Some New Nonlinear Weakly Singular Inequalities and Applications to Volterra-Type Difference Equation

Abstract

Some new nonlinear weakly singular difference inequalities are discussed, which generalize some known weakly singular inequalities and can be used in the analysis of nonlinear Volterra-type difference equations with weakly singular kernel. An application to the upper bound of solutions of a nonlinear difference equation is also presented.

1. Introduction

The discrete version of the well-known Gronwall-Bellman inequality is an important tool in the development of the theory of difference equations as well as the analysis of the numerical schemes of differential equations. A great deal of interest has been given to these inequalities, and many results on their generalizations have been found; for example, see [1–4]. Among them, one of the fundamental cases is Pachpatte’s result [3] for the difference inequality: 𝑢(𝑛)≤𝑎(𝑛)+^𝑛−1_𝑠=0𝑓(𝑠)𝑢(𝑠). (1)

In particular, due to the study of the behavior and numerical solutions for the singular integral equations, some discrete weakly singular integral inequalities also have drawn more and more attention [5–7]. Dixon and McKee [8] investigated the convergence of discretization methods for the Volterra integral and integrodifferential equations, by using the following inequality: 𝑥_𝑖≤𝜓_𝑖+𝑀ℎ^{1−𝛼𝑖−1}_𝑗=0𝑥_𝑗𝑖−𝑗^𝛼,𝑖=1,2,…,𝑁,𝑛>0,𝑁ℎ=𝑇. (2)

Henry [9] presented a linear integral inequality with weakly kernel: 𝑥(𝑡)≤𝑎(𝑡)+^𝑡₀(𝑡−𝑠)^𝛽−1𝑏(𝑠)𝑥(𝑠)𝑑𝑠 (3) to investigate some qualitative properties for a parabolic equation. The corresponding discrete version was discussed by Slodic̆ka [10]. But he studied the case 𝜏_𝑘=𝜏, that is, the case of constant differences. Furthermore, the first formulation of the inequality with a nonlinearity and 𝜏_𝑘 nonconstant was studied in [6], in which the general nonlinear discrete case as follows: 𝑥_𝑛≤𝑎_𝑛+^𝑛−1_𝑘=0𝑡_𝑛−𝑡_𝑘^𝛽−1𝜏_𝑘𝑏_𝑘𝜔𝑥_𝑘 (4) was considered. However, his results are based on the so-called “(𝑞) condition”: (1) 𝜔 satisfies 𝑒^−𝑞𝑡[𝜔(𝑢)]^𝑞≤𝑅(𝑡)𝜔(𝑒^−𝑞𝑡)𝑢^𝑞; (2) there exists 𝑐>0 such that 𝑎_𝑛𝑒^{−𝜏𝑡𝑛}≤𝑐. Recently, a new nonlinear difference inequality: 𝑥^𝛼_𝑛≤𝑎_𝑛+^𝑛−1_𝑘=0𝑡_𝑛−𝑡_𝑘^𝛽−1𝜏_𝑘𝑏_𝑘𝑥^𝜆_𝑘 (5) was discussed by Yang et al. [11]. For other new weakly singular inequalities, lots of work can be found, for example, in [12–22] and references therein.

In this paper, we investigate the new nonlinear weakly singular inequality: 𝑥_𝑛≤𝑎_𝑛+^𝑛−1_𝑘=0𝑡_𝑛−𝑡_𝑘^𝛽−1𝜏_𝑘𝑏_𝑘𝜔𝑥_𝑘, (6) where 0<𝛽≤1, 𝑡₀=0, 𝜏_𝑘=𝑡_𝑘+1−𝑡_𝑘, sup_𝑘∈ℕ𝜏_𝑘=𝜏, and lim_𝑡→∞𝑡_𝑘=∞. Compared to the existing result, our result does not need the so-called “(𝑞) condition” proposed in [6] and can be used to obtain pointwise explicit bounds on solutions for a class of more general weakly singular inequalities of Volterra type. Finally, we also present an application to Volterra-type difference equation with weakly singular kernel.

2. Preliminaries

Let ℝ be the set of real numbers, ℝ₊=(0,∞), and ℕ={0,1,2,…}. 𝐶(𝑋,𝑌) denotes the collection of continuous functions from the set 𝑋 to the set 𝑌. As usual, the empty sum is taken to be 0.

Lemma 1 (Discrete Jensen inequality, [11]). Let 𝐴₁,𝐴₂,…,𝐴_𝑛 be nonnegative real numbers, and let 𝑟>1 be a real number. Then, 𝐴₁+𝐴₂+⋯+𝐴_𝑛^𝑟≤𝑛^𝑟−1𝐴^𝑟₁+𝐴^𝑟₂+⋯+𝐴^𝑟_𝑛. (7)

Lemma 2 (Discrete Hölder inequality, [11]). Let 𝑎_𝑖,𝑏_𝑖(𝑖=1,2,…,𝑛) be nonnegative real numbers, and let 𝑝,𝑞 be positive numbers such that (1/𝑝)+(1/𝑞)=1 (or 𝑝=1,𝑞=∞). Then, ^𝑛_𝑖=1𝑎_𝑖𝑏_𝑖≤^𝑛_𝑖=1𝑎^𝑝_𝑖^1/𝑝𝑛_𝑖=1𝑏^𝑞_𝑖^1/𝑞. (8)
Furthermore, take 𝑝=𝑞=2; then, one gets the discrete Cauchy-Schwarz inequality.

Lemma 3. Suppose that 𝜔(𝑢)∈𝐶(ℝ₊,ℝ₊) is nondecreasing. Let 𝑎_𝑛,𝑐_𝑛 be nonnegative and nondecreasing in 𝑛. If 𝑦_𝑛 is nonnegative such that 𝑦_𝑛≤𝑎_𝑛+𝑐_𝑛^𝑛−1_𝑘=0𝑏_𝑘𝜔𝑦_𝑘,𝑛∈ℕ. (9) Then, 𝑦_𝑛≤Ω⁻¹Ω𝑎_𝑛+𝑐_𝑛^𝑛−1_𝑘=0𝑏_𝑘,0≤𝑛≤𝑀, (10) where Ω(𝑣)=∫^𝑣_𝑣0(1/𝜔(𝑠))𝑑𝑠,𝑣≥𝑣₀, Ω⁻¹ is the inverse function of Ω, and 𝑀 is defined by 𝑀=sup𝑖∶Ω𝑎_𝑖+𝑐_𝑖^𝑖−1_𝑘=0𝑏_𝑘∈DomΩ⁻¹. (11)

3. Main Results

Assume that (A₁) 𝑎_𝑛,𝑏_𝑛 are nonnegative functions for 𝑛∈ℕ, respectively; (A₂) 𝜔(𝑢)∈𝐶(ℝ₊,ℝ₊) is nondecreasing and 𝜔(0)=0.

Define 𝑎_𝑛=max_{0≤𝑘≤𝑛,𝑘∈ℕ}𝑎_𝑘 and 𝜏=max_{0≤𝑘≤𝑛−1,𝑘∈ℕ}𝜏_𝑘, where 𝜏_𝑘 is the variable time step.

Theorem 4. Under assumptions (𝐴₁) and (𝐴₂), if 𝑥_𝑛 is nonnegative such that (6), then(1)for  0<𝛽≤1/2, letting 𝑝=1+𝛽 and 𝑞=(1+𝛽)/𝛽, one has 𝑥_𝑛≤Ω⁻¹Ω2^𝑞−1𝑎^𝑞_𝑛+2^𝑞−1𝜏^{1−(𝑞/𝑝)𝛽2}×𝐾^𝑞/𝑝𝛽𝑒^{𝑞𝜏𝑡𝑛𝑛−1}_𝑘=0𝑒^{−𝑞𝜏𝑡𝑘}𝑏^𝑞_𝑘^1/𝑞, (12) for 0≤𝑛≤𝑁₁, where Ω(𝑢)=∫^𝑢_𝑢0(1/𝜔^𝑞(𝑠^1/𝑞))𝑑𝑠,𝑢≥𝑢₀≥0, Ω⁻¹ is the inverse function of Ω, 𝐾𝛽=1+𝛽^−𝛽2Γ𝛽², (13) and 𝑁₁ is the largest integer number such that Ω2^𝑞−1𝑎^𝑞_𝑛+2^𝑞−1𝜏^{1−(𝑞/𝑝)𝛽2}𝐾^𝑞/𝑝𝛽𝑒^{𝑞𝜏𝑡𝑛}×^𝑛−1_𝑘=0𝑒^{−𝑞𝜏𝑡𝑘}𝑏^𝑞_𝑘∈DomΩ⁻¹; (14)(2)for 1/2<𝛽≤1, letting 𝑝=2 and 𝑞=2, one has 𝑥_𝑛≤Ω⁻¹Ω2𝑎²_𝑛+𝐵𝛽𝜏^2−2𝛽𝑒^{2𝜏𝑡𝑛𝑛−1}_𝑘=0𝑒^{−2𝜏𝑡𝑘}𝑏²_𝑘^1/2, (15) for 0≤𝑛≤𝑁₂, where Ω(𝑢)=∫^𝑢_𝑢0(1/𝜔²(𝑠^1/2))𝑑𝑠,𝑢≥𝑢₀≥0, 𝐵𝛽=4^1−𝛽Γ12𝛽−1,𝛽>2, (16) and 𝑁₂ is the largest integer number such that Ω2𝑎²_𝑛+𝐵𝛽𝜏^2−2𝛽𝑒^{2𝜏𝑡𝑛𝑛−1}_𝑘=0𝑒^{−2𝜏𝑡𝑘}𝑏²_𝑘∈DomΩ⁻¹. (17)

Proof. By definition of 𝑎_𝑛 and assumption (𝐴₁), 𝑎_𝑛 is nonnegative and nondecreasing and 𝑎_𝑛≥𝑎_𝑛. It follows from (6) that 𝑥_𝑛≤𝑎_𝑛+^𝑛−1_𝑘=0𝑡_𝑛−𝑡_𝑘^𝛽−1𝜏_𝑘𝑏_𝑘𝜔𝑥_𝑘. (18)(1)If 0<𝛽≤1/2, using Lemma 2 with the indices 𝑝=1+𝛽,𝑞=(1+𝛽)/𝛽 for (18), we get 𝑥_𝑛≤𝑎_𝑛+^𝑛−1_𝑘=0𝑡_𝑛−𝑡_𝑘^𝛽−1𝜏_𝑘^1/𝑝𝜏_𝑘^1/𝑞𝑒^𝜏𝑡𝑘𝑒^{−𝜏𝑡𝑘}𝑏_𝑘𝜔𝑥_𝑘≤𝑎_𝑛+𝜏^{1/𝑞𝑛−1}_𝑘=0𝑡_𝑛−𝑡_𝑘^𝛽−1𝜏_𝑘^1/𝑝𝑒^𝜏𝑡𝑘𝑒^{−𝜏𝑡𝑘}𝑏_𝑘𝜔𝑥_𝑘≤𝑎_𝑛+𝜏^{1/𝑞𝑛−1}_𝑘=0𝑡_𝑛−𝑡_𝑘^{𝑝(𝛽−1)}𝜏_𝑘𝑒^{𝑝𝜏𝑡𝑘1/𝑝}×^𝑛−1_𝑘=0𝑒^{−𝑞𝜏𝑡𝑘}𝑏^𝑞_𝑘𝜔^𝑞𝑥_𝑘^1/𝑞. (19) By Lemma 1, the inequality above yields 𝑥^𝑞_𝑛≤2^𝑞−1𝑎^𝑞_𝑛+2^𝑞−1𝜏^𝑛−1_𝑘=0𝑡_𝑛−𝑡_𝑘^{𝑝(𝛽−1)}𝜏_𝑘𝑒^{𝑝𝜏𝑡𝑘𝑞/𝑝}×^𝑛−1_𝑘=0𝑒^{−𝑞𝜏𝑡𝑘}𝑏^𝑞_𝑘𝜔^𝑞𝑥_𝑘. (20) Consider that ^𝑛−1_𝑘=0𝑡_𝑛−𝑡_𝑘^{𝑝(𝛽−1)}𝜏_𝑘𝑒^{𝑝𝜏𝑡𝑘}≤^𝑡𝑛₀𝑡_𝑛−𝑠^{𝑝(𝛽−1)}𝑒^𝑝𝜏𝑠𝑑𝑠=𝑒^{𝑝𝜏𝑡𝑛𝑡𝑛}₀𝜂^{𝑝(𝛽−1)}𝑒^{−𝑝𝜏𝜂}=𝑒𝑑𝜂,_{𝑝𝜏𝑡𝑛}𝑝𝜏^{1+𝑝(𝛽−1)𝑝𝜏𝑡𝑛0}𝜎^{𝑝(𝛽−1)𝑒−𝜎}𝑑𝜎≤𝐾𝛽𝜏^−𝛽2𝑒^{𝑝𝜏𝑡𝑛}, (21) where 𝐾(𝛽)=(1+𝛽)^−𝛽2Γ(𝛽²) and Γ(𝑧)=∫^∞₀𝑢^𝑧−1𝑒^−𝑢𝑑𝑢,(𝑅𝑒𝑧>0) is the well-known 𝐺-function. Thus, we have 𝑥^𝑞_𝑛≤2^𝑞−1𝑎^𝑞_𝑛+2^𝑞−1𝜏^{1−(𝑞/𝑝)𝛽2}×𝐾^𝑞/𝑝𝛽𝑒^{𝑞𝜏𝑡𝑛𝑛−1}_𝑘=0𝑒^{−𝑞𝜏𝑡𝑘}𝑏^𝑞_𝑘𝜔^𝑞𝑥_𝑘. (22) Let 𝑣_𝑛=𝑥^𝑞_𝑛, 𝐴_𝑛=2^𝑞−1𝑎^𝑞_𝑛, and 𝐶_𝑛=2^𝑞−1𝜏^{1−(𝑞/𝑝)𝛽2}𝐾^𝑞/𝑝(𝛽)𝑒^{𝑞𝜏𝑡𝑛}. Obviously, 𝐴_𝑛,𝐶_𝑛 are nondecreasing for 𝑛∈ℕ and 𝜔^𝑞(𝑣_𝑘^1/𝑞) satisfies the assumption (𝐴₂). Equation (22) can be rewritten as 𝑣_𝑛≤𝐴_𝑛+𝐶_𝑛^𝑛−1_𝑘=0𝑒^{−𝑞𝜏𝑡𝑘}𝑏^𝑞_𝑘𝜔^𝑞𝑣_𝑘^1/𝑞, (23) which is similar to inequality (9). Using Lemma 3, from (23), we have 𝑣_𝑛≤Ω⁻¹Ω𝐴_𝑛+𝐶_𝑛^𝑛−1_𝑘=0𝑒^{−𝑞𝜏𝑡𝑘}𝑏^𝑞_𝑘, (24) for 0≤𝑛≤𝑁₁, where 𝑁₁ is the largest integer number such that Ω𝐴_𝑛+𝐶_𝑛^𝑛−1_𝑘=0𝑒^{−𝑞𝜏𝑡𝑘}𝑏^𝑞_𝑘∈DomΩ⁻¹. (25) Therefore, by 𝑥_𝑛=𝑣_𝑛^1/𝑞, (12) holds for 0≤𝑛≤𝑁₁.(2)If 1/2<𝛽≤1, applying Cauchy-Schwarz inequality for (18), that is, 𝑝=𝑞=2, we get 𝑥_𝑛≤𝑎_𝑛+^𝑛−1_𝑘=0𝑡_𝑛−𝑡_𝑘^𝛽−1𝜏_𝑘^1/2𝜏_𝑘^1/2𝑒^𝜏𝑡𝑘𝑒^{−𝜏𝑡𝑘}𝑏_𝑘𝜔𝑥_𝑘≤𝑎_𝑛+𝜏^1/2𝑛−1_𝑘=0𝑡_𝑛−𝑡_𝑘^𝛽−1𝜏_𝑘^1/2𝑒^𝜏𝑡𝑘𝑒^{−𝜏𝑡𝑘}𝑏_𝑘𝜔𝑥_𝑘≤𝑎_𝑛+𝜏^1/2𝑛−1_𝑘=0𝑡_𝑛−𝑡_𝑘^2(𝛽−1)𝜏_𝑘𝑒^{2𝜏𝑡𝑘1/2}×^𝑛−1_𝑘=0𝑒^{−2𝜏𝑡𝑘}𝑏²_𝑘𝜔²𝑥_𝑘^1/2. (26) By Lemma 1, the inequality above yields 𝑥²_𝑛≤2𝑎²_𝑛+2𝜏^𝑛−1_𝑘=0𝑡_𝑛−𝑡_𝑘^2(𝛽−1)𝜏_𝑘𝑒^{2𝜏𝑡𝑘}×^𝑛−1_𝑘=0𝑒^{−2𝜏𝑡𝑘}𝑏²_𝑘𝜔²𝑥_𝑘. (27) Because ^𝑛−1_𝑘=0𝑡_𝑛−𝑡_𝑘^2(𝛽−1)𝜏_𝑘𝑒^{2𝜏𝑡𝑘}≤^𝑡𝑛₀𝑡_𝑛−𝑠^2(𝛽−1)𝑒^2𝜏𝑠=𝑒𝑑𝑠_{2𝜏𝑡𝑛}(2𝜏)^{2𝛽−12𝜏𝑡𝑛0}𝜎^{2(𝛽−1)𝑒−𝜎}≤1𝑑𝜎2𝐵𝛽𝜏^1−2𝛽𝑒^{2𝜏𝑡𝑛}, (28) where 𝐵(𝛽)=4^1−𝛽Γ(2𝛽−1),𝛽>1/2, it follows from (27) that 𝑥²_𝑛≤2𝑎²_𝑛+𝐵𝛽𝜏^2−2𝛽𝑒^{2𝜏𝑡𝑛𝑛−1}_𝑘=0𝑒^{−2𝜏𝑡𝑘}𝑏²_𝑘𝜔²𝑥_𝑘. (29) Let 𝑣_𝑛=𝑥²_𝑛, 𝐴_𝑛=2𝑎²_𝑛, and 𝐶_𝑛=𝐵(𝛽)𝜏^2−2𝛽𝑒^{2𝜏𝑡𝑛}. Similarly, 𝐴_𝑛,𝐶_𝑛 also are nondecreasing for 𝑛∈ℕ and 𝜔²(𝑣_𝑘^1/2) also satisfies the assumption (𝐴₂). Equation (29) can be rewritten as 𝑣_𝑛≤𝐴_𝑛+𝐶_𝑛^𝑛−1_𝑘=0𝑒^{−2𝜏𝑡𝑘}𝑏²_𝑘𝜔²𝑣_𝑘^1/2, (30) which also is similar to inequality (9). Using Lemma 3, from (30), we have 𝑣_𝑛≤Ω⁻¹Ω𝐴_𝑛+𝐶_𝑛^𝑛−1_𝑘=0𝑒^{−2𝜏𝑡𝑘}𝑏²_𝑘, (31) for 0≤𝑛≤𝑁₂, and 𝑁₂ is the largest integer number such that Ω𝐴_𝑛+𝐶_𝑛^𝑛−1_𝑘=0𝑒^{−2𝜏𝑡𝑘}𝑏²_𝑘∈DomΩ⁻¹. (32) Clearly, by 𝑥_𝑛=𝑣_𝑛^1/2, (15) also holds for 0≤𝑛≤𝑁₂.

Remark 5. Here, we note that the most significant work in the study of weakly singular inequalities is Medved̆’s method, originally presented in the paper [6] and also applied in the paper [18]. But his result holds under the assumption “𝜔(𝑢) satisfies the condition (q),” that is, “𝑒^−𝑞𝑡[𝜔(𝑢)]^𝑞≤𝑅(𝑡)𝜔(𝑒^−𝑞𝑡𝑢^𝑞), where 𝑅(𝑡) is a continuous, nonnegative function.” In our result, the condition (q) is eliminated.

Corollary 6. Under assumptions (𝐴₁) and (𝐴₂), let 𝜈>0, 𝜇>0(𝜈>𝜇). If 𝑥_𝑛 is nonnegative such that 𝑥^𝜈_𝑛≤𝑎_𝑛+^𝑛−1_𝑘=0𝑡_𝑛−𝑡_𝑘^𝛽−1𝜏_𝑘𝑏_𝑘𝑥^𝜇_𝑘, (33) then(1)if 0<𝛽≤1/2, let 𝑝=1+𝛽 and 𝑞=(1+𝛽)/𝛽, and one gets 𝑥_𝑛≤2^𝑞−1𝑎^𝑞_𝑛^{(𝜈−𝜇)/𝜈}+𝜈−𝜇𝜈2^𝑞−1𝜏^{1−(𝑞/𝑝)𝛽2}×𝐾^𝑞/𝑝𝛽𝑒^{𝑞𝜏𝑡𝑛𝑛−1}_𝑘=0𝑒^{−𝑞𝜏𝑡𝑘}𝑏^𝑞_𝑘^{1/(𝜈−𝜇)𝑞} (34) for 𝑛≥0, where 𝐾(𝛽) is defined as in Theorem 4;(2)if 1/2<𝛽≤1, let 𝑝=𝑞=2, and one gets 𝑥_𝑛≤2𝑎²_𝑛^{(𝜈−𝜇)/𝜈}+𝜈−𝜇𝜈𝐵𝛽𝜏^2−2𝛽×𝑒^{2𝜏𝑡𝑛𝑛−1}_𝑘=0𝑒^{−2𝜏𝑡𝑘}𝑏²_𝑘^{1/2(𝜈−𝜇)}, (35) for 𝑛≥0, where 𝐵(𝛽) is defined as in Theorem 4

Proof. Let 𝑧_𝑛=𝑥^𝜈_𝑛, then 𝑥_𝑛=𝑧_𝑛^1/𝜈 and 𝑥^𝜇_𝑛=𝑧_𝑛^𝜇/𝜈. From (33), we have 𝑧_𝑛≤𝑎_𝑛+^𝑛−1_𝑘=0𝑡_𝑛−𝑡_𝑘^𝛽−1𝜏_𝑘𝑏_𝑘𝑧_𝑘^𝜇/𝜈. (36)
Clearly, 𝜔(𝑧_𝑘)=𝑧_𝑘^𝜇/𝜈 satisfies the assumption (𝐴₂). According to the definition of Ω in Theorem 4, for 0<𝛽≤1/2, letting 𝑢₀=0, we have Ω(𝑢)=^𝑢_𝑢01𝜔^𝑞𝑠^1/𝑞𝑑𝑠=^𝑢₀𝑑𝑠𝑠^𝜇/𝜈=𝜈𝑢𝜈−𝜇^{(𝜈−𝜇)/𝜈},Ω⁻¹(𝑢)=𝜈−𝜇𝜈𝑢^{𝜈/(𝜈−𝜇)},DomΩ⁻¹=[0,∞). (37)(38)
It can be seen easily from (38) that 𝑁₁=∞. Substituting (37) and (38) into (12), we get 𝑧_𝑛≤2^𝑞−1𝑎^𝑞_𝑛^{(𝜈−𝜇)/𝜈}+𝜈−𝜇𝜈2^𝑞−1𝜏^{1−(𝑞/𝑝)𝛽2}×𝐾^𝑞/𝑝𝛽𝑒^{𝑞𝜏𝑡𝑛𝑛−1}_𝑘=0𝑒^{−𝑞𝜏𝑡𝑘}𝑏^𝑞_𝑘^{𝜈/(𝜈−𝜇)𝑞}. (39)
In view of 𝑥_𝑛=𝑧_𝑛^1/𝜈, we can obtain (34). For the case that 1/2<𝛽≤1, in fact, Ω and Ω⁻¹ are the same as (37) and (38), respectively. So, it follows from (37), (38), and (15) that 𝑥_𝑛≤2𝑎²_𝑛^{(𝜈−𝜇)/𝜈}+𝜈−𝜇𝜈𝐵𝛽𝜏^2−2𝛽×𝑒^{2𝜏𝑡𝑛𝑛−1}_𝑘=0𝑒^{−2𝜏𝑡𝑘}𝑏²_𝑘^{1/2(𝜈−𝜇)}, (40) for 𝑛>0.

Remark 7. In [11], Yang et al. investigated inequality (33), under the assumption that 𝑎_𝑛 is nondecreasing. Clearly, our result does not need such condition, and we get a more concise formula.

Remark 8. Letting 𝜈=2 and 𝜇=1, we can get the interesting Henry version of the Ou-Iang-Pachpatte-type difference inequality [3]. Thus, our result is a more general discrete analogue for such inequality.

Corollary 9. Under assumptions (𝐴₁) and (𝐴₂), if 𝑥_𝑛 is nonnegative such that 𝑥_𝑛≤𝑎_𝑛+^𝑛−1_𝑘=0𝑡_𝑛−𝑡_𝑘^𝛽−1𝜏_𝑘𝑏_𝑘𝑥_𝑘, (41) then(1)if 0<𝛽≤1/2, let 𝑝=1+𝛽 and 𝑞=(1+𝛽)/𝛽, and one gets 𝑥_𝑛≤2^{(𝑞−1)/𝑞}𝑎_𝑛exp2^{(𝑞−1)/𝑞}𝜏^{1−(𝑞/𝑝)𝛽2}𝐾^𝑞/𝑝𝛽×𝑒^{𝑞𝜏𝑡𝑛𝑛−1}_𝑘=0𝑒^{−𝑞𝜏𝑡𝑘}𝑏^𝑞_𝑘, (42) for 𝑛≥0, where 𝐾(𝛽) is defined as in Theorem 4;(2)if 1/2<𝛽≤1, let 𝑝=𝑞=2, and one gets 𝑥_𝑛≤√2𝑎_𝑛1exp2𝐵𝛽𝜏^2−2𝛽𝑒^{2𝜏𝑡𝑛𝑛−1}_𝑘=0𝑒^{−2𝜏𝑡𝑘}𝑏²_𝑘, (43) for 𝑛≥0, where 𝐵(𝛽) is defined as in Theorem 4.

Proof. In (41), 𝜔(𝑢)=𝑢 also satisfies the assumption (𝐴₂). Thus, we have Ω(𝑢)=^𝑢_𝑢0𝑑𝑠𝑠𝑢=ln𝑢₀,Ω⁻¹(𝑢)=𝑢₀(𝑢),ΩexpDom⁻¹=[0,∞). (44)
Similarly to the computation in Corollary 6, the estimates (42) and (43) hold, respectively.

4. Application

In this section, we apply our results to discuss the upper bound of solution of a Volterra type difference equation with weakly singular kernel.

Consider the following the inequality: 𝑥_𝑛≤1+^𝑛−1_𝑘=0𝑡_𝑛−𝑡_𝑘^−1/2𝜏_𝑘√𝑥_𝑘. (45)

Obviously, (45) is the special case of inequality (6), then we get 𝑎_𝑛1=1,𝛽=2,𝜔=√𝑢. (46)

Thus, we can take 𝑝=1+𝛽=3/2 and 𝑞=(1+𝛽)/𝛽=3; then, 𝑞/𝑝=2. Moreover, 𝑎_𝑛𝐾𝛽==1,1+𝛽^−𝛽2Γ𝛽²=32^−1/4Γ14,Ω(𝑢)=^𝑢₀𝑑𝑠√𝑠=2√𝑢,Ω⁻¹(𝑢)=𝑢²4. (47)

According to Theorem 4, we obtain 𝑥_𝑛≤Ω⁻¹Ω2^𝑞−1𝑎^𝑞_𝑛+2^𝑞−1𝜏^{1−(𝑞/𝑝)𝛽2}×𝐾^𝑞/𝑝𝛽𝑒^{𝑞𝜏𝑡𝑛𝑛−1}_𝑘=0𝑒^{−𝑞𝜏𝑡𝑘}𝑏^𝑞_𝑘^1/𝑞=Ω⁻¹Ω(4)+4𝜏^1/232^−1/2×Γ²14𝑒^{3𝜏𝑡𝑛𝑛−1}_𝑘=0𝑒^{−3𝜏𝑡𝑘}𝑏³_𝑘^1/3=Ω⁻¹44+3√6𝜏^1/2Γ²14𝑒^{3𝜏𝑡𝑛𝑛−1}_𝑘=0𝑒^{−3𝜏𝑡𝑘}𝑏³_𝑘^1/3=4^−1/344+3√6𝜏^1/2Γ²14𝑒^{3𝜏𝑡𝑛𝑛−1}_𝑘=0𝑒^{−3𝜏𝑡𝑘}𝑏³_𝑘^2/3 (48) for 𝑛>0, which indicates that we get the upper bound of 𝑥_𝑛.

Conflict of Interests

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

Acknowledgments

This work is supported by the Doctoral Program Research Funds of Southwest University of Science and Technology (no. 11zx7129) and the Fundamental Research Funds for the Central Universities (no. skqy201324). The work is also supported by the Applied Basic Research Project of Sichuan Province (no. 2013JY0096).