# A Generalized Nonlinear Sum-Difference Inequality of Product Form

Authors: YongZhou Qin; Wu-Sheng Wang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We establish a generalized nonlinear discrete inequality of product form, which includes both nonconstant terms outside the sums and composite functions of nonlinear function and unknown function without assumption of monotonicity. Upper bound estimations of unknown functions are given by technique of change of variable, amplification method, difference and summation, inverse function, and the dialectical relationship between constants and variables. Using our result we can solve both the discrete inequality in Pachpatte (1995). Our result can be used as tools in the study of difference equations of product form.


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## Body

## 1. Introduction

Being an important tool in the study of existence, uniqueness, boundedness, stability, and other qualitative properties of solutions of differential equations and integral equations, various generalizations of Gronwall inequalities [1, 2] and their applications have attracted great interests of many mathematicians (such as [3–6]). Some recent works can be found, for example, in [7–10] and some references therein. Along with the development of the theory of integral inequalities and the theory of difference equations, more attention is paid to some discrete versions of Gronwall-Bellman type inequalities (such as [3, 4, 11–13]). Some recent works can be found, for example, in [14–24] and some references therein.Pachpatte [4] obtained the explicit bound to the unknown function of the following sum-difference inequality:
(1)u2(n)≤(c12+2∑s=0n-1‍f(s)u(s))(c22+2∑s=0n-1‍g(s)u(s)).
Pachpatte [3] obtained the estimation of the unknown function of the following inequality:
(2)u(n)≤(c1+∑s=0n‍f(s)u(s))(c2+∑s=0n‍g(s)u(s)).
Then, the estimation can be used to study the boundedness, asymptotic behavior, and slow growth of the solutions of the sum-difference equation:
(3)x(n)=k(p1(n)+∑s=0n-1‍f1(n-s)x(s))×(p2(n)+∑s=0n-1‍f2(n-s)x(s)).
However, the bound given on such inequalities in [3, 4] is not directly applicable in the study of certain sum-difference equations. It is desirable to establish new inequalities of the above type, which can be used more effectively in the study of certain classes of sum-difference equations of product form.In this paper, we establish a new integral inequality of product form(4)u(n)≤(p1(n)+∑s=0n-1‍f1(n,s)φ1(u(s)))×(p2(n)+∑s=0n-1‍f2(n,s)φ2(u(s))),∀n∈ℕ0,
wherepi,fi,φi  (i=1,2)may not be monotone. Forφ1,φ2, we employ a technique of monotonization to construct two functions; the second possesses stronger monotonicity than the first. We can demonstrate that inequalities (1) and (2), considered in [3, 4], respectively, can also be solved with our result. Finally, we expound that we can give estimation of solutions of a class of sum-difference equations of product form.


## 2. Main Result and Proof

In this section, we proceed to solve the discrete inequality (4) and present explicit bounds on the embedded unknown function. Letℕ0∶={0,1,2,…},ℕ∶={1,2,…}, andℕab∶={a,a+1,a+2,…,a+n=b}  (a∈ℕ0,n,b∈ℕ). For functionz(n), its difference is defined byΔz=z(n+1)-z(n). Obviously, the linear difference equationΔz(n)=b(n)with the initial conditionz(n0)=0has the solutionz(n)=∑s=n0n-1b(s). For convenience, in the sequel we complementarily define that∑s=n0n0-1b(s)=0.First of all, we monotonize some given functionspi,fi,φiin the sum; let
(5)qi(n)=maxt∈ℕ0npi(t),gi(n,s)=maxt∈ℕ0nfi(t,s),i=1,2,
whereqiandgiare all nondecreasing inn  (i=1,2)and satisfy
(6)qi(n)≥pi(n)≥0,gi(n,s)≥fi(n,s)≥0.
Let
(7)w1(u)=maxs∈[0,u]φi(s),w2(u)≔maxs∈[0,u]{φ2(s)w1(s)}w1(u),
wherewi(u)is nondecreasing inu  (i=1,2)andw2(u)/w1(u)is also nondecreasing inuand satisfies
(8)wi(u)≥φi(u),i=1,2,(9)W1(x)=∫x0x‍dsw1(s),x0>0,(10)W2(x)≔∫x0x‍w1(W1-1(s))dsw2(W1-1(s)),x0>0,(11)W3(x)≔∫x0x‍dsw1(W1-1(W2-1(s))),x0>0,
whereW1-1,W2-1denote the inverse function ofW1,W2, respectively.Theorem 1.
Letf,gbe nonnegative and given functions onℕ0. Suppose thatuis a nonnegative and unknown function. Then, the discrete inequality (4) gives
(12)u(n)≤W1-1(W2-1(W3-1(A(n)+B(n)))),∀n∈ℕ0b,
whereW1,W2,W3are defined by (9), (10), and (11), respectively,W1-1,W2-1,W3-1denote the inverse functions ofW1,W2,W3, respectively,
(13)A(n)=W3(W2(W1(q1(n)q2(n))+∑t=0n-1‍q2(n)g1(n,t))+∑s=0n-1‍q1(n)g2(n,t)),B(n)=∑t=0n-1‍(g2(n,t)∑s=0t‍g1(n,s)+g1(n,t)∑s=0t-1‍g2(n,s)),
andbis the largest natural number such that
(14)b=Max{n∈ℕ0:A(n)+B(n)∈Dom(W3-1),W3-1(A(n)+B(n))∈Dom(W2-1),W2-1(W3-1(A(n)+B(n)))∈Dom(W1-1)}.Proof.
Using (5), (6), (7), and (8), we observe that
(15)u(n)≤(p1(n)+∑s=0n-1‍f1(n,s)φ1(u(s)))×(p2(n)+∑s=0n-1‍f2(n,s)φ2(u(s)))≤(q1(n)+∑s=0n-1‍g1(n,s)w1(u(s)))×(q2(n)+∑s=0n-1‍g2(n,s)w2(u(s)))≤(q1(T)+∑s=0n-1‍g1(T,s)w1(u(s)))×(q2(T)+∑s=0n-1‍g2(T,s)w2(u(s))),∀n∈ℕ0T,
whereT∈ℕ0Jis chosen arbitrarily. Letv(n)denote the function on the right-hand side of (15), namely,
(16)v(n)=(q1(T)+∑s=0n-1‍g1(T,s)w1(u(s)))×(q2(T)+∑s=0n-1‍g2(T,s)w2(u(s))),∀n∈ℕ0T,
which is a nonnegative and nondecreasing function onℕ0Twithv(0)=q1(T)q2(T). Then (4) is equivalent to
(17)u(n)≤v(n),∀n∈ℕ0T.
Using the difference formula
(18)Δx(n)y(n)=x(n+1)Δy(n)+y(n)Δx(n)
and the monotonicity ofwiandv, from (16) and (17), we observe that
(19)Δv(n)=(q1(T)+∑s=0n-1‍g1(T,s)w1(u(s)))×Δ(q2(T)+∑s=0n-1‍g2(T,s)w2(u(s)))+Δ(q1(T)+∑s=0n-1‍g1(T,s)w1(u(s)))×(q2(T)+∑s=0n-1‍g2(T,s)w2(u(s)))=(q1(T)+∑s=0n‍g1(T,s)w1(u(s)))×g2(T,n)w2(u(n))+g1(T,n)w1(u(n))×(q2(T)+∑s=0n-1‍g2(T,s)w2(u(s)))≤(q1(T)+∑s=0n‍g1(T,s)w1(v(s)))×g2(T,n)w2(v(n))+g1(T,n)w1(v(n))×(q2(T)+∑s=0n-1‍g2(T,s)w2(v(s)))=w1(v(n))(q2(T)g1(T,n)w2(v(n))w1(v(n))∑s=0n-1‍+q1(T)g2(T,n)w2(v(n))w1(v(n))+g2(T,n)w2(v(n))w1(v(n))×∑s=0n‍g1(T,s)w1(v(s))+g1(T,n)∑s=0n-1‍g2(T,s)w2(v(s)))≤w1(v(n))(q2(T)g1(T,n)∑s=0n‍+q1(T)g2(T,n)w2(v(n))w1(v(n))+(g2(T,n)∑s=0n‍g1(T,s)+g1(T,n)∑s=0n-1‍g2(T,s))∑s=0n‍×w2(v(n))),∀n∈ℕ0T,
for alln∈ℕ0T. From (19), we have
(20)Δv(n)w1(v(n))≤q2(T)g1(T,n)+q1(T)g2(T,n)w2(v(n))w1(v(n))+(g2(T,n)∑s=0n‍g1(T,s)+g1(T,n)×∑s=0n-1‍g2(T,s))w2(v(n)),∀n∈ℕ0T.
On the other hand, by the mean value theorem for integrals, for arbitrarily given integersn,n+1∈ℕ0T, there existsξin the open interval(v(n),v(n+1))such that
(21)W1(v(n+1))-W1(v(n))=∫v(n)v(n+1)‍dsw1(s)=Δv(n)w1(ξ)≤Δv(n)w1(v(n)),∀n∈ℕ0T,
whereW1is defined by (9). From (20) and (21), we have
(22)W1(v(n+1))-W1(v(n))≤q2(T)g1(T,n)+q1(T)g2(T,n)w2(v(n))w1(v(n))+(g2(T,n)∑s=0n‍g1(T,s)+g1(T,n)∑s=0n-1‍g2(T,s))w2(v(n)),
for alln∈ℕ0T. By settingn=tin (22) and substitutingt=0,1,2,…,n-1successively, we obtain
(23)W1(v(n))≤W1(v(0))+∑t=0n-1‍q2(T)g1(T,t)+∑t=0n-1‍q1(T)g2(T,t)w2(v(t))w1(v(t))+∑t=0n-1‍(g2(T,t)∑s=0t‍g1(T,s)+g1(T,t)∑s=0t-1‍g2(T,s))w2(v(t))≤W1(v(0))+∑t=0T-1‍q2(T)g1(T,t)+∑t=0n-1‍q1(T)g2(T,t)w2(v(t))w1(v(t))+∑t=0n-1‍(g2(T,t)∑s=0t‍g1(T,s)+g1(T,t)∑s=0t-1‍g2(T,s))w2(v(t)),∀n∈ℕ0T.
Letx(n)denote the function on the right-hand side of (23); namely,
(24)x(n)=W1(v(0))+∑t=0T-1‍q2(T)g1(T,t)+∑t=0n-1‍q1(T)g2(T,t)w2(v(t))w1(v(t))+∑t=0n-1‍(g2(T,t)∑s=0t‍g1(T,s)+g1(T,t)∑s=0t-1‍g2(T,s))w2(v(t)),∀n∈ℕ0T.
Thenx(0)=W1(v(0))+∑t=0T-1‍q2(T)g1(T,t),xis a nonnegative and nondecreasing function onℕ0T, and (23) is equivalent to
(25)v(n)≤W1-1(x(n)),∀n∈ℕ0T.
From (24), we obtain
(26)Δx(n)=q1(T)g2(T,n)w2(v(n))w1(v(n))+(g2(T,n)∑s=0n‍g1(T,s)+g1(T,n)∑s=0n-1‍g2(T,s))w2(v(n))≤q1(T)g2(T,n)w2(W1-1(x(n)))w1(W1-1(x(n)))+(g2(T,n)∑s=0n‍g1(T,s)+g1(T,n)∑s=0n-1‍g2(T,s))w2(W1-1(x(n))),∀n∈ℕ0T.
From (26), we have
(27)w1(W1-1(x(n)))Δx(n)w2(W1-1(x(n)))≤q1(T)g2(T,n)+(g2(T,n)∑s=0n‍g1(T,s)+g1(T,n)∑s=0n-1‍g2(T,s))w1(W1-1(x(n))),∀n∈ℕ0T.
Once again, performing the same procedure as in (21), (22), and (23), (27) gives
(28)W2(x(n))≤W2(x(0))+∑s=0n-1‍q1(T)g2(T,t)+∑s=0n-1‍(g2(T,t)∑s=0t‍g1(T,s)+g1(T,t)∑s=0t-1‍g2(T,s))×w1(W1-1(x(t)))≤W2(x(0))+∑s=0T-1‍q1(T)g2(T,t)+∑s=0n-1‍(g2(T,t)∑s=0t‍g1(T,s)+g1(T,t)∑s=0t-1‍g2(T,s))×w1(W1-1(x(t))),∀n∈ℕ0T,
whereW2is defined by (10). Letz(n)denote the function on the right-hand side of (28); namely,
(29)z(n)=W2(x(0))+∑s=0T-1‍q1(T)g2(T,t)+∑s=0n-1‍(g2(T,t)∑s=0t‍g1(T,s)+g1(T,t)∑s=0t-1‍g2(T,s))w1(W1-1(x(t))),∀n∈ℕ0T.
Thenz(0)=W2(x(0))+∑s=0T-1‍q1(T)g2(T,t),zis a nonnegative and nondecreasing function onℕ0T, and (28) is equivalent to
(30)x(n)≤W2-1(z(n)),∀n∈ℕ0T.
From (29) and (30), we obtain
(31)Δz(n)=(g2(T,n)∑s=0n‍g1(T,s)+g1(T,n)∑s=0n-1‍g2(T,s))w1(W1-1(x(n)))≤(g2(T,n)∑s=0n‍g1(T,s)+g1(T,n)∑s=0n-1‍g2(T,s))w1(W1-1(W2-1(z(n)))),
for alln∈ℕ0T. From (31), we have
(32)Δz(n)w1(W1-1(W2-1(z(n))))≤g2(T,n)∑s=0n‍g1(T,s)+g1(T,n)∑s=0n-1‍g2(T,s),∀n∈ℕ0T.
Once again, performing the same procedure as in (21), (22), and (23), (32) gives
(33)W3(z(n))≤W3(z(0))+∑t=0n-1‍(g2(T,t)∑s=0t‍g1(T,s)+g1(T,t)∑s=0t-1‍g2(T,s)),∀n∈ℕ0T.
Using (17), (25), and (30), from (33) we have
(34)u(n)≤v(n)≤W1-1(x(n))≤W1-1(W2-1(z(n)))≤W1-1(W2-1∑s=0t‍(W3-1∑s=0t‍(W3(z(0))∑s=0t‍+∑t=0n-1‍(g2(T,t)∑s=0t‍g1(T,s)+g1(T,t)∑s=0t-1‍g2(T,s)))))≤W1-1(W2-1∑s=0T-1‍(W3-1∑s=0T-1‍(W3(W2(x(0))∑s=0T-1‍+∑s=0T-1‍q1(T)g2(T,t))+∑t=0n-1‍(g2(T,t)∑s=0t‍g1(T,s)+g1(T,t)∑s=0t-1‍g2(T,s)))))≤W1-1(W2-1∑t=0T-1‍(W3-1∑t=0T-1‍(W3∑t=0T-1(W2(W1(q1(T)q2(T))∑t=0T-1‍‍(W2∑t=0T-1‍(W1∑t=0T-1‍(q1(T)q2(T))+∑t=0T-1‍q2(T)g1(T,t))+∑s=0T-1‍q1(T)g2(T,t))+∑t=0n-1‍(g2(T,t)∑s=0t‍g1(T,s)+g1(T,t)∑s=0t-1‍g2(T,s))))),∀n∈ℕ0T.
Asn=T, (34) yields
(35)u(T)≤W1-1(W2-1∑t=0T-1‍(W3-1∑t=0T-1‍(W3∑t=0T-1‍(W2∑t=0T-1‍(W1∑t=0T-1‍(q1(T)q2(T))+∑t=0T-1‍q2(T)g1(T,t))+∑s=0T-1‍q1(T)g2(T,t))+∑t=0T-1‍(g2(T,t)∑s=0t‍g1(T,s)+g1(T,t)∑s=0t-1‍g2(T,s))))).
SinceT∈ℕ, andT≤bis chosen arbitrarily in (35), the estimation (12) is derived. This completes the proof of Theorem 1.


## 3. Application

We consider a sum-difference equation of product form(36)x(n)=(a(n)+∑s=0n-1‍f(n,s)φ1(x(s)))×(b(n)+∑s=0n-1‍g(n,s)φ2(x(s))),∀n∈ℕ0.
From (36), we have
(37)|x(n)|≤(a(n)+∑s=0n-1‍f(n,s)φ1(|x(s)|))×(b(n)+∑s=0n-1‍g(n,s)φ2(|x(s)|)),∀n∈ℕ0.
Letu(n)=|x(n)|,p1(n)=|a(n)|,p2(n)=|b(n)|,f1(n,s)=|f(n,s)|, andf2(n,s)=|g(n,s)|in (37); then (37) is the inequality of the form (4). Applying our result we get the estimation of solution of the sum-difference equations of product form (36).



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