Advances in Dynamical Systems and Applications. ISSN 0973-5321 Volume 3 Number 1 (2008), pp. 1–13 © Research India Publications http://www.ripublication.com/adsa.htm
Nonlinear Dynamic Integral Inequalities in two Independent Variables on Time Scale Pairs1 Douglas R. Anderson Concordia College, Department of Mathematics and Computer Science, Moorhead, MN 56562, USA E-mail:
[email protected]
Abstract We extend from the discrete case some nonlinear inequalities in two independent variables to a more general dynamic setting involving pairs of time scales. These inequalities provide explicit bounds on unknown functions, which can be useful in the study of qualitative properties of solutions of certain classes of dynamic equations on time scales. Several examples are provided. AMS subject classification: 26D15, 26D20, 39A12. Keywords: Dynamic equations, time scales, integral inequalities, Pachpatte inequalities.
1.
Introduction
Since Hilger first accomplished the unification and extension of differential equations, difference equations, q-difference equations, and so on to the encompassing theory of dynamic equations on time scales in his Ph.D. thesis [7], his time-scale calculus has made steady inroads in explaining the interconnections that exist among the various differential and difference theories, and in extending our understanding to a new, more general, robust, and overarching theory. The purpose of this note is to illustrate this new understanding by extending some discrete inequalities by Meng and Ji [10] and Meng and Li [11] to arbitrary time scales; see also some related time-scale inequalities in Agarwal, Bohner, and Peterson [1], and Akin-Bohner, Bohner, and Akin [2]. In particular, we 1
Dedicated to Professor Allan C. Peterson on the occasion of his 65th birthday. Received August 28, 2007; Accepted September 10, 2007 Communicated by Elvan Akin-Bohner
2
Douglas R. Anderson
establish some general nonlinear dynamic inequalities on general time scales involving functions of two independent variables; these inequalities may be applied to the analysis of certain classes of partial dynamic equations on time scales, introduced by Jackson [8]. Throughout this work a knowledge and understanding of time scales and time-scale notation is assumed; for an excellent introduction to the calculus on time scales, see Bohner and Peterson [5,6] for a general overview, the paper introducing nabla derivatives by Atici and Guseinov [3], and the introduction of multiple integrals on time scales by Bohner and Guseinov [4]. For more on discrete inequalities, see the work done by Pachpatte [12, 13, 15, 16].
2.
Preliminary Lemmas and Examples
˜ be arbitrary, unbounded time scales, and let t0 ∈ T. Before we arrive at the Let T and T main results in this section, we need the following lemmas. Note that if T = R, then Lemma 2.1 below is an inequality from Pachpatte [14, Lemma 2.1], and if T = Z, it is the discrete inequality [14, Lemma 2.5]. Lemma 2.1. Let T be an unbounded time scale with t, t0 ∈ T. (I) Suppose u, a, b ∈ Crd and u, a, b ≥ 0, where a is nondecreasing and not identically zero. If t b(τ )u(τ )τ (2.1) u(t) ≤ a(t) + t0
for all t ∈ [t0 , ∞)T , then
u(t) ≤ a(t)eb (t, t0 )
(2.2)
for all t ∈ [t0 , ∞)T . (II) Suppose u, a, b ∈ Cld and u, a, b ≥ 0, where a is nonincreasing and not identically zero. If ∞ u(t) ≤ a(t) +
b(τ )u(τ )∇τ
(2.3)
t
for all t ∈ [t0 , ∞)T , then u(t) ≤ a(t)eˆ−b (t, ∞)
(2.4)
for all t ∈ [t0 , ∞)T , where eˆ is the nabla exponential function [6, Definition 3.10]. Proof. The statement and proof of (I) are from [2, Corollary 3.11], so we focus on (II). Define ∞ w(t) := −
b(τ )u(τ )∇τ. t
Then w∇ (t) = b(t)u(t) implies from assumption (2.3) that w∇ (t) ≤ b(t)a(t) − b(t)w(t).
Nonlinear Dynamic Integral Inequalities in two Independent Variables
3
Since by assumption b ≥ 0, we have that −b ∈ R+ ν and eˆ−b (t, t0 ) > 0 for all t ∈ T by [6, Thm 3.22 (i)]. Employing the nabla quotient rule, we have ∇ w/eˆ−b (·, t0 ) (t) = w∇ (t) + b(t)w(t) /eˆ−b (ρ(t), t0 ); integrating both sides from t to ∞ and using lim w(t) = 0, we obtain t→∞
−w(t)/eˆ−b (t, t0 ) = ≤
∞
t ∞ t
w∇ (τ ) + b(τ )w(τ ) /eˆ−b (ρ(τ ), t0 )∇τ
b(τ )a(τ )/eˆ−b (ρ(τ ), t0 )∇τ.
Thus, using [6, Thm 3.15 (iv) & (v)], we have ∞ ∞ b(τ )u(τ )∇τ ≤ b(τ )a(τ )eˆ−b (t, ρ(τ ))∇τ, t
t
so that
∞
u(t) ≤ a(t) + a(t) t
Let
∞
f (t) = 1 + t
Then
lim f (t) = 1,
t→∞
b(τ )eˆ−b (t, ρ(τ ))∇τ.
b(τ )eˆ−b (t, ρ(τ ))∇τ.
and
f ∇ (t) = −b(t)[f (t)],
which means that f (t) = eˆ−b (t, ∞). The result follows. t b(s)ds = exp Example 2.2. If T = R, then eˆ−b (t, ∞) = exp − ∞
t
∞ b(s)ds ,
and Lemma 2.1 (II) reads as follows: Suppose u, a, b ∈C and u, a, b ≥ 0, where a is nonincreasing and not identically zero. If ∞ b(s)u(s)ds u(t) ≤ a(t) + t
for all t ∈ [t0 , ∞)T , then
u(t) ≤ a(t) exp
∞
b(s)ds t
for all t ∈ [t0 , ∞)T . Example 2.3. Consider the quantum equations case. Let r > 1, and take T = r Z := {0} ∪ {r n }n∈Z .
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Douglas R. Anderson
Replace t by r t , t0 by r t0 , and τ by r τ . Then ∞
1 + (r − 1)r m−1 b(r m ) ,
eˆ−b (t, ∞) =
m=t+1
and Lemma 2.1 (II) reads as follows: Suppose u, a, b are functions and u, a, b ≥ 0, where a is nonincreasing and not identically zero. If u(r ) ≤ a(r ) + (r − 1) t
t
∞
r s−1 b(r s )u(r s )
s=t+1
for all integers t ∈ {t0 , t0 + 1, t0 + 2, . . .}, then ∞
1 + (r − 1)r m−1 b(r m )
u(r ) ≤ a(r ) t
t
m=t+1
for all integers t ∈ {t0 , t0 + 1, t0 + 2, . . .}. Lemma 2.4. [10, Lemma 2] Assume that p ≥ q > 0 and a ≥ 0. Then a q/p ≤
q q−p p − q q/p k k p a+ p p
for any k > 0.
3.
Nonlinear Inequalities on Time Scales
Proceeding from the foundational lemmas above, all of the resulting theorems and corresponding proofs in this section are modelled after those given in the special case of ˜ = Z recently presented by Meng and Ji [10] and Meng and Li [11]; see also T = T Ma and Cheung [9]. We now state and prove our main nonlinear integral inequalities in two independent variables on time scale pairs. Note that not only do these theorems extend [10,11] from the difference equations case to arbitrary time scales, they also allow for different time scales for each independent variable. Theorem 3.1. Let u(t, s), a(t, s), b(t, s), c(t, s), d(t, s) and e(t, s) be nonnegative ˜ that are right-dense continuous for t ∈ T and leftfunctions defined for (t, s) ∈ T × T ˜ with a(t, s) not identically zero; let p, q, r be constants dense continuous for s ∈ T, with p ≥ q > 0 and p ≥ r > 0. If t ∞ p c(τ, η)uq (τ, η) + d(τ, η)ur (τ, η) u (t, s) ≤ a(t, s) + b(t, s) t0 s ˜ + e(τ, η) ∇ητ (3.1)
Nonlinear Dynamic Integral Inequalities in two Independent Variables
5
˜ then for (t, s) ∈ [t0 , ∞)T × T, 1/p , u(t, s) ≤ a(t, s) + b(t, s)f (t, s)ey(·,s) (t, t0 )
˜ (t, s) ∈ [t0 , ∞)T × T,
(3.2)
where q q−p p − q q/p p k + a(τ, η) k c(τ, η) f (t, s) = p p t0 s p − r r/p r r−p ˜ +d(τ, η) k + a(τ, η) k p + e(τ, η) ∇ητ p p t
∞
∞ q
and
y(t, s) :=
p
s
k
q−p p
r r−p p ˜ c(t, η) + k d(t, η) b(t, η)∇η p
(3.3)
(3.4)
˜ and any k > 0. for (t, s) ∈ [t0 , ∞)T × T Proof. Define the function z(t, s) via t ∞ ˜ c(τ, η)uq (τ, η) + d(τ, η)ur (τ, η) + e(τ, η) ∇ητ. z(t, s) := t0
(3.5)
s
Then (3.1) can be rewritten as u(t, s) ≤ [a(t, s) + b(t, s)z(t, s)]1/p , and we have z(t, s) ≤
t t0
∞
c(τ, η) [a(τ, η) + b(τ, η)z(τ, η)]q/p
s
˜ + d(τ, η) [a(τ, η) + b(τ, η)z(τ, η)]r/p + e(τ, η) ∇ητ. Using Lemma 2.4, for any k > 0 we have t ∞
p − q q/p q q−p p k k c(τ, η) z(t, s) ≤ [a(τ, η) + b(τ, η)z(τ, η)] + p p t0 s r r−p + d(τ, η) k p [a(τ, η) + b(τ, η)z(τ, η)] p p − r r/p ˜ + k + e(τ, η) ∇ητ p t ∞ r r−p q q−p p p = f (t, s) + k c(τ, η)b(τ, η) + k d(τ, η)b(τ, η) p p t0 s ˜ × z(τ, η)∇ητ,
(3.6)
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Douglas R. Anderson
where f (t, s) is given above in (3.3). Note that f (t, s) ≥ 0 is continuous, nondecreasing ˜ Let > 0 be given. Then in t and nonincreasing in s, for (t, s) ∈ [t0 , ∞)T × T. f (t, s) + > 0, and t ∞ q q−p z(t, s) ≤1+ k p c(τ, η)b(τ, η) f (t, s) + p t0 s r r−p z(τ, η) ˜ + k p d(τ, η)b(τ, η) ∇ητ. p f (τ, η) + For convenience, if we set w(t, s) := 1 +
t
∞ q
q−p p
c(τ, η)b(τ, η) r r−p z(τ, η) ˜ p + k d(τ, η)b(τ, η) ∇ητ, f (τ, η) + p t0
then
s
p
k
z(t, s) ≤ w(t, s). f (t, s) +
(3.7)
(3.8)
From (3.7), a delta derivative with respect to t yields ∞ r r−p z(t, η) ˜ q q−p t p p k c(t, η)b(t, η) + k d(t, η)b(t, η) ∇η w (t, s) = p p f (t, η) + s ∞ r−p r q q−p ˜ k p c(t, η)b(t, η) + k p d(t, η)b(t, η) w(t, η)∇η ≤ p p s
∞ r−p r q q−p ˜ w(t, s), k p c(t, η)b(t, η) + k p d(t, η)b(t, η) ∇η ≤ p p s whence for y(t, s) given in (3.4), w(t, s) ≤ ey(·,s) (t, t0 ). From (3.8) we obtain z(t, s) ≤ [f (t, s) + ] ey(·,s) (t, t0 ), so that by (3.6), the limit as → 0 yields inequality (3.2). Theorem 3.2. Let u(t, s), a(t, s), b(t, s), c(t, s), d(t, s) and e(t, s) be nonnegative left˜ with a(t, s) not identically zero, dense continuous functions defined for (t, s) ∈ T × T and let p, q, r be constants with p ≥ q > 0 and p ≥ r > 0. If ∞ ∞ p c(τ, η)uq (τ, η) u (t, s) ≤ a(t, s) + b(t, s) t s r ˜ (3.9) + d(τ, η)u (τ, η) + e(τ, η) ∇η∇τ
Nonlinear Dynamic Integral Inequalities in two Independent Variables ˜ then for (t, s) ∈ T × T, 1/p u(t, s) ≤ a(t, s) + b(t, s)f † (t, s)eˆ−y(·,s) (t, ∞) , where
q q−p p − q q/p p k + a(τ, η) k c(τ, η) f (t, s) = p p t s p − r r/p r r−p ˜ +d(τ, η) k + a(τ, η) k p + e(τ, η) ∇η∇τ p p
†
∞ ∞
˜ (t, s) ∈ T × T,
7
(3.10)
(3.11)
˜ for any k > 0, and y(t, s) is given in (3.4) for (t, s) ∈ T × T. Proof. Similar to the proof of Theorem 3.1, define the function z(t, s) via ∞ ∞ ˜ c(τ, η)uq (τ, η) + d(τ, η)ur (τ, η) + e(τ, η) ∇η∇τ. z(t, s) := t
(3.12)
s
Then (3.9) can be rewritten as u(t, s) ≤ [a(t, s) + b(t, s)z(t, s)]1/p , and we have
(3.13)
∞ ∞
c(τ, η) [a(τ, η) + b(τ, η)z(τ, η)]q/p
z(t, s) ≤ t
s
˜ + d(τ, η) [a(τ, η) + b(τ, η)z(τ, η)]r/p + e(τ, η) ∇η∇τ. Using Lemma 2.4, for any k > 0 we see that ∞ ∞
p − q q q−p q/p k p [a(τ, η) + b(τ, η)z(τ, η)] + k c(τ, η) z(t, s) ≤ p p t s r r−p k p [a(τ, η) + b(τ, η)z(τ, η)] + d(τ, η) p p − r r/p ˜ k + e(τ, η) ∇η∇τ + p ∞ ∞ r r−p q q−p † p p k c(τ, η)b(τ, η) + k d(τ, η)b(τ, η) = f (t, s) + p p t s ˜ × z(τ, η)∇η∇τ, where f † (t, s) is given above in (3.11). Note that f † (t, s) > 0 is continuous and ˜ Then nonincreasing for (t, s) ∈ T × T. ∞ ∞ z(t, s) q q−p ≤ 1 + k p c(τ, η)b(τ, η) f † (t, s) p t s z(τ, η) ˜ r r−p p ∇η∇τ. + k d(τ, η)b(τ, η) p f † (τ, η)
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Douglas R. Anderson
For convenience, if we set w(t, s) := 1 + t
∞ ∞ q s
p
k
q−p p
c(τ, η)b(τ, η)
z(τ, η) ˜ r r−p p ∇η∇τ, + k d(τ, η)b(τ, η) p f † (τ, η)
(3.14)
then z(t, s) ≤ w(t, s). f † (t, s)
(3.15)
From (3.15), a nabla derivative with respect to t yields r r−p z(t, η) ˜ p k c(t, η)b(t, η) + k d(t, η)b(t, η) ∇η −w (t, s) = p p f † (t, η) s ∞ r r−p q q−p p p ˜ k c(t, η)b(t, η) + k d(t, η)b(t, η) w(t, η)∇η ≤ p p s
∞ r r−p q q−p p p ˜ w(t, s), k c(t, η)b(t, η) + k d(t, η)b(t, η) ∇η ≤ p p s ∇t
∞ q
q−p p
whence for y(t, s) given in (3.4), w(t, s) ≤ eˆ−y(·,s) (t, ∞). From (3.15) we obtain z(t, s) ≤ f † (t, s)eˆ−y(·,s) (t, ∞), so that by (3.13), the inequality (3.10) holds. Theorem 3.3. Let u(t, s), a(t, s), b(t, s), c(t, s), d(t, s) and e(t, s) be nonnegative ˜ that are right-dense continuous for t ∈ T and leftfunctions defined for (t, s) ∈ T × T ˜ with a(t, s) not identically zero and nondecreasing in t. dense continuous for s ∈ T, Further, let p, q, r be constants with p ≥ q > 0 and p ≥ r > 0. If
t
b(τ, s)up (τ, s)τ u (t, s) ≤ a(t, s) + t0 t ∞ ˜ + c(τ, η)uq (τ, η) + d(τ, η)ur (τ, η) + e(τ, η) ∇ητ p
t0
(3.16)
s
˜ then for (t, s) ∈ [t0 , ∞)T × T, 1/p 1/p eb(·,s) (t, t0 ), u(t, s) ≤ a(t, s) + f ∗ (t, s)ey ∗ (·,s) (t, t0 )
˜ (t, s) ∈ [t0 , ∞)T × T, (3.17)
Nonlinear Dynamic Integral Inequalities in two Independent Variables
9
where
p − q q/p q q−p f (t, s) = k + a(τ, η) k p p p t0 s r−p p − r r/p r p r/p ˜ + d(τ, η)eb(·,η) (τ, t0 ) + e(τ, η) ∇ητ k + a(τ, η) k p p (3.18) t
∗
∞
q/p c(τ, η)eb(·,η) (τ, t0 )
and ∗
∞ q
y (t, s) :=
p
s
k
q−p p
q/p c(t, η)eb(·,η) (t, t0 ) +
r r−p r/p ˜ (3.19) k p d(t, η)eb(·,η) (t, t0 ) ∇η p
˜ and any k > 0. for (t, s) ∈ [t0 , ∞)T × T Proof. Define the function z(t, s) via z(t, s) := a(t, s) t + t0
∞
˜ c(τ, η)uq (τ, η) + d(τ, η)ur (τ, η) + e(τ, η) ∇ητ
(3.20)
s
˜ Then (3.16) can be rewritten as for (t, s) ∈ [t0 , ∞)T × T. u (t, s) ≤ z(t, s) + p
t
b(τ, s)up (τ, s)τ.
t0
˜ and note that z(t, s) is right-dense continuous, nonnegative, nondecreasing Fix s ∈ T, in t ∈ [t0 , ∞)T , and not identically zero; applying Lemma 2.1 (I) to up (t, s), we obtain up (t, s) ≤ z(t, s)eb(·,s) (t, t0 ). We rewrite this using (3.20) to get up (t, s) ≤ [a(t, s) + w(t, s)] eb(·,s) (t, t0 ), where w(t, s) is defined as w(t, s) :=
t t0
∞
˜ c(τ, η)uq (τ, η) + d(τ, η)ur (τ, η) + e(τ, η) ∇ητ.
(3.21)
s
Then 1/p
u(t, s) ≤ [a(t, s) + w(t, s)]1/p eb(·,s) (t, t0 ),
˜ (t, s) ∈ [t0 , ∞)T × T,
(3.22)
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Douglas R. Anderson
and (3.22)
w(t, s) ≤
t t0
∞
q/p
c(τ, η) [a(τ, η) + w(τ, η)]q/p eb(·,η) (τ, t0 )
s
r/p ˜ + d(τ, η) [a(τ, η) + w(τ, η)]r/p eb(·,η) (τ, t0 ) + e(τ, η) ∇ητ t ∞
Lemma 2.4 q q−p k p [a(τ, η) + w(τ, η)] ≤ c(τ, η) p t0 s p − q q/p q/p k eb(·,η) (τ, t0 ) + p p − r r/p r r−p p k k + d(τ, η) [a(τ, η) + w(τ, η)] + p p r/p ˜ × eb(·,η) (τ, t0 ) + e(τ, η) ∇ητ t
∞
q q−p q/p c(τ, η) k p eb(·,η) (τ, t0 ) p t0 s r r−p r/p ˜ + d(τ, η) k p eb(·,η) (τ, t0 ) w(τ, η)∇ητ, p
(3.18)
∗
= f (t, s) +
˜ Note that f ∗ (t, s) ≥ 0 is continuous, nondefor any k > 0 and (t, s) ∈ [t0 , ∞)T × T. ˜ Continuing in the same creasing in t and nonincreasing in s, for (t, s) ∈ [t0 , ∞)T × T. ∗ way as in the proof of Theorem 3.1, we obtain w(t, s) ≤ f (t, s) + ey ∗ (·,s) (t, t0 ), for y ∗ (t, s) given in (3.19). Consequently, by (3.22), the limit as → 0 yields inequality (3.17). Theorem 3.4. Let u(t, s), a(t, s), b(t, s), c(t, s), d(t, s) and e(t, s) be nonnegative left˜ with a(t, s) not identically zero dense continuous functions defined for (t, s) ∈ T × T and nonincreasing in t. Further, let p, q, r be constants with p ≥ q > 0 and p ≥ r > 0. If u (t, s) ≤ a(t, s) + ∞ + p
t
∞
b(τ, s)up (τ, s)∇τ
t ∞
˜ c(τ, η)uq (τ, η) + d(τ, η)ur (τ, η) + e(τ, η) ∇η∇τ
(3.23)
s
˜ then for (t, s) ∈ T × T, 1/p 1/p u(t, s) ≤ a(t, s) + f¯(t, s)eˆ−y(·,s) (t, t0 ) eˆ−b(·,s) (t, t0 ), ¯
˜ (3.24) (t, s) ∈ T × T,
Nonlinear Dynamic Integral Inequalities in two Independent Variables where
11
q−p p − q q q/p k + a(τ, η) k p f¯(t, s) = p p t s r−p r p − r r/p r/p ˜ k + a(τ, η) k p + e(τ, η) ∇η∇τ + d(τ, η)eˆ−b(·,η) (τ, t0 ) p p (3.25)
and
∞ ∞
q/p c(τ, η)eˆ−b(·,η) (τ, t0 )
y(t, ¯ s) :=
∞ q
p
s
k
q−p p
q/p c(t, η)eˆ−b(·,η) (t, t0 ) +
r r−p r/p p ˜ k d(t, η)eˆ−b(·,η) (t, t0 ) ∇η p (3.26)
˜ and any k > 0. for (t, s) ∈ T × T Proof. The proof is similar to that provided for Theorem 3.3, using Lemma 2.1 (II) and proceeding as in the proof of Theorem 3.2.
4. Applications The examples in this section are extensions from Z of examples found in [10] to arbitrary
˜ time scale pairs T, T , including the continuum and quantum cases. Example 4.1. Let p > 0 be a constant real number, and consider the dynamic equation ˜ (both unbounded above) given via on arbitrary time scales T and T ∞ ∞ ˜ ˜ h(τ, η, u(τ, η))∇η∇τ, (t, s) ∈ T × T, (4.1) up (t, s) = a(t, s) + t
s
˜ that is left-dense continuous on where a(t, s) is a function defined for (t, s) ∈ T × T ˜ ˜ ˜ and continuous on T × T, and h : T × T × R → R is left-dense continuous on T × T R. Suppose (4.2) |h(t, s, x)| ≤ c(t, s)|x|q , ˜ and q where c(t, s) is a nonnegative left-dense continuous function for (t, s) ∈ T × T, ˜ we have is a real constant with p ≥ q > 0. Then for (t, s) ∈ T × T 1/p |u(t, s)| ≤ |a(t, s)| + f † (t, s)eˆ−y(·,s) (t, ∞) ∞ q q−p ˜ for y(t, s) := k p c(t, η)∇η, p s where ∞ ∞
q q−p p − q q/p † p ˜ k + |a(τ, η)| k c(τ, η) ∇η∇τ, f (t, s) = p p t s ˜ (t, s) ∈ T × T
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Douglas R. Anderson
for any k > 0. Proof. If u(t, s) is a solution of (4.1), then ∞ p |u(t, s)| ≤ |a(t, s)| +
∞
˜ c(τ, η)|u(τ, η)|q ∇η∇τ
s
t
using (4.1) and (4.2). Since the hypotheses of Theorem 3.2 are met, the result follows from there. Example 4.2. Consider the partial dynamic equation on an arbitrary time scale pair ˜ (both unbounded above) given via (T, T) ˜
˜ (t, s) ∈ T × T,
u∇t ∇s (t, s) = a(t, s) + h(t, s, u(t, s)),
(4.3)
with boundary conditions u(t, ∞) = γ (t),
u(∞, s) = β(s),
u(∞, ∞) = α,
(4.4)
˜ h : T×T ˜× where a(t, s) is a left-dense continuous function defined for (t, s) ∈ T × T, ˜ R → R is left-dense continuous on T × T and continuous on R, α is a real constant, and ˜ → R are left-dense continuous. Suppose γ : T → R and β : T |h(t, s, v) − h(t, s, x)| ≤ c(t, s)|v − x|q ,
(4.5)
˜ and q where c(t, s) is a nonnegative left-dense continuous function for (t, s) ∈ T × T, is a real constant with 1 ≥ q > 0. If v and x are two solutions of (4.3), (4.4), then for ˜ we have (t, s) ∈ T × T ∞ ∞ q ˜ c(τ, η)∇η∇τ, |v(t, s) − x(t, s)| ≤ (1 − q)k eˆ−y(·,s) (t, ∞) t
for any k > 0, where y(t, s) is given by y(t, s) := qk
q−1
∞
s
˜ c(t, η)∇η
s
˜ for (t, s) ∈ T × T. Proof. If u(t, s) is a solution of (4.3), (4.4), then u(t, s) can be written as ∞ ∞ ˜ a(τ, η) + h τ, η, u(τ, η) ∇η∇τ u(t, s) = γ (t) + β(s) − α + s
t
˜ using (4.1) and (4.2). If v and x are two solutions of (4.3), (4.4), then for (t, s) ∈ T × T we have ∞ ∞ ˜ c(τ, η) |v(τ, η) − x(τ, η)|q ∇η∇τ |v(t, s) − x(t, s)| ≤ t
s
using (4.5). Since the hypotheses of Theorem 3.2 are met, the result again follows from there.
Nonlinear Dynamic Integral Inequalities in two Independent Variables
13
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