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Advances in Difference Equations

, 2012:228 | Cite as

Some Volterra-Fredholm type nonlinear discrete inequalities involving four iterated infinite sums

  • Bin ZhengEmail author
  • Bosheng Fu
Open Access
Research
Part of the following topical collections:
  1. Progress in Functional Differential and Difference Equations

Abstract

Some new generalized Volterra-Fredholm type nonlinear discrete inequalities involving four iterated infinite sums are established in this paper. To illustrate the validity of the established inequalities, we present some applications for them, in which new explicit bounds for the solutions of certain infinite sum-difference equations are deduced.

MSC:26D15.

Keywords

nonlinear discrete inequalities Volterra-Fredholm type inequalities sum-difference equations bounds 

1 Introduction

In recent years, many researchers have focused on various generalizations of the known Gronwall-Bellman inequality [1, 2], which provide explicit bounds for unknown solutions of certain difference equations, and a lot of such generalized inequalities have been established in the literature [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] including the known Ou-Iang inequality [3]. In [21], Ma generalized the discrete version of Ou-Iang’s inequality in two variables to a Volterra-Fredholm form for the first time, which has proved to be very useful in the study of qualitative as well as quantitative properties of the solutions of certain Volterra-Fredholm type difference equations. But since then, few results on Volterra-Fredholm type discrete inequalities have been established. Recent results in this direction include the works of Zheng [22], Ma [23], Zheng and Feng [24] to our best knowledge. We notice that the Volterra-Fredholm type discrete inequalities in [22, 23, 24] are constructed by an explicit function u p Open image in new window in the left-hand side (see [[22], Theorems 2.5, 2.6], [[23], Theorems 2.1, 2.5, 2.6, 2.7], [[24], Theorems 5, 8, 10, 11]).

Motivated by the works in [22, 23, 24], in this paper, we establish some new generalized Volterra-Fredholm type discrete inequalities involving four iterated infinite sums with the right-hand side denoted by an arbitrary function ϕ ( u ) Open image in new window, which are of more general forms. To illustrate the usefulness of the established results, we also present some applications for them and study the boundedness of the solutions of certain Volterra-Fredholm type infinite sum-difference equations.

Throughout this paper, ℝ denotes the set of real numbers and R + = [ 0 , ) Open image in new window, and ℤ denotes the set of integers, while N 0 Open image in new window denotes the set of nonnegative integers. In the next of this paper, let Ω : = ( [ m 0 , ] × [ n 0 , ] ) Z 2 Open image in new window, where m 0 , n 0 Z Open image in new window, and let l 1 , l 2 Z Open image in new window be two constants. If U is a lattice, then we denote the set of all ℝ-valued functions on U by ( U ) Open image in new window and denote the set of all R + Open image in new window-valued functions on U by + ( U ) Open image in new window. Finally, for a function f + ( U ) Open image in new window, we have s = m 0 m 1 f = 0 Open image in new window provided m 0 > m 1 Open image in new window.

2 Main results

Lemma 2.1 [[22], Lemma 2.1]

Suppose u , a , b + ( Ω ) Open image in new window. If a ( m , n ) Open image in new window is nonincreasing in the first variable, then for ( m , n ) Ω Open image in new window,
u ( m , n ) a ( m , n ) + s = m + 1 b ( s , n ) u ( s , n ) Open image in new window
implies
u ( m , n ) a ( m , n ) s = m + 1 [ 1 + b ( s , n ) ] . Open image in new window
(1)
Lemma 2.2 Suppose u , a , H + ( Ω ) Open image in new window, b + ( Ω 2 ) Open image in new window, and H, a are nonincreasing in every variable with H ( m , n ) > 0 Open image in new window, while b is nonincreasing in the third variable. φ , ϕ C ( R + , R + ) Open image in new window are strictly increasing with φ ( r ) > 0 Open image in new window, ϕ ( r ) > 0 Open image in new window for r > 0 Open image in new window. If for ( m , n ) Ω Open image in new window, u ( m , n ) Open image in new window satisfies the following inequality:
u ( m , n ) H ( m , n ) + s = m + 1 t = n + 1 b ( s , t , m , n ) φ ( ϕ 1 ( u ( s , t ) + a ( s , t ) ) ) , Open image in new window
(2)
then we have
u ( m , n ) G 1 [ G ( H ( m , n ) ) + s = m + 1 t = n + 1 b ( s , t , m , n ) ] , Open image in new window
(3)
where
G ( z ) = z 0 z 1 φ [ ϕ 1 ( z + a ( m , n ) ) ] d z , z z 0 > 0 . Open image in new window
(4)
Proof Fix ( m 1 , n 1 ) Ω Open image in new window, and let ( m , n ) ( [ m 1 , ] × [ n 1 , ] ) Ω Open image in new window. Then we have
u ( m , n ) H ( m 1 , n 1 ) + s = m + 1 t = n + 1 b ( s , t , m , n ) φ [ ϕ 1 ( u ( s , t ) + a ( s , t ) ) ] . Open image in new window
(5)
Let the right-hand side of (5) be v ( m , n ) Open image in new window. Then
u ( m , n ) v ( m , n ) , ( m , n ) ( [ m 1 , ] × [ n 1 , ] ) Ω , Open image in new window
(6)
On the other hand, according to the mean-value theorem for integrals, there exists ξ such that v ( m , n ) ξ v ( m 1 , n ) Open image in new window, and
v ( m , n ) v ( m 1 , n ) 1 φ ( ϕ 1 ( z + a ( m , n ) ) ) d z = v ( m 1 , n ) v ( m , n ) φ ( ϕ 1 ( ξ + a ( m , n ) ) ) v ( m 1 , n ) v ( m , n ) φ ( ϕ 1 ( v ( m , n ) + a ( m , n ) ) ) . Open image in new window
(8)
So, combining (7) and (8), we have
v ( m , n ) v ( m 1 , n ) 1 φ ( ϕ 1 ( z + a ( m , n ) ) ) d z = G ( v ( m 1 , n ) ) G ( v ( m , n ) ) s = m t = n + 1 b ( s , t , m 1 , n ) s = m + 1 t = n + 1 b ( s , t , m , n ) , Open image in new window
(9)
where G is defined in (4). Set m = η Open image in new window in (9); a summation with respect to η from m + 1 Open image in new window to ∞ yields
G ( v ( m , n ) ) G ( v ( , n ) ) s = m + 1 t = n + 1 b ( s , t , m , n ) 0 = s = m + 1 t = n + 1 b ( s , t , m , n ) . Open image in new window
Noticing v ( , n ) = H ( m 1 , n 1 ) Open image in new window and G is increasing, it follows that
v ( m , n ) G 1 [ G ( H ( m 1 , n 1 ) ) + s = m + 1 t = n + 1 b ( s , t , m , n ) ] . Open image in new window
(10)
Combining (6) and (10), we obtain
Setting m = m 1 Open image in new window, n = n 1 Open image in new window in (11) yields
u ( m 1 , n 1 ) G 1 [ G ( a ( m 1 , n 1 ) ) + s = m 1 + 1 t = n 1 + 1 b ( s , t , m 1 , n 1 ) ] . Open image in new window
(12)

Since ( m 1 , n 1 ) Open image in new window is selected from Ω arbitrarily, then substituting ( m 1 , n 1 ) Open image in new window with ( m , n ) Open image in new window in (12), we get the desired inequality (3). □

Theorem 2.3 Suppose u + ( Ω ) Open image in new window, b i , c i + ( Ω 2 ) Open image in new window, i = 1 , 2 , , l 1 Open image in new window, d i , e i + ( Ω 2 ) Open image in new window, i = 1 , 2 , , l 2 Open image in new window with b i Open image in new window, c i Open image in new window, d i Open image in new window, e i Open image in new window nonincreasing in the last two variables, and there is at least one function among d i Open image in new window, e i Open image in new window, i = 1 , 2 , , l 2 Open image in new window not equivalent to zero, a, φ, ϕ are defined as in Lemma  2.2. If for ( m , n ) Ω Open image in new window, u ( m , n ) Open image in new window satisfies
ϕ ( u ( m , n ) ) a ( m , n ) + i = 1 l 1 s = m + 1 t = n + 1 [ b i ( s , t , m , n ) φ ( u ( s , t ) ) + ξ = s η = t c i ( ξ , η , m , n ) φ ( u ( ξ , η ) ) ] + i = 1 l 2 s = M + 1 t = N + 1 [ d i ( s , t , m , n ) ϕ ( u ( s , t ) ) + ξ = s η = t e i ( ξ , η , m , n ) ϕ ( u ( ξ , η ) ) ] , Open image in new window
(13)
then
u ( m , n ) ϕ 1 { a ( m , n ) + G 1 { G ( T 1 [ s = m 0 M 1 t = n 0 N 1 B ( s , t , M , N ) ] ) + s = m + 1 t = n + 1 B ( s , t , m , n ) } } Open image in new window
(14)
provided that T is increasing, where G is defined in (4), and
Proof Denote
v ( m , n ) = i = 1 l 1 s = m + 1 t = n + 1 [ b i ( s , t , m , n ) φ ( u ( s , t ) ) + ξ = s η = t c i ( ξ , η , m , n ) φ ( u ( ξ , η ) ) ] + i = 1 l 2 s = M + 1 t = N + 1 [ d i ( s , t , m , n ) ϕ ( u ( s , t ) ) + ξ = s η = t e i ( ξ , η , m , n ) ϕ ( u ( ξ , η ) ) ] . Open image in new window
Then we have
u ( m , n ) ϕ 1 ( a ( m , n ) + v ( m , n ) ) . Open image in new window
(19)
So,
v ( m , n ) i = 1 l 1 s = m + 1 t = n + 1 { b i ( s , t , m , n ) φ [ ϕ 1 ( a ( s , t ) + v ( s , t ) ) ] + ξ = s η = t c i ( ξ , η , m , n ) φ [ ϕ 1 ( a ( ξ , η ) + v ( ξ , η ) ) ] } + i = 1 l 2 s = M + 1 t = N + 1 [ d i ( s , t , m , n ) ( a ( s , t ) + v ( s , t ) ) + ξ = s η = t e i ( ξ , η , m , n ) ( a ( ξ , η ) + v ( ξ , η ) ) ] = H ( m , n ) + i = 1 l 1 s = m + 1 t = n + 1 { b i ( s , t , m , n ) φ [ ϕ 1 ( a ( s , t ) + v ( s , t ) ) ] + ξ = s η = t c i ( ξ , η , m , n ) φ [ ϕ 1 ( a ( ξ , η ) + v ( ξ , η ) ) ] } , Open image in new window
(20)
where H ( m , n ) = J ( m , n ) + i = 1 l 2 s = M + 1 t = N + 1 [ d i ( s , t , m , n ) v ( s , t ) + ξ = s η = t e i ( ξ , η , m , n ) v ( ξ , η ) ] Open image in new window, and J ( m , n ) Open image in new window is defined in (17). Then using H ( m , n ) Open image in new window is nonincreasing in every variable, we obtain
v ( m , n ) H ( M , N ) + i = 1 l 1 s = m + 1 t = n + 1 { b i ( s , t , m , n ) φ [ ϕ 1 ( a ( s , t ) + v ( s , t ) ) ] } + ξ = s η = t c i ( ξ , η , m , n ) φ [ ϕ 1 ( a ( ξ , η ) + v ( ξ , η ) ) ] H ( M , N ) + i = 1 l 1 s = m + 1 t = n + 1 [ b i ( s , t , m , n ) + ξ = s η = t c i ( ξ , η , m , n ) ] × φ [ ϕ 1 ( a ( s , t ) + v ( s , t ) ) ] = H ( M , N ) + s = m + 1 t = n + 1 B ( s , t , m , n ) φ [ ϕ 1 ( a ( s , t ) + v ( s , t ) ) ] , Open image in new window
(21)

where B ( s , t , m , n ) Open image in new window is defined in (16).

Since there is at least one function among d i Open image in new window, e i Open image in new window, i = 1 , 2 , , l 2 Open image in new window not equivalent to zero, then H ( M , N ) > 0 Open image in new window. On the other hand, as b i ( s , t , m , n ) Open image in new window, c i ( s , t , m , n ) Open image in new window are both nonincreasing in the last two variables, then B ( s , t , m , n ) Open image in new window is also nonincreasing in the last two variables, and by a suitable application of Lemma 2.2, we obtain
v ( m , n ) G 1 [ G ( H ( M , N ) ) + s = m + 1 t = n + 1 B ( s , t , m , n ) ] . Open image in new window
(22)
Furthermore, by the definitions of H ( m , n ) Open image in new window, μ 1 Open image in new window, μ 2 Open image in new window and (22), we have
H ( M , N ) = J ( M , N ) + i = 1 l 2 s = M + 1 t = N + 1 { d i ( s , t , M , N ) v ( s , t ) + ξ = s η = t e i ( ξ , η , M , N ) v ( ξ , η ) } J ( M , N ) + v ( M , N ) i = 1 l 2 s = M + 1 t = N + 1 { d i ( s , t , M , N ) + ξ = s η = t e i ( ξ , η , M , N ) } = μ 1 + μ 2 v ( M , N ) μ 1 + μ 2 G 1 [ G ( H ( M , N ) ) + s = M + 1 t = N + 1 B ( s , t , M , N ) ] , Open image in new window
and
G ( H ( M , N ) μ 1 μ 2 ) G ( H ( M , N ) ) + s = M + 1 t = N + 1 B ( s , t , M , N ) , Open image in new window
which is rewritten as
T ( H ( M , N ) ) s = M + 1 t = N + 1 B ( s , t , M , N ) , Open image in new window
where T is defined in (15). By T is increasing, we have
H ( M , N ) T 1 [ s = M + 1 t = N + 1 B ( s , t , M , N ) ] . Open image in new window
(23)

Combining (19), (22) and (23), we get the desired result. □

Corollary 2.4 Suppose g 1 i , g 2 i , b 1 i , c 1 i + ( Ω ) Open image in new window, i = 1 , 2 , , l 1 Open image in new window with g 1 i Open image in new window, g 2 i Open image in new window nonincreasing in every variable, d 1 i , e 1 i + ( Ω ) Open image in new window, i = 1 , 2 , , l 2 Open image in new window, u, a, φ, ϕ are defined as in Theorem  2.3. If for ( m , n ) Ω Open image in new window, u ( m , n ) Open image in new window satisfies
ϕ ( u ( m , n ) ) a ( m , n ) + i = 1 l 1 g 1 i ( m , n ) s = m + 1 t = n + 1 [ b 1 i ( s , t ) φ ( u ( s , t ) ) + ξ = s η = t c 1 i ( ξ , η ) φ ( u ( ξ , η ) ) ] + i = 1 l 2 g 2 i ( m , n ) s = M + 1 t = N + 1 [ d 1 i ( s , t ) ϕ ( u ( s , t ) ) + ξ = s η = t e 1 i ( ξ , η ) ϕ ( u ( ξ , η ) ) ] , Open image in new window
then
u ( m , n ) ϕ 1 { a ( m , n ) + G 1 { G ( T 1 [ s = M + 1 t = N + 1 B ( s , t , M , N ) ] ) + s = m + 1 t = n + 1 B ( s , t , m , n ) } } Open image in new window
provided that T is increasing, where G, T are defined in Theorem  2.3, and
B ( s , t , m , n ) = i = 1 l 1 g 1 i ( m , n ) [ b i ( s , t ) + ξ = s η = t c i ( ξ , η ) ] , J ( m , n ) = i = 1 l 2 g 2 i ( m , n ) s = M + 1 t = N + 1 [ d i ( s , t ) a ( s , t ) + ξ = s η = t e i ( ξ , η ) a ( ξ , η ) ] , μ 1 = J ( M , N ) , μ 2 = i = 1 l 2 g 2 i ( m , n ) s = M + 1 t = N + 1 [ d i ( s , t ) + ξ = s η = t e i ( ξ , η ) ] . Open image in new window

The proof for Corollary 2.4 can be completed by setting b i ( s , t , m , n ) = g 1 i ( m , n ) b 1 i ( s , t ) Open image in new window, c i ( s , t , m , n ) = g 1 i ( m , n ) c 1 i ( s , t ) Open image in new window, d i ( s , t , m , n ) = g 2 i ( m , n ) d 1 i ( s , t ) Open image in new window, e i ( s , t , m , n ) = g 2 i ( m , n ) e 1 i ( s , t ) Open image in new window in Theorem 2.3.

Theorem 2.5 Suppose w + ( Ω ) Open image in new window, u, a, b i Open image in new window, c i Open image in new window, d i Open image in new window, e i Open image in new window, φ, ϕ are defined as in Theorem  2.3. Furthermore, assume φ ϕ 1 Open image in new window is submultiplicative, that is, φ ( ϕ 1 ( α β ) ) φ ( ϕ 1 ( α ) ) φ ( ϕ 1 ( β ) ) Open image in new window α , β R + Open image in new window. If for ( m , n ) Ω Open image in new window, u ( m , n ) Open image in new window satisfies
ϕ ( u ( m , n ) ) a ( m , n ) + s = m + 1 w ( s , n ) ϕ ( u ( s , n ) ) + i = 1 l 1 s = m + 1 t = n + 1 [ b i ( s , t , m , n ) φ ( u ( s , t ) ) + ξ = s η = t c i ( ξ , η , m , n ) φ ( u ( ξ , η ) ) ] + i = 1 l 2 s = M + 1 t = N + 1 [ d i ( s , t , m , n ) ϕ ( u ( s , t ) ) + ξ = s η = t e i ( ξ , η , m , n ) ϕ ( u ( ξ , η ) ) ] , Open image in new window
(24)
then
u ( m , n ) ϕ 1 { { a ( m , n ) + G 1 { G ( T 1 [ s = M + 1 t = N + 1 B ¯ ( s , t , M , N ) ] ) + i = 1 l 1 s = m + 1 t = n + 1 B ¯ ( s , t , m , n ) } } w ¯ ( m , n ) } Open image in new window
(25)
Proof Denote
z ( m , n ) = a ( m , n ) + i = 1 l 1 s = m + 1 t = n + 1 [ b i ( s , t , m , n ) φ ( u ( s , t ) ) + ξ = s η = t c i ( ξ , η , m , n ) φ ( u ( ξ , η ) ) ] + i = 1 l 2 s = M + 1 t = N + 1 [ d i ( s , t , m , n ) ϕ ( u ( s , t ) ) + ξ = s η = t e i ( ξ , η , m , n ) ϕ ( u ( ξ , η ) ) ] . Open image in new window
Then we have
ϕ ( u ( m , n ) ) z ( m , n ) + s = m + 1 w ( s , n ) ϕ ( u ( s , n ) ) . Open image in new window
(33)
Obviously, z ( m , n ) Open image in new window is nonincreasing in the first variable. So, by Lemma 2.1, we obtain
ϕ ( u ( m , n ) ) z ( m , n ) s = m + 1 [ 1 + w ( s , n ) ] = z ( m , n ) w ¯ ( m , n ) , Open image in new window
where w ¯ ( m , n ) Open image in new window is defined in (31). Define
v ( m , n ) = i = 1 l 1 s = m + 1 t = n + 1 [ b i ( s , t , m , n ) φ ( u ( s , t ) ) + ξ = s η = t c i ( ξ , η , m , n ) φ ( u ( ξ , η ) ) ] + i = 1 l 2 s = M + 1 t = N + 1 [ d i ( s , t , m , n ) ϕ ( u ( s , t ) ) + ξ = s η = t e i ( ξ , η , m , n ) ϕ ( u ( ξ , η ) ) ] . Open image in new window
Then we obtain
u ( m , n ) ϕ 1 [ ( a ( m , n ) + v ( m , n ) ) w ¯ ( m , n ) ] , Open image in new window
(34)
and furthermore, using φ ϕ 1 Open image in new window is submultiplicative, (34) and Lemma 2.2, we have
v ( m , n ) i = 1 l 1 s = m + 1 t = n + 1 { b i ( s , t , m , n ) φ [ ϕ 1 ( ( a ( s , t ) + v ( s , t ) ) w ¯ ( s , t ) ) ] + ξ = s η = t c i ( ξ , η , m , n ) φ [ ϕ 1 ( ( a ( ξ , η ) + v ( ξ , η ) ) w ¯ ( ξ , η ) ) ] } + i = 1 l 2 s = M + 1 t = N + 1 { d i ( s , t , m , n ) [ a ( s , t ) + v ( s , t ) ] w ¯ ( s , t ) + ξ = s η = t e i ( ξ , η , m , n ) [ a ( ξ , η ) + v ( ξ , η ) ] w ¯ ( ξ , η ) } i = 1 l 1 s = m + 1 t = n + 1 { b i ( s , t , m , n ) φ [ ϕ 1 ( a ( s , t ) + v ( s , t ) ) ] φ [ ϕ 1 ( w ¯ ( s , t ) ) ] + ξ = s η = t c i ( ξ , η , m , n ) φ [ ϕ 1 ( a ( ξ , η ) + v ( ξ , η ) ) ] φ [ ϕ 1 ( w ¯ ( ξ , η ) ) ] } + i = 1 l 2 s = M + 1 t = N + 1 { d i ( s , t , m , n ) [ a ( s , t ) + v ( s , t ) ] w ¯ ( s , t ) + ξ = s η = t e i ( ξ , η , m , n ) [ a ( ξ , η ) + v ( ξ , η ) ] w ¯ ( ξ , η ) } = i = 1 l 1 s = m + 1 t = n + 1 { b ¯ i ( s , t , m , n ) ϕ 1 [ a ( s , t ) + v ( s , t ) ] + ξ = s η = t c ¯ i ( ξ , η , m , n ) ϕ 1 [ a ( ξ , η ) + v ( ξ , η ) ] } + i = 1 l 2 s = M + 1 t = N + 1 { d ¯ i ( s , t , m , n ) [ a ( s , t ) + v ( s , t ) ] + ξ = s η = t e ¯ i ( ξ , η , m , n ) [ a ( ξ , η ) + v ( ξ , η ) ] } = H ¯ ( m , n ) + i = 1 l 1 s = m + 1 t = n + 1 { b ¯ i ( s , t , m , n ) φ [ ϕ 1 ( a ( s , t ) + v ( s , t ) ) ] + ξ = s η = t c i ( ξ , η , m , n ) φ [ ϕ 1 ( a ( ξ , η ) + v ( ξ , η ) ) ] } , Open image in new window
(35)
where H ¯ ( m , n ) = J ¯ ( m , n ) + i = 1 l 2 s = M + 1 t = N + 1 { d ¯ i ( s , t , m , n ) v ( s , t ) + ξ = s η = t e ¯ i ( ξ , η , m , n ) v ( ξ , η ) } Open image in new window, and J ¯ ( m , n ) Open image in new window is defined in (28). Then similar to the process of (21)-(23), we obtain
v ( m , n ) G 1 [ G ( H ¯ ( M , N ) ) + s = m + 1 t = n + 1 B ¯ ( s , t , m , n ) ] , Open image in new window
(36)
and
H ¯ ( M , N ) T 1 [ s = M + 1 t = N + 1 B ¯ ( s , t , M , N ) ] . Open image in new window
(37)

Combining (34), (36) and (37), we get the desired result. □

Theorem 2.6 Suppose u, a, b i Open image in new window, c i Open image in new window, d i Open image in new window, e i Open image in new window, φ, ϕ are defined as in Theorem  2.3. L 1 i , L 2 i , T 1 i , T 2 i : Ω × R + R + Open image in new window, i = 1 , 2 , , l 2 Open image in new window satisfies 0 L j i ( m , n , u ) L j i ( m , n , v ) T j i ( m , n , v ) ( u v ) Open image in new window, j = 1 , 2 Open image in new window for u v 0 Open image in new window. If for ( m , n ) Ω Open image in new window, u ( m , n ) Open image in new window satisfies
ϕ ( u ( m , n ) ) a ( m , n ) + i = 1 l 1 s = m + 1 t = n + 1 [ b i ( s , t , m , n ) φ ( u ( s , t ) ) + ξ = s η = t c i ( ξ , η , m , n ) φ ( u ( ξ , η ) ) ] + i = 1 l 2 s = M + 1 t = N + 1 [ d i ( s , t , m , n ) L 1 i ( s , t , ϕ ( u ( s , t ) ) ) + ξ = s η = t e i ( ξ , η , m , n ) L 2 i ( ξ , η , ϕ ( u ( ξ , η ) ) ) ] , Open image in new window
(38)
then
u ( m , n ) ϕ 1 { a ( m , n ) + G 1 { G ( T 1 [ s = M + 1 t = N + 1 B ˆ ( s , t , M , N ) ] ) + s = m + 1 t = n + 1 B ˆ ( s , t , m , n ) } } Open image in new window
(39)
provided that T is increasing, where G is defined in (4), and
Proof Denote
v ( m , n ) = i = 1 l 1 s = m + 1 t = n + 1 [ b i ( s , t , m , n ) φ ( u ( s , t ) ) + ξ = s η = t c i ( ξ , η , m , n ) φ ( u ( ξ , η ) ) ] + i = 1 l 2 s = M + 1 t = N + 1 [ d i ( s , t , m , n ) L 1 i ( s , t , ϕ ( u ( s , t ) ) ) + ξ = s η = t e i ( ξ , η , m , n ) L 2 i ( ξ , η , ϕ ( u ( ξ , η ) ) ) ] . Open image in new window
Then we have
u ( m , n ) ϕ 1 ( a ( m , n ) + v ( m , n ) ) . Open image in new window
(46)
So,
v ( m , n ) i = 1 l 1 s = m + 1 t = n + 1 { b i ( s , t , m , n ) φ [ ϕ 1 ( a ( s , t ) + v ( s , t ) ) ] + ξ = s η = t c i ( ξ , η , m , n ) φ [ ϕ 1 ( a ( ξ , η ) + v ( ξ , η ) ) ] } + i = 1 l 2 s = M + 1 t = N + 1 { d i ( s , t , m , n ) L 1 i ( s , t , a ( s , t ) + v ( s , t ) ) + ξ = s η = t e i ( ξ , η , m , n ) L 2 i ( ξ , η , a ( ξ , η ) + v ( ξ , η ) ) } = i = 1 l 1 s = m + 1 t = n + 1 { b i ( s , t , m , n ) φ [ ϕ 1 ( a ( s , t ) + v ( s , t ) ) ] + ξ = s η = t c i ( ξ , η , m , n ) φ [ ϕ 1 ( a ( ξ , η ) + v ( ξ , η ) ) ] } + i = 1 l 2 s = M + 1 t = N + 1 { d i ( s , t , m , n ) [ L 1 i ( s , t , a ( s , t ) + v ( s , t ) ) L 1 i ( s , t , a ( s , t ) ) + L 1 i ( s , t , a ( s , t ) ) ] + ξ = s η = t e i ( ξ , η , m , n ) [ L 2 i ( ξ , η , a ( ξ , η ) + v ( ξ , η ) ) L 2 i ( ξ , η , a ( ξ , η ) ) + L 2 i ( ξ , η , a ( ξ , η ) ) ] } i = 1 l 1 s = m + 1 t = n + 1 { b i ( s , t , m , n ) φ [ ϕ 1 ( a ( s , t ) + v ( s , t ) ) ] + ξ = s η = t c i ( ξ , η , m , n ) φ [ ϕ 1 ( a ( ξ , η ) + v ( ξ , η ) ) ] } + i = 1 l 2 s = M + 1 t = N + 1 { d i ( s , t , m , n ) [ T 1 i ( s , t , a ( s , t ) ) v ( s , t ) + L 1 i ( s , t , a ( s , t ) ) ] + ξ = s η = t e i ( ξ , η , m , n ) [ T 2 i ( ξ , η , a ( ξ , η ) ) v ( ξ , η ) + L 2 i ( ξ , η , a ( ξ , η ) ) ] } = H ( m , n ) + i = 1 l 1 s = m + 1 t = n + 1 { b i ( s , t , m , n ) φ [ ϕ 1 ( a ( s , t ) + v ( s , t ) ) ] + ξ = s η = t c i ( ξ , η , m , n ) φ [ ϕ 1 ( a ( ξ , η ) + v ( ξ , η ) ) ] } , Open image in new window
(47)
where H ˆ ( m , n ) = J ˆ ( m , n ) + i = 1 l 2 s = M + 1 t = N + 1 { d ˆ i ( s , t , m , n ) v ( s , t ) + ξ = s η = t e ˆ i ( ξ , η , m , n ) v ( ξ , η ) } Open image in new window, and J ˆ ( m , n ) Open image in new window is defined in (42). Then similar to the process of (21)-(23), we obtain
v ( m , n ) G 1 [ G ( H ˆ ( M , N ) ) + s = m + 1 t = n + 1 B ˆ ( s , t , m , n ) ] , Open image in new window
(48)
and
H ( M , N ) T 1 [ s = M + 1 t = N + 1 B ˆ ( s , t , M , N ) ] . Open image in new window
(49)

Combining (46), (48) and (49), we get the desired result. □

Theorem 2.7 Suppose w + ( Ω ) Open image in new window, u, a, b i Open image in new window, c i Open image in new window, d i Open image in new window, e i Open image in new window, φ, ϕ are defined as in Theorem  2.3, and L j i Open image in new window, T j i Open image in new window, j = 1 , 2 Open image in new window, i = 1 , 2 , , l 2 Open image in new window are defined as in Theorem  2.6. If for ( m , n ) Ω Open image in new window, u ( m , n ) Open image in new window satisfies
ϕ ( u ( m , n ) ) a ( m , n ) + s = m + 1 w ( s , n ) ϕ ( u ( m , n ) ) + i = 1 l 1 s = m + 1 t = n + 1 [ b i ( s , t , m , n ) φ ( u ( s , t ) ) + ξ = s η = t c i ( ξ , η , m , n ) φ ( u ( ξ , η ) ) ] + i = 1 l 2 s = M + 1 t = N + 1 [ d i ( s , t , m , n ) L 1 i ( s , t , ϕ ( u ( s , t ) ) ) + ξ = s η = t e i ( ξ , η , m , n ) L 2 i ( ξ , η , ϕ ( u ( ξ , η ) ) ) ] , Open image in new window
then
u ( m , n ) ϕ 1 { a ( m , n ) + G 1 { G ( T 1 [ s = M + 1 t = N + 1 B ˜ ( s , t , M , N ) ] ) + s = m + 1 t = n + 1 B ˜ ( s , t , m , n ) } } Open image in new window
provided that T is increasing, where G is defined in (4), and

The proof for Theorem 2.7 is similar to the combination of Theorem 2.5 and Theorem 2.6, and we omit the details here.

Remark 2.8 We note that the inequalities established in Theorems 2.3, 2.5-2.7 are essentially different from the results in [22, 23, 24] as the left-hand side of the inequalities established here is an arbitrary function ϕ ( u ) Open image in new window. Furthermore, if we set ϕ ( u ) = u p Open image in new window, a ( m , n ) = 0 Open image in new window, then Theorem 2.5 reduces to [[22], Theorem 2.5].

3 Applications

In this section, we present some applications for the results established above. Similar to the applications in [22, 23, 24], we research a certain Volterra-Fredholm sum-difference equation and derive some new bounds for its solutions.

Example Consider the following Volterra-Fredholm type infinite sum-difference equation:
u p ( m , n ) = s = m + 1 t = n + 1 [ F 1 ( s , t , m , n , u ( s , t ) ) + ξ = s η = t F 2 ( ξ , η , m , n , u ( ξ , η ) ) ] + s = M + 1 t = N + 1 [ G 1 ( s , t , m , n , u ( s , t ) ) + ξ = s η = t G 2 ( ξ , η , m , n , u ( ξ , η ) ) ] , Open image in new window
(50)

where u ( Ω ) Open image in new window, p 1 Open image in new window is an odd number, F i , G i : Ω 2 × R R Open image in new window, i = 1 , 2 Open image in new window.

Theorem 3.1 Suppose u ( m , n ) Open image in new window is a solution of (50), and | F 1 ( s , t , m , n , u ) | f 1 ( s , t , m , n ) | u | p 2 Open image in new window, | F 2 ( s , t , m , n , u ) | f 2 ( s , t , m , n ) | u | p 2 Open image in new window, | G 1 ( s , t , m , n , u ) | g 1 ( s , t , m , n ) | u | p Open image in new window, | G 2 ( s , t , m , n , u ) | g 2 ( s , t , m , n ) | u | p Open image in new window, f i , g i + ( Ω 2 ) Open image in new window, i = 1 , 2 Open image in new window, f i Open image in new window, g i Open image in new window are nondecreasing in the last two variables, and there is at least one function among g 1 Open image in new window, g 2 Open image in new window not equivalent to zero, then we have
u ( m , n ) 4 1 p { μ 1 μ s = M + 1 t = N + 1 B ( s , t , M , N ) + s = m + 1 t = n + 1 B ( s , t , m , n ) } 2 p Open image in new window
(51)
provided that μ < 1 Open image in new window, where
B ( s , t , m , n ) = f 1 ( s , t , m , n ) + ξ = s η = t f 2 ( ξ , η , m , n ) , μ = s = M + 1 t = N + 1 [ g 1 ( s , t , M , N ) + ξ = s η = t g 2 ( ξ , η , M , N ) ] . Open image in new window
Proof From (50) we have
| u ( m , n ) | p s = m + 1 t = n + 1 [ | F 1 ( s , t , m , n , u ( s , t ) ) | + ξ = s η = t | F 2 ( ξ , η , m , n , u ( ξ , η ) ) | ] + s = m 0 M 1 t = n 0 M 1 [ | G 1 ( s , t , m , n , u ( s , t ) ) | + ξ = s η = t | G 2 ( ξ , η , m , n , u ( ξ , η ) ) | ] | a ( m , n ) | + s = m + 1 t = n + 1 [ f 1 ( s , t , m , n ) | u ( s , t ) | p 2 + ξ = s η = t f 2 ( ξ , η , m , n ) | u ( ξ , η ) | p 2 ] + s = M + 1 t = N + 1 [ g 1 ( s , t , m , n ) | u ( s , t ) | p + ξ = s η = t g 2 ( ξ , η , m , n ) | u ( ξ , η ) | p ] . Open image in new window
(52)
Define ϕ ( u ) = u p Open image in new window, φ ( u ) = u p 2 Open image in new window, and
Then by μ < 1 Open image in new window, we have T is strictly increasing, and a suitable application of Theorem 2.3 (with a ( m , n ) = 0 Open image in new window and l 1 = l 2 = 1 Open image in new window) to (52) yields
u ( m , n ) ϕ 1 { G 1 { G ( T 1 [ s = M + 1 t = N + 1 B ( s , t , M , N ) ] ) + s = m + 1 t = n + 1 B ( s , t , m , n ) } } . Open image in new window
(55)

Combining (53)-(55), we can deduce the desired result. □

Theorem 3.2 Suppose u ( m , n ) Open image in new window is a solution of (50), and | F 1 ( s , t , m , n , u ) | f 1 ( s , t , m , n ) | u | p 3 Open image in new window, | F 2 ( s , t , m , n , u ) | f 2 ( s , t , m , n ) | u | p 3 Open image in new window, | G 1 ( s , t , m , n , u ) | g 1 ( s , t , m , n ) L 1 ( s , t , | u | p ) Open image in new window, | G 2 ( s , t , m , n , u ) | g 2 ( s , t , m , n ) L 2 ( s , t , | u | p ) Open image in new window, where f i Open image in new window, g i Open image in new window, i = 1 , 2 Open image in new window are defined as in Theorem  3.1, L 1 , L 2 , T 1 , T 2 : Ω × R + R + Open image in new window satisfies 0 L i ( m , n , u ) L i ( m , n , v ) T i ( m , n , v ) ( u v ) Open image in new window for u v 0 Open image in new window and L i ( m , n , 0 ) = 0 Open image in new window, i = 1 , 2 Open image in new window, then we have
u ( m , n ) { 2 3 [ μ ˆ 2 3 1 μ ˆ 2 3 s = M + 1 t = N + 1 B ˆ ( s , t , M , N ) + s = m + 1 t = n + 1 B ˆ ( s , t , m , n ) ] } 3 2 p Open image in new window
(56)
provided that μ ˆ < 1 Open image in new window, where
B ˆ ( s , t , m , n ) = f 1 ( s , t , m , n ) + ξ = s η = t f 2 ( ξ , η , m , n ) , μ ˆ = s = M + 1 t = N + 1 [ g 1 ( s , t , M , N ) T 1 ( s , t , 0 ) + ξ = s η = t g 2 ( ξ , η , M , N ) T 2 ( ξ , η , 0 ) ] . Open image in new window
Proof From (50) we have
| u ( m , n ) | p s = m + 1 t = n + 1 [ | F 1 ( s , t , m , n , u ( s , t ) ) | + ξ = s η = t | F 2 ( ξ , η , m , n , u ( ξ , η ) ) | ] + s = M + 1 t = N + 1 [ | G 1 ( s , t , m , n , u ( s , t ) ) | + ξ = s η = t | G 2 ( ξ , η , m , n , u ( ξ , η ) ) | ] s = m + 1 t = n + 1 [ f 1 ( s , t , m , n ) | u ( s , t ) | p 3 + ξ = s η = t f 2 ( ξ , η , m , n ) | u ( ξ , η ) | p 3 ] + s = M + 1 t = N + 1 [ g 1 ( s , t , m , n ) L 1 ( s , t , | u ( s , t ) | p ) + ξ = s η = t g 2 ( ξ , η , m , n ) L 2 ( ξ , η , | u ( ξ , η ) | p ) ] s = m + 1 t = n + 1 [ f 1 ( s , t , m , n ) | u ( s , t ) | p 3 + ξ = s η = t f 2 ( ξ , η , m , n ) | u ( ξ , η ) | p 3 ] + s = M + 1 t = N + 1 [ g 1 ( s , t , m , n ) T 1 ( s , t , 0 ) | u ( s , t ) | p + ξ = s η = t g 2 ( ξ , η , m , n ) T 2 ( ξ , η , 0 ) | u ( ξ , η ) | p ] . Open image in new window
(57)
Define ϕ ( u ) = u p Open image in new window, φ ( u ) = u p 3 Open image in new window, and
Then by μ ˆ < 1 Open image in new window, we have T is strictly increasing, and a suitable application of Theorem 2.6 (with ϕ ( u ) = u p Open image in new window, φ ( u ) = u p 3 Open image in new window, a ( m , n ) = 0 Open image in new window and l 1 = l 2 = 1 Open image in new window) to (57) yields
u ( m , n ) ϕ 1 { G 1 { G ( T 1 [ s = M + 1 t = N + 1 B ˆ ( s , t , M , N ) ] ) + s = m + 1 t = n + 1 B ˆ ( s , t , m , n ) } } . Open image in new window
(60)

Combining (58)-(60), we can deduce the desired result. □

Notes

Acknowledgements

The authors thank the referees very much for their careful comments and valuable suggestions on this paper.

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© Zheng and Fu; licensee Springer 2012

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.School of ScienceShandong University of TechnologyZiboChina
  2. 2.School of Journalism and CommunicationWuhan UniversityWuhanChina

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