doi:10.1155/2010/795145
Research Article
Nonlinear Delay Discrete Inequalities and Their Applications to Volterra Type Difference Equations
Yu Wu,
1Xiaopei Li,
2and Shengfu Deng
31Yibin University, Yibin, Sichuan 644007, China
2Department of Mathematics, Zhanjiang Normal University, Zhanjiang, Guangdong 524048, China
3Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China
Correspondence should be addressed to Shengfu Deng,sf [email protected] Received 7 September 2009; Accepted 14 January 2010
Academic Editor: Abdelkader Boucherif
Copyrightq2010 Yu Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Delay discrete inequalities with more than one nonlinear term are discussed, which generalize some known results and can be used in the analysis of various problems in the theory of certain classes of discrete equations. Application examples to show boundedness and uniqueness of solutions of a Volterra type difference equation are also given.
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 1–12 for continuous cases, and 13–20 for discrete cases. In particular, the book 21 written by Pachpatte considered three types of discrete inequalities:
un≤an n−1
s0
fswus, u2n≤an 2
n−1 s0
fsus,
u2n≤an n−1
s0
fswus.
1.1
In this paper, we consider a delay discrete inequality
un≤an m
i1 bin−1 sbi0
fin, swius, n∈N0 1.2
which has m nonlinear terms where N0 {0,1,2, . . .}. We will show that many discrete inequalities like1.1can be reduced to this form. Our main result can be applied to analyze properties of solutions of discrete equations. We also give examples to show boundedness and uniqueness of solutions of a Volterra type difference equation.
2. Main Results
Assume that
C1anis nonnegative forn∈N0anda0>0;
C2bin i1, . . . , mare nondecreasing forn∈N0, the range of eachbibelongs toN0, andbin≤n;
C3allfin, j i1, . . . , mare nonnegative forn, j∈N0;
C4allwi i1, . . . , mare continuous and nondecreasing functions on0,∞and are positive on0,∞. They satisfy the relationshipw1∝w2∝ · · · ∝wmwherewi∝wi1
means thatwi1/wiis nondecreasing on0,∞ see10.
LetWiu u
uidz/wizforu ≥ ui whereui > 0 is a given constant. Then,Wiis strictly increasing so its inverseWi−1is well defined, continuous, and increasing in its corresponding domain. Definebi−1 −1,Δun un1−unandΔ2rn, j rn, j1−rn, j.
Theorem 2.1. Suppose that (C1)–(C4) hold andunis a nonnegative function forn∈N0satisfying 1.2. Then
un≤Wm−1
⎡
⎣Wma0 bmn−1
sbm0
fmn, s n−1
s0
Δ2rmn, s φm
Wm−1−1 rm0, s
⎤
⎦, n≤N1, 2.1
wherean max0≤τ≤n,τ∈N0aτ,fin, j max0≤τ≤n,τ∈N0fiτ, j,rmn, jis determined recursively by
r1
n, j a
j , ri1
n, j
Wir1n,0
bij−1 sbi0
fin, s j−1
s0
Δ2rin, s φi
Wi−1−1ri0, s, i1, . . . , m−1, 2.2
φiu wiu/wi−1u,φ1u w1u,W0 I(Identity), andN1 is the largest positive integer such that
Wia0 biN1−1
sbi0
fiN1, s N1−1
s0
Δ2riN1, s φi
Wi−1−1ri0, s ≤ ∞
ui
dz
wiz, i1, . . . , m. 2.3
Remark 2.2. 1 N1 is defined by 2.3 and N1 ∞ when all wi i 1, . . . , m satisfy ∞
uidz/wiz ∞. Different choices ofuiinWido not affect our resultssee2.
2Ifbin nfori1, . . . , m, then2.1gives the estimate of the following inequality:
un≤an m
i1
n−1 s0
fin, swius, n∈N0 2.4
by replacingbmn−1,bm0,bij−1,bi0, andbiN1−1withn−1, 0,j−1, 0 andN1−1, respectively. Especially, ifb1n nandf1n, s fs, then1.2form1 becomes the first inequality of1.1. Equation2.1shows the same estimate given byb1of Theorem 4.2.3 in the book21.
Lemma 2.3. Δ2rin, j is nonnegative and nondecreasing in n, and rin, j is nonnegative and nondecreasing innandjfori1, . . . , m.
Proof. By the definitions ofan andfin, j, it is easy to check that they are nonnegative and nondecreasing inn, andan ≥anandfin, j≥fin, jfor each fixedjwherei1, . . . , m.
a0>0 inC1implies thatan >0 for alln≤N1. Clearly,
Δ2r1
n1, j
−Δ2r1
n, j 0, Δ2r2
n1, j
−Δ2r2
n, j f1
n1, b1
j
−f1
n, b1
j Δ2r1
n1, j
−Δ2r1
n, j w1
r1
0, j ≥0, 2.5
wherer10, j aj >0 is used, which yields thatΔ2r1n, jandΔ2r2n, jare nondecreasing inn. Assume thatΔ2rln, jis nondecreasing inn. Then
Δ2rl1
n1, j
−Δ2rl1 n, j
fl
n1, bl
j
−fl
n, bl
j Δ2rl
n1, j
−Δ2rl
n, j φl
Wl−1−1 rl
0, j ≥0, 2.6
which implies thatΔ2rl1n, jis nondecreasing inn. By induction,Δ2rin, j i 1, . . . , m are nondecreasing inn. Similarly, we can prove that they are nonnegative by induction again.
Thenrin, j i1, . . . , mare nonnegative and nondecreasing innandj.
Proof ofTheorem 2.1. Take any arbitrary positive integern ≤ N1 and consider the auxiliary inequality
un≤r1n, n m
i1 bin−1 sbi0
fin, swius, n≤n. 2.7
Claim thatunin2.7satisfies
un≤Wm−1
⎡
⎣Wmr1n,0
bmn−1 sbm0
fmn, s n−1
s0
Δ2rmn, s φm
Wm−1−1 rm0, s
⎤
⎦ 2.8
forn≤ {n, N2}whereN2is the largest positive integer such that
Wir1n,0
biN2−1 sbi0
fin, s N2−1
s0
Δ2rin, s φi
Wi−1−1ri0, s ≤ ∞
ui
dz
wiz, 2.9
i1, . . . , m.
Before we prove2.8, notice thatN1≤N2. In fact,rin, n,Δ2rin, n, andfin, nare nondecreasing innbyLemma 2.3. Thus,N2satisfying2.9gets smaller asnis chosen larger.
In particular,N2satisfies the same2.3asN1fornN1ifr1n,0 a0 is applied.
We divide the proof of2.8into two steps by using induction.
Step 1m1. Letzn b1n−1
sb10f1n, sw 1usforn≤nandz0 0. It is clear thatzn is nonnegative and nondecreasing. Observe that2.7is equivalent toun≤r1n, n zn forn≤nand by assumptionsC2andC4andLemma 2.3,
Δzn f1n, b1nw1ub1n≤f1n, b1nw1r1n, b1n zb1n
≤f1n, b1nw1r1n, n zn. 2.10
Sincew1is nondecreasing andr1n, n an >0, we have Δzn Δ2r1n, n
w1r1n, n zn ≤f1n, b1n Δ2r1n, n w1r1n, n zn
≤f1n, b1n Δ2r1n, n w1r10, n.
2.11
Then
zn1r1n,n1
znr1n,n
dτ w1τ ≤
zn1r1n,n1
znr1n,n
dτ
w1zn r1n, n
≤ Δzn Δ2r1n, n w1zn r1n, n
≤f1n, b1n Δ2r1n, n w1r10, n,
2.12
and so
znr1n,n
z0r1n,0
dτ w1τ n−1
s0
zs1r1n,s1
zsr1n,s
dτ w1τ
≤n−1
s0
f1n, b 1s n−1
s0
Δ2r1n, s w1r10, s
b1n−1
sb10
f1n, s n−1
s0
Δ2r1n, s w1r10, s.
2.13
The definition ofW1inTheorem 2.1andz0 0 show
W1zn r1n, n≤W1r1n,0 b1n−1
sb10
f1n, s n−1
s0
Δ2r1n, s
w1r10, s, n≤n. 2.14
Equation2.9shows that the right side of2.14is in the domain ofW1−1for alln≤n. Thus the monotonicity ofW1−1implies
un≤zn r1n, n≤W1−1
⎡
⎣W1r1n, 0
b1n−1 sb10
f1n, s n−1
s0
Δ2r1n, s w1r10, s
⎤
⎦ 2.15
forn≤n; that is, 2.8is true form1.
Step 2mk1. Assume that2.8is true formk. Consider
un≤r1n, n k1
i1 bin−1 sbi0
fin, swius, n≤n. 2.16
Let zn k1
i1 bin−1
sbi0fin, swius and z0 0. Then zn is nonnegative and nondecreasing and satisfiesun≤r1n, n znforn≤n. Moreover, we have
Δzn k1
i1
fin, b inwiubin≤k1
i1
fin, binwir1n, b in zbin. 2.17
Since wi and r1 are nondecreasing in their arguments and r1n, n > 0, we have by the assumptionbin≤n
Δzn Δ2r1n, n w1zn r1n,n ≤
k1
i1 fin, b inwizbin r1n, bin
w1zn r1n, n Δ2r1n, n w1r1n, n
≤f1n, b 1n k1
i2
fin, binwizbin r1n, b in
w1zbin r1n, b in Δ2r1n, n w1r10, n
≤f1n, b 1n k
i1
fi1n, b i1nφi1zbi1n r1n, bi1n
Δ2r1n, n w1r10, n
2.18
forn≤nwhereφi1u wi1u/w1ufori1, . . . , k, which gives zn1r1n,n1
znr1n,n
dτ w1τ ≤
zn1r1n,n1
znr1n,n
dτ
w1zn r1n, n
≤ Δzn Δ2r1n, n w1r1n, n zn
≤f1n, b1n Δ2r1n, n w1r10, n k
i1
fi1n, bi1nφi1zbi1n r1n, bi1n.
2.19
Therefore, znr1n,n
z0r1n,0
dτ
w1τ ≤b1n−1
sb10
f1n, s n−1
s0
Δ2r1n, s w1r10, s
k
i1
n−1 s0
fi1n, bi1sφi1zbi1s r1n, bi1s,
2.20
that is,
W1zn r1n, n≤W1r1n,0 b1n−1
sb10
f1n, s n−1
s0
Δ2r1n, s w1r10, s
k
i1 bi1n−1 sbi10
fi1n, sφi1zs r1n, s,
2.21
or equivalently
ξn≤c1n, n k
i1 bi1n−1 sbi10
fi1n, sφi1
W1−1ξs
, n≤n, 2.22
the same as2.7formkwhereξn W1zn r1n, nand
c1n, n W1r1n,0
b1n−1 sb10
f1n, s n−1
s0
Δ2r1n, s
w1r10, s. 2.23
From the assumption C4, each φi1W1−1, i 1, . . . , k, is continuous and nondecreasing on 0,∞ and is positive on0,∞ sinceW1−1 is continuous and nondecreasing on0,∞.
Moreover,φ2W1−1∝φ3W1−1∝ · · · ∝φk1W1−1. By the inductive assumption, we have
ξn≤Φ−1k1
⎡
⎣Φk1c1n, 0 bk1n−1
sbk10
fk1n, s n−1
s0
Δ2ckn, s ψk1
Φ−1k ck0, s
⎤
⎦ 2.24
forn≤ min{n, N3}whereΦi1u u
ui1dz/φi1W1−1z,u > 0,Φ1 I Identity,ui1 W1ui1,Φ−1i1 is the inverse ofΦi1,ψi1u φi1W1−1u/φiW1−1u wi1W1−1u/
wiW1−1u, i1, . . . , k,
ci1n, n Φi1c1n, 0 bi1n−1
sbi10
fi1n, s n−1
s0
Δ2cin, s ψi1
Φ−1i ci0, s, 2.25
i 1, . . . , k−1,andN3is the largest positive integer such that
Φi1c1n,0 bi1N3−1
sbi10
fi1n, s N3−1
s0
Δ2cin, s ψi1
Φ−1i ci0, s
≤ W1∞
ui1
dz φi1
W1−1z, i1, . . . , k.
2.26
Note that
Φiu u
ui
dz φi
W1−1z u
W1ui
w1
W1−1z dz wi
W1−1z
W−1
1 u
ui
dz
wiz Wi◦W1−1u, i2, . . . , k1,
ψi1
Φ−1i u
wi1 W1−1
Φ−1i u wi
W1−1
Φ−1i u wi1 W1−1
W1
Wi−1u wi
W1−1 W1
Wi−1u wi1
Wi−1u wi
Wi−1u φi1
Wi−1u
, i1, . . . , k1.
2.27
Thus, we have from2.24that
un≤r1n, n zn W1−1ξn
≤Wk1−1
⎡
⎣Wk1
W1−1c1n,0
bk1n−1 sbk10
fk1n, s n−1
s0
Δ2ckn, s φk1
Wk−1ck0, s
⎤
⎦
≤Wk1−1
⎡
⎣Wk1r1n,0
bk1n−1 sbk10
fk1n, s n−1
s0
Δ2ckn, s φk1
Wk−1ck0, s
⎤
⎦
2.28
forn≤min{n, N 3}since c1n,0 W1r1n,0.
In the following, we prove thatcin, n ri1n, nby induction again.
It is clear thatc1n, n r2n, nfori1. Suppose thatcln, n rl1n, n foril. We have
cl1n, n Φl1c1n, 0 bl1n−1
sbl10
fl1n, s n−1
s0
Δ2cln, s ψl1
Φ−1l cl0, s Wl1r1n,0 bl1n−1
sbl10
fl1n, s n−1
s0
Δ2rl1n, s φl1
Wl−1rl10, s rl2n, n,
2.29
wherec1n,0 W1r1n,0is applied. It implies that it is true foril1. Thus,cin, n ri1n, nfori1, . . . , k.
Equation2.26becomes
Wi1r1n,0
bi1N3−1 sbi10
fi1n, s N3−1
s0
Δ2ri1n, s φi1
Wi−1ri10, s
≤ W1∞
ui1
dz φi1
W1−1z W1∞
ui1
w1
W1−1z wi1
W1−1zdz ∞
ui1
dz wi1z
2.30
fori1, . . . , k. It implies thatN2N3. Thus,2.28becomes
un≤Wk1−1
⎡
⎣Wk1r1n,0 bk1n−1
sbk10
fk1n, s n−1
s0
Δ2rk1n, s φk1
Wk−1rk10, s
⎤
⎦ 2.31
forn≤n. It shows that 2.8is true formk1. Thus, the claim is proved.
Now we prove2.1. Replacingnbynin2.8, we have
un ≤Wm−1
⎡
⎣Wmr1n,0 bmn−1
sbm0
fmn, s n−1
s0
Δ2rmn, s φm
Wm−1−1 rm0, s
⎤
⎦. 2.32
Since2.8is true for anyn≤N1, we replacenbynand get
un≤Wm−1
⎡
⎣Wmr1n,0 bmn−1
sbm0
fmn, s n−1
s0
Δ2rmn, s φm
Wm−1−1 rm0, s
⎤
⎦. 2.33
This is exactly2.1sincer1n,0 a0. This proves Theorem 2.1.
Remark 2.4. Ifan 0 for alln∈N0, thena0 0. Letr1,u1n, j:r1n, j u1whereu1>0 is given inW1u u
u1dz/w1z. Using the same arguments as in2.11wherer1n, jis replaced with the positiver1,u1n, j, we haveΔ2r1,u1n, s 0 and2.14becomes
W1zn r1,u1n, n≤W1r1,u1n,0 b1n−1
sb10
f1n, s
W1u1
b1n−1 sb10
f1n, s b1n−1
sb10
f1n, s,
2.34
that is,
un≤zn r1,u1n, n zn u1
≤W1−1
⎡
⎣b1n−1
sb10
f1n, s
⎤
⎦, n≤n, 2.35
which is the same as2.15with a complementary definition thatW10 0. From 1 of Remark 2.2, the estimate of2.35is independent ofu1. Then we similarly obtain2.1and all riare defined by the same formula2.2where we defineWi0 0 fori1, . . . , m.
3. Some Corollaries
In this section, we applyTheorem 2.1and obtain some corollaries.
Assume that ϕ ∈ CR,R is a strictly increasing function with ϕ∞ ∞ where R 0,∞. Consider the inequality
ϕun≤an m
i1 bin−1 sbi0
fin, swius, n∈N0. 3.1
Corollary 3.1. Suppose thatC1)–(C4) hold. Ifunin3.1is nonnegative forn∈N0, then
un≤ϕ−1
⎡
⎢⎣Wm−1
⎛
⎜⎝Wma0 bmn−1
sbm0
fmn, s n−1
s0
Δ2rmn, s φm
Wm−1−1 rm0, s
⎞
⎟⎠
⎤
⎥⎦ 3.2
for n ≤ N1 where Wiu u
uidz/wiϕ−1z, Wi−1 is the inverse of Wi,W0 IIdentity, φiu wiϕ−1u/wi−1ϕ−1u,φ1u w1ϕ−1u, and other related functions are defined as inTheorem 2.1by replacingwiuwithwiϕ−1u.
Proof. Letξn ϕun. Then3.1becomes
ξn≤an m
i1 bin−1
sbi0
fin, swi
ϕ−1ξs
, n∈N0. 3.3
Note thatwiϕ−1usatisfyC4fori1, . . . , m. UsingTheorem 2.1, we obtain the estimate aboutξnby replacingwiuwithwiϕ−1u. Then use the fact thatun ϕ−1ξnand we getCorollary 3.1.
Ifϕu upwherep >0, then3.1reads
upn≤an m
i1 bin−1
sbi0
fin, swius, n∈N0. 3.4
Directly usingCorollary 3.1, we have the following result.
Corollary 3.2. Suppose thatC1)–(C4) hold. Ifunin3.4is nonnegative forn∈N0, then
un≤
⎡
⎢⎣Wm−1
⎛
⎜⎝Wma0 bmn−1
sbm0
fmn, s n−1
s0
Δ2rmn, s φm
Wm−1−1 rm0, s
⎞
⎟⎠
⎤
⎥⎦
1/p
3.5
forn ≤ N1 whereWiu u
uidz/wiz1/p,Wiis the inverse ofWi,W0 IIdentity,φiu wiu1/p/wi−1u1/p,φ1u w1u1/p, and other related functions are defined as inTheorem 2.1 by replacingwiuwithwiu1/p.
Ifm1,p2,b1n n,3.4becomes the second inequality of1.1withf1n, s 2fsandw1u u, and the third inequality of1.1withf1n, s fsandw1u wu, which are discussed in the book21. Equation3.5yields the same estimates of Theorem 4.2.4 in the book21.
4. Applications to Volterra Type Difference Equations
In this section, we applyTheorem 2.1to study boundedness and uniqueness of solutions of a nonlinear delay difference equation of the form
yn βn b1n−1
sb10
F
n, s, ys
b2n−1
sb20
H
n, s, ys
, n∈N0, 4.1
where y : N0 → R is an unknown function,β maps from N0 to R,F and H map from N0×N0×R to R, andbisatisfies the assumptionC2fori1,2.
Theorem 4.1. Suppose thatβ0/0 and the functionsFandHin4.1satisfy the conditions
F
n, s, y≤f1n, sy, H
n, s, y≤f2n, sy,
4.2
wheref1, f2:N0×N0 → 0,∞. Ifynis a solution of4.1onN0, then
yn≤a0 exp
⎡
⎢⎣
b2n−1 sb20
f2n, s n−1
s0
f1n, b1s
Δas/
as
hs
⎤
⎥⎦, 4.3
where
as max
0≤τ≤s,τ∈N0
βτ, f1n, s max
0≤τ≤n,τ∈N0
f1τ, s, f2n, s max
0≤τ≤n,τ∈N0
f2τ, s,
hn
a0 1
2
b1n−1 sb10
f10, s 1 2
n−1
s0
Δas as .
4.4
Proof. Using4.1and4.2, the solutionynsatisfies
un≤an b1n−1
sb10
f1n, sw1us b2n−1
sb20
f2n, sw2us, n∈N0, 4.5
where
un yn, an βn, w1u √
u, w2u u. 4.6
Clearly,an >0 for alln∈N0sinceβ0/0. For positive constantsu1, u2, we have
W1u u
u1
dz
w1z 2√ u−√
u1
, W1−1u u 2 √
u1
2
, W2u
u
u2
dz
w2z ln u
u2, W2−1u u2expu, r1
n, j a
j
>0, r1n,0 a0,
r2
n, j 2
a0 −√ u1
b1j−1 sb10
f1n, s j−1
s0
Δas as , Δ2r2
n, j f1
n, b1
j Δa
j a
j, φ2u w2u w1u √
u.
4.7
It is obvious thatw1andw2satisfyC4. ApplyingTheorem 2.1gives
un≤a0 exp
⎡
⎢⎣
b2n−1 sb20
f2n, s n−1
s0
f1n, b1s
Δas/
as
hs
⎤
⎥⎦ 4.8
which implies4.3.
Theorem 4.2. Suppose thatβ0/0 and the functionsFandHin4.1satisfy the conditions F
n, s, y1
−F
n, s, y2≤f1n, sy1−y2, H
n, s, y1
−H
n, s, y2≤f2n, sy1−y2,
4.9
wheref1, f2:N0×N0 → 0,∞. Then4.1has at most one solution onN0. Proof. Lety1nandy2nbe two solutions of4.1onN0. From4.9, we have
|un| ≤b1n−1
sb10
f1n, sw1us b2n−1
sb20
f2n, sw2us, n∈N0, 4.10
whereun |y1n−y2n|,an 0,w1u √
uandw2u u. ApplingTheorem 2.1, Remark 2.4, and the notationWi0 0 fori1,2, we obtain thatun 0 which implies that the solution is unique.
Acknowledgments
This paper was supported by Guangdong Provincial natural science Foundation07301595.
The authors would like to thank Professor Boling Guo for his great help.
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