Some New Gronwall-Bellman-Type Inequalities on Time Scales and Their Applications

Abstract

We establish some new Gronwall-Bellman-type inequalities on time scales. These inequalities are of new forms compared with other Gronwall-Bellman-type inequalities established so far in the literature. Based on them, new bounds for unknown functions are derived. For illustrating the validity of the inequalities established, we present some applications for them, in which the boundedness for solutions for some certain dynamic equations on time scales is researched.

1. Introduction

As is known, various integral and differential inequalities play an important role in the research of boundedness, global existence, and stability of solutions of differential and integral equations as well as difference equations. Among the investigations for inequalities, generalization of the Gronwall-Bellman-inequality [1, 2] is a hot topic, as such inequalities provide explicit bounds for unknown functions concerned. During the past decades, many Gronwall-Bellman-type inequalities have been discovered (e.g., see [3–18]). Recently, with the development of the theory of time scales [19], many integral inequalities on time scales have been established, for example, [20–32], which have proved to be very effective in the analysis of qualitative as well as quantitative analysis of solutions of dynamic equations. But for some certain dynamic equations, for example, 𝑢^𝑝(𝑡)^Δ=𝐹𝑡,𝑢(𝑡),𝑢(𝑡),^∞_𝑡𝑊(𝜉,𝑢(𝜉))Δ𝜉, (1) or 𝑢^𝑝(𝑡)=𝐶+^∞_𝑡𝐹𝑠,𝑢(𝑠),𝑢(𝑠),^∞_𝑠𝑊((𝜉+𝜉,𝑢))Δ𝜉Δ𝑠_∞𝑇𝐹𝑠,𝑢(𝑠),𝑢(𝑠),^∞_𝑠𝑊(𝜉,𝑢(𝜉))Δ𝜉Δ𝑠, (2) it is inadequate to research the boundedness of their solutions by use of the existing results in the literature. So it is necessary to seek new approach to fulfill such analysis for them.

Based on the analysis above, in this paper, we establish some new Gronwall-Bellman-type inequalities on time scales, which are designed so as to be used as a handy tool to research the boundedness of the solutions of the equations mentioned above. By use of the established inequalities, some new bounds for the solutions for the two equations are derived.

In the rest of the paper, ℝ denotes the set of real numbers, and ℝ₊=[0,∞). 𝕋 denotes an arbitrary time scale, and 𝕋₀=[𝑡₀,∞)⋂𝕋, where 𝑡₀∈𝕋. On 𝕋 we define the forward and backward jump operators 𝜎∈(𝕋,𝕋) and 𝜌∈(𝕋,𝕋) such that 𝜎(𝑡)=inf{𝑠∈𝕋,𝑠>𝑡} and 𝜌(𝑡)=sup{𝑠∈𝕋,𝑠<𝑡}. The graininess 𝜇∈(𝕋,ℝ₊) is defined by 𝜇(𝑡)=𝜎(𝑡)−𝑡. Obviously, 𝜇(𝑡)=0 if 𝕋=ℝ, while 𝜇(𝑡)=1 if 𝕋=ℤ. A point 𝑡∈𝕋 with 𝑡>inf𝕋 is said to be left dense if 𝜌(𝑡)=𝑡, right dense if 𝜎(𝑡)=𝑡, left scattered if 𝜌(𝑡)<𝑡, and right scattered if 𝜎(𝑡)>𝑡. The set 𝕋^𝜅 is defined to be 𝕋 if 𝕋 does not have a left scattered maximum; otherwise it is 𝕋 without the left scattered maximum.

The following definitions and theorems in the theory of time scales are known to us.

Definition 1. A function 𝑓∈(𝕋,ℝ) is called rd-continuous if it is continuous in right-dense points and if the left-sided limits exist in left-dense points, while 𝑓 is called regressive if 1+𝜇(𝑡)𝑓(𝑡)≠0. 𝐶_rd denotes the set of rd-continuous functions, while ℜ denotes the set of all regressive and rd-continuous functions, and ℜ⁺={𝑓∣𝑓∈ℜ,1+𝜇(𝑡)𝑓(𝑡)>0,∀𝑡∈𝕋}.

Definition 2. For some 𝑡∈𝕋^𝜅 and a function 𝑓∈(𝕋,ℝ), the delta derivative of 𝑓 at 𝑡 is denoted by 𝑓^Δ(𝑡) (provided that it exists) with the property such that, for every 𝜀>0, there exists a neighborhood 𝔘 of 𝑡 satisfying 𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡)(𝜎(𝑡)−𝑠)≤𝜀|𝜎(𝑡)−𝑠|∀𝑠∈𝔘. (3)

Definition 3. If 𝐹^Δ(𝑡)=𝑓(𝑡) and 𝑡∈𝕋^𝜅, then 𝐹 is called an antiderivative of 𝑓, and the Cauchy integral of 𝑓 is defined by ^𝑏_𝑎𝑓(𝑡)Δ𝑡=𝐹(𝑏)−𝐹(𝑎), (4) where 𝑎,𝑏∈𝕋^𝜅.

Definition 4. For 𝑝∈ℜ, the exponential function is defined by 𝑒_𝑝(𝑡,𝑠)=exp^𝑡_𝑠𝜉_𝜇(𝜏)𝑝(𝜏)Δ𝜏for𝑠,𝑡∈𝕋. (5)

Definition 5. If sup_𝑡∈𝕋𝑡=∞ and 𝑝∈ℜ, we define 𝑒_𝑝(∞,𝑠)=exp^∞_𝑠𝜉_𝜇(𝜏)𝑝(𝜏)Δ𝜏for𝑡∈𝕋. (6)

Theorem 6 (see [20, Theorem 2.1]). If 𝑓,𝑔∈(𝕋,ℝ) and 𝑡∈𝕋^𝜅, then (i)𝑓^Δ(𝑡)=𝑓(𝜎(𝑡))−𝑓(𝑡)𝜇(𝑡)𝑖𝑓𝜇(𝑡)≠0,lim_𝑠→𝑡𝑓(𝑡)−𝑓(𝑠)(𝑡)𝑡−𝑠𝑖𝑓𝜇=0. (7)(ii)If 𝑓,𝑔 are delta differential at 𝑡, then 𝑓𝑔 is also delta differential at 𝑡, and 𝑓𝑔^Δ(𝑡)=𝑓^Δ(𝑡)𝑔(𝑡)+𝑓(𝜎(𝑡))𝑔^Δ(𝑡). (8)

Theorem 7 (see [20, Theorem 2.2]). If 𝑎,𝑏,𝑐∈𝕋,𝛼∈ℝ, and 𝑓,𝑔∈𝐶_rd, then one has the following:(i)∫^𝑏_𝑎[𝑓(𝑡)+𝑔(𝑡)]Δ𝑡=∫^𝑏_𝑎𝑓(𝑡)Δ𝑡+∫^𝑏_𝑎𝑔(𝑡)Δ𝑡; (ii)∫^𝑏_𝑎(𝛼𝑓)(𝑡)Δ𝑡=𝛼∫^𝑏_𝑎𝑓(𝑡)Δ𝑡; (iii)∫^𝑏_𝑎𝑓(𝑡)Δ𝑡=−∫^𝑎_𝑏𝑓(𝑡)Δ𝑡; (iv)∫^𝑏_𝑎𝑓(𝑡)Δ𝑡=∫^𝑐_𝑎𝑓(𝑡)Δ𝑡+∫^𝑏_𝑐𝑓(𝑡)Δ𝑡; (v)∫^𝑎_𝑎𝑓(𝑡)Δ𝑡=0; (vi)if 𝑓(𝑡)≥0 for all 𝑎≤𝑡≤𝑏, then ∫^𝑏_𝑎𝑓(𝑡)Δ𝑡≥0.

Theorem 8 (see [29, Theorem 5.2]). If 𝑝∈ℜ, then the following conclusions hold:(i)𝑒_𝑝(𝑡,𝑡)≡1 and 𝑒₀(𝑡,𝑠)≡1;(ii)𝑒_𝑝(𝜎(𝑡),𝑠)=(1+𝜇(𝑡)𝑝(𝑡))𝑒_𝑝(𝑡,𝑠);(iii)if 𝑝∈ℜ⁺, then 𝑒_𝑝(𝑡,𝑠)>0 for all 𝑠,𝑡∈𝕋;(iv)if 𝑝∈ℜ⁺, then ⊖𝑝∈ℜ⁺;(v)𝑒_𝑝(𝑡,𝑠)=1/(𝑒_𝑝(𝑠,𝑡))=𝑒_⊖𝑝(𝑠,𝑡),where ⊖𝑝=−𝑝/(1+𝜇𝑝).

Remark 9. If 𝑠=∞, then Theorem 8 ((iii), (v)) still holds.

Theorem 10 (see [29, Theorem 5.1]). If 𝑝∈ℜ and fix 𝑡₀∈𝕋, then the exponential function  𝑒_𝑝(𝑡,𝑡₀) is the unique solution of the following initial value problem: 𝑦^Δ(𝑡)(𝑡)𝑦(𝑡),𝑦𝑡=𝑝₀=1. (9)

2. Main Results

Lemma 11. Suppose that 𝑢,𝑎,𝑚∈𝐶_rd(𝕋₀,ℝ₊), 𝑚(𝑡)=−𝑚(𝑡)𝑏(𝑡), and 𝑚∈ℜ₊. Then 𝑢(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^∞_𝑡𝑚(𝑠)𝑢(𝑠)Δ𝑠,𝑡∈𝕋₀, (10) implies 𝑢(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^∞_𝑡𝑒𝑚(𝑡,𝜎(𝑠))𝑚(𝑠)𝑎(𝑠)Δ𝑠,𝑡∈𝕋₀. (11)

Proof. Denote that 𝑣(𝑡)=∫^∞_𝑡𝑚(𝑠)𝑢(𝑠)Δ𝑠. Then 𝑢(𝑡)(𝑡)(𝑡)𝑣(𝑡),𝑣≤𝑎+𝑏_Δ(𝑡)(𝑡)𝑢(𝑡)(𝑡)𝑏(𝑡)𝑣(𝑡)(𝑡)𝑎(𝑡)=𝑚(𝑡)𝑣(𝑡)(𝑡)𝑎(𝑡).=−𝑚≥−𝑚−𝑚−𝑚 (12) Since 𝑚∈ℜ⁺, then from Theorem 8 (iv), we have ⊖𝑚∈ℜ⁺, and furthermore from Theorem 8 (iii), we obtain 𝑒_⊖𝑚(𝑡,𝛼)>0 for all 𝛼∈𝕋₀.
According to Theorem 6 (ii), 𝑣(𝑡)𝑒_⊖𝑚(𝑡,𝛼)^Δ=𝑒_⊖𝑚(𝑡,𝛼)^Δ𝑣(𝑡)+𝑒_⊖𝑚(𝜎(𝑡),𝛼)𝑣^Δ(𝑡). (13) On the other hand, from Theorem 10, we have 𝑒_⊖𝑚(𝑡,𝛼)^Δ=(⊖𝑚)(𝑡)𝑒_⊖𝑚(𝑡,𝛼). (14) So combining (13), (14) and Theorem 8, it follows that 𝑣(𝑡)𝑒_⊖𝑚(𝑡,𝛼)^Δ=(⊖𝑚)(𝑡)𝑒_⊖𝑚(𝑡,𝛼)𝑣(𝑡)+𝑒_⊖𝑚(𝜎(𝑡),𝛼)𝑣^Δ(𝑡)=𝑒_⊖𝑚(𝜎(𝑡)(⊖𝑚𝑡),𝛼))((𝑡⊖𝑚𝑡)𝑣(𝑡)1+𝜇)()(+𝑣^Δ(𝑡)=𝑒_⊖𝑚(𝜎(𝑡),𝛼)𝑣^Δ(𝑡)−𝑚(𝑡)𝑣(𝑡). (15) Substituting 𝑡 with 𝑠, an integration for (15) with respect to 𝑠 from 𝛼 to ∞ yields 𝑣(∞)𝑒_⊖𝑚(∞,𝛼)−𝑣(𝛼)𝑒_⊖𝑚()=𝛼,𝛼_∞𝛼𝑒_⊖𝑚(𝜎(𝑠),𝛼)𝑣^Δ(𝑠)−𝑚(𝑠)𝑣(𝑠)Δ𝑠. (16) By 𝑣(∞)=0 and 𝑒_⊖𝑚(𝛼,𝛼)=1, it follows that −𝑣(𝛼)≥−^∞_𝛼𝑒_⊖𝑚(𝜎(𝑠),𝛼)𝑚(𝑠)𝑎(𝑠)Δ𝑠=−^∞_𝛼𝑒𝑚(𝛼,𝜎(𝑠))𝑚(𝑠)𝑎(𝑠)Δ𝑠, (17) which is followed by 𝑣(𝛼)≤^∞_𝛼𝑒𝑚(𝛼,𝜎(𝑠))𝑚(𝑠)𝑎(𝑠)Δ𝑠. (18) Since 𝛼∈𝕋₀ is arbitrary, after substituting 𝛼 with 𝑡, we obtain the desired inequality.

Lemma 12. Under the conditions of Lemma 11, furthermore, if 𝑏(𝑡)≡1 and 𝑎(𝑡) is nonincreasing on 𝕋₀, then 𝑢(𝑡)≤𝑎(𝑡)𝑒_−𝑚(𝑡,∞),𝑡∈𝕋₀. (19)

Proof. Since 𝑏(𝑡)≡1 and 𝑎(𝑡) is nondecreasing on 𝕋₀, then 𝑚=𝑚 and 𝑢(𝑡)≤𝑎(𝑡)+^∞_𝑡𝑒_−𝑚((𝑠𝑎(𝑠)𝑚(𝑠)(𝑡𝑡,𝜎))Δ𝑠≤𝑎)1+^∞_𝑡𝑒_−𝑚(𝑡,𝜎(𝑠))𝑚(𝑠)Δ𝑠. (20) From [33, Theorems 2.39 and 2.36 (i)], we have ^∞_𝑡𝑒_−𝑚(𝑡,𝜎(𝑠))𝑚(𝑠)Δ𝑠=lim_𝑥→∞^𝑡_𝑥𝑒_−𝑚(𝑡,𝜎(𝑠))(−𝑚(𝑠))Δ𝑠=lim_𝑥→∞𝑒_−𝑚(𝑡,𝑥)−𝑒_−𝑚(𝑡,𝑡)=𝑒_−𝑚(𝑡,∞)−1. (21) Combining the above information, we can obtain the desired inequality.

Lemma 13 (see [4]). Assume that 𝑎≥0,𝑝≥𝑞≥0, and 𝑝≠0; then for any 𝐾>0, the following inequality holds: 𝑎^𝑞/𝑝≤𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑎+𝑝−𝑞𝑝𝐾^𝑞/𝑝. (22)
First one will study the following Gronwall-Bellman-type inequality on time scales: 𝑢^𝑝(𝑡)(𝑡)(𝑡)×≤𝑎+𝑏_∞𝑡𝑚(𝑠)+𝑓(𝑠)𝑢^𝑝(𝑠)+𝑔(𝑠)𝑢^𝑞(𝑠)+^∞_𝑠ℎ(𝜉)𝑢^𝑟(𝜉)Δ𝜉Δ𝑠. (23)

Theorem 14. Suppose that 𝑢,𝑓,𝑔,ℎ,𝑚,𝑎,𝑏∈𝐶_rd(𝕋₀,ℝ₊),𝑎,𝑏 are nonincreasing, 𝑝,𝑞, and 𝑟 are constants, 𝑝≥𝑞≥0,𝑝≥𝑟≥0, and 𝑝≠0. If for 𝑡∈𝕋₀, 𝑢(𝑡) satisfies (23), then 𝑢(𝑡)≤𝑎(𝑡)+𝑏(𝑡)𝐻₂(𝑡)𝑒_−𝐻3(𝑡,∞)𝐻₁(𝑡)^1/𝑝,𝑡∈𝕋₀, (24) Provided that 𝑓,−𝐻₃∈ℜ₊, where 𝐻₁(𝑡)=1+𝑏(𝑡)^∞_𝑡𝑒𝑓((𝑠𝑓(𝑠)𝑓(𝑡)(𝑡)𝑏(𝑡),𝐻𝑡,𝜎))Δ𝑠,=−𝑓₂(𝑡)=^∞_𝑡𝑚(𝑠)(𝑠𝑞+𝑔)𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑎(𝑠)+𝑝−𝑞𝑝𝐾^𝑞/𝑝×𝐻₁(𝑠)^𝑞/𝑝Δ𝑠+^∞_𝑡^∞_𝑠ℎ(𝜉𝑟)𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑎(𝜉)+𝑝−𝑟𝑝𝐾^𝑟/𝑝×𝐻₁(𝜉)^𝑟/𝑝𝐻Δ𝜉Δ𝑠,∀𝐾>0,₃(𝑡)(𝑡)𝑞=𝑔𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑏(𝑡)𝐻₁(𝑡)^𝑞/𝑝+^∞_𝑡ℎ(𝜉)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑏(𝜉)𝐻₁(𝜉)^𝑟/𝑝Δ𝜉,∀𝐾>0. (25)(26)(27)

Proof. Let the right side of (23) be 𝑣(𝑡). Then 𝑢(𝑡)≤𝑣^1/𝑝(𝑡),𝑡∈𝕋₀,𝑣(𝑡)(𝑡)(𝑡)×≤𝑎+𝑏_∞𝑡𝑚(𝑠)+𝑓(𝑠)𝑣(𝑠)+𝑔(𝑠)𝑣^𝑞/𝑝(𝑠)+^∞_𝑠ℎ(𝜉)𝑣^𝑟/𝑝(𝜉)Δ𝜉Δ𝑠. (28)(29) Let 𝑐(𝑡)(𝑡)(𝑡)×=𝑎+𝑏_∞𝑡𝑚(𝑠)+𝑔(𝑠)𝑣^𝑞/𝑝(𝑠)+^∞_𝑠ℎ(𝜉)𝑣^𝑟/𝑝(𝜉)Δ𝜉Δ𝑠. (30) Then we have 𝑣(𝑡)≤𝑐(𝑡)+𝑏(𝑡)^∞_𝑡𝑓(𝑠)𝑣(𝑠)Δ𝑠,𝑡∈𝕋₀. (31) From Lemma 11, considering 𝑐(𝑡) is nonincreasing on 𝕋₀, we can obtain 𝑣(𝑡)≤𝑐(𝑡)+𝑏(𝑡)^∞_𝑡𝑒𝑓((𝑠𝑐(𝑠)𝑓(𝑠)(𝑡(𝑡)𝑡,𝜎))Δ𝑠≤𝑐)1+𝑏^∞_𝑡𝑒𝑓((𝑠𝑓(𝑠)(𝑡)𝐻𝑡,𝜎))Δ𝑠=𝑐₁(𝑡),𝑡∈𝕋₀, (32) where 𝐻₁(𝑡),𝑓(𝑡) are defined in (25).
Let 𝑦(𝑡)=^∞_𝑡𝑚(𝑠)+𝑔(𝑠)𝑣^𝑞/𝑝(𝑠)+^∞_𝑠ℎ(𝜉)𝑣^𝑟/𝑝(𝜉)Δ𝜉Δ𝑠. (33) Then 𝑐(𝑡)=𝑎(𝑡)+𝑏(𝑡)𝑦(𝑡). (34) From Lemma 13, we have 𝑎(𝑡)+𝑏(𝑡)𝑦(𝑡)^𝑞/𝑝≤𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑎(𝑡)+𝑏(𝑡)𝑦(𝑡)+𝑝−𝑞𝑝𝐾^𝑞/𝑝,𝑎(𝑡)+𝑏(𝑡)𝑦(𝑡)^𝑟/𝑝≤𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑎(𝑡)+𝑏(𝑡)𝑦(𝑡)+𝑝−𝑟𝑝𝐾^𝑟/𝑝,∀𝐾>0. (35) A combination of (32), (33), (34), and (35) yields 𝑦(𝑡)≤^∞_𝑡𝑚(𝑠)+𝑔(𝑠)𝑐(𝑠)𝐻₁(𝑠)^𝑞/𝑝+^𝑠_𝑡0ℎ(𝜉)𝑐(𝜉)𝐻₁(𝜉)^𝑟/𝑝Δ𝜉Δ𝑠≤^∞_𝑡𝑚(𝑠)+𝑔(𝑠)𝑎(𝑠)+𝑏(𝑠)𝑦(𝑠)𝐻₁(𝑠)^𝑞/𝑝+^∞_𝑠ℎ(𝜉)𝑎(𝜉)+𝑏(𝜉)𝑦(𝜉)𝐻₁(𝜉)^𝑟/𝑝≤Δ𝜉Δ𝑠_∞𝑡𝑚(𝑠)(𝑠𝑞+𝑔)𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑎(𝑠)(𝑠)𝑦(𝑠)++𝑏𝑝−𝑞𝑝𝐾^𝑞/𝑝𝐻₁(𝑠)^𝑞/𝑝+Δ𝑠_{∞𝑡∞𝑠}ℎ(𝜉𝑟)𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑎(𝜉)(𝜉)𝑦(𝜉++𝑏)𝑝−𝑟𝑝𝐾^𝑟/𝑝𝐻₁(𝜉)^𝑟/𝑝=Δ𝜉Δ𝑠_∞𝑡𝑚(𝑠)(𝑠𝑞+𝑔)𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑎(𝑠)+𝑝−𝑞𝑝𝐾^𝑞/𝑝×𝐻₁(𝑠)^𝑞/𝑝+^∞_𝑠ℎ(𝜉𝑟)𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑎(𝜉)+𝑝−𝑟𝑝𝐾^𝑟/𝑝𝐻₁(𝜉)^𝑟/𝑝+Δ𝜉Δ𝑠_∞𝑡𝑔(𝑠)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑏(𝑠)𝐻₁(𝑠)^𝑞/𝑝𝑦(𝑠)+^∞_𝑠ℎ(𝜉)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}×𝑏(𝜉)𝐻₁(𝜉)^𝑟/𝑝𝑦(𝜉)Δ𝜉Δ𝑠≤𝐻₂(𝑡)+^∞_𝑡𝐻₃(𝑠)𝑦(𝑠)Δ𝑠, (36) where 𝐻₂(𝑡),𝐻₃(𝑡) are defined in (26) and (27), respectively. By 𝐻₂(𝑡) being nonincreasing on 𝕋₀ and −𝐻₃∈ℜ₊, according to Lemma 12, we have 𝑦(𝑡)≤𝐻₂(𝑡)𝑒_−𝐻3(𝑡,∞),𝑡∈𝕋₀. (37) Combining (32), (34), and (37), we obtain 𝑣(𝑡)≤𝑎(𝑡)+𝑏(𝑡)𝐻₂(𝑡)𝑒_−𝐻3(𝑡,∞)𝐻₁(𝑡). (38) From (28), (38), we can obtain the desired inequality (13).

Based on Theorem 14, we will establish two Volterra-Fredholm type delay integral inequalities on time scales in the following two theorems.

Theorem 15. Suppose that 𝑢,𝑓,𝑔,ℎ,𝑚,𝑝,𝑞, and 𝑟 are defined as in Theorem 14 with −𝑓∈ℜ₊, 𝐶>0 is a constant, and 𝑇∈𝕋₀ is a fixed number. If for 𝑡∈[𝑇,∞)⋂𝕋, 𝑢(𝑡) satisfies the following inequality: 𝑢^𝑝(𝑡)≤𝐶+^∞_𝑡𝑚(𝑠)+𝑓(𝑠)𝑢^𝑝(𝑠)+𝑔(𝑠)𝑢^𝑞(𝑠)+^∞_𝑠ℎ(𝜉)𝑢^𝑟(𝜉)+Δ𝜉Δ𝑠_∞𝑇𝑚(𝑠)+𝑓(𝑠)𝑢^𝑝(𝑠)+𝑔(𝑠)𝑢^𝑞(𝑠)+^∞_𝑠ℎ(𝜉)𝑢^𝑟(𝜉)Δ𝜉Δ𝑠, (39) and furthermore, (1+𝐻₂₁𝐻(𝑇)𝑒₃(𝑇,∞))𝑒_−𝑓(𝑇,∞)<2, then for 𝑡∈[𝑇,∞)⋂𝕋, 𝑢(𝑡)≤𝐶+𝐻₂₂(𝑇)𝑒₋𝐻₃()𝑇,∞𝑒_𝑓(𝑇,∞)2−1+𝐻₂₁(𝑇)𝑒₋𝐻₃()𝑇,∞𝑒_𝑓()×𝐻𝑇,∞1+₂₁(𝑡)𝑒₋𝐻₃(+𝐻𝑡,∞)₂₂(𝑡)𝑒₋𝐻₃(𝑡,∞)𝑒_−𝑓(𝑡,∞)^1/𝑝, (40) provided that −𝐻₃∈ℜ₊, where 𝐻₁(𝑡)=1+^∞_𝑡𝑒𝑓((𝑠𝑓(𝑠)𝑓(𝑡)(𝑡)𝑏(𝑡),𝐻𝑡,𝜎))Δ𝑠,=−𝑓₂₁(𝑡)=^∞_𝑡𝑔(𝑠)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝐻₁(𝑠)^𝑞/𝑝+^∞_𝑠ℎ(𝜉)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}𝐻₁(𝜉)^𝑟/𝑝𝐻Δ𝜉Δ𝑠,∀𝐾>0,₂₂(𝑡)=^∞_𝑡𝑚(𝑠)+𝑔(𝑠)𝑝−𝑞𝑝𝐾^𝑞/𝑝𝐻₁(𝑠)^𝑞/𝑝+^∞_𝑠ℎ(𝜉)𝑝−𝑟𝑝𝐾^𝑟/𝑝𝐻₁(𝜉)^𝑟/𝑝𝐻Δ𝜉Δ𝑠,∀𝐾>0,₃(𝑡)(𝑡)𝑞=𝑔𝑝𝐾^{(𝑞−𝑝)/𝑝}𝐻₁(𝑡)^𝑞/𝑝+^∞_𝑡ℎ(𝜉)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}𝐻₁(𝜉)^𝑟/𝑝Δ𝜉,∀𝐾>0. (41)(42)(43)(44)

Proof. Let the right side of (39) be 𝑣(𝑡). Then 𝑢(𝑡)≤𝑣^1/𝑝(𝑡),𝑡∈[𝑇,∞)𝕋. (45) Considering 𝑣(∞)=𝐶+∫^∞_𝑇[𝑚(𝑠)+𝑓(𝑠)𝑢^𝑝(𝑠)+𝑔(𝑠)𝑢^𝑞(𝑠)+∫^∞_𝑠ℎ(𝜉)𝑢^𝑟(𝜉)Δ𝜉]Δ𝑠, it follows that 𝑣(𝑡)(∞)+=𝑣_∞𝑡𝑚(𝑠)+𝑓(𝑠)𝑢^𝑝(𝑠)+𝑔(𝑠)𝑢^𝑞(𝑠)+^∞_𝑠ℎ(𝜉)𝑢^𝑟(𝜉)(∞)+Δ𝜉Δ𝑠≤𝑣_∞𝑡𝑚(𝑠)(𝑠)𝑣(𝑠)(𝑠)𝑣+𝑓+𝑔^𝑞/𝑝(𝑠)+^∞_𝑠ℎ(𝜉)𝑣^𝑟/𝑝(𝜉)[)Δ𝜉Δ𝑠,𝑡∈𝑇,∞𝕋. (46) We notice that the structure of (46) is just similar to that of (29). So following the same manner as the process of (29)–(38) in Theorem 14 (i.e., 𝑣(∞) takes the place of 𝑎(𝑡) in Theorem 14, and let 𝑏(𝑡)≡1 in Theorem 14), considering 𝐻₁(𝑡)=1+∫^∞_𝑡𝑒_−𝑓(𝑡,𝜎(𝑠))𝑓(𝑠)Δ𝑠=𝑒_−𝑓(𝑡,∞), we can obtain 𝑣(𝑡)≤𝑣(∞)1+𝐻₂₁(𝑡)𝑒₋𝐻₃()+𝐻𝑡,∞₂₂(𝑡)𝑒₋𝐻₃(𝑡,∞)𝑒_−𝑓(),[)𝑡,∞𝑡∈𝑇,∞𝕋, (47) where 𝐻₂₁(𝑡),𝐻₂₂(𝑡), and 𝐻₃(𝑡) are defined in (42), (43), and (44), respectively. Setting 𝑡=𝑇 in (47), we obtain 𝑣(𝑇)≤𝑣(∞)1+𝐻₂₁(𝑇)𝑒₋𝐻₃()+𝐻𝑇,∞₂₂(𝑇)𝑒₋𝐻₃(𝑇,∞)𝑒_−𝑓(𝑇,∞). (48) As 2𝑣(∞)−𝐶=𝑣(𝑇), it follows that (∞)(𝑇)≤𝑣(∞𝐻2𝑣−𝐶=𝑣)1+₂₁(𝑇)𝑒₋𝐻₃(+𝐻𝑇,∞)₂₂(𝑇)𝑒₋𝐻₃(𝑇,∞)𝑒_−𝑓(𝑇,∞); (49) that is, 𝑣(∞)≤𝐶+𝐻₂₂(𝑇)𝑒₋𝐻₃()𝑇,∞𝑒_−𝑓(𝑇,∞)2−1+𝐻₂₁(𝑇)𝑒₋𝐻₃()𝑇,∞𝑒_−𝑓().𝑇,∞ (50) Combining (45), (47), and (50), we can obtain the desired inequality (39).

Theorem 16. Suppose that 𝑢,𝑓,ℎ,𝑝, and 𝑟 are defined as in Theorem 14 with −𝑓∈ℜ₊, 𝐶>0 is a constant, 𝑇∈𝕋₀ is a fixed number, 𝐿∈𝐶(𝕋₀×ℝ₊,ℝ₊), and 0≤𝐿(𝑠,𝑥)−𝐿(𝑠,𝑦)≤𝑀(𝑠,𝑦)(𝑥−𝑦) for 𝑥≥𝑦≥0, where 𝑀∈𝐶(𝕋₀×ℝ₊,ℝ₊). If for 𝑡∈[𝑇,∞)⋂𝕋, 𝑢(𝑡) satisfies the following inequality: 𝑢^𝑝(𝑡)+≤𝐶_∞𝑡𝑓(𝑠)𝑢^𝑝(𝑠)((𝑠++𝐿𝑠,𝑢))_∞𝑠ℎ(𝜉)𝑢^𝑟(𝜉)+Δ𝜉Δ𝑠_∞𝑇𝑓(𝑠)𝑢^𝑝(𝑠)((𝑠++𝐿𝑠,𝑢))_∞𝑠ℎ(𝜉)𝑢^𝑟(𝜉)Δ𝜉Δ𝑠, (51) with the initial condition (11), and furthermore, 𝑒₋𝐻₂(𝑇,∞)𝑒_−𝑓(𝑇,∞)<2, then 𝑢(𝑡)≤𝐶+𝐻₁(𝑇)𝑒₋𝐻₂(𝑇,∞)𝑒_−𝑓(𝑇,∞)2−𝑒₋𝐻₂(𝑇,∞)𝑒_−𝑓()+𝐻𝑇,∞¹(𝑡)×𝑒₋𝐻₂(𝑡,∞)𝑒_−𝑓(𝑡,∞)^1/𝑝,𝑡∈[𝑇,∞)𝕋, (52) provided that −𝐻₂∈ℜ₊, where 𝐻₁(𝑡)=^∞_𝑡𝐿𝑠,𝑝−1𝑝𝐾^1/𝑝𝑒_−𝑓(𝑠,∞)^1/𝑝+Δ𝑠_{∞𝑡∞𝑠}ℎ(𝜉)𝑝−𝑟𝑝𝐾^𝑟/𝑝𝑒_−𝑓(𝜉,∞)^𝑟/𝑝𝐻Δ𝜉Δ𝑠,∀𝐾>0,₂(𝑡)=𝑀𝑡,𝑝−1𝑝𝐾^1/𝑝𝑒_−𝑓(𝑡,∞)^1/𝑝×1𝑝𝐾^{(1−𝑝)/𝑝}𝑒_−𝑓(𝑡,∞)^1/𝑝+^∞_𝑡ℎ(𝜉)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑒_−𝑓(𝜉,∞)^𝑟/𝑝Δ𝜉,∀𝐾>0. (53)(54)

Proof. Let the right side of (51) be 𝑣(𝑡). Then 𝑢(𝑡)≤𝑣^1/𝑝(𝑡),𝑡∈[𝑇,∞)𝕋. (55) Furthermore, considering 𝑣(∞)=𝐶+∫^∞_𝑇[𝑓(𝑠)𝑢^𝑝(𝑠)+𝐿(𝑠,𝑢(𝑠))+∫^∞_𝑠ℎ(𝜉)𝑢^𝑟(𝜉)Δ𝜉]Δ𝑠, we have 𝑣(𝑡)=𝑣(∞)+^∞_𝑡𝑓(𝑠)𝑢^𝑝(𝑠)((𝑠++𝐿𝑠,𝑢))_∞𝑠ℎ(𝜉)𝑢^𝑟(𝜉)(∞)+Δ𝜉Δ𝑠≤𝑣^∞_𝑡𝑓(𝑠)𝑣(𝑠)+𝐿𝑠,𝑣^1/𝑝(𝑠)+^∞_𝑠ℎ(𝜉)𝑣^𝑟/𝑝(𝜉)Δ𝜉Δ𝑠. (56) Let 𝑐(𝑡)=^∞_𝑡𝐿𝑠,𝑣^1/𝑝(𝑠)+^∞_𝑠ℎ(𝜉)𝑣^𝑟/𝑝(𝜉)Δ𝜉Δ𝑠. (57) Then 𝑣(𝑡)≤𝑣(∞)+𝑐(𝑡)+^∞_𝑡𝑓(𝑠)𝑣(𝑠)[)Δ𝑠,𝑡∈𝑇,∞𝕋. (58) Considering 𝑐(𝑡) is nonincreasing on 𝕋₀, by Lemma 12 we obtain 𝑣(𝑡)≤(𝑣(∞)+𝑐(𝑡))𝑒_−𝑓(),[)𝑡,∞𝑡∈𝑇,∞𝕋. (59) Combining (57) and (59), it follows that 𝑐(𝑡)≤^∞_𝑡𝐿𝑠,(𝑣(∞)+𝑐(𝑠))𝑒_−𝑓(𝑠,∞)^1/𝑝+^∞_𝑠ℎ(𝜉)(𝑣(∞)+𝑐(𝜉))𝑒_−𝑓(𝜉,∞)^𝑟/𝑝[)Δ𝜉Δ𝑠,𝑡∈𝑇,∞𝕋. (60) On the other hand, from Lemma 13 one can see that the following inequalities hold: (𝑣(∞)+𝑐(𝑡))^1/𝑝≤1𝑝𝐾^{(1−𝑝)/𝑝}(𝑣(∞)+𝑐(𝑡))+𝑝−1𝑝𝐾^1/𝑝,(𝑣(∞)+𝑐(𝑡))^𝑟/𝑝≤𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}(𝑣(∞)+𝑐(𝑡))+𝑝−𝑟𝑝𝐾^𝑟/𝑝,∀𝐾>0. (61) So combining (60) and (61), we have 𝑣(∞)(𝑡)(∞)++𝑐≤𝑣_∞𝑡𝐿1𝑠,𝑝𝐾^{(1−𝑝)/𝑝}(𝑣(∞)(𝑠++𝑐))𝑝−1𝑝𝐾^1/𝑝𝑒_−𝑓(𝑠,∞)^1/𝑝+Δ𝑠_{∞𝑡∞𝑠}ℎ(𝜉𝑟)𝑝𝐾^{(𝑟−𝑝)/𝑝}(𝑣(∞)(𝜉++𝑐))𝑝−𝑟𝑝𝐾^𝑟/𝑝×𝑒_−𝑓(𝜉,∞)^𝑟/𝑝(∞)+Δ𝜉Δ𝑠=𝑣^∞_𝑡𝐿1𝑠,𝑝𝐾^{(1−𝑝)/𝑝}(𝑣(∞)(𝑠++𝑐))𝑝−1𝑝𝐾^1/𝑝×𝑒_−𝑓(𝑠,∞)^1/𝑝−𝐿𝑠,𝑝−1𝑝𝐾^1/𝑝𝑒_−𝑓(𝑠,∞)^1/𝑝+𝐿𝑠,𝑝−1𝑝𝐾^1/𝑝𝑒_−𝑓(𝑠,∞)^1/𝑝+Δ𝑠_{∞𝑡∞𝑠}ℎ(𝜉𝑟)𝑝𝐾^{(𝑟−𝑝)/𝑝}(𝑣(∞)(𝜉++𝑐))𝑝−𝑟𝑝𝐾^𝑟/𝑝×𝑒_−𝑓(𝜉,∞)^𝑟/𝑝(∞)+Δ𝜉Δ𝑠≤𝑣^∞_𝑡𝑀𝑠,𝑝−1𝑝𝐾^1/𝑝𝑒_−𝑓(𝑠,∞)^1/𝑝×1𝑝𝐾^{(1−𝑝)/𝑝}(𝑣(∞)(𝑠×𝑒+𝑐))_−𝑓(𝑠,∞)^1/𝑝+𝐿𝑠,𝑝−1𝑝𝐾^1/𝑝×𝑒_−𝑓(𝑠,∞)^1/𝑝+Δ𝑠_{∞𝑡∞𝑠}ℎ(𝜉)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑒_−𝑓(𝜉,∞)^𝑟/𝑝×(𝑣(∞)(𝑠Δ𝜉+𝑐))Δ𝑠+^∞_𝑡^∞_𝑠ℎ(𝜉)𝑝−𝑟𝑝𝐾^𝑟/𝑝𝑒_−𝑓(𝜉,∞)^𝑟/𝑝(∞)+𝐻Δ𝜉Δ𝑠=𝑣₁(𝑡)+^∞_𝑡𝐻₂(𝑠)(𝑣(∞)+𝑐(𝑠))Δ𝑠, (62)
where 𝐻₁(𝑡),𝐻₂(𝑡) are defined in (53) and (54), respectively.
Considering 𝐻₁(𝑡) is nonincreasing on 𝕋₀, then by Lemma 12, we have 𝑣(∞)+𝑐(𝑡)≤𝑣(∞)+𝐻₁(𝑡)𝑒₋𝐻₂(),[)𝑡,∞𝑡∈𝑇,∞𝕋. (63) Combining (59) and (63), we obtain 𝑣(𝑡)≤𝑣(∞)+𝐻₁(𝑡)𝑒₋𝐻₂(𝑡,∞)𝑒_−𝑓(),[)𝑡,∞𝑡∈𝑇,∞𝕋. (64) Setting 𝑡=𝑇 in (64), considering 2𝑣(∞)−𝐶=𝑣(𝑇), we obtain (∞)(𝑇)≤𝑣(∞)+𝐻2𝑣−𝐶=𝑣₁(𝑇)𝑒₋𝐻₂(𝑇,∞)𝑒_−𝑓(𝑇,∞), (65) which is followed by 𝑣(∞)≤𝐶+𝐻₁(𝑇)𝑒₋𝐻₂(𝑇,∞)𝑒_−𝑓(𝑇,∞)2−𝑒₋𝐻₂(𝑇,∞)𝑒_−𝑓().𝑇,∞ (66) Then combining (55), (64), and (66), we can obtain the desired inequality (52).

3. Applications

In this section, we apply the results established above to analysis of boundedness of solutions for certain dynamic equations.

Example 17. Consider the following dynamic equation: 𝑢^𝑝(𝑡)^Δ=𝐹𝑡,𝑢(𝑡),𝑢(𝑡),^∞_𝑡𝑊(𝜉,𝑢(𝜉))Δ𝜉,𝑡∈𝕋₀, (67) where 𝑢∈𝐶_rd(𝕋₀,ℝ), 𝐶 is a constant with 𝐶=𝑢^𝑝(∞), and 𝑝>0 is a constant.

Theorem 18. Suppose that 𝑢(𝑡) is a solution of (66)-(67) and assume that |𝐹(𝑡,𝑥,𝑦,𝑧)|≤𝑓(𝑡)|𝑥|^𝑝+𝑔(𝑡)|𝑦|^𝑞+|𝑧| and |𝑊(𝑡,𝑥)|≤ℎ(𝑡)|𝑥|^𝑟, where 𝑓,𝑔,ℎ,𝑞, and 𝑟 are defined the same as in Theorem 14; then the following inequality holds: |𝑢(𝑡)|≤|𝐶|+𝐻₂(𝑡)𝑒_−𝐻3(𝑡,∞)𝐻₁(𝑡)^1/𝑝,𝑡∈𝕋₀, (68) where 𝐻₁(𝑡)=1+^∞_𝑡𝑒_−𝑓((𝑠𝑓(𝑠)𝐻𝑡,𝜎))Δ𝑠,₂(𝑡)=^∞_𝑡𝑔(𝑠𝑞)𝑝𝐾^{(𝑞−𝑝)/𝑝}|𝐶|+𝑝−𝑞𝑝𝐾^𝑞/𝑝×𝐻₁(𝑠)^𝑞/𝑝+^∞_𝑠ℎ(𝜉𝑟)𝑝𝐾^{(𝑟−𝑝)/𝑝}|𝐶|+𝑝−𝑟𝑝𝐾^𝑟/𝑝×𝐻₁(𝜉)^𝑟/𝑝𝐻Δ𝜉Δ𝑠,∀𝐾>0,₃(𝑡)(𝑡)𝑞=𝑔𝑝𝐾^{(𝑞−𝑝)/𝑝}𝐻₁(𝑡)^𝑞/𝑝+^∞_𝑡ℎ(𝜉)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}𝐻₁(𝜉)^𝑟/𝑝Δ𝜉,∀𝐾>0. (69)

Proof. The equivalent integral equation of (56) can be denoted by 𝑢^𝑝(𝑡)=𝐶+^∞_𝑡𝐹𝑠,𝑢(𝑠),𝑢(𝑠),^∞_𝑠𝑊(𝜉,𝑢(𝜉))Δ𝜉Δ𝑠. (70) Then we have 𝑢^𝑝(𝑡≤|𝐶|+)_∞𝑡𝐹𝑠,𝑢(𝑠),𝑢(𝑠),^∞_𝑠𝑊((𝜉≤|𝐶|+𝜉,𝑢))Δ𝜉Δ𝑠_∞𝑡𝑓(𝑠)|𝑢(𝑠)|^𝑝+𝑔(𝑠)|𝑢(𝑠)|^𝑞+^∞_𝑠𝑊((𝜉≤|𝐶|+𝜉,𝑢))Δ𝜉Δ𝑠_∞𝑡𝑓(𝑠)|𝑢(𝑠)|^𝑝+𝑔(𝑠)|𝑢(𝑠)|^𝑞+^∞_𝑠ℎ(𝜉)𝑢(𝜉)^𝑟Δ𝜉Δ𝑠. (71) A suitable application of Theorem 14 (i.e., |𝐶| takes the place of a(𝑡) in Theorem 14, and 𝑏(𝑡)≡1 in Theorem 14) yields (68).

Remark 19. Under the conditions of Theorem 18, considering 𝐻₁(𝑡)=1+∫^∞_𝑡𝑒_−𝑓(𝑡,𝜎(𝑠))𝑓(𝑠)Δ𝑠=𝑒_−𝑓(𝑡,∞), furthermore we have the following estimate: |𝑢(𝑡)|≤|𝐶|+𝐻₂(𝑡)𝑒_−𝐻3(𝑡,∞)𝑒_−𝑓(𝑡,∞)^1/𝑝,𝑡∈𝕋₀. (72)

Example 20. Consider the following dynamic equation: 𝑢^𝑝(𝑡)=𝐶+^∞_𝑡𝐹𝑠,𝑢(𝑠),𝑢(𝑠),^∞_𝑠𝑊((𝜉+𝜉,𝑢))Δ𝜉Δ𝑠_∞𝑇𝐹𝑠,𝑢(𝑠),𝑢(𝑠),^∞_𝑠𝑊((𝜉[)𝜉,𝑢))Δ𝜉Δ𝑠,𝑡∈𝑇,∞𝕋, (73) where 𝑢∈𝐶_rd(𝕋₀,ℝ), 𝑇∈𝕋₀, 𝐶 is a constant with 𝐶=𝑢^𝑝(∞), and 𝑝>0 is a constant.

Theorem 21. Suppose that 𝑢(𝑡) is a solution of (73) and assume that |𝐹(𝑡,𝑥,𝑦,𝑧)|≤𝑓(𝑡)|𝑥|^𝑝+𝐿(𝑡,|𝑦|)+|𝑧| and |𝑊(𝑡,𝑥)|≤ℎ(𝑡)|𝑥|^𝑟, where 𝑓,ℎ,𝑟, and 𝐿 are defined the same as in Theorem 16. Then the following inequality holds: 𝑢(𝑡)≤|𝐶|+𝐻₁(𝑇)𝑒₋𝐻₂(𝑇,∞)𝑒_−𝑓(𝑇,∞)2−𝑒₋𝐻₂(𝑇,∞)𝑒_−𝑓()+𝐻𝑇,∞¹(𝑡)×𝑒₋𝐻₂(𝑡,∞)𝑒_−𝑓(𝑡,∞),𝑡∈[𝑇,∞]𝕋, (74) provided that 𝑒₋𝐻₂(𝑇,∞)𝑒_−𝑓(𝑇,∞)<2 and −𝑓,−𝐻₂∈ℜ⁺, where 𝐻₁(𝑡),𝐻₂(𝑡) are defined as in Theorem 16.

Proof. From (73), we have 𝑢^𝑝(𝑡)≤|𝐶|+^∞_𝑡𝐹𝑠,𝑢(𝑠),𝑢(𝑠),^𝑠_∞𝑊((𝜉+𝜉,𝑢))Δ𝜉Δ𝑠_∞𝑇𝐹𝑠,𝑢(𝑠),𝑢(𝑠),^𝑠_∞𝑊((𝜉≤|𝐶|+𝜉,𝑢))Δ𝜉Δ𝑠_∞𝑡𝑓(𝑠)|𝑢(𝑠)|^𝑝(|𝑢(𝑠++𝐿𝑠,)|)_𝑠∞𝑊((𝜉+𝜉,𝑢))Δ𝜉Δ𝑠_∞𝑇𝑓(𝑠)|𝑢(𝑠)|^𝑝(|𝑢(𝑠++𝐿𝑠,)|)_𝑠∞𝑊((𝜉≤|𝐶|+𝜉,𝑢))Δ𝜉Δ𝑠_∞𝑡𝑓(𝑠)|𝑢(𝑠)|^𝑝(|𝑢(𝑠++𝐿𝑠,)|)_𝑠∞ℎ(𝜉)𝑢(𝜉)^𝑟Δ𝑠+^∞_𝑇𝑓(𝑠)|𝑢(𝑠)|^𝑝(|𝑢(𝑠++𝐿𝑠,)|)_𝑠∞ℎ(𝜉)𝑢(𝜉)^𝑟Δ𝑠. (75) Then under the condition 𝑒₋𝐻₂(𝑇,∞)𝑒_−𝑓(𝑇,∞)<2, a suitable application of Theorem 16 yields (74).

Acknowledgments

This work was partially supported by Natural Science Foundation of Shandong Province (China) (Grant no. ZR2013AQ009) and Doctoral initializing Foundation of Shandong University of Technology (China) (Grant no. 4041-413030).