Bounds of Double Integral Dynamic Inequalities in Two Independent Variables on Time Scales

Abstract

Our aim in this paper is to establish some explicit bounds of the unknown function in a certain class of nonlinear dynamic inequalities in two independent variables on time scales which are unbounded above. These on the one hand generalize and on the other hand furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of partial dynamic equations on time scales. Some examples are considered to demonstrate the applications of the results.

1. Introduction

During the past decade, a number of dynamic inequalities have been established by some authors which are motivated by some applications, for example, when studying the behavior of solutions of certain classes of dynamic equations, the bounds provided by earlier inequalities are inadequate in applications and some new and specific type of dynamic inequalities on time scales are required. The general idea is to prove a result for a dynamic inequality where the domain of the unknown function is a so-called time scale 𝕋, which may be an arbitrary closed subset of the real numbers ℝ. In [1, Theorem 6.1], it is proved that if 𝑢, 𝑓, and 𝑝∈𝐶_rd and 𝑝∈ℛ⁺, then𝑢^Δ(𝑡)≤𝑓(𝑡)+𝑝(𝑡)𝑢(𝑡),∀𝑡∈𝑡₀,∞_𝕋,(1.1) implies𝑢(𝑡)≤𝑢𝑡₀𝑒_𝑝𝑡,𝑡₀+^𝑡𝑡₀𝑒_𝑝(𝑡,𝜎(𝑠))𝑓(𝑠)Δ𝑠,∀𝑡∈𝑡₀,∞_𝕋,(1.2) where ℛ⁺∶={𝑓∈ℛ∶1+𝜇(𝑡)𝑓(𝑡)>0,𝑡∈𝕋} and ℛ is the class of rd-continuous and regressive functions. A function 𝑓∶𝕋→ℝ is said to be right-dense continuous (rd-continuous) provided 𝑓 is continuous at right-dense points and at left-dense points in 𝕋, left-hand limits exist and are finite. The set of all such rd-continuous functions is denoted by 𝐶_rd(𝕋). The graininess function 𝜇 for a time scale 𝕋 is defined by 𝜇(𝑡)∶=𝜎(𝑡)−𝑡, and, for any function 𝑓∶𝕋→ℝ, the notation 𝑓^𝜎(𝑡) denotes 𝑓(𝜎(𝑡)), where 𝜎(𝑡) is the forward jump operator defined by 𝜎(𝑡)∶=inf{𝑠∈𝕋∶𝑠>𝑡}. We say that a function 𝑓∶𝕋→ℝ is regressive provided 1+𝜇(𝑡)𝑓(𝑡)≠0,𝑡∈𝕋. The set of all regressive functions on a time scale 𝕋 forms an Abelian group under the addition ⊕ defined by 𝑝⊕𝑞∶=𝑝+𝑞+𝜇𝑝𝑞. Throughout this paper, we will assume that sup𝕋=∞ and define the time scale interval [𝑡₀,∞)_𝕋 by [𝑡₀,∞)_𝕋∶=[𝑡₀,∞)∩𝕋. The exponential function 𝑒_𝑝(𝑡,𝑠) on time scales is defined by𝑒_𝑝(𝑡,𝑠)=exp^𝑡𝑠𝜉_𝜇(𝜏)(𝑝(𝜏))Δ𝜏,for𝑡,𝑠∈𝕋,(1.3) where 𝜉_ℎ(𝑧) is the cylinder transformation, which is given by𝜉_ℎ(𝑧)=𝑧lo,g(1ℎ+ℎ𝑧),ℎ=ℎ≠0.0,(1.4) Alternatively, for 𝑝∈ℛ, one can define the exponential function 𝑒_𝑝(⋅,𝑡₀) to be the unique solution of the IVP 𝑥^Δ=𝑝(𝑡)𝑥, with 𝑥(𝑡₀)=1. If 𝑝∈ℛ, then 𝑒_𝑝(𝑡,𝑠) is real-valued and nonzero on 𝕋. If 𝑝∈ℛ⁺, then 𝑒_𝑝(𝑡,𝑡₀) is always positive, 𝑒_𝑝(𝑡,𝑡)=1, and 𝑒₀(𝑡,𝑠)=1. Note that𝑒_𝑝𝑒𝑡,𝑡₀=exp^𝑡𝑡₀𝑝(𝑠)𝑑𝑠,if𝕋=ℝ,_𝑝𝑒𝑡,𝑡₀=^𝑡−1_𝑠=𝑡0(1+𝑝(𝑠)),if𝕋=ℕ,_𝑝𝑡,𝑡₀=^𝑡−1_𝑠=𝑡0(1+(𝑞−1)𝑠𝑝(𝑠)),if𝕋=𝑞^ℕ0.(1.5) The book on the subject of time scales by Bohner and Peterson [1] summarizes and organizes much of time scale calculus. The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus (see [2]), that is, when 𝕋=ℝ,𝕋=ℕ, and 𝕋=𝑞^ℕ0={𝑞^𝑡∶𝑡∈ℕ₀}, where 𝑞>1.

In this paper, we will refer to the (delta) integral which we can define as follows: If 𝐺^Δ(𝑡)=𝑔(𝑡), then the Cauchy (delta) integral of 𝑔 is defined by ∫^𝑡𝑎𝑔(𝑠)Δ𝑠∶=𝐺(𝑡)−𝐺(𝑎). It can be shown (see [1]) that if 𝑔∈𝐶_rd(𝕋), then the Cauchy integral 𝐺(𝑡)∶=∫^𝑡𝑡₀𝑔(𝑠)Δ𝑠 exists, 𝑡₀∈𝕋, and satisfies 𝐺^Δ(𝑡)=𝑔(𝑡), 𝑡∈𝕋. There are applications of dynamic equations on time scales to quantum mechanics, electrical engineering, neural networks, heat transfer, and combinatorics. A recent cover story article in New Scientist [3] discusses several possible applications. Also, in [1, Theorem 6.4], it is proved that if 𝑢,𝑎, and 𝑝∈𝐶_rd and 𝑝∈ℛ⁺, then𝑢(𝑡)≤𝑎(𝑡)+^𝑡𝑡₀𝑝(𝑠)𝑢(𝑠)Δ𝑠,∀𝑡∈𝑡₀,∞_𝕋,(1.6) implies that𝑢(𝑡)≤𝑎(𝑡)+^𝑡𝑡₀𝑒_𝑝(𝑡,𝜎(𝑠))𝑎(𝑠)𝑝(𝑠)Δ𝑠,∀𝑡∈𝑡₀,∞_𝕋.(1.7) Since (1.7) provides an explicit bound to the unknown function 𝑢(𝑡) and a tool to the study of many qualitative as well as quantitative properties of solutions of dynamic equations, it has become one of the very few classic and most influential results in the theory and applications of dynamic inequalities. Because of its fundamental importance, over the years, many generalizations and analogous results of (1.7) have been established. Since the discovery of the inequalities (1.1)–(1.7), much work has been done, and many papers which deal with various generalizations and extensions have appeared in the literature, we refer the reader to [4–9] and the references cited therein. On the other hand, a few authors have focused on the theory of partial dynamic equations on time scales [10–15]. However, to the best of author’s knowledge, only [16–19] have studied integral inequalities useful in the theory of partial dynamic equations on time scales. Before, we give a brief summary of some of the results of dynamic inequalities in two independent variables, we present some basic definitions about calculus in two variables on time scales (for more details, we refer to [12]).

Let 𝕋₁ and 𝕋₂ be two time scales with at least two points, and consider the time scale intervals Ω₁=[𝑡₀,∞)∩𝕋₁ and Ω₂=[𝑠₀,∞)∩𝕋₂ for 𝑡₀∈𝕋₁ and 𝑠₀∈𝕋₂. Let 𝜎₁,𝜌₁,Δ₁ and 𝜎₂,𝜌₂,Δ₂ denote the forward jump operators, backward jump operators, and the delta differentiation operator, respectively, on 𝕋₁ and 𝕋₂. We say that a real valued function 𝑓 on 𝕋₁×𝕋₂ at (𝑡,𝑠)∈Ω≡Ω₁×Ω₂ has a Δ₁ partial derivative 𝑓^Δ1(𝑡,𝑠) with respect to 𝑡 if for each 𝜖>0 there exists a neighborhood 𝑈_𝑡 of 𝑡 such that||𝑓𝜎₁(𝑡),𝑠−𝑓(𝜂,𝑠)−𝑓^Δ1(𝑡,𝑠)𝜎₁(𝑡)−𝜂||≤𝜖||𝜎(𝑡)−𝜂||,∀𝜂∈𝑈_𝑡.(1.8) In this case, we say 𝑓^Δ1(𝑡,𝑠) is the (partial delta) derivative of 𝑓(𝑡,𝑠) at 𝑡. We say that a real valued function 𝑓 on 𝕋₁×𝕋₂ at (𝑡,𝑠)∈Ω₁×Ω₂ has a Δ₂ partial derivative 𝑓^Δ2(𝑡,𝑠) with respect to 𝑠 if for each 𝜖>0 there exists a neighborhood 𝑈_𝑠 of 𝑠 such that||𝑓𝑡,𝜎₂(𝑠)−𝑓(𝑡,𝜉)−𝑓^Δ2(𝑡,𝑠)𝜎₂(𝑡)−𝜉||≤𝜖||𝜎(𝑡)−𝜉||,∀𝜉∈𝑈_𝑠.(1.9) In this case, we say 𝑓^Δ2(𝑡,𝑠) is the (partial delta) derivative of 𝑓(𝑡,𝑠) at 𝑠. The function 𝑓 is called rd-continuous in 𝑡 if for every 𝛼₂∈𝕋₂ the function 𝑓(𝑡,𝛼₂) is rd-continuous on 𝕋₁. The function 𝑓 is called rd-continuous in 𝑠 if for every 𝛼₁∈𝕋₁ the function 𝑓(𝛼₁,𝑠) is rd-continuous on 𝕋₂. Now, we are ready to present some results for dynamic inequalities in two independent variables on times scales which are related to the main results in our paper. In [18], the authors proved that if 𝑎,𝑓, and 𝑢 are positive rd-continuous functions and 𝑎 is nonnegative and nondecreasing in each of its variables, then𝑢(𝑥,𝑦)≤𝑎(𝑥,𝑦)+^𝑥𝑥₀^𝑦𝑦₀𝑓(𝑠,𝑡)𝑢(𝑠,𝑡)Δ𝑡Δ𝑠,(1.10) for all (𝑥,𝑦)∈[𝑥₀,∞)_𝕋×[𝑦₀,∞)_𝕋, implies𝑢(𝑥,𝑦)≤𝑎(𝑥,𝑦)𝑒_𝐹𝑥,𝑥₀,where𝐹=^𝑦𝑦₀𝑓(𝑥,𝑡)Δ𝑡.(1.11) In [19], the author proved that if 𝑎,𝑏,𝑔,ℎ, and 𝑢 are positive continuous real functions defined on 𝕋×𝕋 and 𝛾>1 is a real constant, then𝑢^𝛾(𝑥,𝑦)≤𝑎(𝑥,𝑦)+𝑏(𝑥,𝑦)^𝑥𝑥₀^𝑦𝑦₀𝑔(𝑠,𝑡)𝑢^𝛾(𝑠,𝑡)+ℎ(𝑠,𝑡)𝑢(𝑠,𝑡)Δ𝑡Δ𝑠,(1.12) implies𝑢(𝑥,𝑦)≤𝑎(𝑥,𝑦)+𝑏(𝑥,𝑦)𝑚(𝑥,𝑦)𝑒_𝐺𝑡,𝑡₀^1/𝛾,(1.13) for all (𝑥,𝑦)∈[𝑥₀,∞)_𝕋×[𝑦₀,∞)_𝕋, where𝑚(𝑥,𝑦)=^𝑥𝑥₀𝐺(𝑠^𝑦𝑦₀,𝑡𝑎)(=𝑠,𝑡)_𝑦𝑦0𝑔(𝑠,𝑡)+𝛾−𝛾1+𝑎(𝛾𝑠,𝑡)ℎ(𝑠,𝑡)Δ𝑡Δ𝑠,𝑔(𝑥,𝑡)+ℎ(𝑥𝛾,𝑡)𝑏(𝑥,𝑡)Δ𝑡.(1.14) In this paper, we are concerned with bounds of the double integral nonlinear dynamic inequality in two independent variables𝑢^𝛾(𝑡,𝑠)≤𝑎(𝑡,𝑠)+𝑏(𝑡,𝑠)^𝑡𝑡₀^𝑠𝑠₀𝑓(𝜏,𝜂)𝑢^𝛿(𝜏,𝜂)+𝑔(𝜏,𝜂)𝑢^𝛼(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏,(1.15) for all (𝑡,𝑠)∈[𝑡₀,∞)_𝕋×[𝑠₀,∞)_𝕋. For (1.15), we will assume the following hypotheses:(𝐻)𝑢,𝑎,𝑏,𝑓,and𝑔arerd-continuouspositivefunctionsonΩ¹𝛼,𝛿,𝜆,and𝛾arepositiveconstants.×Ω₂,(1.16)

The main aim in this paper is to establish some explicit bounds of the unknown function 𝑢(𝑡,𝑠) of the inequality (1.15). Our results not only complement the results in [18, 19] but also improve the results in [19], in the sense that the results can be applied in the cases when 𝛾≤1. The main results will be proved by employing the Bernoulli inequality [20, Bernoulli’s inequality]:(1+𝑥)^𝛾≤1+𝛾𝑥,for0<𝛾≤1,𝑥>−1,(1.17) the Young inequality [20]:𝑎𝑏≤𝑎^𝑝𝑝+𝑏𝑞^𝑞,where𝑎,𝑏≥0,𝑝>1,1𝑝+1𝑞=1,(1.18) and the algebraic inequalities [20]:(𝑎(𝑎++𝑏)𝑏^𝜆)_𝜆≤2^𝜆−1𝑎^𝜆+𝑏^𝜆,for𝑎,𝑏≥0,𝜆≥1,(1.19)≤𝑎^𝜆+𝑏^𝜆,for𝑎,𝑏≥0,0≤𝜆≤1.(1.20) Some examples are considered to illustrate the main results.

2. Main Results

Before, we stated and proved the main results and we proved some Lemmas which play important roles in the proofs of the main results. We will assume that the equations or the inequalities possess such nontrivial solutions.

Lemma 2.1. Let 𝕋 be an unbounded time scale with (𝑡₀,𝑠₀) and (𝑡,𝑠)∈𝕋×𝕋. Let 𝑔_𝑖∶ℝ→ℝ for 𝑖=1,2,…,𝑛 be functions with 𝑔_𝑖(𝑥₁(𝑡,𝑠))≤𝑔_𝑖(𝑥₂(𝑡,𝑠)) for 𝑖=1,2,…,𝑛, where 𝑥_𝑖(𝑡,𝑠)∶𝕋×𝕋→ℝ for 𝑖=1,2, whenever 𝑥₁≤𝑥₂. Let 𝑣,𝑤∶𝕋×𝕋→ℝ be differentiable with 𝑣^Δ𝑡Δ𝑠(𝑡,𝑠)≤^𝑛_𝑖=1𝑎_𝑖(𝑡,𝑠)𝑔_𝑖(𝑣(𝑡,𝑠)),𝑤^Δ𝑡Δ𝑠(𝑡,𝑠)≥^𝑛_𝑖=1𝑎_𝑖(𝑡,𝑠)𝑔_𝑖(𝑤(𝑡,𝑠)),(2.1) for all (𝑡,𝑠)∈𝕋×𝕋. Then, 𝑣(𝑡₀,𝑠₀)<𝑤(𝑡₀,𝑠₀) implies 𝑣(𝑡,𝑠)≤𝑤(𝑡,𝑠) for all (𝑡,𝑠)∈[𝑡₀,∞)_𝕋×[𝑠₀,∞)_𝕋.

Proof. The proof is by induction and similar to the proof of Theorem 6.9 in [1] and hence is omitted.

Lemma 2.2. Let 𝕋 be an unbounded time scale with (𝑡₀,𝑠₀) and (𝑡,𝑠)∈𝕋×𝕋. Suppose that 𝑔_𝑖∶ℝ→ℝ is nondecreasing for 𝑖=1,2,…,𝑛 and 𝑦∶𝕋×𝕋→ℝ⁺ is such that 𝑔_𝑖(𝑦) is rd-continuous. Let 𝑝_𝑖 be rd-continuous and positive for 𝑖=1,2,…,𝑛 and 𝑓∶𝕋×𝕋→ℝ⁺ differentiable. Then, 𝑦(𝑡,𝑠)≤𝑓(𝑡,𝑠)+^𝑛_𝑖=1^𝑡𝑡₀^𝑠𝑠₀𝑝_𝑖(𝜂,𝜏)𝑔_𝑖(𝑦(𝜂,𝑠))Δ𝜂Δ𝜏,∀(𝑡,𝑠)∈𝕋×𝕋,(2.2) implies 𝑦(𝑡,𝑠)≤𝑥(𝑡,𝑠) for all (𝑡,𝑠)∈[𝑡₀,∞)_𝕋×[𝑠₀,∞)_𝕋, where 𝑥(𝑡,𝑠) solves the initial value problem 𝑥^Δ𝑡Δ𝑠(𝑡,𝑠)=𝑓^Δ𝑡Δ𝑠(𝑡,𝑠)+^𝑛_𝑖=1𝑥𝑡𝑝_𝑖(𝑡,𝑠)𝑔_𝑖(𝑥(𝑡,𝑠)),₀,𝑠₀>𝑓𝑡₀,𝑠₀>0.(2.3)

Proof. Let 𝑣(𝑡,𝑠)∶=𝑓(𝑡,𝑠)+^𝑛_𝑖=1^𝑡𝑡₀^𝑠𝑡₀𝑝_𝑖(𝜏,𝜂)𝑔_𝑖(𝑦(𝜏,𝜂))Δ𝜂Δ𝜏,(2.4) for all (𝑡,𝑠)∈[𝑡₀,∞)_𝕋×[𝑠₀,∞)_𝕋. Then, 𝑣^Δ𝑡Δ𝑠(𝑡,𝑠)∶=𝑓^Δ𝑡Δ𝑠(𝑡,𝑠)+^𝑛_𝑖=1𝑝_𝑖(𝑡,𝑠)𝑔_𝑖(𝑦(𝑡,𝑠)),(2.5) for all (𝑡,𝑠)∈[𝑡₀,∞)_𝕋×[𝑠₀,∞)_𝕋 and 𝑦(𝑡,𝑠)≤𝑣(𝑡,𝑠) so that 𝑣^Δ𝑡Δ𝑠(𝑡,𝑠)≤𝑓^Δ𝑡Δ𝑠(𝑡,𝑠)+^𝑛_𝑖=1𝑝_𝑖(𝑡,𝑠)𝑔_𝑖(𝑣(𝑡,𝑠)),(2.6) for all (𝑡,𝑠)∈[𝑡₀,∞)_𝕋×[𝑠₀,∞)_𝕋. Since 𝑣(𝑡₀,𝑠₀)=𝑓(𝑡₀,𝑠₀)<𝑥₀=𝑥(𝑡₀,𝑠₀), the comparison Lemma 2.1 yields 𝑣(𝑡,𝑠)≤𝑥(𝑡,𝑠) for all (𝑡,𝑠)≥(𝑡₀,𝑠₀), where 𝑥(𝑡,𝑠) solves the initial value problem (2.3). Hence, since 𝑦(𝑡,𝑠)≤𝑣(𝑡,𝑠), we obtain 𝑦(𝑡,𝑠)≤𝑥(𝑡,𝑠). The proof is complete.

Now, we are ready to state and prove the main results in this paper. First, we consider the case when 𝜆≥1 and 𝛼,𝛿≤𝛾. For simplicity, we introduce the following notations:𝐹(𝑡,𝑠)∶=2^2(𝜆−1)^𝑡𝑡₀^𝑠𝑠₀𝑓^𝜆+2_2(𝜆−1)(𝜏,𝜂)𝑎^𝛿/𝛾(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏^𝑡𝑡₀^𝑠𝑠₀𝑔^𝜆𝐺(𝑡,𝑠)∶=2_2(𝜆−1)(𝜏,𝜂)𝑎^𝛼/𝛾(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏,𝑓^𝜆(𝑡,𝑠)𝛿𝛾𝑎^{(𝛿/𝛾)−1}(𝑡,𝑠)^𝜆+𝑔^𝜆(𝑡,𝑠)𝛼𝛾𝑎^{(𝛼/𝛾)−1}(𝑡,𝑠)^𝜆.(2.7)

Theorem 2.3. Let 𝕋 be an unbounded time scale with (𝑡₀,𝑠₀)∈𝕋×𝕋. Assume that (𝐻) holds, 𝜆,𝛾≥1  and 𝛼,𝛿≤𝛾. Then, 𝑢^𝛾(𝑡,𝑠)≤𝑎(𝑡,𝑠)+𝑏(𝑡,𝑠)^𝑡𝑡₀^𝑠𝑠₀𝑓(𝜏,𝜂)𝑢^𝛿(𝜏,𝜂)+𝑔(𝜏,𝜂)𝑢^𝛼(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏,(2.8) for (𝑡,𝑠)∈Ω,  implies that 𝑢(𝑡,𝑠)≤𝑎^1/𝛾(𝑡,𝑠)+1𝛾𝑎^(1/𝛾)−1(𝑡,𝑠)𝑏(𝑡,𝑠)𝑤(𝑡,𝑠),∀(𝑡,𝑠)∈Ω,(2.9) where 𝑤(𝑡) solves the initial value problem 𝑤^Δ𝑡Δ𝑠(𝑡,𝑠)=𝐹^Δ𝑡Δ𝑠(𝑡,𝑠)+𝑏^𝜆(𝑡,𝑠)𝐺(𝑡,𝑠)𝑤^𝜆(𝑡,𝑠),𝑤𝑡₀,𝑠₀>0.(2.10)

Proof. Define a function 𝑦(𝑡,𝑠) by 𝑦(𝑡,𝑠)∶=^𝑡𝑡₀^𝑠𝑡₀𝑓(𝜏,𝜂)𝑢^𝛿(𝜏,𝜂)+𝑔(𝜏,𝜂)𝑢^𝛼(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏.(2.11) This reduces (2.8) to 𝑢^𝛾(𝑡,𝑠)≤𝑎(𝑡,𝑠)+𝑏(𝑡,𝑠)𝑦(𝑡,𝑠),for(𝑡,𝑠)∈Ω.(2.12) This implies that 𝑢(𝑡,𝑠)≤(𝑎(𝑡,𝑠)+𝑏(𝑡,𝑠)𝑦(𝑡,𝑠))^1/𝛾,for(𝑡,𝑠)∈Ω.(2.13) Applying the inequality (1.17) (noting that 1/𝛾≤1), we see that 𝑢(𝑡,𝑠)≤𝑎^1/𝛾(𝑡,𝑠)+1𝛾𝑎^(1/𝛾)−1(𝑡,𝑠)𝑏(𝑡,𝑠)𝑦(𝑡,𝑠),for(𝑡,𝑠)∈Ω.(2.14) From (2.13), we obtain 𝑢^𝛼(𝑡,𝑠)≤𝑎^𝛼/𝛾(𝑡,𝑠)1+𝑏(𝑡,𝑠)𝑦(𝑡,𝑠)𝑎(𝑡,𝑠)^𝛼/𝛾,for(𝑡,𝑠)∈Ω.(2.15) Applying inequality (1.17) on (2.15) (where 𝛼≤𝛾), we obtain for (𝑡,𝑠)∈Ω that 𝑢^𝛼(𝑡,𝑠)≤𝑎^𝛼/𝛾(𝑡,𝑠)1+𝛼=𝑎_𝛼/𝛾𝛾𝑎𝑏((𝑡𝑡,,𝑠𝑠))𝑦(𝑡,𝑠)(𝑡,𝑠)+𝛼𝛾𝑎^{(𝛼/𝛾)−1}(𝑡,𝑠)𝑏(𝑡,𝑠)𝑦(𝑡,𝑠).(2.16) Also, from (2.13), we obtain 𝑢^𝛿(𝑡,𝑠)≤𝑎^𝛿/𝛾(𝑡,𝑠)1+𝑏(𝑡,𝑠)𝑦(𝑡,𝑠)𝑎(𝑡,𝑠)^𝛿/𝛾,for(𝑡,𝑠)∈Ω.(2.17) Applying inequality (1.17) on (2.17) (where 𝛿≤𝛾), we have for (𝑡,𝑠)∈Ω that 𝑢^𝛿(𝑡,𝑠)≤𝑎^𝛿/𝛾(𝑡,𝑠)1+𝛿=𝑎_𝛿/𝛾𝛾𝑎𝑏((𝑡𝑡,,𝑠𝑠))𝑦(𝑡,𝑠)(𝑡,𝑠)+𝛿𝛾𝑎^{(𝛿/𝛾)−1}(𝑡,𝑠)𝑏(𝑡,𝑠)𝑦(𝑡,𝑠).(2.18) Combining (2.11), (2.16), and (2.18) and applying the inequality (1.19) (noting that 𝜆≥1), we have 𝑦(𝑡,𝑠)=^𝑡𝑡₀^𝑠𝑡₀𝑓(𝜏,𝜂)𝑢^𝛿≤2_𝜆−1(𝜏,𝜂)+𝑔(𝜏,𝜂)𝑢^𝛼(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏^𝑡𝑡₀^𝑠𝑡₀𝑓(𝜏,𝜂)𝑢^𝛿+2_𝜆−1(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏^𝑡𝑡₀𝑔(𝜏,𝜂)𝑢^𝛼≤2_𝜆−1(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏^𝑡𝑡₀^𝑠𝑡₀𝑓^𝜆(𝜏,𝜂)𝑎^𝛿/𝛾(𝜏,𝜂)+𝛿𝛾𝑎^{(𝛿/𝛾)−1}+2_𝜆−1(𝜏,𝜂)𝑏(𝜏,𝜂)𝑦(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏^𝑡𝑡₀^𝑠𝑡₀𝑔^𝜆(𝜏,𝜂)𝑎^𝛼/𝛾(𝜏,𝜂)+𝛼𝛾𝑎^{(𝛼/𝛾)−1}(𝜏,𝜂)𝑏(𝜏,𝜂)𝑦(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏.(2.19) This implies that 𝑦(𝑡,𝑠)≤2^2(𝜆−1)^𝑡𝑡₀^𝑠𝑡₀+2_2(𝜆−1)𝑓^𝜆(𝜏,𝜂)𝑎^(𝛿/𝛾)(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏^𝑡𝑡₀^𝑠𝑡₀𝑓^𝜆(𝜏,𝜂)𝛿𝛾𝑎^{(𝛿/𝛾)−1}+2_2(𝜆−1)(𝜏,𝜂)𝑏(𝜏,𝜂)^𝜆𝑦^𝜆(𝜏,𝜂)Δ𝜂Δ𝜏^𝑡𝑡₀^𝑠𝑡₀+2_2(𝜆−1)𝑔^𝜆(𝜏,𝜂)𝑎^𝛼/𝛾(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏^𝑡𝑡₀^𝑠𝑡₀𝑔^𝜆(𝜏,𝜂)𝛼𝛾𝑎^{(𝛼/𝛾)−1}=𝐹(𝑡,𝑠)+_𝑡𝑡0(𝜏,𝜂)𝑏(𝜏,𝜂)^𝜆𝑦^𝜆(𝜏,𝜂)Δ𝜂Δ𝜏^𝑠𝑡₀𝐺(𝜏,𝜂)𝑦^𝜆(𝜏,𝜂)Δ𝜂Δ𝜏,for(𝑡,𝑠)∈Ω.(2.20) Now an application of Lemma 2.2 (with 𝑛=1 and 𝑔(𝑦)=𝑦^𝜆) gives that 𝑦(𝑡,𝑠)<𝑤(𝑡,𝑠),for(𝑡,𝑠)∈Ω,(2.21) where 𝑤(𝑡,𝑠) solves the initial value problem (2.10). Substituting (2.21) into (2.14), we obtain the desired inequality (2.9). The proof is complete.

Theorem 2.4. Let 𝕋 be an unbounded time scale with (𝑡₀,𝑠₀)∈𝕋×𝕋. Assume that (𝐻) holds, 𝜆,𝛾≥1 and 𝛼,𝛿≤𝛾. Then, (2.8) implies 𝑢(𝑡,𝑠)≤𝑎^1/𝛾(𝑡,𝑠)+𝑏^1/𝛾(𝑡,𝑠)𝑤^1/𝛾(𝑡,𝑠),∀(𝑡,𝑠)∈Ω,(2.22) where 𝑤(𝑡) solves the initial value problem 𝑤^Δ𝑡Δ𝑠(𝑡,𝑠)=𝐹^Δ𝑡Δ𝑠(𝑡,𝑠)+𝐺𝑤₁(𝑡𝑡,𝑠)𝑤^{𝜆(𝛿/𝛾)}(𝑡,𝑠)+𝐺₂(𝑡,𝑠)𝑤^{𝜆(𝛼/𝛾)}(𝑡,𝑠),₀,𝑠₀>0,(2.23) where 𝐹(𝑡) is defined as in (2.7) and 𝐺₁(𝑡,𝑠)∶=2^2(𝜆−1)𝑓^𝜆(𝑡,𝑠)𝑏^{𝜆𝛿/𝛾}(𝑡,𝑠),𝐺₂∶=2^2(𝜆−1)𝑔^𝜆(𝑡,𝑠)𝑏^{𝜆𝛼/𝛾}(𝑡,𝑠).(2.24)

Proof. Define a function 𝑦(𝑡,𝑠) by (2.11) and proceed as in the proof of Theorem 2.3 to obtain 𝑢(𝑡,𝑠)≤(𝑎(𝑡,𝑠)+𝑏(𝑡,𝑠)𝑦(𝑡,𝑠))^1/𝛾,for(𝑡,𝑠)∈Ω.(2.25) Applying the inequality (1.20), we see that 𝑢(𝑡,𝑠)≤𝑎^1/𝛾(𝑡,𝑠)+𝑏^1/𝛾(𝑡,𝑠)𝑦^1/𝛾(𝑡,𝑠),for(𝑡,𝑠)∈Ω.(2.26) From (2.25), we obtain 𝑢^𝛼(𝑡,𝑠)≤(𝑎(𝑡,𝑠)+𝑏(𝑡,𝑠)𝑦(𝑡,𝑠))^𝛼/𝛾,for(𝑡,𝑠)∈Ω.(2.27) Applying inequality (1.20) on (2.27) (where 𝛼≤𝛾), we obtain for (𝑡,𝑠)∈Ω that 𝑢^𝛼(𝑡,𝑠)≤𝑎^𝛼/𝛾(𝑡,𝑠)+𝑏^𝛼/𝛾(𝑡,𝑠)𝑦^𝛼/𝛾(𝑡,𝑠).(2.28) Also, from (2.25), we have by (1.20) that 𝑢^𝛿(𝑡,𝑠)≤𝑎^𝛿/𝛾(𝑡,𝑠)+𝑏^𝛿/𝛾(𝑡,𝑠)𝑦^𝛿/𝛾(𝑡,𝑠),for(𝑡,𝑠)∈Ω.(2.29) Combining (2.11), (2.28), and (2.29) and applying the inequality (1.19) (noting that 𝜆≥1), we have 𝑦(𝑡,𝑠)=^𝑡𝑡₀^𝑠𝑡₀𝑓(𝜏,𝜂)𝑢^𝛿≤2_𝜆−1(𝜏,𝜂)+𝑔(𝜏,𝜂)𝑢^𝛼(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏^𝑡𝑡₀^𝑠𝑡₀𝑓(𝜏,𝜂)𝑢^𝛿+2_𝜆−1(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏^𝑡𝑡₀𝑔(𝜏,𝜂)𝑢^𝛼≤2_𝜆−1(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏^𝑡𝑡₀^𝑠𝑡₀+2_𝜆−1𝑓^𝜆(𝜏,𝜂)𝑎^𝛿/𝛾(𝜏,𝜂)+𝑏^𝛿/𝛾(𝜏,𝜂)𝑦^𝛿/𝛾(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏^𝑡𝑡₀^𝑠𝑡₀𝑔^𝜆(𝜏,𝜂)𝑎^𝛼/𝛾(𝜏,𝜂)+𝑏^𝛼/𝛾(𝜏,𝜂)𝑦^𝛼/𝛾(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏.(2.30) This implies that 𝑦(𝑡,𝑠)≤2^2(𝜆−1)^𝑡𝑡₀^𝑠𝑡₀+2_2(𝜆−1)𝑓^𝜆(𝜏,𝜂)𝑎^𝛿/𝛾(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏^𝑡𝑡₀^𝑠𝑡₀+2_2(𝜆−1)𝑔^𝜆(𝜏,𝜂)𝑎^𝛼/𝛾(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏^𝑡𝑡₀^𝑠𝑡₀+2_2(𝜆−1)𝑓^𝜆(𝜏,𝜂)𝑏^𝛿/𝛾(𝜏,𝜂)^𝜆𝑦^{𝜆(𝛿/𝛾)}(𝜏,𝜂)Δ𝜂Δ𝜏^𝑡𝑡₀^𝑠𝑡₀=𝐹(𝑡,𝑠)+_𝑡𝑡0𝑔^𝜆(𝜏,𝜂)𝑏^𝛼/𝛾(𝜏,𝜂)^𝜆𝑦^{𝜆(𝛼/𝛾)}(𝜏,𝜂)Δ𝜂Δ𝜏^𝑠𝑡₀𝐺₁(𝜏,𝜂)𝑦^{𝜆(𝛿/𝛾)}(𝜏,𝜂)+𝐺₂(𝜏,𝜂)𝑦^{𝜆(𝛼/𝛾)}(𝜏,𝜂)Δ𝜂Δ𝜏,(2.31) for (𝑡,𝑠)∈Ω. Now, an application of Lemma 2.2 (with 𝑛=2,𝑔₁(𝑦)=𝑦^{𝜆(𝛿/𝛾)}, and 𝑔₂(𝑦)=𝑦^{𝜆(𝛼/𝛾)}) gives that 𝑦(𝑡,𝑠)<𝑤(𝑡,𝑠),for(𝑡,𝑠)∈Ω,(2.32) where 𝑤(𝑡,𝑠) solves the initial value problem (2.23). Substituting (2.32) into (2.26), we obtain the desired inequality (2.22). The proof is complete.

As in the proof of Theorem 2.3 by employing the inequality (1.20) instead of the inequality (1.19), we can obtain an explicit bound for 𝑢(𝑡) when 0≤𝜆≤1. This will be presented below in Theorem 2.5 without proof since the proof is similar to the proof of Theorem 2.3. For simplicity, we introduce the following notations:𝐹₁(𝑡,𝑠)∶=^𝑡𝑡₀^𝑠𝑠₀𝑓^𝜆+(𝜏,𝜂)𝑎^{𝜆(𝛿/𝛾)}(𝜏,𝜂)Δ𝜂Δ𝜏_𝑡𝑡0^𝑠𝑠₀𝑔^𝜆𝐺(𝜏,𝜂)𝑎^{𝜆(𝛼/𝛾)}(𝜏,𝜂)Δ𝜂Δ𝜏,₃(𝑡,𝑠)∶=𝑓^𝜆(𝑡,𝑠)𝛿𝛾𝑎^{(𝛿/𝛾)−1}(𝑡,𝑠)^𝜆+𝑔^𝜆(𝑡,𝑠)𝛼𝛾𝑎^{(𝛼/𝛾)−1}(𝑡,𝑠)^𝜆.(2.33)

Theorem 2.5. Let 𝕋 be an unbounded time scale with (𝑡₀,𝑠₀)∈𝕋×𝕋. Assume that (𝐻) holds, 𝛾≥1,0<𝜆≤1,𝛿≤𝛾, and 𝛼≤𝛾. Then, (2.8) implies that 𝑢(𝑡,𝑠)≤𝑎^1/𝛾(𝑡,𝑠)+1𝛾𝑎^(1/𝛾)−1(𝑡,𝑠)𝑏(𝑡,𝑠)𝑧(𝑡,𝑠),for(𝑡,𝑠)∈Ω,(2.34) where 𝑧(𝑡) solves the initial value problem 𝑧^Δ𝑡Δ𝑡(𝑡)=𝐹^Δ𝑡Δ𝑠₁(𝑡)+𝐺₃(𝑡,𝑠)𝑏^𝜆(𝑡,𝑠)𝑧^𝜆(𝑡,𝑠),𝑧𝑡₀,𝑠₀>0.(2.35)

In the following, we apply the Young inequality (1.18) to find a new explicit upper bound for 𝑢(𝑡) of (2.8) when 𝜆≥1 and 0≤𝜆≤1. First, we consider the case when 𝜆≥1 and assume that 𝜆(𝛼/𝛾)<1 and 𝜆(𝛿/𝛾)<1.

Theorem 2.6. Let 𝕋 be an unbounded time scale with (𝑡₀,𝑠₀)∈𝕋×𝕋. Assume that (𝐻) holds, 𝛾,𝜆≥1and 𝛼,𝛿≤𝛾 such that (𝜆𝛼/𝛾)<1 and (𝜆𝛿/𝛾)<1. Then, (2.8) implies that 𝑢(𝑡,𝑠)≤𝑎^1/𝛾(𝑡,𝑠)+𝑏^1/𝛾(𝑡,𝑠)𝐹^31/𝛾(𝑡,𝑠),∀(𝑡,𝑠)∈Ω,(2.36) where 𝐹₃(𝑡,𝑠)∶=𝐹₀(𝑡,𝑠)+𝑒_{𝛽(𝑠−𝑠0)}𝑡,𝑡₀,𝛽=𝜆𝛼𝐹₀𝛾+𝛾𝛿,(𝑡,𝑠)∶=𝐹(𝑡,𝑠)+(𝛾−𝜆𝛿)𝛾^𝑡𝑡₀𝐺₁+(𝛾−𝜆𝛼)(𝜏,𝜂)^{𝛾/(𝛾−𝜆𝛿)}Δ𝜂Δ𝜏𝛾^𝑡𝑡₀𝐺₂(𝜏,𝜂)^{𝛾/(𝛾−𝜆𝛼)}Δ𝜂Δ𝜏,(2.37) and 𝐹,𝐺₁, and 𝐺₂ are defined as in (2.7) and (2.24).

Proof. Define a function 𝑦(𝑡,𝑠) by (2.11) and proceed as in the proof of Theorem 2.4 to obtain 𝑦(𝑡,𝑠)≤𝐹𝑢((𝑡𝑡,,𝑠𝑠))+≤𝑎^1/𝛾_𝑡𝑡0(𝑡,𝑠)+𝑏^1/𝛾(𝑡,𝑠)𝑦^1/𝛾(𝑡,𝑠),for(𝑡,𝑠)∈Ω,(2.38)^𝑠𝑡₀𝐺₁(𝜏,𝜂)𝑦^{𝜆(𝛿/𝛾)}(𝜏,𝜂)+𝐺₂(𝜏,𝜂)𝑦^{𝜆(𝛼/𝛾)}(𝜏,𝜂)Δ𝜂Δ𝜏,(2.39) where 𝐹,𝐺₁, and 𝐺₂ are defined as in (2.7) and (2.24). Applying the Young inequality (1.18) on the term 𝐺₁𝑦^{𝜆(𝛿/𝛾)} with 𝑞=𝛾/𝜆𝛿>1 and 𝑝=𝛾/(𝛾−𝜆𝛿)>1, we see that 𝐺₁𝑦^{𝜆(𝛿/𝛾)}≤(𝛾−𝜆𝛿)𝛾𝐺¹^{𝛾/(𝛾−𝜆𝛿)}+𝜆𝛿𝛾𝑦.(2.40) Again applying the Young inequality (1.18) on the term 𝐺₂𝑦^{𝜆(𝛼/𝛾)} with 𝑞=𝛾/𝜆𝛼>1 and 𝑝=𝛾/(𝛾−𝜆𝛼)>1, we see that 𝐺₂𝑦^{𝜆(𝛼/𝛾)}≤(𝛾−𝜆𝛼)𝛾𝐺²^{𝛾/(𝛾−𝜆𝛼)}+𝜆𝛼𝛾𝑦.(2.41) Substituting (2.40) and (2.41) into (2.39), we have 𝑦(𝑡,𝑠)≤𝐹(𝑡,𝑠)+(𝛾−𝜆𝛿)𝛾^𝑡𝑡₀^𝑠𝑡₀𝐺₁+(𝛾−𝜆𝛼)(𝜏,𝜂)^{𝛾/(𝛾−𝜆𝛿)}Δ𝜂Δ𝜏𝛾^𝑡𝑡₀^𝑠𝑡₀𝐺₂+𝜆𝛼(𝜏,𝜂)^{𝛾/(𝛾−𝜆𝛼)}Δ𝜂Δ𝜏𝛾+𝜆𝛾𝛿^𝑡𝑡₀^𝑠𝑠₀𝑦(𝜏,𝜂)Δ𝜂Δ𝜏,∀(𝑡,𝑠)∈Ω.(2.42) From the definitions of 𝐹₀(𝑡) and 𝛽, we get that 𝑦(𝑡,𝑠)≤𝐹₀(𝑡,𝑠)+𝛽^𝑡𝑡₀^𝑠𝑡₀𝑦(𝜏,𝜂)Δ𝜂Δ𝜏,for(𝑡,𝑠)∈Ω.(2.43) Applying the inequality (1.10) with 𝑓(𝑡,𝑠)=𝛽, we have 𝑦(𝑡,𝑠)<𝐹₀(𝑡,𝑠)+𝑒_{𝛽(𝑠−𝑠0)}𝑡,𝑡₀,∀(𝑡,𝑠)∈Ω.(2.44) Substituting (2.44) into (2.38), we get the desired inequality (2.36). The proof is complete.

Theorem 2.7. Let 𝕋 be an unbounded time scale with (𝑡₀,𝑠₀)∈𝕋×𝕋. Assume that (𝐻) holds, 𝛾≥1,0<𝜆≤1, and 𝛼,𝛿≤𝛾. Then, (2.8) implies that 𝑢(𝑡,𝑠)≤𝑎^1/𝛾(𝑡,𝑠)+1𝛾𝑎^(1/𝛾)−1(𝑡,𝑠)𝑏(𝑡,𝑠)𝐹₄(𝑡,𝑠),for(𝑡,𝑠)∈Ω,(2.45) where 𝐹𝐹₄₂(𝑡,𝑠)∶=𝐹₂(𝑡,𝑠)+𝑒_{𝜆(𝑠−𝑠0)}𝑡,𝑡₀,(𝑡,𝑠)∶=𝐹₁(𝑡,𝑠)+(1−𝜆)^𝑡𝑡₀^𝑠𝑠₀𝐺₃(𝜏,𝜂)^1/(1−𝜆)Δ𝜂Δ𝜏,(2.46) and 𝐹₁ and 𝐺₃ are defined as in (2.33).

Proof. Define a function 𝑦(𝑡,𝑠) by (2.11) and proceed as in the proof of Theorem 2.3 to obtain 𝑢(𝑡,𝑠)≤𝑎^1/𝛾(𝑡,𝑠)+1𝛾𝑎^(1/𝛾)−1𝑦(𝑡,𝑠)≤𝐹₁(𝑡,𝑠)𝑏(𝑡,𝑠)𝑦(𝑡,𝑠),for(𝑡,𝑠)∈Ω,(2.47)(𝑡,𝑠)+^𝑡𝑡₀^𝑠𝑠₀𝐺₃(𝑠)𝑦^𝜆(𝜏,𝜂)Δ𝜂Δ𝜏,for(𝑡,𝑠)∈Ω,(2.48) where 𝐹₁ and 𝐺₃ are defined in (2.33). Applying the Young inequality (1.18) on the term 𝐺₃𝑦^𝜆 with 𝑞=1/𝜆>1 and 𝑝=1/(1−𝜆)>1, we see that 𝐺₃𝑦^𝜆≤(1−𝜆)𝐺₃^1/1−𝜆+𝜆𝑦.(2.49) This and (2.48) imply that 𝑦(𝑡,𝑠)≤𝐹₁(𝑡,𝑠)+(1−𝜆)^𝑡𝑡₀^𝑠𝑠₀𝐺₃+(𝜏,𝜂)^1/(1−𝜆)Δ𝜂Δ𝜏_𝑡𝑡0^𝑠𝑠₀𝜆𝑦(𝜏,𝜂)Δ𝜂Δ𝜏.(2.50) Using the definition of 𝐹₂(𝑡,𝑠), we get that 𝑦(𝑡,𝑠)≤𝐹₂(𝑡,𝑠)+𝜆^𝑡𝑡₀^𝑠𝑠₀𝑦(𝜏,𝜂)Δ𝜂Δ𝜏,for(𝑡,𝑠)∈Ω.(2.51) Applying the inequality (1.10) with 𝑓(𝑡,𝑠)=𝜆, we have 𝑦(𝑡,𝑠)<𝐹₂(𝑡,𝑠)+𝑒_{𝜆(𝑠−𝑠0)}𝑡,𝑡₀,∀(𝑡,𝑠)∈Ω.(2.52) Substituting (2.52) into (2.47), we get the desired inequality (2.45). The proof is complete.

Next in the following, we consider the case when 𝛾≤1 and establish some new explicit bounds of the unknown function 𝑢(𝑡,𝑠) of (2.8).

Theorem 2.8. Let 𝕋 be an unbounded time scale with (𝑡₀,𝑠₀)∈𝕋×𝕋. Assume that (𝐻) holds, 𝜆≤1,𝛾≤1, and 𝛼𝜆,𝛿𝜆≤𝛾. Then, (2.8) implies that 𝑢^𝛾(𝑡,𝑠)≤𝑎(𝑡,𝑠)+𝑏(𝑡,𝑠)𝐻(𝑡,𝑠)+𝑒_{𝛽1(𝑠−𝑠0)}𝑡,𝑡₀,for(𝑡,𝑠)∈Ω,(2.53) where 𝛽₁=(𝜆𝛼/𝛾)+(𝜆𝛿/𝛾) and 𝐻(𝑡,𝑠)=2^{𝜆𝛿((1/𝛾)−1)}^𝑡𝑡₀^𝑠𝑡₀+2_{𝜆𝛼((1/𝛾)−1)}𝑓^𝜆(𝜏,𝜂)𝑎^{𝜆𝛿/𝛾}(𝜏,𝜂)Δ𝜂Δ𝜏^𝑡𝑡₀^𝑠𝑡₀+2_{𝜆𝛿((1/𝛾)−1)}𝑔^𝜆(𝜏,𝜂)𝑎^{𝜆𝛼/𝛾}(𝜏,𝜂)Δ𝜂Δ𝜏(𝛾−𝜆𝛿)𝛾^𝑡𝑡₀^𝑠𝑡₀𝑓^𝜆+2_{𝜆𝛼((1/𝛾)−1)}(𝜏,𝜂)𝑏^{𝜆𝛿/𝛾}(𝜏,𝜂)^{𝛾/(𝛾−𝜆𝛿)}Δ𝜂Δ𝜏(𝛾−𝜆𝛼)𝛾^𝑡𝑡₀^𝑠𝑡₀𝑔^𝜆(𝜏,𝜂)𝑏^{𝜆𝛼/𝛾}(𝜏,𝜂)^{𝛾/(𝛾−𝜆𝛼)}Δ𝜂Δ𝜏.(2.54)

Proof. Define a function 𝑦(𝑡,𝑠) by 𝑦(𝑡,𝑠)∶=^𝑡𝑡₀^𝑠𝑡₀𝑓(𝜏,𝜂)𝑢^𝛿(𝜏,𝜂)+𝑔(𝜏,𝜂)𝑢^𝛼(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏.(2.55) This reduces (2.8) to 𝑢^𝛾(𝑡,𝑠)≤𝑎(𝑡,𝑠)+𝑏(𝑡,𝑠)𝑦(𝑡,𝑠),for(𝑡,𝑠)∈Ω.(2.56) This implies that 𝑢(𝑡,𝑠)≤(𝑎(𝑡,𝑠)+𝑏(𝑡,𝑠)𝑦(𝑡,𝑠))^1/𝛾,for(𝑡,𝑠)∈Ω.(2.57) Applying the inequality (1.19) (noting that 𝛾≤1), we see that 𝑢(𝑡,𝑠)≤2^(1/𝛾)−1𝑎^1/𝛾(𝑡,𝑠)+𝑏^1/𝛾(𝑡,𝑠)𝑦^1/𝛾(𝑡,𝑠),for(𝑡,𝑠)∈Ω.(2.58) From (2.58), we obtain 𝑢^𝛼(𝑡,𝑠)≤2^{𝛼((1/𝛾)−1)}𝑎^1/𝛾(𝑡,𝑠)+𝑏^1/𝛾(𝑡,𝑠)𝑦^1/𝛾(𝑡,𝑠)^𝛼,for(𝑡,𝑠)∈Ω.(2.59) Also, from (2.58), we obtain 𝑢^𝛿(𝑡,𝑠)≤2^{𝛿((1/𝛾)−1)}𝑎^1/𝛾(𝑡,𝑠)+𝑏^1/𝛾(𝑡,𝑠)𝑦^1/𝛾(𝑡,𝑠)^𝛿,for(𝑡,𝑠)∈Ω.(2.60) Combining (2.55), (2.59), and (2.60) and applying the inequality (1.19) (noting that 𝜆≤1), we have 𝑦(𝑡,𝑠)=^𝑡𝑡₀^𝑠𝑡₀𝑓(𝜏,𝜂)𝑢^𝛿≤(𝜏,𝜂)+𝑔(𝜏,𝜂)𝑢^𝛼(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏_𝑡𝑡0^𝑠𝑡₀𝑓(𝜏,𝜂)𝑢^𝛿+(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏_𝑡𝑡0≤2𝑔(𝜏_{𝜆𝛿((1/𝛾)−1)},𝜂)𝑢^𝛼(𝜏,𝜂)^𝜆Δ𝜂Δ𝜏^𝑡𝑡₀^𝑠𝑡₀+2_{𝜆𝛼((1/𝛾)−1)}𝑓^𝜆(𝜏,𝜂)𝑎^1/𝛾(𝑡,𝑠)+𝑏^1/𝛾(𝑡,𝑠)𝑦^1/𝛾(𝑡,𝑠)^𝜆𝛿Δ𝜂Δ𝜏^𝑡𝑡₀^𝑠𝑡₀𝑔^𝜆(𝜏,𝜂)𝑎^1/𝛾(𝑡,𝑠)+𝑏^1/𝛾(𝑡,𝑠)𝑦^1/𝛾(𝑡,𝑠)^𝛼𝜆Δ𝜂Δ𝜏.(2.61) This implies (noting that 𝛼𝜆≤1 and 𝛿𝜆≤1) that 𝑦(𝑡,𝑠)≤2^{𝜆𝛿((1/𝛾)−1)}^𝑡𝑡₀^𝑠𝑡₀+2_{𝜆𝛿((1/𝛾)−1)}𝑓^𝜆(𝜏,𝜂)𝑎^{𝜆𝛿/𝛾}(𝜏,𝜂)Δ𝜂Δ𝜏^𝑡𝑡₀^𝑠𝑡₀+2_{𝜆𝛼((1/𝛾)−1)}𝑓^𝜆(𝜏,𝜂)𝑏^{𝜆𝛿/𝛾}(𝜏,𝜂)𝑦^{𝜆𝛿/𝛾}(𝜏,𝜂)Δ𝜂Δ𝜏^𝑡𝑡₀^𝑠𝑡₀+2_{𝜆𝛼((1/𝛾)−1)}𝑔^𝜆(𝜏,𝜂)𝑎^{𝜆𝛼/𝛾}(𝜏,𝜂)Δ𝜂Δ𝜏^𝑡𝑡₀^𝑠𝑡₀𝑔^𝜆(𝜏,𝜂)𝑏^{𝜆𝛼/𝛾}(𝜏,𝜂)𝑦^{𝜆𝛼/𝛾}(𝜏,𝜂)Δ𝜂Δ𝜏.(2.62) Applying the Young inequality (1.18) on the term 𝐻₁𝑦^{𝜆(𝛿/𝛾)} with 𝑞=𝛾/𝜆𝛿>1 and 𝑝=𝛾/(𝛾−𝜆𝛿)>1, we see that 𝐻₁𝑦^{𝜆(𝛿/𝛾)}≤(𝛾−𝜆𝛿)𝛾𝐻¹^{𝛾/(𝛾−𝜆𝛿)}+𝜆𝛿𝛾𝑦,(2.63) where 𝐻₁=𝑓^𝜆𝑏^{𝜆𝛿/𝛾}. Again applying the Young inequality (1.18) on the term 𝐻₂𝑦^{𝜆(𝛼/𝛾)} with 𝑞=𝛾/𝜆𝛼>1 and 𝑝=𝛾/(𝛾−𝜆𝛼)>1, we see that 𝐻₂𝑦^{𝜆(𝛼/𝛾)}≤(𝛾−𝜆𝛼)𝛾𝐻²^{𝛾/(𝛾−𝜆𝛼)}+𝜆𝛼𝛾𝑦,(2.64) where 𝐻₂=𝑔^𝜆𝑏^{𝜆𝛼/𝛾}. Combining (2.62)–(2.64), we have 𝑦(𝑡,𝑠)≤𝐻(𝑡,𝑠)+𝛽₁^𝑡𝑡₀^𝑠𝑡₀𝑦(𝜏,𝜂)Δ𝜂Δ𝜏.(2.65) Applying the inequality (1.10) on (2.65) with 𝑓(𝑡,𝑠)=𝛽₁, we have 𝑦(𝑡,𝑠)<𝐻(𝑡,𝑠)+𝑒_{𝛽1(𝑠−𝑠0)}𝑡,𝑡₀,∀(𝑡,𝑠)∈Ω.(2.66) Substituting (2.66) into (2.56), we get the desired inequality (2.45). The proof is complete.

Remark 2.9. The above results can be extended to the general inequality 𝑢^𝛾(𝑡,𝑠)≤𝑎(𝑡,𝑠)+𝑏(𝑡,𝑠)^𝑡𝑡₀^𝑠𝑠₀𝐿(𝜏,𝜂,𝑢(𝜏,𝜂))Δ𝜂Δ𝜏,(2.67) when 𝐿(𝜏,𝜂,𝑢)−𝐿(𝜏,𝜂,𝑣)≤𝐿₁(𝜏,𝜂,𝑣)(𝑢−𝑣)^𝛿+𝐿₂(𝜏,𝜂,𝑣)(𝑢−𝑣)^𝛼^𝜆.(2.68) In fact, by using the new substitution 𝑦(𝑡,𝑠)=^𝑡𝑡₀^𝑠𝑠₀𝐿(𝜏,𝜂,𝑢(𝜏,𝜂))Δ𝜂Δ𝜏,(2.69) the inequality (2.67) can be written as 𝑢^𝛾(𝑡,𝑠)≤𝑎(𝑡,𝑠)+𝑏(𝑡,𝑠)𝑦(𝑡,𝑠),(2.70) and then, since 𝛾≥1, we have 𝑢(𝑡,𝑠)≤𝐴(𝑡,𝑠)+𝐵(𝑡,𝑠)𝑦(𝑡,𝑠),(2.71) where 𝐴(𝑡,𝑠)=𝑎^1/𝛾(𝑡,𝑠) and 𝐵(𝑡,𝑠)=𝑎^(1/𝛾)−1(𝑡,𝑠)𝑏(𝑡,𝑠). This implies that 𝑦(𝑡,𝑠)=^𝑡𝑡₀^𝑠𝑠₀≤𝐿(𝜏,𝜂,𝑢(𝜏,𝜂))Δ𝜂Δ𝜏_𝑡𝑡0^𝑠𝑠₀+[𝐿(𝜏,𝜂,𝐴(𝜏,𝜂)+𝐵(𝜏,𝜂)𝑦(𝜏,𝜂))−𝐿(𝜏,𝜂,𝐴(𝜏,𝜂))]Δ𝜂Δ𝜏_𝑡𝑡0^𝑠𝑠₀𝐿(𝜏,𝜂,𝐴(𝜏,𝜂))Δ𝜂Δ𝜏.(2.72) Using (2.68) in the last inequality, we get an inequality similar to the inequality (2.8) and then follow the proof of the above results to find some explicit bounds of (2.8). The details are left to the reader.

3. Applications

In this section, we present some applications of our main results.

Example 3.1. Consider the partial dynamic equation on time scales (𝑢^𝛾(𝑡,𝑠))^Δ𝑡Δ𝑠=𝐻(𝑡,𝑠,𝑢(𝑡,𝑠))+ℎ(𝑡,𝑠),(𝑡,𝑠)∈Ω≡𝑡₀,∞_𝕋×𝑠₀,∞_𝕋,(3.1) with initial boundary conditions 𝑢𝑡,𝑠₀=𝑎(𝑡)>,𝑢𝑡₀,𝑠=𝑏(𝑠)>,𝑎𝑡₀=𝑏𝑠₀=0,(3.2) where 𝛾≥1 is a constant and 𝐻 and ℎ are rd-continuous functions on Ω,𝑎∶[𝑡₀,∞)_𝕋→ℝ⁺ and 𝑏∶[𝑡₀,∞)_𝕋→ℝ⁺ are rd-continuous functions. Assume that ||𝐻(𝑡,𝑠,𝑢)||≤𝑓(𝑡,𝑠)||𝑢(𝑡,𝑠)||^𝛿+𝑔(𝑡,𝑠)||𝑢(𝑡,𝑠)||^𝛼,(3.3) where 𝑓(𝑡,𝑠) and 𝑔(𝑡,𝑠) are nonnegative rd-continuous functions for (𝑡,𝑠)∈Ω and 𝛼,𝛿<𝛾. If 𝑢(𝑡,𝑠) is a solution of (3.1)-(3.2), then 𝑢(𝑡,𝑠) satisfies ||𝑢(𝑡,𝑠)||≤𝑎^1/𝛾(𝑡,𝑠)+𝐴^1/𝛾(𝑡,𝑠),∀(𝑡,𝑠)∈Ω,(3.4) where 𝑎(𝑡,𝑠)=𝑎^𝛾(𝑡)+𝑏^𝛾(𝑠)+^𝑡𝑡₀^𝑠𝑠₀𝐴(𝑡,𝑠)∶=𝐻₀||ℎ(𝜏,𝜂)||Δ𝜂Δ𝜏,(𝑡,𝑠)+𝑒_{𝛽(𝑠−𝑠0)}𝑡,𝑡₀,𝛽=𝛼𝐻₀𝛾+𝛾𝛿,(𝑡,𝑠)=^𝑡𝑡₀^𝑠𝑠₀𝑓(𝜏,𝜂)𝑎^𝛿/𝛾+(𝜏,𝜂)Δ𝜂Δ𝜏_𝑡𝑡0^𝑠𝑠₀𝑔(𝜏,𝜂)𝑎^𝛼/𝛾+(𝛾−𝛿)(𝜏,𝜂)Δ𝜂Δ𝜏𝛾^𝑡𝑡₀^𝑠𝑠₀+(𝛾−𝛼)(𝑓(𝜏,𝑠))^{𝛾/(𝛾−𝛿)}Δ𝜂Δ𝜏𝛾^𝑡𝑡₀^𝑠𝑠₀(𝑔(𝜏,𝜂))^{𝛾/(𝛾−𝛼)}Δ𝜂Δ𝜏.(3.5) In fact, the solution of (3.1)-(3.2) satisfies ||𝑢(𝑡,𝑠)||^𝛾=𝑎^𝛾(𝑡)+𝑏^𝛾(𝑡)+^𝑡𝑡₀^𝑠𝑠₀ℎ(𝜏,𝜂)Δ𝜂Δ𝜏+^𝑡𝑡₀^𝑠𝑠₀𝐻(𝜏,𝜂,𝑢(𝜏,𝜂))Δ𝜂Δ𝜏,(3.6) for (𝑡,𝑠)∈Ω.Therefore, ||𝑢(𝑡,𝑠)||^𝛾≤𝑎(𝑡,𝑠)+^𝑡𝑡₀^𝑠𝑠₀||𝐻(𝜏,𝜂,𝑢(𝜏,𝜂))||Δ𝜂Δ𝜏,for(𝑡,𝑠)∈Ω.(3.7) It follows from (3.3) and (3.7) that ||𝑢(𝑡,𝑠)||^𝛾≤𝑎(𝑡,𝑠)+^𝑡𝑡₀^𝑠𝑠₀𝑓(𝜏,𝑠)||𝑢(𝜏,𝜂)||^𝛿+𝑔(𝜏,𝜂)||𝑢(𝜏,𝜂)||^𝛼Δ𝜂Δ𝜏,(3.8) for (𝑡,𝑠)∈Ω. Applying Theorem 2.6 on (3.8) with 𝜆=1 and 𝑏(𝑡,𝑠)=1, we obtain (3.4).

Example 3.2. Consider the equation 𝑢^𝛾(𝑡,𝑠)=𝐻(𝑡,𝑠,𝑢(𝑡,𝑠))+ℎ(𝑡,𝑠),(𝑡,𝑠)∈Ω≡𝑡₀,∞_𝕋×𝑠₀,∞_𝕋,(3.9) where 𝛾≥1 is a constant and 𝐻 and ℎare rd-continuous on Ω,𝑎∶[𝑡₀,∞)_𝕋→ℝ⁺ and 𝑏∶[𝑡₀,∞)_𝕋→ℝ⁺ are rd-continuous functions. Assume that ||𝐻(𝑡,𝑠,𝑢)||≤𝑓(𝑡,𝑠)||𝑢(𝑡,𝑠)||^𝛿+𝑔(𝑡,𝑠)||𝑢(𝑡,𝑠)||^𝛼,(3.10) where 𝑓(𝑡,𝑠) and 𝑔(𝑡,𝑠) are nonnegative rd-continuous functions for (𝑡,𝑠)∈Ω and 𝛼,𝛿<𝛾. If 𝑢(𝑡,𝑠) is a solution of (3.1)-(3.2), then 𝑢(𝑡,𝑠) satisfies ||𝑢(𝑡,𝑠)||≤||ℎ(𝑡,𝑠)||^1/𝛾+1𝛾||ℎ(𝑡,𝑠)||^(1/𝛾)−1𝐵^1/𝛾(𝑡,𝑠),∀(𝑡,𝑠)∈Ω,(3.11) where 𝐵(𝑡,𝑠)∶=𝐹^∗(𝑡,𝑠)+(1−𝜆)^𝑡𝑡₀^𝑡𝑠₀𝐺^∗𝐹_∗(𝜏,𝜂)^1/(1−𝜆)Δ𝜂Δ𝜏+𝑒_{𝜆(𝑠−𝑠0)}𝑡,𝑡₀,(𝑡,𝑠)∶=^𝑡𝑡₀^𝑠𝑠₀𝑓^𝜆𝐺(𝜏,𝜂)𝑎^{𝜆(𝛿/𝛾)}(𝜏,𝜂)+𝑔^𝜆(𝜏,𝜂)𝑎^{𝜆(𝛼/𝛾)}(𝜏,𝜂)Δ𝜂Δ𝜏,_∗(𝑡,𝑠)∶=𝑓^𝜆(𝑡,𝑠)𝛿𝛾𝑎^{(𝛿/𝛾)−1}(𝑡,𝑠)^𝜆+𝑔^𝜆(𝑡,𝑠)𝛼𝛾𝑎^{(𝛼/𝛾)−1}(𝑡,𝑠)^𝜆.(3.12) In fact, the solution of (3.9) satisfies ||𝑢(𝑡,𝑠)||^𝛾≤||ℎ(𝑡,𝑠)||+^𝑡𝑡₀^𝑠𝑠₀||𝐻(𝜏,𝜂,𝑢(𝜏,𝜂))||Δ𝜂Δ𝜏,for(𝑡,𝑠)∈Ω.(3.13) It follows from (3.10) and (3.13) that ||𝑢(𝑡,𝑠)||^𝛾≤||ℎ(𝑡,𝑠)||+^𝑡𝑡₀^𝑠𝑠₀𝑓(𝜏,𝑠)||𝑢(𝜏,𝜂)||^𝛿+𝑔(𝜏,𝜂)||𝑢(𝜏,𝜂)||^𝛼^𝜆Δ𝜂Δ𝜏,(3.14) for (𝑡,𝑠)∈Ω. Applying Theorem 2.7 on (3.14) with 𝑏(𝑡,𝑠)=1, we obtain (3.11).

Example 3.3. Assume that 𝕋=ℝ and consider the partial differential equation 𝜕𝜕𝑠𝑢^𝛾−1(𝑡,𝑠)𝜕𝜕𝑡𝑢(𝑡,𝑠)+𝐻(𝑡,𝑠,𝑢(𝑡,𝑠))=ℎ(𝑡,𝑠),(𝑡,𝑠)∈Ω^∗,(3.15) where Ω^∗=[0,∞)×[0,∞), with initial boundary conditions 𝑢(𝑡,0)=𝑎(𝑡)>0,𝑢(0,𝑠)=𝑏(𝑠)>0,𝑎(0)=𝑏(0)=0.(3.16) Assume that 𝛾≥1 is a constant and 𝐻∶[0,∞)×[0,∞)×ℝ→ℝ and ℎ∶[0,∞)_ℝ×[0,∞)_ℝ→ℝ, 𝑎∶ℝ→ℝ⁺ and 𝑏∶ℝ→ℝ⁺ are continuous functions. Also, we assume that ||𝐻(𝑡,𝑠,𝑢)||≤𝑓(𝑡,𝑠)||𝑢(𝑡,𝑠)||^𝛿+𝑔(𝑡,𝑠)||𝑢(𝑡,𝑠)||^𝛼,(3.17) where 𝑓(𝑡,𝑠) and 𝑔(𝑡,𝑠) are nonnegative continuous functions for (𝑡,𝑠)∈Ω^∗ and 𝛼,𝛿<𝛾. If 𝑢(𝑡,𝑠) is a solution of (3.1)-(3.2), then 𝑢(𝑡,𝑠) satisfies ||𝑢(𝑡,𝑠)||≤𝑎^1/𝛾(𝑡,𝑠)+𝛾^1/𝛾𝐵^1/𝛾(𝑡,𝑠),∀(𝑡,𝑠)∈Ω^∗,(3.18) where 𝑎(𝑡,𝑠)=𝑎^𝛾(𝑡)+𝑏^𝛾(𝑠)+𝛾^𝑡𝑡₀^𝑠𝑠₀𝐵(𝑡,𝑠)∶=𝐻₀||ℎ(𝜏,𝜂)||Δ𝜂Δ𝜏,(𝑡,𝑠)+𝑒_{𝛽(𝑠−𝑠0)}𝑡,𝑡₀,𝛽=𝛼𝐻₀𝛾+𝛾𝛿,(𝑡,𝑠)=^𝑡𝑡₀^𝑠𝑠₀𝑓(𝜏,𝜂)𝑎^𝛿/𝛾(𝜏,𝜂)Δ𝜂Δ𝜏+^𝑡𝑡₀^𝑠𝑠₀𝑔(𝜏,𝜂)𝑎^𝛼/𝛾+(𝛾−𝛿)(𝜏,𝜂)Δ𝜂Δ𝜏𝛾^𝑡𝑡₀^𝑠𝑠₀+(𝛾−𝛼)(𝑓(𝜏,𝑠))^{𝛾/(𝛾−𝛿)}Δ𝜂Δ𝜏𝛾^𝑡𝑡₀^𝑠𝑠₀(𝑔(𝜏,𝜂))^{𝛾/(𝛾−𝛼)}Δ𝜂Δ𝜏.(3.19) In fact, the solution formula of (3.15)-(3.16), after integration twice, is given by ||𝑢(𝑡,𝑠)||^𝛾−𝑎^𝛾(𝑡)−𝑏^𝛾(𝑡)+𝛾^𝑡𝑡₀^𝑠𝑠₀=𝛾_𝑡𝑡0𝐻(𝜏,𝜂,𝑢(𝜏,𝜂))Δ𝜂Δ𝜏^𝑠𝑠₀ℎ(𝜏,𝜂)Δ𝜂Δ𝜏,for(𝑡,𝑠)∈Ω^∗.(3.20) Therefore, ||𝑢(𝑡,𝑠)||^𝛾≤𝑎(𝑡,𝑠)+𝛾^𝑡𝑡₀^𝑠𝑠₀||𝐻(𝜏,𝜂,𝑢(𝜏,𝜂))||Δ𝜂Δ𝜏,for(𝑡,𝑠)∈Ω^∗.(3.21) It follows from (3.17) and (3.21) that ||𝑢(𝑡,𝑠)||^𝛾≤𝑎(𝑡,𝑠)+𝛾^𝑡𝑡₀^𝑠𝑠₀𝑓(𝜏,𝑠)||𝑢(𝜏,𝜂)||^𝛿+𝑔(𝜏,𝜂)||𝑢(𝜏,𝜂)||^𝛼Δ𝜂Δ𝜏,(3.22) for (𝑡,𝑠)∈Ω^∗. Applying Theorem 2.6 on (3.22) with 𝜆=1 and 𝑏(𝑡,𝑠)=𝛾, we obtain (3.18).

Acknowledgments

The author is very grateful to the referees for their valuable remarks and comments which significantly contributed to the quality of the paper. This project was supported by King Saud University, Dean-ship of Scientific Research, College of Science Research Centre.