A Generalized Nonlinear Sum-Difference Inequality of Product Form

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.

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: 𝑢²(𝑛)≤𝑐²₁+2^𝑛−1_𝑠=0𝑓(𝑠)𝑢(𝑠)𝑐²₂+2^𝑛−1_𝑠=0𝑔(𝑠)𝑢(𝑠). (1) Pachpatte [3] obtained the estimation of the unknown function of the following inequality: 𝑢(𝑛)≤𝑐₁+^𝑛_𝑠=0𝑓(𝑠)𝑢(𝑠)𝑐₂+^𝑛_𝑠=0𝑔(𝑠)𝑢(𝑠). (2) Then, the estimation can be used to study the boundedness, asymptotic behavior, and slow growth of the solutions of the sum-difference equation: 𝑥(𝑛)=𝑘𝑝₁(𝑛)+^𝑛−1_𝑠=0𝑓₁()𝑥(𝑠×𝑝𝑛−𝑠)₂(𝑛)+^𝑛−1_𝑠=0𝑓₂(𝑛−𝑠)𝑥(𝑠). (3) 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 𝑢(𝑛)≤𝑝₁(𝑛)+^𝑛−1_𝑠=0𝑓₁(𝑛,𝑠)𝜑₁(𝑢(𝑠×𝑝))₂(𝑛)+^𝑛−1_𝑠=0𝑓₂(𝑛,𝑠)𝜑₂(𝑢(𝑠)),∀𝑛∈ℕ₀, (4) where𝑝_𝑖,𝑓_𝑖,𝜑_𝑖  (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,…}, andℕ^𝑏_𝑎∶={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)=∑^𝑛−1_𝑠=𝑛0𝑏(𝑠). For convenience, in the sequel we complementarily define that∑^𝑛0−1_𝑠=𝑛0𝑏(𝑠)=0.

First of all, we monotonize some given functions𝑝_𝑖,𝑓_𝑖,𝜑_𝑖in the sum; let 𝑞_𝑖(𝑛)=max_{𝑡∈ℕ𝑛0}𝑝_𝑖(𝑡),𝑔_𝑖(𝑛,𝑠)=max_{𝑡∈ℕ𝑛0}𝑓_𝑖(𝑡,𝑠),𝑖=1,2, (5) where𝑞_𝑖and𝑔_𝑖are all nondecreasing in𝑛  (𝑖=1,2)and satisfy 𝑞_𝑖(𝑛)≥𝑝_𝑖(𝑛)≥0,𝑔_𝑖(𝑛,𝑠)≥𝑓_𝑖(𝑛,𝑠)≥0. (6) Let 𝑤₁(𝑢)=max_{[]𝑠∈0,𝑢}𝜑_𝑖(𝑠),𝑤₂(𝑢)∶=max_{[]𝑠∈0,𝑢}𝜑₂(𝑠)𝑤₁(𝑠)𝑤₁(𝑢), (7) where𝑤_𝑖(𝑢)is nondecreasing in𝑢  (𝑖=1,2)and𝑤₂(𝑢)/𝑤₁(𝑢)is also nondecreasing in𝑢and satisfies 𝑤_𝑖(𝑢)≥𝜑_𝑖(𝑢)𝑊,𝑖=1,2,₁(𝑥)=^𝑥_𝑥0𝑑𝑠𝑤₁(𝑠),𝑥₀𝑊>0,₂(𝑥)∶=^𝑥_𝑥0𝑤₁𝑊₁⁻¹(𝑠)𝑑𝑠𝑤₂𝑊₁⁻¹(𝑠),𝑥₀𝑊>0,₃(𝑥)∶=^𝑥_𝑥0𝑑𝑠𝑤₁𝑊₁⁻¹𝑊₂⁻¹(𝑠),𝑥₀>0, (8)(9)(10)(11) 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 𝑢(𝑛)≤𝑊₁⁻¹𝑊₂⁻¹𝑊₃⁻¹(𝐴(𝑛)+𝐵(𝑛)),∀𝑛∈ℕ^𝑏₀, (12) where𝑊₁,𝑊₂,𝑊₃are defined by (9), (10), and (11), respectively,𝑊₁⁻¹,𝑊₂⁻¹,𝑊₃⁻¹denote the inverse functions of𝑊₁,𝑊₂,𝑊₃, respectively, 𝐴(𝑛)=𝑊₃𝑊₂𝑊₁𝑞₁(𝑛)𝑞₂(𝑛)+^𝑛−1_𝑡=0𝑞₂(𝑛)𝑔₁(+𝑛,𝑡)_{𝑛−1𝑠=0}𝑞₁(𝑛)𝑔₂(),𝐵(𝑛)=𝑛,𝑡_{𝑛−1𝑡=0}𝑔₂(𝑛,𝑡)^𝑡_𝑠=0𝑔₁(𝑛,𝑠)+𝑔₁(𝑛,𝑡)^𝑡−1_𝑠=0𝑔₂(𝑛,𝑠), (13) and𝑏is the largest natural number such that 𝑏=Max𝑛∈ℕ₀∶𝐴(𝑛)+𝐵(𝑛)∈Dom𝑊₃⁻¹,𝑊₃⁻¹(𝐴(𝑛)+𝐵(𝑛))∈Dom𝑊₂⁻¹,𝑊₂⁻¹𝑊₃⁻¹(𝐴(𝑛)+𝐵(𝑛))∈Dom𝑊₁⁻¹. (14)

Proof. Using (5), (6), (7), and (8), we observe that 𝑢(𝑛)≤𝑝₁(𝑛)+^𝑛−1_𝑠=0𝑓₁(𝑛,𝑠)𝜑₁(𝑢(𝑠×𝑝))₂(𝑛)+^𝑛−1_𝑠=0𝑓₂(𝑛,𝑠)𝜑₂(𝑢(𝑠≤𝑞))₁(𝑛)+^𝑛−1_𝑠=0𝑔₁(𝑛,𝑠)𝑤₁(𝑢(𝑠×𝑞))₂(𝑛)+^𝑛−1_𝑠=0𝑔₂(𝑛,𝑠)𝑤₂(𝑢(𝑠≤𝑞))₁(𝑇)+^𝑛−1_𝑠=0𝑔₁(𝑇,𝑠)𝑤₁(𝑢(𝑠×𝑞))₂(𝑇)+^𝑛−1_𝑠=0𝑔₂(𝑇,𝑠)𝑤₂(𝑢(𝑠)),∀𝑛∈ℕ^𝑇₀, (15) where𝑇∈ℕ^𝐽₀is chosen arbitrarily. Let𝑣(𝑛)denote the function on the right-hand side of (15), namely, 𝑣(𝑛)=𝑞₁(𝑇)+^𝑛−1_𝑠=0𝑔₁(𝑇,𝑠)𝑤₁(𝑢(𝑠×𝑞))₂(𝑇)+^𝑛−1_𝑠=0𝑔₂(𝑇,𝑠)𝑤₂(𝑢(𝑠)),∀𝑛∈ℕ^𝑇₀, (16) which is a nonnegative and nondecreasing function onℕ^𝑇₀with𝑣(0)=𝑞₁(𝑇)𝑞₂(𝑇). Then (4) is equivalent to 𝑢(𝑛)≤𝑣(𝑛),∀𝑛∈ℕ^𝑇₀. (17) Using the difference formula Δ𝑥(𝑛)𝑦(𝑛)=𝑥(𝑛+1)Δ𝑦(𝑛)+𝑦(𝑛)Δ𝑥(𝑛) (18) and the monotonicity of𝑤_𝑖and𝑣, from (16) and (17), we observe that Δ𝑣(𝑛)=𝑞₁(𝑇)+^𝑛−1_𝑠=0𝑔₁(𝑇,𝑠)𝑤₁(𝑢(𝑠𝑞))×Δ₂(𝑇)+^𝑛−1_𝑠=0𝑔₂(𝑇,𝑠)𝑤₂(𝑢(𝑠𝑞))+Δ₁(𝑇)+^𝑛−1_𝑠=0𝑔₁(𝑇,𝑠)𝑤₁(𝑢(𝑠×𝑞))₂(𝑇)+^𝑛−1_𝑠=0𝑔₂(𝑇,𝑠)𝑤₂(𝑢(𝑠=𝑞))₁(𝑇)+^𝑛_𝑠=0𝑔₁(𝑇,𝑠)𝑤₁(𝑢(𝑠))×𝑔₂(𝑇,𝑛)𝑤₂(𝑢(𝑛))+𝑔₁(𝑇,𝑛)𝑤₁(𝑢(𝑛×𝑞))₂(𝑇)+^𝑛−1_𝑠=0𝑔₂(𝑇,𝑠)𝑤₂(𝑢(𝑠≤𝑞))₁(𝑇)+^𝑛_𝑠=0𝑔₁(𝑇,𝑠)𝑤₁(𝑣(𝑠))×𝑔₂(𝑇,𝑛)𝑤₂(𝑣(𝑛))+𝑔₁(𝑇,𝑛)𝑤₁(𝑣(𝑛×𝑞))₂(𝑇)+^𝑛−1_𝑠=0𝑔₂(𝑇,𝑠)𝑤₂(𝑣(𝑠))=𝑤₁(𝑣(𝑛))𝑞₂(𝑇)𝑔₁(𝑇,𝑛)+𝑞₁(𝑇)𝑔₂()𝑤𝑇,𝑛²(𝑣(𝑛))𝑤₁(𝑣(𝑛))+𝑔₂()𝑤𝑇,𝑛²(𝑣(𝑛))𝑤₁(𝑣(𝑛×))_𝑛𝑠=0𝑔₁(𝑇,𝑠)𝑤₁(𝑣(𝑠))+𝑔₁(𝑇,𝑛)^𝑛−1_𝑠=0𝑔₂(𝑇,𝑠)𝑤₂(𝑣(𝑠))≤𝑤₁(𝑣(𝑛))𝑞₂(𝑇)𝑔₁(𝑇,𝑛)+𝑞₁(𝑇)𝑔₂()𝑤𝑇,𝑛²(𝑣(𝑛))𝑤₁(𝑣(𝑛+𝑔))₂(𝑇,𝑛)^𝑛_𝑠=0𝑔₁(𝑇,𝑠)+𝑔₁(𝑇,𝑛)^𝑛−1_𝑠=0𝑔₂(𝑇,𝑠)×𝑤₂(𝑣(𝑛)),∀𝑛∈ℕ^𝑇₀, (19) for all𝑛∈ℕ^𝑇₀. From (19), we have Δ𝑣(𝑛)𝑤₁(𝑣(𝑛))≤𝑞₂(𝑇)𝑔₁(𝑇,𝑛)+𝑞₁(𝑇)𝑔₂()𝑤𝑇,𝑛²(𝑣(𝑛))𝑤₁(𝑣(𝑛+𝑔))₂(𝑇,𝑛)^𝑛_𝑠=0𝑔₁(𝑇,𝑠)+𝑔₁()×𝑇,𝑛_{𝑛−1𝑠=0}𝑔₂(𝑇,𝑠)𝑤₂(𝑣(𝑛)),∀𝑛∈ℕ^𝑇₀. (20) On the other hand, by the mean value theorem for integrals, for arbitrarily given integers𝑛,𝑛+1∈ℕ^𝑇₀, there exists𝜉in the open interval(𝑣(𝑛),𝑣(𝑛+1))such that 𝑊₁(𝑣(𝑛+1))−𝑊₁(𝑣(𝑛))=_𝑣(𝑛)^𝑣(𝑛+1)𝑑𝑠𝑤₁(𝑠)=Δ𝑣(𝑛)𝑤₁(𝜉)≤Δ𝑣(𝑛)𝑤₁(𝑣(𝑛)),∀𝑛∈ℕ^𝑇₀, (21) where𝑊₁is defined by (9). From (20) and (21), we have 𝑊₁(𝑣(𝑛+1))−𝑊₁(𝑣(𝑛))≤𝑞₂(𝑇)𝑔₁(𝑇,𝑛)+𝑞₁(𝑇)𝑔₂()𝑤𝑇,𝑛²(𝑣(𝑛))𝑤₁(𝑣(𝑛+𝑔))₂(𝑇,𝑛)^𝑛_𝑠=0𝑔₁(𝑇,𝑠)+𝑔₁(𝑇,𝑛)^𝑛−1_𝑠=0𝑔₂(𝑇,𝑠)𝑤₂(𝑣(𝑛)), (22) for all𝑛∈ℕ^𝑇₀. By setting𝑛=𝑡in (22) and substituting𝑡=0,1,2,…,𝑛−1successively, we obtain 𝑊₁(𝑣(𝑛))≤𝑊₁(𝑣(0+))_{𝑛−1𝑡=0}𝑞₂(𝑇)𝑔₁()+𝑇,𝑡_{𝑛−1𝑡=0}𝑞₁(𝑇)𝑔₂()𝑤𝑇,𝑡²(𝑣(𝑡))𝑤₁(𝑣(𝑡+))_{𝑛−1𝑡=0}𝑔₂(𝑇,𝑡)^𝑡_𝑠=0𝑔₁(𝑇,𝑠)+𝑔₁(𝑇,𝑡)^𝑡−1_𝑠=0𝑔₂(𝑇,𝑠)𝑤₂(𝑣(𝑡))≤𝑊₁(𝑣(0))+^𝑇−1_𝑡=0𝑞₂(𝑇)𝑔₁()+𝑇,𝑡_{𝑛−1𝑡=0}𝑞₁(𝑇)𝑔₂()𝑤𝑇,𝑡²(𝑣(𝑡))𝑤₁(𝑣(𝑡+))_{𝑛−1𝑡=0}𝑔₂(𝑇,𝑡)^𝑡_𝑠=0𝑔₁(𝑇,𝑠)+𝑔₁(𝑇,𝑡)^𝑡−1_𝑠=0𝑔₂(𝑇,𝑠)𝑤₂(𝑣(𝑡)),∀𝑛∈ℕ^𝑇₀. (23) Let𝑥(𝑛)denote the function on the right-hand side of (23); namely, 𝑥(𝑛)=𝑊₁(𝑣(0))+^𝑇−1_𝑡=0𝑞₂(𝑇)𝑔₁()+𝑇,𝑡_{𝑛−1𝑡=0}𝑞₁(𝑇)𝑔₂()𝑤𝑇,𝑡²(𝑣(𝑡))𝑤₁(𝑣(𝑡+))_{𝑛−1𝑡=0}𝑔₂(𝑇,𝑡)^𝑡_𝑠=0𝑔₁(𝑇,𝑠)+𝑔₁(𝑇,𝑡)^𝑡−1_𝑠=0𝑔₂(𝑇,𝑠)𝑤₂(𝑣(𝑡)),∀𝑛∈ℕ^𝑇₀. (24) Then𝑥(0)=𝑊₁(𝑣(0))+∑^𝑇−1_𝑡=0𝑞₂(𝑇)𝑔₁(𝑇,𝑡),𝑥is a nonnegative and nondecreasing function onℕ^𝑇₀, and (23) is equivalent to 𝑣(𝑛)≤𝑊₁⁻¹(𝑥(𝑛)),∀𝑛∈ℕ^𝑇₀. (25) From (24), we obtain Δ𝑥(𝑛)=𝑞₁(𝑇)𝑔₂()𝑤𝑇,𝑛²(𝑣(𝑛))𝑤₁(𝑣(𝑛+𝑔))₂(𝑇,𝑛)^𝑛_𝑠=0𝑔₁(𝑇,𝑠)+𝑔₁(𝑇,𝑛)^𝑛−1_𝑠=0𝑔₂(𝑇,𝑠)𝑤₂(𝑣(𝑛))≤𝑞₁(𝑇)𝑔₂()𝑤𝑇,𝑛²𝑊₁⁻¹(𝑥(𝑛))𝑤₁𝑊₁⁻¹(𝑥(𝑛+𝑔))₂(𝑇,𝑛)^𝑛_𝑠=0𝑔₁(𝑇,𝑠)+𝑔₁(𝑇,𝑛)^𝑛−1_𝑠=0𝑔₂(𝑇,𝑠)𝑤₂𝑊₁⁻¹(𝑥(𝑛)),∀𝑛∈ℕ^𝑇₀. (26) From (26), we have 𝑤₁𝑊₁⁻¹(𝑥(𝑛))Δ𝑥(𝑛)𝑤₂𝑊₁⁻¹(𝑥(𝑛))≤𝑞₁(𝑇)𝑔₂()+𝑔𝑇,𝑛₂(𝑇,𝑛)^𝑛_𝑠=0𝑔₁(𝑇,𝑠)+𝑔₁(𝑇,𝑛)^𝑛−1_𝑠=0𝑔₂(𝑇,𝑠)𝑤₁𝑊₁⁻¹(𝑥(𝑛)),∀𝑛∈ℕ^𝑇₀. (27) Once again, performing the same procedure as in (21), (22), and (23), (27) gives 𝑊₂(𝑥(𝑛))≤𝑊₂(𝑥(0))+^𝑛−1_𝑠=0𝑞₁(𝑇)𝑔₂()+𝑇,𝑡_{𝑛−1𝑠=0}𝑔₂(𝑇,𝑡)^𝑡_𝑠=0𝑔₁(𝑇,𝑠)+𝑔₁(𝑇,𝑡)^𝑡−1_𝑠=0𝑔₂(𝑇,𝑠)×𝑤₁𝑊₁⁻¹(𝑥(𝑡))≤𝑊₂(𝑥(0))+^𝑇−1_𝑠=0𝑞₁(𝑇)𝑔₂()+𝑇,𝑡_{𝑛−1𝑠=0}𝑔₂(𝑇,𝑡)^𝑡_𝑠=0𝑔₁(𝑇,𝑠)+𝑔₁(𝑇,𝑡)^𝑡−1_𝑠=0𝑔₂(𝑇,𝑠)×𝑤₁𝑊₁⁻¹(𝑥(𝑡)),∀𝑛∈ℕ^𝑇₀, (28) where𝑊₂is defined by (10). Let𝑧(𝑛)denote the function on the right-hand side of (28); namely, 𝑧(𝑛)=𝑊₂(𝑥(0))+^𝑇−1_𝑠=0𝑞₁(𝑇)𝑔₂()+𝑇,𝑡_{𝑛−1𝑠=0}𝑔₂(𝑇,𝑡)^𝑡_𝑠=0𝑔₁(𝑇,𝑠)+𝑔₁(𝑇,𝑡)^𝑡−1_𝑠=0𝑔₂(𝑇,𝑠)𝑤₁𝑊₁⁻¹(𝑥(𝑡)),∀𝑛∈ℕ^𝑇₀. (29) Then𝑧(0)=𝑊₂(𝑥(0))+∑^𝑇−1_𝑠=0𝑞₁(𝑇)𝑔₂(𝑇,𝑡),𝑧is a nonnegative and nondecreasing function onℕ^𝑇₀, and (28) is equivalent to 𝑥(𝑛)≤𝑊₂⁻¹(𝑧(𝑛)),∀𝑛∈ℕ^𝑇₀. (30) From (29) and (30), we obtain Δ𝑧(𝑛)=𝑔₂(𝑇,𝑛)^𝑛_𝑠=0𝑔₁(𝑇,𝑠)+𝑔₁(𝑇,𝑛)^𝑛−1_𝑠=0𝑔₂(𝑇,𝑠)𝑤₁𝑊₁⁻¹(𝑥(𝑛≤𝑔))₂(𝑇,𝑛)^𝑛_𝑠=0𝑔₁(𝑇,𝑠)+𝑔₁(𝑇,𝑛)^𝑛−1_𝑠=0𝑔₂(𝑇,𝑠)𝑤₁𝑊₁⁻¹𝑊₂⁻¹(𝑧(𝑛)), (31) for all𝑛∈ℕ^𝑇₀. From (31), we have Δ𝑧(𝑛)𝑤₁𝑊₁⁻¹𝑊₂⁻¹(𝑧(𝑛))≤𝑔₂(𝑇,𝑛)^𝑛_𝑠=0𝑔₁(𝑇,𝑠)+𝑔₁(𝑇,𝑛)^𝑛−1_𝑠=0𝑔₂(𝑇,𝑠),∀𝑛∈ℕ^𝑇₀. (32) Once again, performing the same procedure as in (21), (22), and (23), (32) gives 𝑊₃(𝑧(𝑛))≤𝑊₃(𝑧(0+))_{𝑛−1𝑡=0}𝑔₂(𝑇,𝑡)^𝑡_𝑠=0𝑔₁(𝑇,𝑠)+𝑔₁(𝑇,𝑡)^𝑡−1_𝑠=0𝑔₂(𝑇,𝑠),∀𝑛∈ℕ^𝑇₀. (33) Using (17), (25), and (30), from (33) we have 𝑢(𝑛)≤𝑣(𝑛)≤𝑊₁⁻¹(𝑥(𝑛))≤𝑊₁⁻¹𝑊₂⁻¹(𝑧(𝑛))≤𝑊₁⁻¹𝑊₂⁻¹𝑊₃⁻¹𝑊₃(𝑧(0+))_{𝑛−1𝑡=0}𝑔₂(𝑇,𝑡)^𝑡_𝑠=0𝑔₁(𝑇,𝑠)+𝑔₁(𝑇,𝑡)^𝑡−1_𝑠=0𝑔₂(𝑇,𝑠)≤𝑊₁⁻¹𝑊₂⁻¹𝑊₃⁻¹𝑊₃𝑊₂(𝑥(0+))_{𝑇−1𝑠=0}𝑞₁(𝑇)𝑔₂()+𝑇,𝑡_{𝑛−1𝑡=0}𝑔₂(𝑇,𝑡)^𝑡_𝑠=0𝑔₁(𝑇,𝑠)+𝑔₁(𝑇,𝑡)^𝑡−1_𝑠=0𝑔₂(𝑇,𝑠)≤𝑊₁⁻¹𝑊₂⁻¹𝑊₃⁻¹𝑊₃𝑊₂𝑊₁𝑞₁(𝑇)𝑞₂(𝑇)+^𝑇−1_𝑡=0𝑞₂(𝑇)𝑔₁()+𝑇,𝑡_{𝑇−1𝑠=0}𝑞₁(𝑇)𝑔₂(+𝑇,𝑡)_{𝑛−1𝑡=0}𝑔₂(𝑇,𝑡)^𝑡_𝑠=0𝑔₁(𝑇,𝑠)+𝑔₁(𝑇,𝑡)^𝑡−1_𝑠=0𝑔₂(𝑇,𝑠),∀𝑛∈ℕ^𝑇₀. (34) As𝑛=𝑇, (34) yields 𝑢(𝑇)≤𝑊₁⁻¹𝑊₂⁻¹𝑊₃⁻¹𝑊₃𝑊₂𝑊₁𝑞₁(𝑇)𝑞₂(𝑇)+^𝑇−1_𝑡=0𝑞₂(𝑇)𝑔₁()+𝑇,𝑡_{𝑇−1𝑠=0}𝑞₁(𝑇)𝑔₂()+𝑇,𝑡_{𝑇−1𝑡=0}𝑔₂(𝑇,𝑡)^𝑡_𝑠=0𝑔₁(𝑇,𝑠)+𝑔₁(𝑇,𝑡)^𝑡−1_𝑠=0𝑔₂(𝑇,𝑠). (35) 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 𝑥(𝑛)=𝑎(𝑛)+^𝑛−1_𝑠=0𝑓(𝑛,𝑠)𝜑₁(𝑥(𝑠×𝑏(𝑛)+))_{𝑛−1𝑠=0}𝑔(𝑛,𝑠)𝜑₂(𝑥(𝑠)),∀𝑛∈ℕ₀. (36) From (36), we have |𝑥(𝑛)|≤𝑎(𝑛)+^𝑛−1_𝑠=0𝑓(𝑛,𝑠)𝜑₁𝑥(𝑠×𝑏(𝑛)+(|)|)_{𝑛−1𝑠=0}𝑔(𝑛,𝑠)𝜑₂(|𝑥(𝑠)|),∀𝑛∈ℕ₀. (37) 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).

Acknowledgments

This research was supported by National Natural Science Foundation of China (Project no. 11161018) and Guangxi Natural Science Foundation (2012GXNSFAA053009).