; nÞÞ w1 ðr 1 ðm; ~ n ~ ; m; nÞ þ zðm; nÞÞ w1 ðzðm; nÞ þ r 1 ðm;
k X n1 ~ n ~ ; m; nÞ X D r ðm; ~f 1 ðm; ~f iþ1 ðm; ~ iþ1 ðzðm; tÞ þ r 1 ðm; ~ n ~; m; tÞ þ 3 1 ~ n ~ ; m; tÞ/ ~ n ~ ; m; tÞÞ: þ w1 ðr 1 ð0; 0; m; 0ÞÞ i¼1 t¼0 t¼0
n1 X
Therefore,
Z
~ n ~ ;m;nÞ zðm;nÞþr 1 ðm;
~ n ~ ;0;nÞ zð0;nÞþr 1 ðm;
m 1 X n1 m 1 k m 1 X n1 X X X ~ n ~ ; s; nÞ X ds D3 r 1 ðm; ~f 1 ðm; ~f ðm; ~ iþ1 ðzðs; tÞ ~ n ~ ; s; tÞ þ ~ ; s; tÞ/ 6 þ iþ1 ~ n w1 ðr 1 ð0; 0; s; 0ÞÞ i¼1 s¼0 t¼0 w1 ðsÞ s¼0 t¼0 s¼0
~ n ~ ; s; tÞÞ; þ r 1 ðm; that is
~ n ~ ; m; nÞÞ 6 W 1 ðr1 ðm; ~ n ~ ; 0; nÞÞ þ W 1 ðzðm; nÞ þ r 1 ðm;
m1 n1 XX
~f 1 ðm; ~ n ~ ; s; tÞ þ
s¼0 t¼0
þ
k X m1 X n1 X
m1 X s¼0
~ n ~; s; nÞ D3 r 1 ðm; w1 ðr 1 ð0; 0; s; 0ÞÞ
~f ðm; ~ iþ1 ðzðs; tÞ þ r1 ðm; ~ ; s; tÞ/ ~ n ~ ; s; tÞÞ; iþ1 ~ n
s¼0 t¼0
i¼1
or equivalently
~ n ~ ; m; nÞ þ nðm; nÞ 6 c1 ðm;
k m1 n1 X XX i¼1
~f ðm; ~ iþ1 ðW 1 ðnðs; tÞÞÞ ~; s; tÞ/ iþ1 ~ n 1
s¼0 t¼0
~ and 0 6 n 6 n ~ the same as (2.5) for k where for 0 6 m 6 m
~ n ~ ; m; nÞÞ; nðm; nÞ ¼ W 1 ðzðm; nÞ þ r1 ðm; ~ n ~ ; m; nÞ ¼ W 1 ðr 1 ðm; ~ n ~ ; 0; nÞÞ þ c1 ðm;
m1 X n1 X s¼0 t¼0
~f 1 ðm; ~ n ~ ; s; tÞ þ
m 1 X s¼0
~ n ~ ; s; nÞ D3 r 1 ðm; : w1 ðr 1 ð0; 0; s; 0ÞÞ
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~ iþ1 ðW 1 Þ, i = 1, . . . , k, is continuous and nondecreasing on [0, 1) and is positive on (0, 1) From the assumption (C3), each / 1 1 ~ 2 W 1 / / / ~ kþ1 W 1 . By the inductive ~ 1 W 1 / / since W is continuous and nondecreasing on [0, 1). Moreover, / 1
1
1
1
assumption, we have
2 1 4 kþ1
~ n ~ ; 0; nÞÞ þ Ukþ1 ðc1 ðm;
nðm; nÞ 6 U
m 1 X n1 X
~f ðm; ~ ; s; tÞ þ kþ1 ~ n
m 1 X
s¼0 t¼0
s¼0
~ ; M 3 g and 0 6 n 6 minfn ~ ; N 3 g where Uiþ1 ðuÞ ¼ for 0 6 m 6 minfn
U1 iþ1 is the inverse of Uiþ1 ; wiþ1 ðuÞ ¼
~ ðW 1 ðuÞÞ / iþ1 1 ~ ðW 1 ðuÞÞ / i
1
~ n ~ ; m; nÞ ¼ Uiþ1 ðc1 ðm; ~ n ~ ; 0; nÞÞ þ ciþ1 ðm;
¼
wiþ1 ðW 1 1 ðuÞÞ wi ðW 1 1 ðuÞÞ
m1 n1 XX
3 ~ n ~ ; s; nÞ D3 ck ðm; 5 wkþ1 U1 ðc ð0; 0; s; 0ÞÞ k k
Ru
dz ~ iþ1 / ~ ðW 1 ðzÞÞ ; u iþ1 1
ð2:11Þ
~ iþ1 ¼ W 1 ðuiþ1 Þ; u > 0; U1 ¼ I (Identity), u
, i = 1, . . . , k
~f ðm; ~ ; s; tÞ þ iþ1 ~ n
m1 X
s¼0 t¼0
s¼0
~ n ~ ; s; nÞ D c ðm; 3 i ; 1 wiþ1 Ui ðci ð0; 0; s; 0ÞÞ
i ¼ 1; . . . ; k 1;
and M3 and N3 are positive integers satisfying
~ n ~ ; 0; N3 ÞÞ þ Uiþ1 ðc1 ðm;
M 3 1 N 3 1 X X s¼0
~f iþ1 ðm; ~ n ~ ; s; tÞ þ
t¼0
M 3 1 X s¼0
~ n ~ ; s; N3 Þ D3 ci ðm; 6 wiþ1 U1 ðc ð0; 0; s; 0ÞÞ i i
Z
W 1 ð1Þ
~ iþ1 u
dz ; ~ iþ1 W 1 ðzÞ / 1
i ¼ 1; . . . ; k: ð2:12Þ
Note that
Ui ðuÞ ¼
Z
u
~i u
dz
¼ ~ i W 1 ðzÞ / 1
Z
u
W 1 ðui Þ
Z W 1 w1 W 1 1 ðuÞ 1 ðzÞ dz dz ¼ ¼ W i W 1 1 ðuÞ; 1 w ðzÞ i u wi W 1 ðzÞ i
i ¼ 2; . . . ; k þ 1
and
wiþ1
wiþ1 W 1 U1 ðuÞ wiþ1 W 1 W 1 W 1 ðuÞ wiþ1 W 1 ðuÞ 1 1 i i i ¼ ¼ ¼ /iþ1 W 1 U1 ðuÞ ; i i ðuÞ ¼ U1 W 1 W 1 wi W 1 wi W 1 wi W 1 1 1 i ðuÞ i ðuÞ i ðuÞ
i ¼ 1; . . . ; k:
~ n ~ ; 0; nÞ ¼ W 1 ðr 1 ðm; ~ n ~ ; 0; nÞÞ and we have from (2.11) that Thus, use the fact c1 ðm;
~ n ~ ; m; nÞ þ zðm; nÞ ¼ W 1 uðm; nÞ 6 r 1 ðm; 1 ðnðm; nÞÞ 2 m1 X n1 m1 X X 1 ~f ðm; 4 ~ ~ ~; s; tÞ þ 6 W 1 kþ1 ~ n kþ1 W kþ1 W 1 ðc 1 ðm; n; 0; nÞÞ þ
3 ~ n ~ ; s; nÞ D3 ck ðm; 5 1 s¼0 t¼0 s¼0 /kþ1 W k ðc k ð0; 0; s; 0ÞÞ 2 3 m1 X n1 m1 X X ~ n ~ ; s; nÞ D3 ck ðm; 1 4 ~ ~ n ~ ; 0; nÞÞ þ ~ n ~ ; s; tÞ þ 5 f kþ1 ðm; 6 W kþ1 W kþ1 ðr 1 ðm; 1 s¼0 t¼0 s¼0 /kþ1 W k ðc k ð0; 0; s; 0ÞÞ
ð2:13Þ
~ ; M 3 g and 0 6 n 6 minfn ~ ; N 3 g. for 0 6 m 6 minfn ~ n ~ ; m; nÞ ¼ riþ1 ðm; ~ n ~ ; m; nÞ by induction again. In the following, we prove that ci ðm; ~ n ~ ; m; nÞ ¼ r 2 ðm; ~ n ~ ; m; nÞ for i = 1. Suppose that It is clear that c1 ðm;
~ n ~ ; m; nÞ ¼ r lþ1 ðm; ~ n ~ ; m; nÞ cl ðm; for i = l. We have
~ n ~ ; m; nÞ ¼ Ulþ1 ðc1 ðm; ~ n ~ ; 0; nÞÞ þ clþ1 ðm;
m1 n1 XX
~f ðm; ~ ; s; tÞ þ lþ1 ~ n
m1 X
s¼0 t¼0
~ n ~ ; 0; nÞÞ þ ¼ W lþ1 ðr 1 ðm;
m1 X n1 X s¼0 t¼0
s¼0
~f ðm; ~ ; s; tÞ þ lþ1 ~ n
m 1 X s¼0
~ n ~; s; nÞ D c ðm; 3 l 1 wlþ1 Ul ðcl ð0; 0; s; 0ÞÞ ~ n ~ ; s; nÞ D r ðm; ~ n ~ ; m; nÞ; 3 lþ1 ¼ r lþ2 ðm; 1 /lþ1 W l ðr lþ1 ð0; 0; s; 0ÞÞ
~ n ~ ; 0; nÞ ¼ W 1 ðr1 ðm; ~ n ~ ; 0; nÞÞ is applied. It implies that it is true for i = l + 1. Thus, ci ðm; ~ n ~ ; m; nÞ ¼ r iþ1 ðm; ~ n ~ ; m; nÞ for where c1 ðm; i = 1, . . . , k. Eq. (2.12) becomes
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~ n ~ ; 0; N3 ÞÞ þ W iþ1 ðr 1 ðm;
M 3 1 N 3 1 X X s¼0
¼
Z
W 1 ð1Þ
~ iþ1 u
~f ðm; ~ ; s; tÞ þ iþ1 ~ n
t¼0
M 3 1 X s¼0
Z 1 w1 W 1 1 ðzÞ dz dz ¼ 1 w ðzÞ iþ1 uiþ1 wiþ1 W 1 ðzÞ
~ n ~ ; s; N 3 Þ D r ðm; 3 iþ1 6 1 /iþ1 W i ðr iþ1 ð0; 0; s; 0ÞÞ
Z
W 1 ð1Þ
~ iþ1 u
dz ~ /iþ1 W 1 1 ðzÞ ð2:14Þ
for i = 1, . . . , k. It implies that we may choose M3 = M2 and N3 = N2. Thus, (2.13) becomes
2
uðm; nÞ 6
4 ~ ~ W 1 kþ1 W kþ1 ðr 1 ðm; n; 0; nÞÞ
þ
m1 X n1 X
~f ðm; ~; s; tÞ þ kþ1 ~ n
s¼0 t¼0
m 1 X s¼0
3 ~ n ~ ; s; nÞ D3 r kþ1 ðm; 5 /kþ1 W 1 k ðr kþ1 ð0; 0; s; 0ÞÞ
~ and 0 6 n 6 n ~ . It shows that (2.6) is true for k + 1. Thus, the claim is proved. for 0 6 m 6 m ~ and n ~ in (2.6), respectively, we have Now we prove (2.1). Replacing m and n by m
2 ~ n ~Þ 6 uðm;
4W k ðr 1 ðm; ~ n ~ ; 0; n ~ ÞÞ W 1 k
þ
~ ~ 1 m1 n X X
~f ðm; ~ ; s; tÞ þ k ~ n
s¼0 t¼0
~ m1 X
~ n ~ ; s; n ~Þ D3 rk ðm;
3
5: /k W 1 k1 ðr k ð0; 0; s; 0ÞÞ
s¼0
~ 6 M 1 and n ~ 6 N 1 , we replace m ~ and n ~ by m and n and get Since (2.6) is true for any m
2 uðm; nÞ 6
4W k ðr 1 ðm; n; 0; nÞÞ W 1 k
þ
m1 X n1 X
~f ðm; n; s; tÞ þ k
s¼0 t¼0
m1 X
D3 rk ðm; n; s; nÞ
3
5: /k W 1 k1 ðr k ð0; 0; s; 0ÞÞ
s¼0
~ð0; tÞ. This proves Theorem 2.1. h This is exactly (2.1) since r 1 ðm; n; 0; tÞ ¼ a Remark 2.2. ~ð0; 0Þ ¼ 0. Let (1) If a(m, n) = 0 for all m, n 2 N0, then a
r1;u1 ðm; n; s; tÞ ¼ r 1 ðm; n; s; tÞ þ u1 ; where u1 > 0 is given in W 1 ðuÞ ¼ r1;u1 ðm; n; s; tÞ, (2.9) becomes
Ru
dz . u1 w1 ðzÞ
Using the same arguments as in (2.8) where r1(m, n, s, t) is replaced with the positive
~ n ~ ; m; nÞÞ 6 W 1 ðr 1;u1 ðm; ~ n ~ ; 0; nÞÞ þ W 1 ðzðm; nÞ þ r 1;u1 ðm;
m1 X n1 X
~f 1 ðm; ~ n ~ ; s; tÞ ¼ W 1 ðu1 Þ þ
s¼0 t¼0
¼
m1 n1 XX
m1 n1 XX
~f 1 ðm; ~ n ~ ; s; tÞ
s¼0 t¼0
~f 1 ðm; ~ n ~; s; tÞ;
s¼0 t¼0
i.e.
" ~ n ~; m; nÞ ¼ zðm; nÞ þ u1 6 W 1 uðm; nÞ 6 zðm; nÞ þ r1;u1 ðm; 1
m1 n1 XX
# ~f 1 ðm; ~ n ~; s; tÞ ;
~ 06n6n ~; 0 6 m 6 m;
ð2:15Þ
s¼0 t¼0
which is the same as (2.10) with a complementary definition that W1(0) = 0. From Remark 2.1, the estimate of (2.15) is independent of u1. Then we similarly obtain (2.1) and all ri are defined by the same formula (2.3) where we define Wi(0) = 0 for i = 1, . . . , k. (2) If we fix m and vary n in the proof, then u(m, n) satisfies
2 4W k ða ~ðm; 0ÞÞ þ uðm; nÞ 6 W 1 k
m 1 X n1 X
~f ðm; n; s; tÞ þ k
s¼0 t¼0
n1 X t¼0
D4 r k ðm; n; m; tÞ
3
5; /k W 1 k1 ðr k ð0; 0; 0; tÞÞ
0 6 m 6 M1 ; 0 6 n 6 N1 ;
where rk(m, n, s, t) is determined by
~ðs; tÞ; r1 ðm; n; s; tÞ ¼ a riþ1 ðm; n; s; tÞ ¼ W i ðr1 ðm; n; s; 0ÞÞ þ
s1 X t1 X
s¼0 g¼0
~f ðm; n; s; gÞ þ i
t1 X
g¼0
D r ðm; n; s; gÞ 4 i ; /i W 1 i1 ðr i ð0; 0; 0; gÞÞ
i ¼ 1; . . . ; k 1;
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D4 r i ðm; n; s; tÞ ¼ ri ðm; n; s; t þ 1Þ r i ðm; n; s; tÞ;
i ¼ 1; . . . ; k;
M1 and N1 are positive integers satisfying
~ðM1 ; 0ÞÞ þ W i ða
M 1 1 N 1 1 X X s¼0
~f ðM 1 ; N1 ; s; tÞ þ i
t¼0
NX 1 1 t¼0
D4 ri ðM1 ; N1 ; M 1 ; tÞ 6 /i W 1 i1 ðr i ð0; 0; 0; tÞÞ
Z
1 ui
dz ; wi ðzÞ
i ¼ 1; . . . ; k
~ðm; nÞ is nondecreasing in n by and other functions are defined in Theorem 2.1. Here we replace the condition in (C1) that D1 a ~ðm; nÞ ¼ a ~ðm; n þ 1Þ a ~ðm; nÞ is nondecreasing in m. the condition that D2 a 3. Some corollaries Assume that u 2 C(R+, R+) is a strictly increasing function with u(1) = 1 where R+ = [0, 1). Consider the inequality
uðuðm; nÞÞ 6 aðm; nÞ þ
k m1 n1 X XX i¼1
fi ðm; n; s; tÞwi ðuðs; tÞÞ;
m; n 2 N0 :
ð3:1Þ
s¼0 t¼0
Corollary 3.1. Suppose that (C1)–(C3) hold. If u(m, n) in (3.1) is nonnegative for m, n 2 N0, then
2
uðm; nÞ 6 u
0
1 4 f 1 @ f ~ Wk W k ðað0; nÞÞ
þ
m1 X n1 X
~f ðm; n; s; tÞ þ k
s¼0 t¼0
m 1 X s¼0
13 D3 r k ðm; n; s; nÞ A5 f 1 ðrk ð0; 0; s; 0ÞÞ ^k W / k1
1 dz f i ðuÞ ¼ R u fi; W f 0 ¼ I (Identity), / f i is the inverse of W ^ i ðuÞ ¼ wi ðu ðuÞÞ ; for 0 6 m 6 M1 and 0 6 n 6 N1 where W ;W ui wi ðu1 ðzÞÞ wi1 ðu1 ðuÞÞ ^ 1 ðuÞ ¼ w1 u1 ðuÞ , and other related functions are defined as in Theorem 2.1 by replacing wi(u) with wi(u1(u)). /
Proof of Corollary 3.1. Let n(m, n) = u(u(m, n)). Then (3.1) becomes
nðm; nÞ 6 aðm; nÞ þ
k m1 n1 X XX i¼1
fi ðm; n; s; tÞwi u1 ðnðs; tÞÞ ;
m; n 2 N0 :
s¼0 t¼0
Note that wi(u1(u)) satisfy (C3) for i = 1, . . . , k. Using Theorem 2.1, we obtain the estimate about n(m, n) by replacing wi(u) with wi(u1(u)). Then use the fact that u(m, n) = u1(n(m, n)) and we get Corollary 3.1. h If u(u) = up where p > 0, then (3.1) reads
up ðm; nÞ 6 aðm; nÞ þ
k m 1 X n1 X X i¼1
fi ðm; n; s; tÞwi ðuðs; tÞÞ;
m; n 2 N0 :
ð3:2Þ
s¼0 t¼0
Directly using Corollary 3.1, we have the following result. Corollary 3.2. Suppose that (C1)–(C3) hold. If u(m, n) in (3.2) is nonnegative for m, n 2 N0, then
2
0
f k ða f 1 @ W ~ð0; nÞÞ þ uðm; nÞ 6 4 W
m1 X n1 X
k
~f ðm; n; s; tÞ þ k
m1 X
s¼0 t¼0
s¼0
131=p D3 rk ðm; n; s; nÞ A5 f 1 ðr k ð0; 0; s; 0ÞÞ ^k W / k1
1=p R f i ðuÞ ¼ u dz ; W fi; W f 0 ¼ I (Identity), / f i is the inverse of W ^ i ðuÞ ¼ wi ðu Þ ; / ^ 1 ðuÞ ¼ for 0 6 m 6 M1 and 0 6 n 6 N1 where W ui wi ðz1=p Þ wi1 ðu1=p Þ 1=p w1 u , and other related functions are defined as in Theorem 2.1 by replacing wi(u) with wi(u1/p).
4. Applications to Volterra type difference equations In this section, we apply Theorem 2.1 to study boundedness and uniqueness of solutions of a nonlinear delay difference equation of the form
yðm; nÞ ¼ bðm; nÞ þ
m1 n1 XX s¼0 t¼0
Fðm; n; s; t; yðs; tÞÞ þ
m1 n1 XX
Hðm; n; s; t; yðs; tÞÞ
s¼0 t¼0
for m, n 2 N0 where y : N20 ! R is an unknown function, b maps from N20 to R, and F and H map from N40 R to R. Theorem 4.1. Suppose that b(0, 0) – 0 and the functions F and H in (4.1) satisfy the conditions
ð4:1Þ
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pffiffiffiffiffiffi jFðm; n; s; t; yÞj 6 f1 ðm; n; s; tÞ jyj; jHðm; n; s; t; yÞj 6 f2 ðm; n; s; tÞjyj; where f1 ; f2 :
N40
ð4:2Þ
! ½0; 1Þ. If y(m, n) is a solution of (4.1) on
2
N20 ,
then
3 Pn1 ~ ~ðs;nÞ D1 a p ffiffiffiffiffiffiffiffi m 1 X n1 m 1 t¼0 f 1 ðm; n; s; tÞ þ X X ~ðs;nÞ7 a ~f 2 ðm; n; s; tÞ þ ~ð0; nÞ exp 6 jyðm; nÞj 6 a 4 5; hðsÞ s¼0 t¼0 s¼0
ð4:3Þ
where
~ðm; nÞ ¼ a
max
06s6m;06g6n;s;g2N0
~f 1 ðm; n; s; tÞ ¼ ~f 2 ðm; n; s; tÞ ¼ hðsÞ ¼
jbðs; gÞj;
max
f1 ðs; g; s; tÞ;
max
f2 ðs; g; s; tÞ;
06s6m;06g6n;s;g2N0 06s6m;06g6n;s;g2N0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi s1 1X D1 a~ðs; 0Þ ~ð0; 0Þ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : a 2 s¼0 a ~ðs; 0Þ
Proof. Using (4.1) and (4.2), the solution y(m, n) satisfies
uðm; nÞ 6 aðm; nÞ þ
m1 n1 XX
f1 ðm; n; s; tÞw1 ðuðs; tÞÞ þ
s¼0 t¼0
m1 n1 XX
f2 ðm; n; s; tÞw2 ðuðs; tÞÞ;
m; n 2 N0 ;
ð4:4Þ
s¼0 t¼0
where
uðm; nÞ ¼ jyðm; nÞj;
aðm; nÞ ¼ jbðm; nÞj;
w1 ðuÞ ¼
pffiffiffi u;
w2 ðuÞ ¼ u:
~ðm; nÞ > 0 for all m, n 2 N0 since b(0, 0) – 0. For positive constants u1, u2, we have Clearly, a
W 1 ðuÞ ¼
Z
u
u1 u
Z
pffiffiffi pffiffiffiffiffi dz ¼ 2 u u1 ; w1 ðzÞ
W 1 1 ðuÞ ¼
u 2
þ
pffiffiffiffiffi2 u1 ;
dz u ¼ ln ; W 1 W 2 ðuÞ ¼ 2 ðuÞ ¼ u2 expðuÞ; u2 u2 w2 ðzÞ ~ðs; tÞ > 0; r 1 ðm; n; 0; tÞ ¼ a ~ð0; tÞ; r 1 ðm; n; s; tÞ ¼ a qffiffiffiffiffiffiffiffiffiffiffiffiffi X s1 X t1 s1 X pffiffiffiffiffi D1 a~ðs; tÞ ~f 1 ðm; n; s; gÞ þ ~ð0; tÞ u1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; a r 2 ðm; n; s; tÞ ¼ 2 ~ðs; 0Þ a s¼0 g¼0 s¼0
D3 r2 ðm; n; s; tÞ ¼
t1 X
D1 a~ðs; tÞ ~f 1 ðm; n; s; gÞ þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ~ðs; 0Þ a g¼0
/2 ðuÞ ¼
w2 ðuÞ pffiffiffi ¼ u: w1 ðuÞ
It is obvious that w1 and w2 satisfy (C3). Applying Theorem 2.1 gives
2
m1 X n1 m1 X X ~f 2 ðm; n; s; tÞ þ ~ð0; nÞ exp 6 uðm; nÞ 6 a 4 s¼0 t¼0
s¼0
3 Pn1 ~ ~ðs;nÞ D1 a p ffiffiffiffiffiffiffiffi g¼0 f 1 ðm; n; s; gÞ þ a~ðs;0Þ 7 5; hðsÞ
ð4:5Þ
which implies (4.3). h Theorem 4.2. Suppose that b(0, 0) – 0 and the functions F and H in (4.1) satisfy the conditions
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jFðm; n; s; t; y1 Þ Fðm; n; s; t; y2 Þj 6 f1 ðn; sÞ jy1 y2 j; jHðm; n; s; t; y1 Þ Hðm; n; s; t; y2 Þj 6 f2 ðn; sÞjy1 y2 j;
ð4:6Þ
where f1 ; f2 : N40 ! ½0; 1Þ. Then (4.1) has at most one solution on N20 . Proof. Let y1(m, n) and y2(m, n) be two solutions of (4.1) on N20 . From (4.6), we have
uðm; n; s; tÞ 6
m 1 X n1 X s¼0 t¼0
f1 ðm; n; s; tÞw1 ðuðs; tÞÞ þ
m 1 X n1 X s¼0 t¼0
f2 ðm; n; s; tÞw2 ðuðs; tÞÞ;
m; n 2 N0 ;
ð4:7Þ
S. Deng / Applied Mathematics and Computation 217 (2010) 2217–2225
2225
pffiffiffi where u(m, n) = jy1(m, n) y2(m, n)j, a(m, n) = 0, w1 ðuÞ ¼ u and w2(u) = u. Apply Theorem 2.1, (1) of Remark 2.2 and the notation Wi(0) = 0 for i = 1, 2, we obtain that u(m, n) = 0 which implies that the solution is unique. h Acknowledgement This paper is supported by China Postdoctoral Science Foundation (20100470108). References [1] R.P. Agarwal, On an integral inequality in n independent variables, J. Math. Anal. Appl. 85 (1982) 192–196. [2] R.P. Agarwal, S. Deng, W. Zhang, Generalization of a retarded Gronwall-like inequality and its applications, Appl. Math. Comput. 165 (2005) 599–612. [3] I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hung. 7 (1956) 81–94. [4] S. Borysenko, G. Matarazzo, M. Pecoraro, A generalization of Bihari’s lemma for discontinuous functions and its application to the stability problem of differential equations with impulse disturbance, Georgian Math. J. 13 (2006) 229–238. [5] W. Cheung, Some new nonlinear inequalities and applications to boundary value problems, Nonlinear Anal. 64 (2006) 2112–2128. [6] W. Cheung, Some discrete nonlinear inequalities and applications to boundary value problems for difference equations, J. Difference Equ. Appl. 10 (2004) 213–223. [7] W. Cheung, J. Ren, Discrete non-linear inequalities and applications to boundary value problems, J. Math. Anal. Appl. 319 (2006) 708–724. [8] S.K. Choi, S. Deng, N.J. Koo, W. Zhang, Nonlinear integral inequalities of Bihari-type without class H, Math. Inequal. Appl. 8 (2005) 643–654. [9] S. Deng, Nonlinear delay discrete inequalities and their applications to Volterra type difference equations, submitted for publication. [10] S. Deng, C. Prather, Nonlinear discrete inequalities of Bihari-type, submitted for publication. [11] L. Horväth, Generalizations of special Bihari type integral inequalities, Math. Inequal. Appl. 8 (2005) 441–449. [12] N. Lungu, On some Gronwall–Bihari–Wendorff-type inequalities, Semin. Fixed Point Theory Cluj-Napoca 3 (2002) 249–254. [13] B.G. Pachpatte, On generalizations of Bihari’s inequality, Soochow J. Math. 31 (2005) 261–271. [14] B.G. Pachpatte, Integral inequalities of the Bihari type, Math. Inequal. Appl. 5 (2002) 649–657. [15] B.G. Pachpatte, On Bihari like integral and discrete inequalities, Soochow J. Math. 17 (1991) 213–232. [16] V.N. Phat, J.Y. Park, On the Gronwall inequality and asymptotic stability of nonlinear discrete systems with multiple delays, Dyn. Syst. Appl. 10 (2001) 577–588. [17] M. Pinto, Integral inequalities of Bihari-type and applications, Funkcial. Ekvac. 33 (1990) 387–403. [18] J. Popenda, R.P. Agarwal, Discrete Gronwall inequalities in many variables, Comput. Math. Appl. 38 (1999) 63–70. [19] Sh. Salem, K.R. Raslan, Some new discrete inequalities and their applications, JIPAM J. Inequal. Pure Appl. Math. 5 (2004) (article 2, 9 pp.). [20] F. Wong, C. Yeh, C. Hong, Gronwall inequalities on time scales, Math. Inequal. Appl. 9 (2006) 75–86. [21] K. Zheng, Y. Wu, S. Zhong, Discrete nonlinear integral inequalities in two variables and their applications, Appl. Math. Comput. 207 (2009) 140–147. [22] W. Zhang, S. Deng, Projected Gronwall–Bellman’s inequality for integrable functions, Math. Comput. Modell. 34 (2001) 393–402.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Nonlinear discrete inequalities with two variables and their applications Shengfu Deng Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, PR China
a r t i c l e
i n f o
a b s t r a c t An error in the proof of Theorem 1 by Zheng et al. [K. Zheng, Y. Wu, S. Zhong, Discrete nonlinear integral inequalities in two variables and their applications, Appl. Math. Comput. 207 (2009) 140–147] is reported. This paper gives the right proof under some additional condition. Application examples to show boundedness and uniqueness of solutions of a Volterra type difference equation are also given. Ó 2010 Elsevier Inc. All rights reserved.
Keywords: Discrete inequality Nonlinear Volterra type Boundedness Uniqueness
1. Introduction Gronwall–Bellman inequalities and their various linear and nonlinear generalizations play very important roles in the discussion of existence, uniqueness, continuation, boundedness, and stability properties of solutions of differential equations and difference equations. The literature on such inequalities and their applications is vast. For example, see [1–5,8,11– 14,17,22] for continuous cases, and [6,7,9,10,15,16,18–21] for discrete cases. Recently, Zheng et al. [21] considered the following discrete inequality
uðm; nÞ 6 aðm; nÞ þ
k m 1 X n1 X X
fi ðm; n; s; tÞwi ðuðs; tÞÞ;
m; n 2 N0
ð1:1Þ
s¼0 t¼0
i¼1
with k nonlinear terms where N0 = {0, 1, 2, . . . }. In order to prove their Theorem 1, they constructed an auxiliary inequality
~1 Þ þ ~ 1; n uðm; nÞ 6 b1 ðm
k m 1 X n1 X X i¼1
~f ðm ~ 1 ; s; t Þwi ðuðs; tÞÞ; i ~ 1; n
~ 06n6n ~ 0 6 m 6 m;
ð1:2Þ
s¼0 t¼0
~ and n ~ so that the term b1 ðm ~ 1 Þ is a constant with respect to variables m and n. They gave a ~ 1; n for some positive integers m claim for the inequality (1.2). Firstly, they proved that their claim is true for k = 1. Then they assumed that their claim is also true for k. For k + 1, (1.2) was changed into
nðm; nÞ 6 c1 ðm; nÞ þ
k X m1 X n1 X i¼1
~f ðm ~ 1 ; s; t Þ/i ðnðs; tÞÞ: iþ1 ~ 1 ; n
ð1:3Þ
s¼0 t¼0
They said that (1.3) has the same form as (1.2) so they applied their inductive assumption. However, this is not right because ~ 1 Þ in (1.2) is a constant with respect to m and n. In this paper, we require ~ 1; n c1(m, n) in (1.3) is a function of m and n but b1 ðm that a(m, n) satisfies some additional condition, and give not only the right proof but examples to show boundedness and uniqueness of solutions of a Volterra type difference equation. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.07.022
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2. Main results ~ðm; nÞ ¼ max06s6m;06g6n;s;g2N0 aðs; gÞ and define D1u(m, n) = u(m + 1, n) u(m, n). Assume that: Let a ~ðm; nÞ is nondecreasing in n; (C1) a (m, n) is nonnegative for m, n 2 N0 and a(0, 0) > 0; D1 a (C2) all fi(m, n, s, t) (i = 1, . . . , k) are nonnegative for m, n, s, t 2 N0; (C3) all wi (i = 1, . . . , k) are continuous and nondecreasing functions on [0, 1) and are positive on (0, 1). They satisfy the w relationship w1 / w3 / / wm where wi / wi+1 means that wiþ1i is nondecreasing on (0, 1) (see [17]). Let W i ðuÞ ¼
Ru
for u P ui where ui > 0 is a given constant. Then, Wi is strictly increasing so its inverse W 1 is well dei
dz ui wi ðzÞ
fined, continuous and increasing in its corresponding domain. Theorem 2.1. Suppose that (C1)–(C3) hold and u (m, n) is a nonnegative function for m, n 2 N0 satisfying (1.1) . Then
2 uðm; nÞ 6
4W k ða ~ð0; nÞÞ W 1 k
þ
m1 X n1 X
~f ðm; n; s; tÞ þ k
s¼0 t¼0
m 1 X s¼0
D3 rk ðm; n; s; nÞ
3
5; /k W 1 k1 ðr k ð0; 0; s; 0ÞÞ
0 6 m 6 M1 ; 0 6 n 6 N1 ;
ð2:1Þ
where
~f ðm; n; s; tÞ ¼ i
max
06s6m;06g6n;s;g2N0
fi ðs; g; s; tÞ;
ð2:2Þ
rk(m, n, s, t) is determined recursively by
~ðs; tÞ; r 1 ðm; n; s; tÞ ¼ a r iþ1 ðm; n; s; tÞ ¼ W i ðr 1 ðm; n; 0; tÞÞ þ
s1 X t1 X
~f ðm; n; s; gÞ i
s¼0 g¼0
þ
s1 X
s¼0
ð2:3Þ
D3 r i ðm; n; s; tÞ
; /i W 1 i1 ðr i ð0; 0; s; 0ÞÞ
D3 ri ðm; n; s; tÞ ¼ r i ðm; n; s þ 1; tÞ r i ðm; n; s; tÞ; /i ðuÞ ¼
wi ðuÞ , wi1 ðuÞ
i ¼ 1; . . . ; k 1; i ¼ 1; . . . ; k;
/1(u) = w1(u), W0 = I (Identity), and M1 and N1 are positive integers satisfying
~ð0; N1 ÞÞ þ W i ða
M 1 1 N 1 1 X X s¼0
M 1 1 X
~f ðM1 ; N1 ; s; tÞ þ i
t¼0
Remark 2.1. If all wi(i = 1, . . . , k) satisfy not affect our results (see [2]).
s¼0
R1 ui
dz wi ðzÞ
D r ðM ; N ; s; N1 Þ 3 i 1 1 6 /i W 1 i1 ðr i ð0; 0; s; 0ÞÞ
Z
1
ui
dz ; wi ðzÞ
i ¼ 1; . . . ; k:
ð2:4Þ
¼ 1, then we may choose N1 = 1 and M1 = 1. Different choices of ui in Wi do
Lemma 2.1. For i = 1, . . . , k, D3ri(m, n, s, t) is nonnegative and nondecreasing in m, n and t, and ri(m, n, s, t) is nonnegative and nondecreasing in its arguments. ~ðm; nÞ and ~f i ðm; n; s; tÞ, it is easy to check that they are nonnegative and nondecreasing in m and Proof. By the definitions of a ~ðm; nÞ P aðm; nÞ and ~f i ðm; n; s; tÞ P fi ðm; n; s; tÞ where i = 1, . . . , k. a(0, 0) > 0 in (C1) implies that a ~ðm; nÞ > 0 for all n, and a ~ðs; 0Þ > 0, we have m 6 M1 and n 6 N1. Clearly, using the fact r 1 ð0; 0; s; 0Þ ¼ a
D3 r 1 ðm þ 1; n; s; tÞ D3 r1 ðm; n; s; tÞ ¼ 0;
D3 r 2 ðm þ 1; n; s; tÞ D3 r2 ðm; n; s; tÞ ¼
t1 X
~f 1 ðm þ 1; n; s; jÞ
t1 X
j¼0
j¼0
~f 1 ðm; n; s; jÞ þ D3 r 1 ðm þ 1; n; s; tÞ D3 r 1 ðm; n; s; tÞ P 0; w1 ðr 1 ð0; 0; s; 0ÞÞ
which yield that D3r1(m, n, s, t) and D3r2(m, n, s, t) are nondecreasing in m. Assume that D3rl(m, n, s, t) is nondecreasing in m. Then
D3 r lþ1 ðm þ 1; n; s; tÞ D3 r lþ1 ðm; n; s; tÞ ¼
t1 X j¼0
~f ðm þ 1; n; s; jÞ l
t1 X
1; n; s; tÞ D3 r l ðm; n; s; tÞ ~f ðm; n; s; jÞ þ D3 r l ðm þ P 0; l 1 / W j¼0 l l1 ðr l ð0; 0; s; 0ÞÞ
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S. Deng / Applied Mathematics and Computation 217 (2010) 2217–2225
which implies that D3rl+1(m, n, s, j) is nondecreasing in m. By induction, D3ri(m, n, s, j) is nondecreasing in m. Similarly, since ~ðm; nÞ is nondecreasing in n, we can prove that D3ri(m, n, s, j) is also nondecreasing in n and t, and nonnegative by inducD1 a tion again. Then ri(m, n, s, t) is nonnegative and nondecreasing in its arguments. h ~ 6 N 1 . Consider an auxiliary inequality ~ and n ~ where m ~ 6 M1 ; n Proof of Theorem 2.1. Take any arbitrary positive integers m
~ n ~ ; m; nÞ þ uðm; nÞ 6 r 1 ðm;
k m1 n1 X XX
~f ðm; ~ ; s; tÞwi ðuðs; tÞÞ i ~ n
ð2:5Þ
s¼0 t¼0
i¼1
~ 06n6n ~ . Claim that u(m, n) in (2.5) satisfies for 0 6 m 6 m;
2 4W k ðr 1 ðm; ~ n ~ ; 0; nÞÞ þ uðm; nÞ 6 W 1 k
m1 X n1 X
~f ðm; ~ ; s; tÞ þ k ~ n
m1 X
s¼0 t¼0
s¼0
~ n ~; s; nÞ D3 r k ðm;
3
5 /k W 1 k1 ðr k ð0; 0; s; 0ÞÞ
ð2:6Þ
~ M 2 g and 0 6 n 6 minfn ~ ; N 2 g where M2 and N2 are positive integers satisfying for 0 6 m 6 minfm;
~ n ~ ; 0; N 2 ÞÞ þ W i ðr 1 ðm;
M 2 1 N 2 1 X X s¼0
~f ðm; ~; s; tÞ þ i ~ n
t¼0
M 2 1 X
~ n ~ ; s; N2 Þ D r ðm; 3 i 6 1 /i W i1 ðr i ð0; 0; s; 0ÞÞ
s¼0
Z
1 ui
dz ; wi ðzÞ
i ¼ 1; . . . ; k:
ð2:7Þ
~ n ~ ; m; nÞ; D3 r i ðm; ~ n ~ ; m; nÞ and ~f i ðm; ~ n ~ ; m; nÞ Before we prove (2.6), notice that we may choose M1 6 M2 and N1 6 N2. In fact, r i ðm; ~ and n ~ by Lemma 2.1. Thus, M2 and N2 satisfying (2.7) get smaller as m ~ and n ~ are chosen larger. In are nondecreasing in m ~ ¼ M 1 and n ~ ¼ N1 . particular, M2 and N2 satisfy the same (2.4) as M1 and N1 for m We divide the proof of (2.6) into two steps by using a mathematical introduction. Step 1. k = 1. P Pn1 ~ ~ ~ Let zðm; nÞ ¼ m1 s¼0 t¼0 f 1 ðm; n; s; tÞw1 ðuðs; tÞÞ and z(0, n) = 0. It is clear that z(m, n) is nonnegative and nondecreasing in m ~ n ~ ; m; nÞ þ zðm; nÞ for 0 6 m 6 m; ~ 06n6n ~ and by Lemma 2.1 and n. Observe that (2.5) is equivalent to uðm; nÞ 6 r 1 ðm;
D1 zðm; nÞ ¼
n1 X
~f 1 ðm; ~ n ~ ; m; tÞw1 ðuðm; tÞÞ 6
n1 X
t¼0
~f 1 ðm; ~ n ~ ; m; tÞw1 ðr 1 ðm; ~ n ~ ; m; tÞ þ zðm; tÞÞ
t¼0
~ n ~ ; m; nÞ þ zðm; nÞÞ 6 w1 ðr1 ðm;
n1 X
~f 1 ðm; ~ n ~; m; tÞ:
t¼0
~ n ~ ; m; nÞ > 0, we have Since w1 is nondecreasing and r1 ðm; n1 ~ n ~ ; m; nÞ X ~ n ~; m; nÞ D1 zðm; nÞ þ D3 r 1 ðm; D3 r 1 ðm; ~f 1 ðm; ~ n ~ ; m; tÞ þ 6 ~ n ~ ; m; nÞÞ ~ n ~ ; m; nÞÞ w1 ðzðm; nÞ þ r 1 ðm; w ð zðm; nÞ þ r 1 1 ðm; t¼0 n1 X
~ n ~ ; m; nÞ D r ðm; ~f 1 ðm; ~ n ~ ; m; tÞ þ 3 1 : w ðr ð0; 0; m; 0ÞÞ 1 1 t¼0
6
ð2:8Þ
Then
Z
~ n ~ ;mþ1;nÞ zðmþ1;nÞþr 1 ðm;
~ n ~ ;m;nÞ zðm;nÞþr 1 ðm;
ds 6 w1 ðsÞ 6
Z
~ n ~ ;mþ1;nÞ zðmþ1;nÞþr 1 ðm;
~ n ~ ;m;nÞ zðm;nÞþr1 ðm;
~ n ~; m; nÞ ds D1 zðm; nÞ þ D3 r 1 ðm; 6 ~ n ~ ; m; nÞÞ w1 ðzðm; nÞ þ r1 ðm; ~ n ~ ; m; nÞÞ w1 ðzðm; nÞ þ r 1 ðm;
n1 X
~ n ~ ; m; nÞ D r ðm; ~f 1 ðm; ~ n ~ ; m; tÞ þ 3 1 ; w1 ðr 1 ð0; 0; m; 0ÞÞ t¼0
so
Z
~ n ~ ;m;nÞ zðm;nÞþr 1 ðm;
~ n ~ ;0;nÞ zð0;nÞþr 1 ðm;
m1 X ds ¼ w1 ðsÞ s¼0
Z
~ n ~ ;sþ1;nÞ zðsþ1;nÞþr 1 ðm;
~ n ~ ;s;nÞ zðs;nÞþr1 ðm;
m1 n1 m1 X D3 r 1 ðm; XX ~ n ~ ; s; nÞ ds ~f 1 ðm; ~ n ~; s; tÞ þ 6 : w w1 ðsÞ s¼0 t¼0 ðr ð0; 0; s; 0ÞÞ 1 1 s¼0
The definition of W1 in Theorem 2.1 and z(0, n) = 0 show
~ n ~ ; m; nÞÞ 6 W 1 ðr 1 ðm; ~ n ~; 0; nÞÞ þ W 1 ðzðm; nÞ þ r 1 ðm;
m 1 X n1 X
~f 1 ðm; ~ n ~ ; s; tÞ þ
s¼0 t¼0
~ 06n6n ~: 0 6 m 6 m; Eq. (2.7) shows that the right side of (2.9) is in the domain of W 1 1 implies
m 1 X s¼0
~ n ~ ; s; nÞ D3 r 1 ðm; ; w1 ðr 1 ð0; 0; s; 0ÞÞ ð2:9Þ
W 1 1
~ and 0 6 n 6 n ~ . Thus the monotonicity of for all 0 6 m 6 m
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" ~ n ~ ; m; nÞ 6 W 1 ~ n ~; 0; nÞÞ þ uðm; nÞ 6 zðm; nÞ þ r1 ðm; W 1 ðr 1 ðm; 1
m1 X n1 X
~f 1 ðm; ~ n ~ ; s; tÞ þ
s¼0 t¼0
m1 X s¼0
~ n ~ ; s; nÞ D3 r 1 ðm; w1 ðr 1 ð0; 0; s; 0ÞÞ
# ð2:10Þ
~ and 0 6 n 6 n ~ , i.e., (2.6) is true for k = 1. for 0 6 m 6 m Step 2. k + 1. Assume that (2.6) is true for k. Consider
~ n ~ ; m; nÞ þ uðm; nÞ 6 r 1 ðm;
kþ1 m 1 X n1 X X
~f ðm; ~ ; s; tÞwi ðuðs; tÞÞ i ~ n
s¼0 t¼0
i¼1
Pkþ1 Pm1 Pn1 ~ ~ and 0 6 n 6 n ~ . Let zðm; nÞ ¼ i¼1 ~ ~ for 0 6 m 6 m s¼0 t¼0 f i ðm; n; s; tÞwi ðuðs; tÞÞ and z(0, n) = 0. Then z(m, n) is nonnegative ~ n ~ ; m; nÞ þ zðm; nÞ for 0 6 m 6 m ~ and 0 6 n 6 n ~ . Moreover, we and nondecreasing in m and n and satisfies uðm; nÞ 6 r 1 ðm; have
D1 zðm; nÞ ¼
kþ1 X n1 X i¼1
~f i ðm; ~ n ~; m; tÞwi ðuðm; tÞÞ 6
t¼0
kþ1 X n1 X i¼1
~f i ðm; ~ n ~ ; m; tÞwi ðr 1 ðm; ~ n ~; m; tÞ þ zðm; tÞÞ:
t¼0
~ n ~ ; m; nÞ > 0, we have by Lemma 2.1 Since wi is nondecreasing and r1 ðm;
Pkþ1 Pn1 ~ ~ ~ ~ ~ ~ n ~ ; m; nÞ D3 r 1 ðm; i¼1 t¼0 f i ðm; n; m; tÞwi ðzðm; tÞ þ r 1 ðm; n; m; tÞÞ þ ~ ~ ~ ~; m; nÞÞ w1 ðzðm; nÞ þ r 1 ðm; n; m; nÞÞ w1 ðr 1 ðm; n
~ n ~; m; nÞ D1 zðm; nÞ þ D3 r 1 ðm; 6 ~ ~ w1 ðzðm; nÞ þ r1 ðm; n; m; nÞÞ
n1 X
kþ1 X n1 ~ n ~ ; m; nÞ X ~ n ~ ; m; tÞÞ D r ðm; w ðzðm; tÞ þ r1 ðm; ~f 1 ðm; ~f ðm; ~ n ~ ; m; tÞ þ 3 1 ~ ; m; tÞ i þ i ~ n ~ ~; m; tÞÞ w w ðr ð0; 0; m; 0ÞÞ ðzðm; tÞ þ r ð m; n 1 1 1 1 t¼0 i¼2 t¼0
6
k X n1 ~ n ~ ; m; nÞ X D r ðm; ~f 1 ðm; ~f ðm; ~ iþ1 ðzðm; tÞ þ r 1 ðm; ~ n ~ ; m; tÞ þ 3 1 ~ ; m; tÞ/ ~ n ~ ; m; tÞÞ þ iþ1 ~ n w1 ðr 1 ð0; 0; m; 0ÞÞ i¼1 t¼0 t¼0
n1 X
6
~ iþ1 ðuÞ ¼ wiþ1 ðuÞ for i = 1, . . . , k, which gives ~ and 0 6 n 6 n ~ where / for 0 6 m 6 m w1 ðuÞ
Z
~ n ~ ;mþ1;nÞ zðmþ1;nÞþr1 ðm;
~ n ~ ;m;nÞ zðm;nÞþr 1 ðm;
ds 6 w1 ðsÞ 6
Z
~ n ~ ;mþ1;nÞ zðmþ1;nÞþr 1 ðm;
~ n ~ ;m;nÞ zðm;nÞþr1 ðm;
~ n ~ ; m; nÞ ds D1 zðm; nÞ þ D3 r 1 ðm; 6 ~ n ~ ; m; nÞÞ w1 ðr 1 ðm; ~ n ~ ; m; nÞ þ zðm; nÞÞ w1 ðzðm; nÞ þ r 1 ðm;
k X n1 ~ n ~ ; m; nÞ X D r ðm; ~f 1 ðm; ~f iþ1 ðm; ~ iþ1 ðzðm; tÞ þ r 1 ðm; ~ n ~; m; tÞ þ 3 1 ~ n ~ ; m; tÞ/ ~ n ~ ; m; tÞÞ: þ w1 ðr 1 ð0; 0; m; 0ÞÞ i¼1 t¼0 t¼0
n1 X
Therefore,
Z
~ n ~ ;m;nÞ zðm;nÞþr 1 ðm;
~ n ~ ;0;nÞ zð0;nÞþr 1 ðm;
m 1 X n1 m 1 k m 1 X n1 X X X ~ n ~ ; s; nÞ X ds D3 r 1 ðm; ~f 1 ðm; ~f ðm; ~ iþ1 ðzðs; tÞ ~ n ~ ; s; tÞ þ ~ ; s; tÞ/ 6 þ iþ1 ~ n w1 ðr 1 ð0; 0; s; 0ÞÞ i¼1 s¼0 t¼0 w1 ðsÞ s¼0 t¼0 s¼0
~ n ~ ; s; tÞÞ; þ r 1 ðm; that is
~ n ~ ; m; nÞÞ 6 W 1 ðr1 ðm; ~ n ~ ; 0; nÞÞ þ W 1 ðzðm; nÞ þ r 1 ðm;
m1 n1 XX
~f 1 ðm; ~ n ~ ; s; tÞ þ
s¼0 t¼0
þ
k X m1 X n1 X
m1 X s¼0
~ n ~; s; nÞ D3 r 1 ðm; w1 ðr 1 ð0; 0; s; 0ÞÞ
~f ðm; ~ iþ1 ðzðs; tÞ þ r1 ðm; ~ ; s; tÞ/ ~ n ~ ; s; tÞÞ; iþ1 ~ n
s¼0 t¼0
i¼1
or equivalently
~ n ~ ; m; nÞ þ nðm; nÞ 6 c1 ðm;
k m1 n1 X XX i¼1
~f ðm; ~ iþ1 ðW 1 ðnðs; tÞÞÞ ~; s; tÞ/ iþ1 ~ n 1
s¼0 t¼0
~ and 0 6 n 6 n ~ the same as (2.5) for k where for 0 6 m 6 m
~ n ~ ; m; nÞÞ; nðm; nÞ ¼ W 1 ðzðm; nÞ þ r1 ðm; ~ n ~ ; m; nÞ ¼ W 1 ðr 1 ðm; ~ n ~ ; 0; nÞÞ þ c1 ðm;
m1 X n1 X s¼0 t¼0
~f 1 ðm; ~ n ~ ; s; tÞ þ
m 1 X s¼0
~ n ~ ; s; nÞ D3 r 1 ðm; : w1 ðr 1 ð0; 0; s; 0ÞÞ
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~ iþ1 ðW 1 Þ, i = 1, . . . , k, is continuous and nondecreasing on [0, 1) and is positive on (0, 1) From the assumption (C3), each / 1 1 ~ 2 W 1 / / / ~ kþ1 W 1 . By the inductive ~ 1 W 1 / / since W is continuous and nondecreasing on [0, 1). Moreover, / 1
1
1
1
assumption, we have
2 1 4 kþ1
~ n ~ ; 0; nÞÞ þ Ukþ1 ðc1 ðm;
nðm; nÞ 6 U
m 1 X n1 X
~f ðm; ~ ; s; tÞ þ kþ1 ~ n
m 1 X
s¼0 t¼0
s¼0
~ ; M 3 g and 0 6 n 6 minfn ~ ; N 3 g where Uiþ1 ðuÞ ¼ for 0 6 m 6 minfn
U1 iþ1 is the inverse of Uiþ1 ; wiþ1 ðuÞ ¼
~ ðW 1 ðuÞÞ / iþ1 1 ~ ðW 1 ðuÞÞ / i
1
~ n ~ ; m; nÞ ¼ Uiþ1 ðc1 ðm; ~ n ~ ; 0; nÞÞ þ ciþ1 ðm;
¼
wiþ1 ðW 1 1 ðuÞÞ wi ðW 1 1 ðuÞÞ
m1 n1 XX
3 ~ n ~ ; s; nÞ D3 ck ðm; 5 wkþ1 U1 ðc ð0; 0; s; 0ÞÞ k k
Ru
dz ~ iþ1 / ~ ðW 1 ðzÞÞ ; u iþ1 1
ð2:11Þ
~ iþ1 ¼ W 1 ðuiþ1 Þ; u > 0; U1 ¼ I (Identity), u
, i = 1, . . . , k
~f ðm; ~ ; s; tÞ þ iþ1 ~ n
m1 X
s¼0 t¼0
s¼0
~ n ~ ; s; nÞ D c ðm; 3 i ; 1 wiþ1 Ui ðci ð0; 0; s; 0ÞÞ
i ¼ 1; . . . ; k 1;
and M3 and N3 are positive integers satisfying
~ n ~ ; 0; N3 ÞÞ þ Uiþ1 ðc1 ðm;
M 3 1 N 3 1 X X s¼0
~f iþ1 ðm; ~ n ~ ; s; tÞ þ
t¼0
M 3 1 X s¼0
~ n ~ ; s; N3 Þ D3 ci ðm; 6 wiþ1 U1 ðc ð0; 0; s; 0ÞÞ i i
Z
W 1 ð1Þ
~ iþ1 u
dz ; ~ iþ1 W 1 ðzÞ / 1
i ¼ 1; . . . ; k: ð2:12Þ
Note that
Ui ðuÞ ¼
Z
u
~i u
dz
¼ ~ i W 1 ðzÞ / 1
Z
u
W 1 ðui Þ
Z W 1 w1 W 1 1 ðuÞ 1 ðzÞ dz dz ¼ ¼ W i W 1 1 ðuÞ; 1 w ðzÞ i u wi W 1 ðzÞ i
i ¼ 2; . . . ; k þ 1
and
wiþ1
wiþ1 W 1 U1 ðuÞ wiþ1 W 1 W 1 W 1 ðuÞ wiþ1 W 1 ðuÞ 1 1 i i i ¼ ¼ ¼ /iþ1 W 1 U1 ðuÞ ; i i ðuÞ ¼ U1 W 1 W 1 wi W 1 wi W 1 wi W 1 1 1 i ðuÞ i ðuÞ i ðuÞ
i ¼ 1; . . . ; k:
~ n ~ ; 0; nÞ ¼ W 1 ðr 1 ðm; ~ n ~ ; 0; nÞÞ and we have from (2.11) that Thus, use the fact c1 ðm;
~ n ~ ; m; nÞ þ zðm; nÞ ¼ W 1 uðm; nÞ 6 r 1 ðm; 1 ðnðm; nÞÞ 2 m1 X n1 m1 X X 1 ~f ðm; 4 ~ ~ ~; s; tÞ þ 6 W 1 kþ1 ~ n kþ1 W kþ1 W 1 ðc 1 ðm; n; 0; nÞÞ þ
3 ~ n ~ ; s; nÞ D3 ck ðm; 5 1 s¼0 t¼0 s¼0 /kþ1 W k ðc k ð0; 0; s; 0ÞÞ 2 3 m1 X n1 m1 X X ~ n ~ ; s; nÞ D3 ck ðm; 1 4 ~ ~ n ~ ; 0; nÞÞ þ ~ n ~ ; s; tÞ þ 5 f kþ1 ðm; 6 W kþ1 W kþ1 ðr 1 ðm; 1 s¼0 t¼0 s¼0 /kþ1 W k ðc k ð0; 0; s; 0ÞÞ
ð2:13Þ
~ ; M 3 g and 0 6 n 6 minfn ~ ; N 3 g. for 0 6 m 6 minfn ~ n ~ ; m; nÞ ¼ riþ1 ðm; ~ n ~ ; m; nÞ by induction again. In the following, we prove that ci ðm; ~ n ~ ; m; nÞ ¼ r 2 ðm; ~ n ~ ; m; nÞ for i = 1. Suppose that It is clear that c1 ðm;
~ n ~ ; m; nÞ ¼ r lþ1 ðm; ~ n ~ ; m; nÞ cl ðm; for i = l. We have
~ n ~ ; m; nÞ ¼ Ulþ1 ðc1 ðm; ~ n ~ ; 0; nÞÞ þ clþ1 ðm;
m1 n1 XX
~f ðm; ~ ; s; tÞ þ lþ1 ~ n
m1 X
s¼0 t¼0
~ n ~ ; 0; nÞÞ þ ¼ W lþ1 ðr 1 ðm;
m1 X n1 X s¼0 t¼0
s¼0
~f ðm; ~ ; s; tÞ þ lþ1 ~ n
m 1 X s¼0
~ n ~; s; nÞ D c ðm; 3 l 1 wlþ1 Ul ðcl ð0; 0; s; 0ÞÞ ~ n ~ ; s; nÞ D r ðm; ~ n ~ ; m; nÞ; 3 lþ1 ¼ r lþ2 ðm; 1 /lþ1 W l ðr lþ1 ð0; 0; s; 0ÞÞ
~ n ~ ; 0; nÞ ¼ W 1 ðr1 ðm; ~ n ~ ; 0; nÞÞ is applied. It implies that it is true for i = l + 1. Thus, ci ðm; ~ n ~ ; m; nÞ ¼ r iþ1 ðm; ~ n ~ ; m; nÞ for where c1 ðm; i = 1, . . . , k. Eq. (2.12) becomes
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~ n ~ ; 0; N3 ÞÞ þ W iþ1 ðr 1 ðm;
M 3 1 N 3 1 X X s¼0
¼
Z
W 1 ð1Þ
~ iþ1 u
~f ðm; ~ ; s; tÞ þ iþ1 ~ n
t¼0
M 3 1 X s¼0
Z 1 w1 W 1 1 ðzÞ dz dz ¼ 1 w ðzÞ iþ1 uiþ1 wiþ1 W 1 ðzÞ
~ n ~ ; s; N 3 Þ D r ðm; 3 iþ1 6 1 /iþ1 W i ðr iþ1 ð0; 0; s; 0ÞÞ
Z
W 1 ð1Þ
~ iþ1 u
dz ~ /iþ1 W 1 1 ðzÞ ð2:14Þ
for i = 1, . . . , k. It implies that we may choose M3 = M2 and N3 = N2. Thus, (2.13) becomes
2
uðm; nÞ 6
4 ~ ~ W 1 kþ1 W kþ1 ðr 1 ðm; n; 0; nÞÞ
þ
m1 X n1 X
~f ðm; ~; s; tÞ þ kþ1 ~ n
s¼0 t¼0
m 1 X s¼0
3 ~ n ~ ; s; nÞ D3 r kþ1 ðm; 5 /kþ1 W 1 k ðr kþ1 ð0; 0; s; 0ÞÞ
~ and 0 6 n 6 n ~ . It shows that (2.6) is true for k + 1. Thus, the claim is proved. for 0 6 m 6 m ~ and n ~ in (2.6), respectively, we have Now we prove (2.1). Replacing m and n by m
2 ~ n ~Þ 6 uðm;
4W k ðr 1 ðm; ~ n ~ ; 0; n ~ ÞÞ W 1 k
þ
~ ~ 1 m1 n X X
~f ðm; ~ ; s; tÞ þ k ~ n
s¼0 t¼0
~ m1 X
~ n ~ ; s; n ~Þ D3 rk ðm;
3
5: /k W 1 k1 ðr k ð0; 0; s; 0ÞÞ
s¼0
~ 6 M 1 and n ~ 6 N 1 , we replace m ~ and n ~ by m and n and get Since (2.6) is true for any m
2 uðm; nÞ 6
4W k ðr 1 ðm; n; 0; nÞÞ W 1 k
þ
m1 X n1 X
~f ðm; n; s; tÞ þ k
s¼0 t¼0
m1 X
D3 rk ðm; n; s; nÞ
3
5: /k W 1 k1 ðr k ð0; 0; s; 0ÞÞ
s¼0
~ð0; tÞ. This proves Theorem 2.1. h This is exactly (2.1) since r 1 ðm; n; 0; tÞ ¼ a Remark 2.2. ~ð0; 0Þ ¼ 0. Let (1) If a(m, n) = 0 for all m, n 2 N0, then a
r1;u1 ðm; n; s; tÞ ¼ r 1 ðm; n; s; tÞ þ u1 ; where u1 > 0 is given in W 1 ðuÞ ¼ r1;u1 ðm; n; s; tÞ, (2.9) becomes
Ru
dz . u1 w1 ðzÞ
Using the same arguments as in (2.8) where r1(m, n, s, t) is replaced with the positive
~ n ~ ; m; nÞÞ 6 W 1 ðr 1;u1 ðm; ~ n ~ ; 0; nÞÞ þ W 1 ðzðm; nÞ þ r 1;u1 ðm;
m1 X n1 X
~f 1 ðm; ~ n ~ ; s; tÞ ¼ W 1 ðu1 Þ þ
s¼0 t¼0
¼
m1 n1 XX
m1 n1 XX
~f 1 ðm; ~ n ~ ; s; tÞ
s¼0 t¼0
~f 1 ðm; ~ n ~; s; tÞ;
s¼0 t¼0
i.e.
" ~ n ~; m; nÞ ¼ zðm; nÞ þ u1 6 W 1 uðm; nÞ 6 zðm; nÞ þ r1;u1 ðm; 1
m1 n1 XX
# ~f 1 ðm; ~ n ~; s; tÞ ;
~ 06n6n ~; 0 6 m 6 m;
ð2:15Þ
s¼0 t¼0
which is the same as (2.10) with a complementary definition that W1(0) = 0. From Remark 2.1, the estimate of (2.15) is independent of u1. Then we similarly obtain (2.1) and all ri are defined by the same formula (2.3) where we define Wi(0) = 0 for i = 1, . . . , k. (2) If we fix m and vary n in the proof, then u(m, n) satisfies
2 4W k ða ~ðm; 0ÞÞ þ uðm; nÞ 6 W 1 k
m 1 X n1 X
~f ðm; n; s; tÞ þ k
s¼0 t¼0
n1 X t¼0
D4 r k ðm; n; m; tÞ
3
5; /k W 1 k1 ðr k ð0; 0; 0; tÞÞ
0 6 m 6 M1 ; 0 6 n 6 N1 ;
where rk(m, n, s, t) is determined by
~ðs; tÞ; r1 ðm; n; s; tÞ ¼ a riþ1 ðm; n; s; tÞ ¼ W i ðr1 ðm; n; s; 0ÞÞ þ
s1 X t1 X
s¼0 g¼0
~f ðm; n; s; gÞ þ i
t1 X
g¼0
D r ðm; n; s; gÞ 4 i ; /i W 1 i1 ðr i ð0; 0; 0; gÞÞ
i ¼ 1; . . . ; k 1;
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S. Deng / Applied Mathematics and Computation 217 (2010) 2217–2225
D4 r i ðm; n; s; tÞ ¼ ri ðm; n; s; t þ 1Þ r i ðm; n; s; tÞ;
i ¼ 1; . . . ; k;
M1 and N1 are positive integers satisfying
~ðM1 ; 0ÞÞ þ W i ða
M 1 1 N 1 1 X X s¼0
~f ðM 1 ; N1 ; s; tÞ þ i
t¼0
NX 1 1 t¼0
D4 ri ðM1 ; N1 ; M 1 ; tÞ 6 /i W 1 i1 ðr i ð0; 0; 0; tÞÞ
Z
1 ui
dz ; wi ðzÞ
i ¼ 1; . . . ; k
~ðm; nÞ is nondecreasing in n by and other functions are defined in Theorem 2.1. Here we replace the condition in (C1) that D1 a ~ðm; nÞ ¼ a ~ðm; n þ 1Þ a ~ðm; nÞ is nondecreasing in m. the condition that D2 a 3. Some corollaries Assume that u 2 C(R+, R+) is a strictly increasing function with u(1) = 1 where R+ = [0, 1). Consider the inequality
uðuðm; nÞÞ 6 aðm; nÞ þ
k m1 n1 X XX i¼1
fi ðm; n; s; tÞwi ðuðs; tÞÞ;
m; n 2 N0 :
ð3:1Þ
s¼0 t¼0
Corollary 3.1. Suppose that (C1)–(C3) hold. If u(m, n) in (3.1) is nonnegative for m, n 2 N0, then
2
uðm; nÞ 6 u
0
1 4 f 1 @ f ~ Wk W k ðað0; nÞÞ
þ
m1 X n1 X
~f ðm; n; s; tÞ þ k
s¼0 t¼0
m 1 X s¼0
13 D3 r k ðm; n; s; nÞ A5 f 1 ðrk ð0; 0; s; 0ÞÞ ^k W / k1
1 dz f i ðuÞ ¼ R u fi; W f 0 ¼ I (Identity), / f i is the inverse of W ^ i ðuÞ ¼ wi ðu ðuÞÞ ; for 0 6 m 6 M1 and 0 6 n 6 N1 where W ;W ui wi ðu1 ðzÞÞ wi1 ðu1 ðuÞÞ ^ 1 ðuÞ ¼ w1 u1 ðuÞ , and other related functions are defined as in Theorem 2.1 by replacing wi(u) with wi(u1(u)). /
Proof of Corollary 3.1. Let n(m, n) = u(u(m, n)). Then (3.1) becomes
nðm; nÞ 6 aðm; nÞ þ
k m1 n1 X XX i¼1
fi ðm; n; s; tÞwi u1 ðnðs; tÞÞ ;
m; n 2 N0 :
s¼0 t¼0
Note that wi(u1(u)) satisfy (C3) for i = 1, . . . , k. Using Theorem 2.1, we obtain the estimate about n(m, n) by replacing wi(u) with wi(u1(u)). Then use the fact that u(m, n) = u1(n(m, n)) and we get Corollary 3.1. h If u(u) = up where p > 0, then (3.1) reads
up ðm; nÞ 6 aðm; nÞ þ
k m 1 X n1 X X i¼1
fi ðm; n; s; tÞwi ðuðs; tÞÞ;
m; n 2 N0 :
ð3:2Þ
s¼0 t¼0
Directly using Corollary 3.1, we have the following result. Corollary 3.2. Suppose that (C1)–(C3) hold. If u(m, n) in (3.2) is nonnegative for m, n 2 N0, then
2
0
f k ða f 1 @ W ~ð0; nÞÞ þ uðm; nÞ 6 4 W
m1 X n1 X
k
~f ðm; n; s; tÞ þ k
m1 X
s¼0 t¼0
s¼0
131=p D3 rk ðm; n; s; nÞ A5 f 1 ðr k ð0; 0; s; 0ÞÞ ^k W / k1
1=p R f i ðuÞ ¼ u dz ; W fi; W f 0 ¼ I (Identity), / f i is the inverse of W ^ i ðuÞ ¼ wi ðu Þ ; / ^ 1 ðuÞ ¼ for 0 6 m 6 M1 and 0 6 n 6 N1 where W ui wi ðz1=p Þ wi1 ðu1=p Þ 1=p w1 u , and other related functions are defined as in Theorem 2.1 by replacing wi(u) with wi(u1/p).
4. Applications to Volterra type difference equations In this section, we apply Theorem 2.1 to study boundedness and uniqueness of solutions of a nonlinear delay difference equation of the form
yðm; nÞ ¼ bðm; nÞ þ
m1 n1 XX s¼0 t¼0
Fðm; n; s; t; yðs; tÞÞ þ
m1 n1 XX
Hðm; n; s; t; yðs; tÞÞ
s¼0 t¼0
for m, n 2 N0 where y : N20 ! R is an unknown function, b maps from N20 to R, and F and H map from N40 R to R. Theorem 4.1. Suppose that b(0, 0) – 0 and the functions F and H in (4.1) satisfy the conditions
ð4:1Þ
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S. Deng / Applied Mathematics and Computation 217 (2010) 2217–2225
pffiffiffiffiffiffi jFðm; n; s; t; yÞj 6 f1 ðm; n; s; tÞ jyj; jHðm; n; s; t; yÞj 6 f2 ðm; n; s; tÞjyj; where f1 ; f2 :
N40
ð4:2Þ
! ½0; 1Þ. If y(m, n) is a solution of (4.1) on
2
N20 ,
then
3 Pn1 ~ ~ðs;nÞ D1 a p ffiffiffiffiffiffiffiffi m 1 X n1 m 1 t¼0 f 1 ðm; n; s; tÞ þ X X ~ðs;nÞ7 a ~f 2 ðm; n; s; tÞ þ ~ð0; nÞ exp 6 jyðm; nÞj 6 a 4 5; hðsÞ s¼0 t¼0 s¼0
ð4:3Þ
where
~ðm; nÞ ¼ a
max
06s6m;06g6n;s;g2N0
~f 1 ðm; n; s; tÞ ¼ ~f 2 ðm; n; s; tÞ ¼ hðsÞ ¼
jbðs; gÞj;
max
f1 ðs; g; s; tÞ;
max
f2 ðs; g; s; tÞ;
06s6m;06g6n;s;g2N0 06s6m;06g6n;s;g2N0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi s1 1X D1 a~ðs; 0Þ ~ð0; 0Þ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : a 2 s¼0 a ~ðs; 0Þ
Proof. Using (4.1) and (4.2), the solution y(m, n) satisfies
uðm; nÞ 6 aðm; nÞ þ
m1 n1 XX
f1 ðm; n; s; tÞw1 ðuðs; tÞÞ þ
s¼0 t¼0
m1 n1 XX
f2 ðm; n; s; tÞw2 ðuðs; tÞÞ;
m; n 2 N0 ;
ð4:4Þ
s¼0 t¼0
where
uðm; nÞ ¼ jyðm; nÞj;
aðm; nÞ ¼ jbðm; nÞj;
w1 ðuÞ ¼
pffiffiffi u;
w2 ðuÞ ¼ u:
~ðm; nÞ > 0 for all m, n 2 N0 since b(0, 0) – 0. For positive constants u1, u2, we have Clearly, a
W 1 ðuÞ ¼
Z
u
u1 u
Z
pffiffiffi pffiffiffiffiffi dz ¼ 2 u u1 ; w1 ðzÞ
W 1 1 ðuÞ ¼
u 2
þ
pffiffiffiffiffi2 u1 ;
dz u ¼ ln ; W 1 W 2 ðuÞ ¼ 2 ðuÞ ¼ u2 expðuÞ; u2 u2 w2 ðzÞ ~ðs; tÞ > 0; r 1 ðm; n; 0; tÞ ¼ a ~ð0; tÞ; r 1 ðm; n; s; tÞ ¼ a qffiffiffiffiffiffiffiffiffiffiffiffiffi X s1 X t1 s1 X pffiffiffiffiffi D1 a~ðs; tÞ ~f 1 ðm; n; s; gÞ þ ~ð0; tÞ u1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; a r 2 ðm; n; s; tÞ ¼ 2 ~ðs; 0Þ a s¼0 g¼0 s¼0
D3 r2 ðm; n; s; tÞ ¼
t1 X
D1 a~ðs; tÞ ~f 1 ðm; n; s; gÞ þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ~ðs; 0Þ a g¼0
/2 ðuÞ ¼
w2 ðuÞ pffiffiffi ¼ u: w1 ðuÞ
It is obvious that w1 and w2 satisfy (C3). Applying Theorem 2.1 gives
2
m1 X n1 m1 X X ~f 2 ðm; n; s; tÞ þ ~ð0; nÞ exp 6 uðm; nÞ 6 a 4 s¼0 t¼0
s¼0
3 Pn1 ~ ~ðs;nÞ D1 a p ffiffiffiffiffiffiffiffi g¼0 f 1 ðm; n; s; gÞ þ a~ðs;0Þ 7 5; hðsÞ
ð4:5Þ
which implies (4.3). h Theorem 4.2. Suppose that b(0, 0) – 0 and the functions F and H in (4.1) satisfy the conditions
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jFðm; n; s; t; y1 Þ Fðm; n; s; t; y2 Þj 6 f1 ðn; sÞ jy1 y2 j; jHðm; n; s; t; y1 Þ Hðm; n; s; t; y2 Þj 6 f2 ðn; sÞjy1 y2 j;
ð4:6Þ
where f1 ; f2 : N40 ! ½0; 1Þ. Then (4.1) has at most one solution on N20 . Proof. Let y1(m, n) and y2(m, n) be two solutions of (4.1) on N20 . From (4.6), we have
uðm; n; s; tÞ 6
m 1 X n1 X s¼0 t¼0
f1 ðm; n; s; tÞw1 ðuðs; tÞÞ þ
m 1 X n1 X s¼0 t¼0
f2 ðm; n; s; tÞw2 ðuðs; tÞÞ;
m; n 2 N0 ;
ð4:7Þ
S. Deng / Applied Mathematics and Computation 217 (2010) 2217–2225
2225
pffiffiffi where u(m, n) = jy1(m, n) y2(m, n)j, a(m, n) = 0, w1 ðuÞ ¼ u and w2(u) = u. Apply Theorem 2.1, (1) of Remark 2.2 and the notation Wi(0) = 0 for i = 1, 2, we obtain that u(m, n) = 0 which implies that the solution is unique. h Acknowledgement This paper is supported by China Postdoctoral Science Foundation (20100470108). References [1] R.P. Agarwal, On an integral inequality in n independent variables, J. Math. Anal. Appl. 85 (1982) 192–196. [2] R.P. Agarwal, S. Deng, W. Zhang, Generalization of a retarded Gronwall-like inequality and its applications, Appl. Math. Comput. 165 (2005) 599–612. [3] I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hung. 7 (1956) 81–94. [4] S. Borysenko, G. Matarazzo, M. Pecoraro, A generalization of Bihari’s lemma for discontinuous functions and its application to the stability problem of differential equations with impulse disturbance, Georgian Math. J. 13 (2006) 229–238. [5] W. Cheung, Some new nonlinear inequalities and applications to boundary value problems, Nonlinear Anal. 64 (2006) 2112–2128. [6] W. Cheung, Some discrete nonlinear inequalities and applications to boundary value problems for difference equations, J. Difference Equ. Appl. 10 (2004) 213–223. [7] W. Cheung, J. Ren, Discrete non-linear inequalities and applications to boundary value problems, J. Math. Anal. Appl. 319 (2006) 708–724. [8] S.K. Choi, S. Deng, N.J. Koo, W. Zhang, Nonlinear integral inequalities of Bihari-type without class H, Math. Inequal. Appl. 8 (2005) 643–654. [9] S. Deng, Nonlinear delay discrete inequalities and their applications to Volterra type difference equations, submitted for publication. [10] S. Deng, C. Prather, Nonlinear discrete inequalities of Bihari-type, submitted for publication. [11] L. Horväth, Generalizations of special Bihari type integral inequalities, Math. Inequal. Appl. 8 (2005) 441–449. [12] N. Lungu, On some Gronwall–Bihari–Wendorff-type inequalities, Semin. Fixed Point Theory Cluj-Napoca 3 (2002) 249–254. [13] B.G. Pachpatte, On generalizations of Bihari’s inequality, Soochow J. Math. 31 (2005) 261–271. [14] B.G. Pachpatte, Integral inequalities of the Bihari type, Math. Inequal. Appl. 5 (2002) 649–657. [15] B.G. Pachpatte, On Bihari like integral and discrete inequalities, Soochow J. Math. 17 (1991) 213–232. [16] V.N. Phat, J.Y. Park, On the Gronwall inequality and asymptotic stability of nonlinear discrete systems with multiple delays, Dyn. Syst. Appl. 10 (2001) 577–588. [17] M. Pinto, Integral inequalities of Bihari-type and applications, Funkcial. Ekvac. 33 (1990) 387–403. [18] J. Popenda, R.P. Agarwal, Discrete Gronwall inequalities in many variables, Comput. Math. Appl. 38 (1999) 63–70. [19] Sh. Salem, K.R. Raslan, Some new discrete inequalities and their applications, JIPAM J. Inequal. Pure Appl. Math. 5 (2004) (article 2, 9 pp.). [20] F. Wong, C. Yeh, C. Hong, Gronwall inequalities on time scales, Math. Inequal. Appl. 9 (2006) 75–86. [21] K. Zheng, Y. Wu, S. Zhong, Discrete nonlinear integral inequalities in two variables and their applications, Appl. Math. Comput. 207 (2009) 140–147. [22] W. Zhang, S. Deng, Projected Gronwall–Bellman’s inequality for integrable functions, Math. Comput. Modell. 34 (2001) 393–402.