Estimation of Unknown Functions of Iterative Difference Inequalities and Their Applications

Abstract

We establish some new nonlinear retarded finite difference inequalities. The results that we propose here can be used as tools in the theory of certain new classes of finite difference equations in various difference situations. We also give applications of the inequalities to show the usefulness of our results.

1. Introduction

An integral inequality that provides an explicit bound to the unknown function furnishes a handy tool to investigate qualitative properties of solutions of differential and integral equations. One of the best known and widely used inequalities in the study of nonlinear differential equations is Gronwall-Bellman inequality [1, 2], which can be stated as follows. If 𝑢 and 𝑓 are nonnegative continuous functions on an interval [𝑎,𝑏] satisfying 𝑢(𝑡)≤𝑐+^𝑡_𝑎𝑓(𝑠)𝑢(𝑠)𝑑𝑠,𝑡∈[𝑎,𝑏], (1) for some constant 𝑐≥0, then 𝑢(𝑡)≤𝑐exp^𝑡_𝑎𝑓(𝑠)𝑑s,𝑡∈[𝑎,𝑏]. (2) Being an important tool in the study of 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 [3–5]. Some recent works can be found in [6–12] and the references therein. Along with the development of the theory of integral inequalities and the theory of difference equations, more and more attentions are paid to discrete versions of Gronwall-type inequalities; see [13–36] and the references cited therein.

Sugiyama [13] established the most precise and complete discrete analogue of the Gronwall inequality [1] in the following form. Let 𝑢(𝑛) and 𝑓(𝑛) be nonnegative functions defined for 𝑛∈𝐍, and suppose that 𝑓(𝑛)≥0 for every 𝑛∈𝐍. If 𝑢(𝑛)<𝑢₀+^𝑛−1_𝑠=𝑛0𝑓(𝑠)𝑢(𝑠),𝑛∈𝐍, (3) where 𝐍 is the set of points 𝑛₀+𝑘 (𝑘=0,1,2,…), 𝑛₀≥0 is a given integer, and 𝑢₀ is a nonnegative constant, then 𝑢(𝑛)<𝑢₀^𝑛−1_𝑠=𝑛01+𝑓(𝑠),𝑛∈𝐍. (4) Pachpatte [15] established some generalized discrete analogue of the Gronwall inequality in the following form. Let 𝑚(𝑠) be a positive and monotone nondecreasing function on 𝐍, and let 𝑎(𝑠),𝑏(𝑠) be nonnegative functions on 𝐍. If 𝑢(𝑛) satisfies 𝑢(𝑛)≤𝑚(𝑠)+^𝑛−1_𝑠=𝑛0𝑎(𝑠)𝑢(𝑠)+^𝑠−1_𝜏=𝑛0𝑏(𝜏)𝑢(𝜏),∀𝑛∈𝐍, (5) then 𝑢(𝑛)≤𝑃(𝑛)𝑚(𝑠),∀𝑛∈𝐍, (6) where 𝑃(𝑛)=1+^𝑛−1_𝑠=𝑛0𝑎(𝑠)^𝑠−1_𝜏=𝑛0[1+𝑎(𝜏)+𝑏(𝜏)],∀𝑛∈𝐍. (7)

Lemma 1 (see [16]). Suppose that 𝑢₀ is a nonnegative constant and 𝑢(𝑛), 𝑎(𝑛), 𝑏(𝑛), 𝑐(𝑛), and 𝑑(𝑛) are nonnegative functions defined on 𝐍, 1+𝑎(𝑛)−𝑏(𝑛)≥0 for all 𝑛∈𝐍. If 𝑢(𝑛) satisfies the inequality 𝑢(𝑛)≤𝑢₀+^𝑛−1_𝑠=𝑛0𝑎(𝑠)𝑢(𝑠)+^𝑛−1_𝑠=𝑛0𝑏(𝑠)^𝑠−1_𝜏=𝑛0𝑐(𝜏)𝑢(𝜏)^𝜏−1_𝑡=𝑛0𝑑(𝑡)𝑢(𝑡),∀𝑛∈𝐍, (8) then 𝑢(𝑛)≤𝑢₀^𝑛−1_𝑠=𝑛0[(𝑠)(𝑠+1+𝑎−𝑏)]_{𝑛−1𝑠=𝑛0}𝑏(𝑠)𝑅(𝑠)^𝑛−1_𝑡=𝑠+1[1+𝑎(𝑡)−𝑏(𝑡)],∀𝑛∈𝐍, (9) where 𝑅(𝑛)≤𝑢₀∏^𝑛−1_𝑠=𝑛0[1+𝑎(𝑠)+𝑏(𝑠)+𝑐(𝑠)𝑄(𝑠)]1+𝑢₀∑^𝑛−1_𝑠=𝑛0𝑐(𝑠)∏^𝑠_𝑡=𝑛0[(𝑡)(𝑡)(𝑡)𝑄(𝑡,1+𝑎+𝑏+𝑐)]∀𝑛∈𝐍, (10) in which 𝑄(𝑛)≤𝑢₀∏^𝑛−1_𝑠=𝑛0[1+𝑎(𝑠)+𝑏(𝑠)+𝑑(𝑠)]1−𝑢₀∑^𝑛−1_𝑠=𝑛0𝑐(𝑠)∏^𝑠_𝑡=𝑛0[(𝑡)(𝑡)(𝑡,1+𝑎+𝑏+𝑑)]∀𝑛∈𝐍, (11) and ∑^𝑛−1_𝑠=𝑛0𝑐(𝑠)∏^𝑠_𝑡=𝑛0[1+𝑎(𝑡)+𝑏(𝑡)+𝑑(𝑡)]<𝑢₀⁻¹ for all 𝑛∈𝐍.

Lemma 2 (see [14, 18]). Let 𝑤(𝑛,𝑟) be a real-valued function defined for 𝑛∈𝐍, 0≤𝑟<∞ and monotone nondecreasing with respect to 𝑟 for any fixed 𝑛∈𝐍. Let 𝑢(𝑛) be a real-valued function defined for 𝑛∈𝐍 such that Δ𝑢(𝑛)≤𝑤(𝑛,𝑢(𝑛)),∀𝑛∈𝐍. (12) Let 𝑟(𝑛) be a solution of Δ𝑟(𝑛)=𝑤(𝑛,𝑟(𝑛)),𝑟(0)=𝑟₀,∀𝑛∈𝐍, (13) such that 𝑢(0)≤𝑟(0). Then 𝑢(𝑛)≤𝑟(𝑛),∀𝑛∈𝐍. (14)

Pachpatte [18, 19] also established some difference inequalities of product form as follows. Let 𝑢,𝑎,𝑏 be nonnegative functions defined on 𝐍 and let 𝑐 be a nonnegative constant. Let 𝑤(𝑛,𝑟) be a nonnegative function defined for 𝑛∈𝐍, 0≤𝑟<∞ and monotone nondecreasing with respect to 𝑟 for any fixed 𝑛∈𝐍. If 𝑢(𝑛) satisfies 𝑢²(𝑛)≤𝑐²+2^𝑛−1_𝑠=𝑛0𝑢(𝑠)𝑎(𝑠)𝑢(𝑠)+^𝑠−1_𝑡=𝑛0𝑏(𝑡)𝑢(𝑡)+𝑤(𝑠,𝑢(𝑠)),∀𝑛∈𝐍, (15) then 𝑢(𝑛)≤𝑃(𝑛)𝑟(𝑛),∀𝑛∈𝐍, (16) where 𝑃(𝑛) is defined by (7), and 𝑟(𝑛) is a solution of Δ𝑟(𝑛)=𝑤(𝑛,𝑃(𝑛)𝑟(𝑛)),𝑟(0)=𝑐,∀𝑛∈𝐍. (17) Let 𝑢,𝑎,𝑏 be nonnegative functions defined for 𝑛∈𝐍 and let 𝑐 be a nonnegative constant. Let 𝑤(𝑛,𝑟) be a nonnegative function defined for 𝑛∈𝐍, 0≤𝑟<∞ and monotone nondecreasing with respect to 𝑟 for any fixed 𝑛∈𝐍. If 𝑢(𝑛) satisfies 𝑢²(𝑛)≤𝑐²+^𝑛−1_𝑠=𝑛0𝑎(𝑠𝑢()(𝑠×𝑢(𝑠)+)(𝑠+1+𝑢))_{𝑠−1𝜏=𝑛0}𝑏(𝜏)𝑢(𝜏)+𝑤(𝑠,𝑢(𝑠)),∀𝑛∈𝐍, (18) then 𝑢(𝑛)≤𝑃(𝑛)𝑟(𝑛),∀𝑛∈𝐍, (19) where 𝑃(𝑛) is defined by (7), and 𝑟(𝑛) is a solution of the difference equation Δ𝑟(𝑛)=𝑎(𝑛)𝑤(𝑛,𝑃(𝑛)𝑟(𝑛)),𝑟(0)=𝑐,∀𝑛∈𝐍. (20)

Motivated by the results given in [16, 18, 19], in this paper, we discuss new nonlinear finite difference inequalities: 𝑢²(𝑛)≤𝑐²+^𝑛−1_𝑠=𝑛0𝑓(𝑠𝑢()(𝑠×𝑢(𝑠)+)(𝑠+1+𝑢))_{𝑠−1𝑡=𝑛0}𝑔(𝑡)𝑢(𝑡)×^𝑡−1_𝜏=𝑛0ℎ(𝜏)𝑢(𝜏)+𝑤(𝑠,𝑢(𝑠)),∀𝑛∈𝐍. (21) Our inequalities can be used as tools in the study of certain classes of finite difference equations. We also present some immediate applications to show the importance of our results to study the various problems in the theory of finite difference equations.

2. Main Results

Throughout this paper, let 𝐑=(−∞,+∞), 𝐑₊=[0,+∞). Let 𝐍∶={𝑛₀,𝑛₀+1,𝑛₀+2,…} and 𝐍_𝑇∶={𝑛₀,𝑛₀+1,𝑛₀+2,…,𝑇},𝑇∈𝐍. For function 𝑢(𝑛), 𝑛∈𝐍, we define the operator Δ by Δ𝑢(𝑛)=𝑢(𝑛+1)−𝑢(𝑛). Obviously, the linear difference equation Δ𝑢(𝑛)=𝑓(𝑛) with the initial condition 𝑢(𝑛₀)=0 has the solution 𝑢(𝑛)=∑^𝑛−1_𝑠=𝑛0𝑓(𝑠). For convenience, in the sequel we complementarily define that ∑^𝑛0−1_𝑠=𝑛0𝑓(𝑠)=0 and ∏^𝑛0−1_𝑠=𝑛0𝑓(𝑠)=1.

Theorem 3. Let 𝛽>0 be a constant, 𝑥,𝑦 positive functions defined on 𝐍, 𝑥 a monotone increasing function, and 𝑦 a monotone decreasing function. Let 𝜙 be a nonnegative function defined on 𝐍 such that 𝑥(𝑛+1)𝑦(𝑛+1)−𝑥(𝑛)𝑦(𝑛)≤𝜙(𝑛)𝑥(𝑛)𝑦(𝑛+1)^𝛽,∀𝑛∈𝑁. (22)(i)Suppose 𝛽>1. If 1+(1−𝛽)[𝑥(𝑛₀)𝑦(𝑛₀)]^𝛽−1∑^𝑛−1_𝑠=𝑛0𝜙(𝑠)>0, then 𝑥(𝑛)≤𝑥𝑛₀𝑦𝑛₀𝑦(𝑛)1+1−𝛽𝑥𝑛₀𝑦𝑛₀^𝛽−1∑^𝑛−1_𝑠=𝑛0𝜙(𝑠)^1/(𝛽−1),∀𝑛∈𝑁. (23)(ii)Suppose 0<𝛽<1. Then 𝑥(𝑛)≤𝑦⁻¹(𝑛)𝑥𝑛₀𝑦𝑛₀^1−𝛽+^𝑛−1_𝑠=𝑛01−𝛽𝜙(𝑠)^1/(1−𝛽),∀𝑛∈𝑁. (24)

Proof. (i) We apply mean value theorem for differentiation to the function 𝐹(𝑧)=𝑧^1−𝛽1−𝛽,𝑧>0, (25) and then there exists 𝜉 between 𝑥(𝑛)𝑦(𝑛) and 𝑥(𝑛+1)𝑦(𝑛+1) such that 𝑥(𝑛+1)𝑦(𝑛+1)^1−𝛽−𝑥(𝑛)𝑦(𝑛)^1−𝛽=1−𝛽𝜉^−𝛽𝑥(𝑛+1)𝑦(𝑛+1)−𝑥(𝑛)𝑦(𝑛). (26) Because 𝑥(𝑛) is monotone increasing and 𝑦(𝑛) is monotone decreasing and −𝛽<0, we see that [𝑥(𝑛)𝑦(𝑛+1)]^−𝛽≥[𝑥(𝑛+1)𝑦(𝑛+1)]^−𝛽 and [𝑥(𝑛)𝑦(𝑛+1)]^−𝛽≥[𝑥(𝑛)𝑦(𝑛)]^−𝛽. So for all values of 𝜉 between 𝑥(𝑛)𝑦(𝑛) and 𝑥(𝑛+1)𝑦(𝑛+1) we have 𝑥(𝑛)𝑦(𝑛+1)^−𝛽≥𝜉^−𝛽. (27) From (22) and (27), we have 𝜉^−𝛽𝑥()𝑦()(𝑛)𝑦(𝑛≤𝑥(𝑛)𝑦(𝑛+1𝑛+1−𝑥)𝑛+1)^−𝛽𝜙(𝑛)𝑥(𝑛)𝑦(𝑛+1)^𝛽=𝜙(𝑛). (28) Since 1−𝛽<0, from (26) and (28) we have 𝑥(𝑛+1)𝑦(𝑛+1)^1−𝛽−𝑥(𝑛)𝑦(𝑛)^1−𝛽≥1−𝛽𝜙(𝑛). (29) Taking 𝑛=𝑠 in (29) and summing up over 𝑠 from 𝑛₀ to 𝑛−1, we obtain 𝑥(𝑛)𝑦(𝑛)^1−𝛽≥𝑥𝑛₀𝑦𝑛₀^1−𝛽+^𝑛−1_𝑠=𝑛01−𝛽𝜙(𝑠). (30) From (30), we obtain our required estimation (23).
(ii) Now by following the same steps as in the proof of (i) before (29) we have 𝑥(𝑛+1)𝑦(𝑛+1)^1−𝛽−𝑥(𝑛)𝑦(𝑛)^1−𝛽≤1−𝛽𝜙(𝑛), (31) because 1−𝛽>0. Taking 𝑛=𝑠 in (31) and summing up over 𝑠 from 𝑛₀ to 𝑛−1, we obtain 𝑥(𝑛)𝑦(𝑛)^1−𝛽≤𝑥𝑛₀𝑦𝑛₀^1−𝛽+^𝑛−1_𝑠=𝑛01−𝛽𝜙(𝑠). (32) From (32), we obtain our required estimation (24).

Theorem 4. Let 𝑚(𝑠) be a positive and monotone nondecreasing function defined on 𝐍 and 𝑓(𝑠),𝑔(𝑠),ℎ(𝑠) nonnegative functions defined on 𝐍. If 𝑢(𝑛) satisfies 𝑢(𝑛)≤𝑚(𝑛)+^𝑛−1_𝑠=𝑛0𝑓(𝑠)𝑢(𝑠)+^𝑛−1_𝑠=𝑛0𝑓(𝑠)^𝑠−1_𝑡=𝑛0𝑔(𝑡)𝑢(𝑡)×^𝑡−1_𝜏=𝑛0ℎ(𝜏)𝑢(𝜏),∀𝑛∈𝐍, (33) then 𝑢(𝑛)≤𝑚(𝑛)+^𝑛−1_𝑠=𝑛0𝑓(𝑠)𝑈(𝑠),∀𝑛∈𝐍, (34) where 𝑈(𝑛)≤𝑚(𝑛)∏^𝑛−1_𝑠=𝑛01+𝑓(𝑠)+𝑔(𝑠)𝑉(𝑠)1+𝑚(𝑛)∑^𝑛−1_𝑠=𝑛0𝑔(𝑠)∏^𝑠_𝑡=𝑛0(𝑡)(𝑡)𝑉(𝑡,1+𝑓+𝑔)∀𝑛∈𝐍, (35) in which 𝑉(𝑛)=𝑚(𝑛)∏^𝑛−1_𝑠=𝑛01+𝑓(𝑠)+ℎ(𝑠)1−𝑚(𝑛)∑^𝑛−1_𝑠=𝑛0𝑔(𝑠)∏^𝑠_𝑡=𝑛0(𝑡)(𝑡,1+𝑓+ℎ)∀𝑛∈𝐍, (36) and ∑^𝑛−1_𝑠=𝑛0𝑔(𝑠)∏^𝑠_𝑡=𝑛0[1+𝑓(𝑡)+ℎ(𝑡)]<𝑚⁻¹(𝑛) for all 𝑛∈𝐍.

Proof. Fix 𝑇∈𝐍, where 𝑇 is chosen arbitrarily, since 𝑚(𝑡) is a nonnegative and monotone nondecreasing function, from (33), we have 𝑢(𝑛)≤𝑚(𝑇)+^𝑛−1_𝑠=𝑛0𝑓(𝑠)𝑢(𝑠)+^𝑛−1_𝑠=𝑛0𝑓(𝑠)^𝑠−1_𝑡=𝑛0𝑔(𝑡)𝑢(𝑡)×^𝑡−1_𝜏=𝑛0ℎ(𝜏)𝑢(𝜏),∀𝑛∈𝑁_𝑇. (37) Define a function 𝑧(𝑛) by the right-hand side of (37). Then 𝑧(𝑛) is a positive and monotone nondecreasing function defined on 𝐍. We have 𝑧𝑛₀=𝑚(𝑇),𝑢(𝑛)≤𝑧(𝑛),∀𝑛∈𝐍. (38) Using the definitions of the operator Δ and 𝑧, we obtain Δ𝑧(𝑛)=𝑓(𝑛)𝑢(𝑛)+^𝑛−1_𝑡=𝑛0𝑔(𝑡)𝑢(𝑡)^𝑡−1_𝜏=𝑛0ℎ(𝜏)𝑢(𝜏)≤𝑓(𝑛)𝑧(𝑛)+^𝑛−1_𝑡=𝑛0𝑔(𝑡)𝑧(𝑡)^𝑡−1_𝜏=𝑛0ℎ(𝜏)𝑧(𝜏),∀𝑛∈𝑁_𝑇. (39) Let 𝑧₁(𝑛)=𝑧(𝑛)+^𝑛−1_𝑡=𝑛0𝑔(𝑡)𝑧(𝑡)^𝑡−1_𝜏=𝑛0ℎ(𝜏)𝑧(𝜏),∀𝑛∈𝑁_𝑇. (40) Then 𝑧₁𝑛₀=𝑧𝑛₀=𝑚(𝑇),𝑧(𝑛)≤𝑧₁(𝑛),∀𝑛∈𝑁_𝑇. (41) It follows that Δ𝑧₁(𝑛)=Δ𝑧(𝑛)+𝑔(𝑛)𝑧(𝑛)^𝑛−1_𝜏=𝑛0ℎ(𝜏)𝑧(𝜏(𝑛)𝑧)≤𝑓₁(𝑛)+𝑔(𝑛)𝑧₁(𝑛)^𝑛−1_𝜏=𝑛0ℎ(𝜏)𝑧₁(𝜏),∀𝑛∈𝑁_𝑇. (42) Adding 𝑔(𝑛)𝑧²₁(𝑛) to both sides of the above inequality we have Δ𝑧₁(𝑛)+𝑔(𝑛)𝑧²₁(𝑛)≤𝑓(𝑛)𝑧₁(𝑛)+𝑔(𝑛)𝑧₁(𝑛)×𝑧₁(𝑛)+^𝑛−1_𝜏=𝑛0ℎ(𝜏)𝑧₁(𝜏),∀𝑛∈𝑁_𝑇. (43) Put 𝑧₂(𝑛)=𝑧₁(𝑛)+^𝑛−1_𝜏=𝑛0ℎ(𝜏)𝑧₁(𝜏),∀𝑛∈𝑁_𝑇, (44) and then 𝑧₁(𝑛)≤𝑧₂(𝑛), 𝑧₂(𝑛₀)=𝑧₁(𝑛₀)=𝑚(𝑇) and Δ𝑧₂(𝑛)=Δ𝑧₁(𝑛)+ℎ(𝑛)𝑧₁(𝑛)≤𝑓(𝑛)𝑧₂(𝑛)+𝑔(𝑛)𝑧²₂(𝑛)+ℎ(𝑛)𝑧₂(𝑛),∀𝑛∈𝑁_𝑇. (45) We see that the inequality 𝑧₂(𝑛+1)−1+𝑓(𝑛)+ℎ(𝑛)𝑧₂(𝑛)≤𝑔(𝑛)𝑧²₂(𝑛),∀𝑛∈𝑁_𝑇. (46) Define a function 𝑃₁(𝑛)=^𝑛−1_𝑠=𝑛01+𝑓(𝑠)+ℎ(𝑠)⁻¹,∀𝑛∈𝑁_𝑇. (47) Multiplying by 𝑃₁(𝑛+1) to both sides of (46) we obtain 𝑧₂(𝑛+1)𝑃₁(𝑛+1)−𝑧₂(𝑛)𝑃₁(𝑛)≤𝑃₁⁻¹(𝑛+1)𝑔(𝑛)𝑧₂(𝑛)𝑃₁(𝑛+1)²,∀𝑛∈𝑁_𝑇. (48) Let 𝑥(𝑛)=𝑧₂(𝑛), 𝑦(𝑛)=𝑃₁(𝑛), 𝜙(𝑛)=𝑃₁⁻¹(𝑛+1)𝑔(𝑛), and 𝛽=2. Because 𝑧₂(𝑛) is monotone increasing, 𝑃₁(𝑛) is monotone decreasing and 2>0; applying Theorem 3 to (48) we obtain 𝑧₂(𝑛)≤𝑧₂𝑛₀𝑃₁⁻¹(𝑛)1−𝑧₂𝑛₀∑^𝑛−1_𝑠=𝑛0𝑔(𝑠)𝑃₁⁻¹()=𝑚(𝑇)∏𝑠+1_{𝑛−1𝑠=𝑛0}1+𝑓(𝑠)+ℎ(𝑠)1−𝑚(𝑇)∑^𝑛−1_𝑠=𝑛0𝑔(𝑠)∏^𝑠_𝑡=𝑛0(𝑡)(𝑡,1+𝑓+ℎ)∀𝑛∈𝑁_𝑇, (49) where 𝑃₁(𝑛₀)=1, 𝑧₂(𝑛₀)=𝑚(𝑇) are used. Define a function 𝑉 of the right-hand side of (49). Substituting (49) in (43) we obtain 𝑧₁(𝑛+1)−1+𝑓(𝑛)+𝑔(𝑛)𝑉(𝑛)𝑧₁(𝑛)≤−𝑔(𝑛)𝑧²₁(𝑛),∀𝑛∈𝑁_𝑇. (50) Performing the same derivation as in (46)–(49), we obtain from (50) that 𝑧₁(𝑛)≤𝑚(𝑇)∏^𝑛−1_𝑠=𝑛01+𝑓(𝑠)+𝑔(𝑠)𝑉(𝑠)1+𝑚(𝑇)∑^𝑛−1_𝑠=𝑛0𝑔(𝑠)∏^𝑠_𝑡=𝑛0(𝑡)(𝑡)𝑉(𝑡,1+𝑓+𝑔)∀𝑛∈𝑁_𝑇. (51) Define a function 𝑈 of the right-hand side of (51). Substituting (51) in (39) we obtain Δ𝑧(𝑛)=𝑓(𝑛)𝑈(𝑛),∀𝑛∈𝑁_𝑇. (52) Using (38), from (52) it follows that 𝑢(𝑛)≤𝑚(𝑇)+^𝑛−1_𝑠=𝑛0𝑓(𝑠)𝑈(𝑠),∀𝑛∈𝑁_𝑇. (53) Since 𝑇∈𝐍 is arbitrary, from (53), we get the required estimate (35).

Theorem 5. Let 𝑢,𝑓,𝑔,ℎ be nonnegative functions defined for 𝑛∈𝐍 and 𝑐 a nonnegative constant. Let 𝑤(𝑛,𝑟) be a real-valued function defined for 𝑛∈𝐍, 0≤𝑟<∞, and monotone nondecreasing with respect to 𝑟 for any fixed 𝑛∈𝐍. If 𝑢(𝑛) satisfies (21), then 𝑢(𝑛)≤𝑣(𝑛)+^𝑛−1_𝑠=𝑛0𝑓(𝑠)𝑊₁(𝑠),∀𝑛∈𝐍, (54) where 𝑊₁(𝑛)≤𝑣(𝑛)∏^𝑛−1_𝑠=𝑛01+𝑓(𝑠)+𝑔(𝑠)𝑊₂(𝑠)1+𝑣(𝑛)∑^𝑛−1_𝑠=𝑛0𝑔(𝑠)∏^𝑠_𝑡=𝑛01+𝑓(𝑡)+𝑔(𝑡)𝑊₂(𝑡,)∀𝑛∈𝐍, (55) in which 𝑊₂(𝑛)=𝑣(𝑛)∏^𝑛−1_𝑠=𝑛01+𝑓(𝑠)+ℎ(𝑠)1−𝑣(𝑛)∑^𝑛−1_𝑠=𝑛0𝑔(𝑠)∏^𝑠_𝑡=𝑛0(𝑡)(𝑡,1+𝑓+ℎ)∀𝑛∈𝐍, (56) and 𝑣(𝑛) is a solution of the difference equation Δ𝑟(𝑛)=𝑓(𝑛)𝑤𝑛,𝑟(𝑛)+^𝑛−1_𝑠=𝑛0𝑓(𝑠)𝑊₃(𝑠,𝑟(0))=𝑐,∀𝑛∈𝐍, (57) where 𝑊₃(𝑛)≤𝑟(𝑛)∏^𝑛−1_𝑠=𝑛01+𝑓(𝑠)+𝑔(𝑠)𝑊₄(𝑠)1+𝑟(𝑛)∑^𝑛−1_𝑠=𝑛0𝑔(𝑠)∏^𝑠_𝑡=𝑛01+𝑓(𝑡)+𝑔(𝑡)𝑊₄(𝑡,)∀𝑛∈𝐍, (58) in which 𝑊₄(𝑛)=𝑟(𝑛)∏^𝑛−1_𝑠=𝑛01+𝑓(𝑠)+ℎ(𝑠)1−𝑟(𝑛)∑^𝑛−1_𝑠=𝑛0𝑔(𝑠)∏^𝑠_𝑡=𝑛0(𝑡)(𝑡,1+𝑓+ℎ)∀𝑛∈𝐍, (59) and ∑^𝑛−1_𝑠=𝑛0𝑔(𝑠)∏^𝑠_𝑡=𝑛0[1+𝑓(𝑡)+ℎ(𝑡)]<𝑣⁻¹(𝑛) for all 𝑛∈𝐍.

Proof. We first assume that 𝑐>0 and define a function 𝑧(𝑛) by the right-hand side of (21). Then 𝑧(𝑛) is a positive and monotone nondecreasing function defined on 𝐍. We have 𝑧(0)=𝑐²,𝑢(𝑛)≤𝑧(𝑛),∀𝑛∈𝐍. (60) Using the definitions of the operator Δ and 𝑧, we obtain (𝑛)(𝑛𝑢()(𝑛×𝑢(𝑛)+Δ𝑧=𝑓)(𝑛+1+𝑢))_{𝑛−1𝑡=𝑛0}𝑔(𝑡)𝑢(𝑡)^𝑡−1_𝜏=𝑛0ℎ(𝜏)𝑢(𝜏)((𝑛(𝑛)+𝑤𝑛,𝑢))≤𝑓𝑧(𝑛+1)+𝑧(𝑛)×𝑧(𝑛)+^𝑛−1_𝑡=𝑛0𝑔(𝑡)𝑧(𝑡)×^𝑡−1_𝜏=𝑛0ℎ(𝜏)𝑧(𝜏)+𝑤𝑛,𝑧(𝑛),∀𝑛∈𝐍. (61) From (61) it follows that the inequality Δ𝑧(𝑛)=Δ𝑧(𝑛)𝑧(𝑛+1)+𝑧(𝑛)≤𝑓(𝑛)𝑧(𝑛)+^𝑛−1_𝑡=𝑛0𝑔(𝑡)𝑧(𝑡)×^𝑡−1_𝜏=𝑛0ℎ(𝜏)𝑧(𝜏)+𝑤𝑛,𝑧(𝑛) (62) holds for all 𝑛∈𝐍. Setting 𝑛=𝑠 in (62) and substituting 𝑠=𝑛₀,1,2,…,𝑛−1, successively, we get 𝑧(𝑛)≤𝑐+^𝑛−1_𝑠=𝑛0𝑓(𝑠)×𝑧(𝑠)+^𝑠−1_𝑡=𝑛0𝑔(𝑡)𝑧(𝑡)×^𝑡−1_𝜏=𝑛0ℎ(𝜏)𝑧(𝜏)+𝑤𝑠,𝑧(𝑠),∀𝑛∈𝐍. (63) Define a function 𝑧₁(𝑛) by 𝑧₁(𝑛)=𝑐+^𝑛−1_𝑠=𝑛0𝑓(𝑠)𝑤𝑠,𝑧(𝑠),∀𝑛∈𝐍. (64) Then 𝑧₁(𝑛)=𝑐 and Δ𝑧₁(𝑛)=𝑓(𝑛)𝑤𝑛,𝑧(𝑛),∀𝑛∈𝐍. (65) Using (64), the inequality (63) can be written as 𝑧(𝑛)≤𝑧₁(𝑛)+^𝑛−1_𝑠=𝑛0𝑓(𝑠)𝑧(𝑠)+^𝑠−1_𝑡=𝑛0𝑔(𝑡)𝑧(𝑡)×^𝑡−1_𝜏=𝑛0ℎ(𝜏)𝑧(𝜏),∀𝑛∈𝐍. (66) Since 𝑧₁(𝑛) is positive and monotone nondecreasing for 𝑛∈𝐍, 𝑓(𝑠),𝑔(𝑠),ℎ(𝑠) satisfy the conditions in Theorem 4. Now an application of Theorem 4 to (66) yields 𝑧(𝑛)≤𝑧₁(𝑛)+^𝑛−1_𝑠=𝑛0𝑓(𝑠)𝑊₁(𝑠),∀𝑛∈𝐍, (67) where 𝑊₁(𝑛)≤𝑧₁(𝑛)∏^𝑛−1_𝑠=𝑛01+𝑓(𝑠)+𝑔(𝑠)𝑊₂(𝑠)1+𝑧₁(𝑛)∑^𝑛−1_𝑠=𝑛0𝑔(𝑠)∏^𝑠_𝑡=𝑛01+𝑓(𝑡)+𝑔(𝑡)𝑊₂(𝑡,)∀𝑛∈𝐍, (68) in which 𝑊₂(𝑛)=𝑧₁(𝑛)∏^𝑛−1_𝑠=𝑛01+𝑓(𝑠)+ℎ(𝑠)1−𝑧₁(𝑛)∑^𝑛−1_𝑠=𝑛0𝑔(𝑠)∏^𝑠_𝑡=𝑛0(𝑡)(𝑡,1+𝑓+ℎ)∀𝑛∈𝐍. (69) Since 𝑤(𝑛,𝑟) is monotone nondecreasing with respect to 𝑟 for any fixed 𝑛∈𝐍, from (65) and (67), we have Δ𝑧₁(𝑛)≤𝑓(𝑛)𝑤𝑛,𝑧₁(𝑛)+^𝑛−1_𝑠=𝑛0𝑓(𝑠)𝑊₁(𝑠),∀𝑛∈𝐍. (70) Now as a suitable application of Lemma 2, we obtain 𝑧₁(𝑛)≤𝑣(𝑛),∀𝑛∈𝐍, (71) where 𝑣(𝑛) is a solution of (57). Using (60), (67), and (71), we obtain our required estimation (54).
If 𝑐 is nonnegative, we can carry out the above procedure with 𝑐+𝜖 instead of 𝑐 where 𝜖 is an arbitrary small number. Letting 𝜖→0, we obtain (54).

3. Application to Finite Difference Equations

In this section, we consider the following difference equation: (𝑛)(𝑛)×𝐹(𝑛),Δ𝑥=𝑓𝑛,𝑥^𝑛−1_𝑡=𝑛0𝑔(𝑡)𝑥(𝑡)×^𝑡−1_𝜏=𝑛0𝐻(𝑡,𝜏,𝑥(𝜏))+𝐾(𝑛,𝑥(𝑛)),∀𝑛∈𝐍,𝑥𝑛₀=𝑥₀, (72)(73) where 𝐹, 𝐻, 𝐾 are real-valued functions defined, respectively, on 𝐍×𝐑², 𝐍²×𝐑, 𝐍×𝐑, 𝑓 is as defined in Theorem 5, and 𝑥₀ is a constant. We assume that |𝐾((𝑛(|𝑥(𝑛,|𝐻((𝑡≤𝑛,𝑥))|≤𝑤𝑛,)|)𝑛,𝑡,𝑥))|^𝑛−1_𝑡=𝑛0ℎ(𝑡𝑥(𝑡,𝐹(𝑛)(𝑛)≤|𝑥(𝑛+𝑦(𝑛,)|)|𝑛,𝑥,𝑦)|) (74) where 𝑔, ℎ, 𝑤 are as defined in Theorem 5. Using the definitions of the operator Δ, from (72), we see that the inequality 𝑥()(𝑛)(𝑛)𝐹(𝑛),𝑛+1−𝑥=𝑓𝑛,𝑥^𝑛−1_𝑡=𝑛0𝑔(𝑡)𝑥(𝑡)×^𝑡−1_𝜏=𝑛0𝐻(𝑡,𝜏,𝑥(𝜏))+𝐾(𝑛,𝑥(𝑛)) (75) holds for all 𝑛∈𝐍. It follows that 𝑥²(𝑛+1)−𝑥²(𝑛)(𝑛𝑥()(𝑛×𝐹(𝑛),=𝑓)[𝑛+1−𝑥)]𝑛,𝑥^𝑛−1_𝑡=𝑛0𝑔(𝑡)𝑥(𝑡)^𝑡−1_𝜏=𝑛0𝐻((𝜏((𝑛𝑡,𝜏,𝑥))+𝐾𝑛,𝑥)),∀𝑛∈𝐍. (76) From (76), we have 𝑥²(𝑛)=𝑥²𝑛₀+^𝑛−1_𝑠=𝑛0𝑓(𝑠𝑥()(𝑠×𝐹(𝑠),)[𝑠+1−𝑥)]𝑠,𝑥^𝑠−1_𝑡=𝑛0𝑔(𝑡)𝑥(𝑡)×^𝑡−1_𝜏=𝑛0𝐻((𝜏((𝑠𝑡,𝜏,𝑥))+𝐾𝑠,𝑥)),∀𝑛∈𝐍. (77) Using the conditions (74), we obtain |𝑥(𝑛)|²=𝑥²𝑛₀+^𝑛−1_𝑠=𝑛0𝑓(𝑠𝑥(−|𝑥(𝑠×|𝑥(𝑠+)[|𝑠+1)|)|])|^𝑠−1_𝑡=𝑛0𝑔(𝑡𝑥(𝑡×)|)|_{𝑡−1𝜏=𝑛0}ℎ(𝜏𝑥(𝜏(|𝑥(𝑠)|)|+𝑤𝑠,)|),∀𝑛∈𝐍. (78) Now an application of Theorem 5 to (78) yields the estimation of the difference equation (72), that is, |𝑥(𝑛)|≤𝑣(𝑛)+^𝑛−1_𝑠=𝑛0𝑓(𝑠)𝑊₅(𝑠),∀𝑛∈𝐍, (79) where 𝑊₅(𝑛)≤𝑣(𝑛)∏^𝑛−1_𝑠=𝑛01+𝑓(𝑠)+𝑔(𝑠)𝑊₆(𝑠)1+𝑣(𝑛)∑^𝑛−1_𝑠=𝑛0𝑔(𝑠)∏^𝑠_𝑡=𝑛01+𝑓(𝑡)+𝑔(𝑡)𝑊₆(𝑡,)∀𝑛∈𝐍, (80) in which 𝑊₆(𝑛)=𝑣(𝑛)∏^𝑛−1_𝑠=𝑛01+𝑓(𝑠)+ℎ(𝑠)1−𝑣(𝑛)∑^𝑛−1_𝑠=𝑛0𝑔(𝑠)∏^𝑠_𝑡=𝑛0(𝑡)(𝑡,1+𝑓+ℎ)∀𝑛∈𝐍, (81) and 𝑣(𝑛) is a solution of the difference equation Δ𝑟(𝑛)=𝑓(𝑛)𝑤𝑛,𝑟(𝑛)+^𝑛−1_𝑠=𝑛0𝑓(𝑠)𝑊₇(𝑠,𝑟(0)=𝑥₀),∀𝑛∈𝐍, (82) where 𝑊₇(𝑛)≤𝑟(𝑛)∏^𝑛−1_𝑠=𝑛01+𝑓(𝑠)+𝑔(𝑠)𝑊₈(𝑠)1+𝑟(𝑛)∑^𝑛−1_𝑠=𝑛0𝑔(𝑠)∏^𝑠_𝑡=𝑛01+𝑓(𝑡)+𝑔(𝑡)𝑊₈(𝑡,)∀𝑛∈𝐍, (83) in which 𝑊₈(𝑛)=𝑟(𝑛)∏^𝑛−1_𝑠=𝑛01+𝑓(𝑠)+ℎ(𝑠)1−𝑟(𝑛)∑^𝑛−1_𝑠=𝑛0𝑔(𝑠)∏^𝑠_𝑡=𝑛0(𝑡)(𝑡,1+𝑓+ℎ)∀𝑛∈𝐍, (84) and ∑^𝑛−1_𝑠=𝑛0𝑔(𝑠)∏^𝑠_𝑡=𝑛0[1+𝑓(𝑡)+ℎ(𝑡)]<𝑣⁻¹(𝑛) for all 𝑛∈𝐍.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by National Natural Science Foundation of China (Project no. 11161018,11171079), the NSF of Guangxi Zhuang Autonomous Region (no. 2012GXNSFAA053009), and the SRF of the Education Department of Guangxi Zhuang Autonomous Region (nos. 201106LX599 and 201106LX591).