Existence of Positive Solutions to Singular 𝑝-Laplacian General Dirichlet Boundary Value Problems with Sign Changing Nonlinearity

Abstract

By using the well-known Schauder fixed point theorem and upper and lower solution method, we present some existence criteria for positive solution of an 𝑚-point singular 𝑝-Laplacian dynamic equation on time scales with the sign changing nonlinearity. These results are new even for the corresponding differential (𝕋=ℝ) and difference equations (𝕋=ℤ), as well as in general time scales setting. As an application, an example is given to illustrate the results.

1. Introduction

Initiated by Hilger in his Ph.D. thesis [1] in 1988, the theory of time scales has been improved greatly ever since, especially in the unification of the theory of differential equations in the continuous case and the theory of finite difference equations in the discrete case. For the time being, it remains active and attracts many distinguished researchers' attention. The reason is two sided. On the one hand, the calculus on time scales not only can unify differential and difference equations, but also can provide accurate information of phenomena that manifest themselves partly in continuous time and partly in discrete time. On the other hand, it is also widely applied to the research of biology, heat transfer, stock market, wound healing and epidemic models [2–6], and so forth. For instance, Hoffacker et al. have used the theory to model how students suffering from the eating disorder bulimia are influenced by their college friends. With the theory on time scales, they can model how the number of sufferers changes during the continuous college term as well as during long breaks [5]. Hence, the dynamic equations on time scales are worth studying theoretically and practically [3, 5, 7].

Here and hereafter, we denote 𝜑_𝑝(𝑢) is 𝑝-Laplacian operator, that is, 𝜑_𝑝(𝑢)=|𝑢|^𝑝−2𝑢 for 𝑝>1 and (𝜑_𝑝)⁻¹=𝜑_𝑞, where 1/𝑝+1/𝑞=1. We make the blanket assumption that 0,𝑇 are points in 𝕋, by an interval (0,𝑇)_𝕋 we always mean (0,𝑇)∩𝕋. Other types of interval are defined similarly.

Recent research results indicate that considerable work has been made in the existence problems of solutions of boundary value problems on time scales, for details, see [8–16] and the references therein. In particular, some of them are considered the existence of positive solutions of 𝑝-Laplacian boundary value problems on time scales, see [17–22]. The main tools used in these papers are the various fixed point theorems in cones. Very recently, when the nonlinear term 𝑓 is allowed to change sign, Su et al. [23–25] proved the existence of positive solutions to 𝑝-Laplacian dynamic equations with sign changing nonlinearity on time scales.

Motivated by references [23–25], we consider the following 𝑚-point singular 𝑝-Laplacian boundary value problem on time scales of the form (𝜑_𝑝(𝑢^Δ(𝑡)))^∇+𝑞(𝑡)𝑓(𝑡,𝑢(𝑡))=0,𝑡∈(0,𝑇)_𝕋,𝑢(0)=0,𝑢(𝑇)=^𝑚_𝑖=1𝜓_𝑖(𝑢^Δ(𝜉_𝑖)),𝑚∈ℕ, (1.1)(1.2) where 𝑓(𝑡,𝑢)∶(0,𝑇)_𝕋×(0,∞)→ℝ is continuous and 𝜓_𝑖∶ℝ→ℝ are continuous, nondecreasing and 𝜓_𝑖 may be nonlinear, 0≤𝜉₁<𝜉₂<⋯<𝜉_𝑚≤𝑇. The singularity may occur at 𝑢=0,𝑡=0 and 𝑡=𝑇, and the nonlinearity is allowed to change sign. In particular, the boundary condition (1.2) includes the Dirichlet boundary condition. We obtain some new existence criteria for positive solutions of the boundary value problem (1.1) and (1.2) by using the upper and lower method. Our results are new even for the corresponding differential (𝕋=ℝ) and difference equations (𝕋=ℤ), as well as in general time scales setting. As an application, an example is given to illustrate these results. In particular, our results improve and generalize some known results of Agarwal et al. [26], O'Regan [27] (𝑝=2) and Lü et al. [28] when 𝕋=ℝ; include the results of Lü et al. [29] when 𝕋=ℝ; extend and include the results of Jiang et al. [30] in the case of 𝕋=ℤ.

For the convenience of statements, now we present some basic definitions and lemmas concerning the calculus on time scales that one needs to read this manuscript, which can be found in [3, 7]. One of other excellent sources on dynamical systems on time scales is from the book in [31].

Definition 1.1 (see [3, 7]). A time scale 𝕋 is a nonempty closed subset of ℝ. It follows that the jump operators 𝜎,𝜌∶𝕋→𝕋 defined by 𝜎(𝑡)=inf{𝜏∈𝕋∶𝜏>𝑡},𝜌(𝑡)=sup{𝜏∈𝕋∶𝜏<𝑡} (1.3) (supplemented by inf∅∶=sup𝕋 and sup∅∶=inf𝕋 ) are well defined. The point 𝑡∈𝕋 is left-dense, left-scattered, right-dense, right-scattered if 𝜌(𝑡)=𝑡,𝜌(𝑡)<𝑡,𝜎(𝑡)=𝑡,𝜎(𝑡)>𝑡, respectively. If 𝕋 has a right-scattered minimum 𝑚, define 𝕋_𝜅=𝕋−{𝑚}; otherwise, set 𝕋_𝜅=𝕋. If 𝕋 has a left-scattered maximum 𝑀, define 𝕋^𝜅=𝕋−{𝑀}; otherwise, set 𝕋^𝜅=𝕋. The forward graininess is 𝜇(𝑡)∶=𝜎(𝑡)−𝑡. Similarly, the backward graininess is 𝜈(𝑡)∶=𝑡−𝜌(𝑡).

Definition 1.2 (see [7]). We say that a function 𝑓∶𝕋→ℝ is right-increasing at a point 𝑡₀∈𝕋⧵{max𝕋} provided the following conditions hold.(i)If 𝑡₀ is right-scattered, then 𝑓(𝜎(𝑡₀))>𝑓(𝑡₀).(ii)If 𝑡₀ is right-dense, then there is a neighborhood 𝑈 of 𝑡₀ such that 𝑓(𝑡)>𝑓(𝑡₀) for all 𝑡∈𝑈 with 𝑡>𝑡₀.Similarly, we say that 𝑓 is right-decreasing if above in (i), 𝑓(𝜎(𝑡₀))<𝑓(𝑡₀) and (ii), 𝑓(𝑡)<𝑓(𝑡₀).

Definition 1.3 (see [3]). A function 𝑓∶𝕋→ℝ is called predifferentiable with (region of differential) 𝐷 provided the following conditions hold:(i)𝑓 is continuous on 𝕋;(ii)𝐷⊂𝕋^𝜅;(iii)𝕋^𝜅⧵𝐷 is countable and contains no right-scattered elements of 𝕋;(iv)𝑓 is differentiable at each 𝑡∈𝐷.

Next, we list some lemmas which will be used in the sequel.

Lemma 1.4 (see [3, 7]). Suppose 𝑓∶𝕋→ℝ is a function and let 𝑡∈𝕋^𝜅, then one has the following:(i)If 𝑓 is differentiable at 𝑡, then 𝑓 is continuous at 𝑡.(ii)If 𝑓 is continuous at 𝑡 and 𝑡 is right-scattered, then 𝑓 is differentiable at 𝑡 with 𝑓^Δ(𝑡)=𝑓(𝜎(𝑡))−𝑓(𝑡)=𝜇(𝑡)𝑓(𝜎(𝑡))−𝑓(𝑡).𝜎(𝑡)−𝑡 (1.4)(iii)If 𝑓 is right-dense, then 𝑓 is differentiable at 𝑡 if and only one the limit lim_𝑠→𝑡=𝑓(𝑡)−𝑓(𝑠)𝑡−𝑠 (1.5) exists as a finite number. In this case 𝑓^Δ(𝑡)=lim_𝑠→𝑡𝑓(𝑡)−𝑓(𝑠).𝑡−𝑠 (1.6)(iv)If 𝑓 is differentiable at 𝑡, then 𝑓(𝜎(𝑡))=𝑓(𝑡)+𝜇(𝑡)𝑓^Δ(𝑡)=𝑓(𝑡)+(𝜎(𝑡)−𝑡)𝑓^Δ(𝑡). (1.7)

Lemma 1.5 (see [7]). Suppose 𝑓∶𝕋→ℝ is differentiable at 𝑡₀∈𝕋⧵{max𝕋}. If 𝑓 assumes its local right-minimum at 𝑡₀, then 𝑓^Δ(𝑡₀)≥0. If 𝑓 assumes its local right-maximum at 𝑡₀, then 𝑓^Δ(𝑡₀)≤0.

Lemma 1.6 ((Mean Value Theorem) [7]). Let 𝑓 be a continuous function on [𝑎,𝑏] that is differentiable on [𝑎,𝑏). Then there exist 𝜉^∗,𝜏^∗∈[𝑎,𝑏) such that 𝑓^Δ(𝜏^∗)≤𝑓(𝑏)−𝑓(𝑎)𝑏−𝑎≤𝑓^Δ(𝜉^∗). (1.8)

Lemma 1.7 (see [3]). Suppose 𝑓 and 𝑔 are pre-differential with 𝐷. If 𝑈 is a compact interval with endpoints 𝑟,𝑠∈𝕋, then |𝑓(𝑠)−𝑓(𝑟)|≤{sup_{𝑡∈𝑈𝜅∩𝐷∩𝕋}|𝑓^Δ(𝑡)|}|𝑠−𝑟|.

Now, we can obtain the following lemma which is similar to Lemma 1.7. The proofs are similar to the proofs of Lemma 1.7 by a slight modification and we omit the proofs.

Lemma 1.8. Suppose 𝑓(𝑡,𝑢) and 𝑔(𝑡,𝑢) are predifferential with 𝐷×(0,+∞). If 𝑈 is a compact interval with endpoints 𝑟,𝑠∈𝕋, then |𝑓(𝑠,𝑢)−𝑓(𝑟,𝑢)|≤{sup_{𝑡∈𝑈𝜅∩𝐷∩𝕋}|𝑓^Δ(𝑡,𝑢)|}|𝑠−𝑟|, here 𝐷⊂𝕋^𝜅.

Throughout this paper, it is assumed that

(H1)𝑓(𝑡,𝑢)∶(0,𝑇)_𝕋×(0,∞)→ℝ is continuous;(H2)𝑞∈𝐶((0,𝑇)_𝕋,(0,∞)) and 𝑞∈𝐶^∇_𝑙𝑑[0,𝑇]_𝕋;(H3)𝜓_𝑖∶ℝ→ℝ are continuous and nondecreasing, here 𝑖=1,2,…,𝑚.

2. Existence Results

Define the Banach space 𝔹=𝐶[0,𝑇]_𝕋 with the norm ‖𝑦‖=sup_{𝑡∈[0,𝑇]𝕋}|𝑦(𝑡)|.

To demonstrate existence of positive solutions to problem (1.1) and (1.2), we first approximate the singular problem by means of a sequence of nonsingular problems, and by using the lower and upper solution for nonsingular problem together with Schauders fixed point theorem, and then we establish the existence of solutions to each approximating problem. Our results are new even for the corresponding differential (𝕋=ℝ) and difference equations (𝕋=ℤ), as well as in general time scales setting. If we consider the corresponding differential equation (𝕋=ℝ) of problem (1.1) and (1.2) in the method mentioned above, we obtain the same existence results to problem (1.1) and (1.2). In the same way, we consider the corresponding difference equation (𝕋=ℝ) of problem (1.1) and (1.2), we obtain the same existence results to problem (1.1) and (1.2). Here, the two same existence results are obtained in different settings by using the essentially same method. Naturally, it is quite necessary to consider the existence results to problem (1.1) and (1.2) in same setting. In this case, we need to solve the problem with the help of calculus on time scales, because it not only can unify differential and difference equations, but also can provide accurate information of phenomena that manifests themselves partly in continuous time and partly in discrete time. For example, we can consider the problem (1.1) and (1.2) on time scales 1𝕋={0}∪{(2)^ℕ1}∪[2,1]∪[2,3]. (2.1) However, if 𝑡 is taken from (2.1), we cannot study the problem (1.1) and (1.2) only in differential case, neither can we study the problem (1.1) and (1.2) only in difference case.

Now we state and prove our main result.

Theorem 2.1. Let 𝑛₀∈{1,2,…} be fixed. Assume that (H1)–(H3) hold and the following conditions are satisfied. (A1)For each 𝑛∈{𝑛₀,𝑛₀+1,…}≡ℕ₁, there is a constant 𝜌_𝑛 such that {𝜌_𝑛} is a strictly monotone decreasing sequence with lim_𝑛→∞𝜌_𝑛=0, and 𝑞(𝑡)𝑓(𝑡,𝜌_𝑛)≥0 for 𝑡∈[1/2^𝑛+1,𝑇]_𝕋;(A2)There exists a function 𝛼∈𝐶[0,𝑇]_𝕋∩𝐶^Δ(0,𝑇]_𝕋,𝜑_𝑝(𝛼^Δ)∈𝐶^∇(0,𝑇)_𝕋 with 𝛼(0)=0,𝛼(𝑇)−∑^𝑚_𝑖=1𝜓_𝑖(𝛼^Δ(𝜉_𝑖))≤0,𝛼>0on(0,𝑇]_𝕋 and −(𝜑_𝑝(𝛼^Δ))^∇≤𝑞(𝑡)𝑓(𝑡,𝛼)for𝑡∈(0,𝑇)_𝕋;(A3)There exists a function 𝛽∈𝐶[0,𝑇]_𝕋∩𝐶^Δ(0,𝑇]_𝕋, 𝜑_𝑝(𝛽^Δ)∈𝐶^∇(0,𝑇)_𝕋 with 𝛽≥𝛼,𝛽≥𝜌_𝑛0for𝑡∈[0,𝑇]_𝕋 and 𝛽(𝑇)−∑^𝑚_𝑖=1𝜓_𝑖(𝛽^Δ(𝜉_𝑖))>0, with −(𝜑_𝑝(𝛽^Δ))^∇≥𝑞(𝑡)𝑓(𝑡,𝛽) for 𝑡∈(0,𝑇)_𝕋, and −(𝜑_𝑝(𝛽^Δ))^∇≥𝑞(𝑡)𝑓(1/2^𝑛0+1,𝛽) for 𝑡∈(0,1/2^𝑛0+1)_𝕋.Then the boundary value problem (1.1) and (1.2) has a positive solution 𝑢∈𝐶[0,𝑇]_𝕋∩𝐶^Δ(0,𝑇]_𝕋,𝜑_𝑝(𝑢^Δ)∈𝐶^∇(0,𝑇)_𝕋 with 𝑢≥𝛼 for 𝑡∈[0,𝑇]_𝕋.

Proof. It follows from the condition (A1) that 1/2^𝑛+1∈(0,𝑇]_𝕋 for each 𝑛∈ℕ₁. That is, (0,𝑇]_𝕋 is not empty. Without loss of generality, fix 𝑛∈ℕ₁. If 𝜉₁>0, then we can suppose that min_{𝑡∈[𝜉1,𝑇]𝕋}𝛼(𝑡)≥𝜌_𝑛, let 𝑡_𝑛∈(0,𝜉₁)_𝕋 be such that 𝛼(𝑡_𝑛)=𝜌_𝑛,𝛼≤𝜌_𝑛for𝑡∈[0,𝑡_𝑛]_𝕋. (2.2) If 𝜉₁=0, then we can suppose that min_{𝑡∈[𝜉2,𝑇]𝕋}𝛼(𝑡)≥𝜌_𝑛, let 𝑡_𝑛∈(0,𝜉₂)_𝕋 be such that (2.2) holds. Define 𝛼_𝑛𝜌(𝑡)=^𝑛if𝑡∈[0,𝑡_𝑛]_𝕋,𝛼if𝑡∈[𝑡_𝑛,𝑇]_𝕋,here𝛼(𝑡_𝑛)=𝜌_𝑛. (2.3) We denote 𝑒_𝑛=[1/2^𝑛+1,𝑇]_𝕋,𝜔_𝑛(𝑡)=max{1/2^𝑛+1,𝑡}for𝑡∈[0,𝑇]_𝕋 and 𝑓_𝑛(𝑡,𝑥)=max{𝑓(𝑡,𝑥),𝑓(𝜔_𝑛(𝑡),𝑥)}. (2.4) Define a sequence ℎ_𝑛0(𝑡,𝑥)=𝑓_𝑛0(𝑡,𝑥) and ℎ_𝑛(𝑡,𝑥)=min{𝑓_𝑛0(𝑡,𝑥),…,𝑓_𝑛(𝑡,𝑥)},𝑛=𝑛₀+1,𝑛₀+2,…. (2.5) Then 𝑓(𝑡,𝑥)≤⋯≤ℎ_𝑛+1(𝑡,𝑥)≤ℎ_𝑛(𝑡,𝑥)≤⋯≤ℎ_𝑛0(𝑡,𝑥)for(𝑡,𝑥)∈(0,𝑇)_𝕋ℎ×(0,∞),_𝑛(𝑡,𝑥)=𝑓(𝑡,𝑥)for(𝑡,𝑥)∈𝑒_𝑛×(0,∞). (2.6)
Consider the 𝑝-Laplacian boundary value problem (𝜑_𝑝(𝑢^Δ(𝑡)))^∇+𝑞(𝑡)ℎ^∗_𝑛0(𝑡,𝑢(𝑡))=0,𝑡∈(0,𝑇)_𝕋,𝑢(0)=𝜌_𝑛0,𝑢(𝑇)−^𝑚_𝑖=1𝜓^∗_𝑖(𝑢^Δ(𝜉_𝑖))=𝜌_𝑛0, (2.7)(2.8) where ℎ^∗_𝑛0ℎ(𝑡,𝑢(𝑡))=^𝑛0(𝑡,𝛼_𝑛0(𝑡))+𝑟(𝛼_𝑛0(𝑡)−𝑢(𝑡)),𝑢(𝑡)≤𝛼_𝑛0ℎ(𝑡),_𝑛0(𝑡,𝑢(𝑡)),𝛼_𝑛0ℎ(𝑡)≤𝑢(𝑡)≤𝛽,_𝑛0(𝑡,𝛽(𝑡))+𝑟(𝛽(𝑡)−𝑢(𝑡)),𝑢(𝑡)≥𝛽,𝜓^∗_𝑖(𝑧_𝑖𝜓)=^𝑖(𝛼^Δ(𝜉_𝑖)),𝑧_𝑖≤𝛼^Δ_𝑛0(𝜉_𝑖)=𝛼^Δ(𝜉_𝑖𝜓),_𝑖(𝑧_𝑖),𝛼^Δ_𝑛0(𝜉_𝑖)≤𝑧_𝑖≤𝛽^Δ(𝜉_𝑖𝜓),_𝑖(𝛽^Δ(𝜉_𝑖)),𝑧_𝑖≥𝛽^Δ(𝜉_𝑖),𝑖=1,…,𝑚, (2.9)(2.10) and 𝑟∶ℝ→[−1,1] is the radial retraction function defined by 𝑟(𝑢)=𝑢𝑢,|𝑢|≤1,|𝑢|,|𝑢|>1. (2.11)
Suppose 𝐶₀[0,𝑇]_𝕋={𝑢∈𝐶[0,𝑇]_𝕋∶𝑢(0)=0}𝐶^Δ_𝜌𝑛0[0,𝑇]_𝕋={𝑢∈𝐶^Δ[0,𝑇]_𝕋∶𝑢(0)=𝜌_𝑛0}. (2.12) We define the mappings 𝐿_𝑝;𝐹∶𝐶^Δ_𝜌𝑛0[0,𝑇]_𝕋→𝐶₀[0,𝑇]_𝕋×ℝ be such that 𝐿_𝑝𝑢(𝑡)=(𝜑_𝑝(𝑢^Δ(𝑡))−𝜑_𝑝(𝑢^Δ(0)),𝑢(𝑇)),𝐹𝑢(𝑡)=(−^𝑡₀𝑞(𝑥)ℎ^∗_𝑛0(𝑥,𝑢(𝑥))∇𝑥,^𝑚_𝑖=1𝜓^∗_𝑖(𝑢^Δ(𝜉_𝑖))+𝜌_𝑛0). (2.13)(2.14) By using the Arzela-Ascoli theorem on time scales [2], we can show that 𝐹 is continuous and compact. By using the (2.7), (2.8), (2.13) and (2.14), we obtain (𝜑_𝑝(𝑢^Δ(𝑡))−𝜑_𝑝(𝑢^Δ(0)),𝑢(𝑇))=(−^𝑡₀𝑞(𝑥)ℎ^∗_𝑛0(𝑥,𝑢(𝑥))∇𝑥,^𝑚_𝑖=1𝜓^∗_𝑖(𝑢^Δ(𝜉_𝑖))+𝜌_𝑛0), (2.15) that is 𝐿_𝑝𝑢(𝑡)=𝐹𝑢(𝑡). (2.16) If 𝐿_𝑝𝑣=(𝑢,𝛾)for𝑢∈𝐶₀[0,𝑇]_𝕋,𝛾=𝜌_𝑛0+^𝑇₀𝜑_𝑞(𝑢(𝑥)−𝑢(𝑇))Δ𝑥, (2.17) then 𝑣(𝑡)=𝜌_𝑛0+∫^𝑡₀𝜑_𝑞(𝑢(𝑥)−𝑢(𝑇))Δ𝑥, hence 𝐿_𝑝⁻¹ exists and is continuous. So 𝑢(𝑡)=𝐿_𝑝⁻¹𝐹𝑢(𝑡). (2.18) It is clear that solving the boundary value problem (2.7) and (2.8) is equivalent to finding a fixed point of 𝑢=𝐿_𝑝⁻¹𝐹𝑢≡𝑁𝑢, where 𝑁=𝐿_𝑝⁻¹𝐹∶𝐶^Δ_𝜌𝑛0[0,𝑇]_𝕋→𝐶^Δ_𝜌𝑛0[0,𝑇]_𝕋 is compact. Schauder's fixed point theorem guarantees that the boundary value problem (2.7) and (2.8) has a solution 𝑢_𝑛0(𝑡)∈𝐶^Δ[0,𝑇]_𝕋 with 𝜑_𝑝(𝑢^Δ_𝑛0(𝑡))∈𝐶^∇(0,𝑇)_𝕋.
We first show that 𝛼_𝑛0(𝑡)≤𝑢_𝑛0(𝑡)for𝑡∈[0,𝑇]_𝕋. (2.19) If (2.19) is not true, the function 𝑢_𝑛0(𝑡)−𝛼_𝑛0(𝑡) has a negative minimum for some 𝜏∈(0,𝑇]_𝕋. We consider two cases, namely, 𝜏∈(0,𝑇)_𝕋 and 𝜏=𝑇.
Case 1. Assume that 𝜏∈(0,𝑇)_𝕋, then we claim (𝜑_𝑝(𝑢^Δ_𝑛0))^∇(𝜏)≥(𝜑_𝑝(𝛼^Δ_𝑛0))^∇(𝜏). (2.20) Since 𝑢_𝑛0(𝑡)−𝛼_𝑛0(𝑡) has a negative minimum for some 𝜏∈(0,𝑇)_𝕋, in view of Definition 1.2, Lemmas 1.4 and 1.5, we have 𝑢^Δ_𝑛0(𝜏)−𝛼^Δ_𝑛0(𝜏)≥0 and there exists a 𝛿 with 𝜏−𝛿∈[0,𝜏)_𝕋 such that 𝑢^Δ_𝑛0(𝑡)−𝛼^Δ_𝑛0(𝑡)≤0for𝑡∈[𝜏−𝛿,𝜏)_𝕋. Thus 𝜑_𝑝(𝑢^Δ_𝑛0(𝑡))−𝜑_𝑝(𝛼^Δ_𝑛0(𝑡))≤𝜑_𝑝(𝑢^Δ_𝑛0(𝜏))−𝜑_𝑝(𝛼^Δ_𝑛0(𝜏))for𝑡∈[𝜏−𝛿,𝜏)_𝕋, (2.21) which leads to 𝜑_𝑝(𝑢^Δ_𝑛0(𝑡))−𝜑_𝑝(𝑢^Δ_𝑛0(𝜏))≥𝜑𝑡−𝜏^𝑝(𝛼^Δ_𝑛0(𝑡))−𝜑_𝑝(𝛼^Δ_𝑛0(𝜏))𝑡−𝜏for𝑡∈[𝜏−𝛿,𝜏)_𝕋. (2.22)
If 𝜏 is left-dense, in view of Lemma 1.4(𝜑_𝑝(𝑢^Δ_𝑛0))^∇(𝜏)=lim_{𝑡∈[𝜏−𝛿,𝜏)→𝜏}𝜑_𝑝(𝑢^Δ_𝑛0(𝑡))−𝜑_𝑝(𝑢^Δ_𝑛0(𝜏))𝑡−𝜏≥lim_{𝑡∈[𝜏−𝛿,𝜏)→𝜏}𝜑_𝑝(𝛼^Δ_𝑛0(𝑡))−𝜑_𝑝(𝛼^Δ_𝑛0(𝜏))𝑡−𝜏=(𝜑_𝑝(𝛼^Δ_𝑛0))^∇(𝜏). (2.23)
If 𝜏 is left-scattered, by Lemma 1.4 and (2.22) we obtain (𝜑_𝑝(𝑢^Δ_𝑛0))^∇𝜑(𝜏)=^𝑝(𝑢^Δ_𝑛0(𝜏))−𝜑_𝑝(𝑢^Δ_𝑛0(𝜌(𝜏)))≥𝜑𝜏−𝜌(𝜏)_𝑝(𝛼^Δ_𝑛0(𝜏))−𝜑_𝑝(𝛼^Δ_𝑛0(𝜌(𝜏)))𝜏−𝜌(𝜏)=(𝜑_𝑝(𝛼^Δ_𝑛0))^∇(𝜏). (2.24) Hence, (2.20) is established.
However, by (2.3), (2.9) and 𝑢_𝑛0(𝜏)<𝛼_𝑛0(𝜏), we obtain (𝜑_𝑝(𝑢^Δ_𝑛0(𝜏)))^∇−(𝜑_𝑝(𝛼^Δ_𝑛0(𝜏)))^∇=−[𝑞(𝜏)ℎ_𝑛0(𝜏,𝛼_𝑛0(𝜏))+𝑞(𝜏)𝑟(𝛼_𝑛0(𝜏)−𝑢_𝑛0(𝜏))+(𝜑_𝑝(𝛼^Δ_𝑛0(𝜏)))^∇]=−[𝑞(𝜏)ℎ_𝑛0(𝜏,𝛼(𝜏))+𝑞(𝜏)𝑟(𝛼(𝜏)−𝑢_𝑛0(𝜏))+(𝜑_𝑝(𝛼^Δ(𝜏)))^∇],𝜏∈[𝑡_𝑛0,𝑇)_𝕋,−[𝑞(𝜏)ℎ_𝑛0(𝜏,𝜌_𝑛0)+𝑞(𝜏)𝑟(𝜌_𝑛0−𝑢_𝑛0(𝜏))],𝜏∈(0,𝑡_𝑛0)_𝕋. (2.25)
Assume that 𝜏∈[1/2^𝑛0+1,𝑇]_𝕋, then 𝑔_𝑛0(𝜏,𝑥)=𝑓(𝜏,𝑥) for 𝑥∈(0,∞), by (A1) and (A2), we have (𝜑_𝑝(𝑢^Δ_𝑛0(𝜏)))^∇−(𝜑_𝑝(𝛼^Δ_𝑛0(𝜏)))^∇=−[𝑞(𝜏)𝑓(𝜏,𝛼(𝜏))+𝑞(𝜏)𝑟(𝛼(𝜏)−𝑢_𝑛0(𝜏))+(𝜑_𝑝(𝛼^Δ(𝜏)))^∇],𝜏∈[𝑡_𝑛0,𝑇)_𝕋,−[𝑞(𝜏)𝑓(𝜏,𝜌_𝑛0)+𝑞(𝜏)𝑟(𝜌_𝑛0−𝑢_𝑛0(𝜏))],𝜏∈(0,𝑡_𝑛0)_𝕋,<0, (2.26) which implies a contraction.
Assume that 𝜏∈(0,1/2^𝑛0+1)_𝕋, then ℎ_𝑛0(𝜏,𝑥)=max{𝑓(1/2^𝑛0+1,𝑥),𝑓(𝜏,𝑥)}, in view of (A1), (A2) and 𝑞(𝜏)>0, we have (𝜑_𝑝(𝑢^Δ_𝑛0(𝜏)))^∇−(𝜑_𝑝(𝛼^Δ_𝑛0(𝜏)))^∇≤−[𝑞(𝜏)𝑓(𝜏,𝛼(𝜏))+𝑞(𝜏)𝑟(𝛼(𝜏)−𝑢_𝑛0(𝜏))+(𝜑_𝑝(𝛼^Δ(𝜏)))^∇],𝜏∈[𝑡_𝑛0,𝑇)_𝕋,1−[𝑞(𝜏)𝑓(2^𝑛0+1,𝜌_𝑛0)+𝑞(𝜏)𝑟(𝜌_𝑛0−𝑢_𝑛0(𝜏))],𝜏∈(0,𝑡_𝑛0)_𝕋,<0. (2.27) which implies a contraction.Case 2. Assume that 𝜏=𝑇. That is, 𝛼_𝑛0(𝑇)−𝑢_𝑛0(𝑇)>0, by (2.3), (2.8) and (2.10) together with 𝛼(𝑇)≤∑^𝑚_𝑖=1𝜓_𝑖(𝛼^Δ(𝜉_𝑖)), we have the following three subcases.
(a) If 𝑢^Δ_𝑛0(𝜉_𝑖)≤𝛼^Δ(𝜉_𝑖)for𝑖=1,2,…,𝑚, then 0<𝛼_𝑛0(𝑇)−𝑢_𝑛0(𝑇)=𝛼(𝑇)−^𝑚_𝑖=1𝜓^∗_𝑖(𝑢^Δ_𝑛0(𝜉_𝑖))−𝜌_𝑛0<^𝑚_𝑖=1𝜓_𝑖(𝛼^Δ(𝜉_𝑖))−^𝑚_𝑖=1𝜓^∗_𝑖(𝑢^Δ_𝑛0(𝜉_𝑖=))_𝑚𝑖=1𝜓_𝑖(𝛼^Δ(𝜉_𝑖))−^𝑚_𝑖=1𝜓_𝑖(𝛼^Δ(𝜉_𝑖))=0, (2.28) this is a contradiction.
(b) If 𝛼^Δ(𝜉_𝑖)<𝑢^Δ_𝑛0(𝜉_𝑖)for𝑖=1,2,…,𝑚. Assume that 𝑢^Δ_𝑛0(𝜉_𝑖)≤𝛽^Δ(𝜉_𝑖)for𝑖=1,2,…,𝑚, then ^𝑚_𝑖=1𝜓^∗_𝑖(𝑢^Δ_𝑛0(𝜉_𝑖))=^𝑚_𝑖=1𝜓_𝑖(𝑢^Δ_𝑛0(𝜉_𝑖)). (2.29)
Assume that 𝛽^Δ(𝜉_𝑖)<𝑢^Δ_𝑛0(𝜉_𝑖)for𝑖=1,2,…,𝑚, then ^𝑚_𝑖=1𝜓^∗_𝑖(𝑢^Δ_𝑛0(𝜉_𝑖))=^𝑚_𝑖=1𝜓_𝑖(𝛽^Δ(𝜉_𝑖)). (2.30)
Assume that there exist sequences {𝑖_𝑙1}={1,2,…,𝑙₁} and {𝑖_𝑘1}={1,2,…,𝑘₁} such that 𝛽^Δ(𝜉_𝑖𝑙1)<𝑢^Δ_𝑛0(𝜉_𝑖𝑙) and 𝑢^Δ_𝑛0(𝜉_𝑖𝑘1)≤𝛽^Δ(𝜉_𝑖𝑘1), here 𝑙₁+𝑘₁=𝑚, then ^𝑚_𝑖=1𝜓^∗_𝑖(𝑢^Δ_𝑛0(𝜉_𝑖))=^𝑘1_𝑖𝑘1=1𝜓_𝑖𝑘1(𝑢^Δ_𝑛0(𝜉_𝑖𝑘1))+^𝑙1_𝑖𝑙1=1𝜓_𝑖𝑙1(𝛽^Δ(𝜉_𝑖𝑙1)). (2.31) Hence, by (2.29), (2.30) and (2.31) together with the monotonicity of 𝜓_𝑖, we have 0<𝛼_𝑛0(𝑇)−𝑢_𝑛0(𝑇)=𝛼(𝑇)−^𝑚_𝑖=1𝜓^∗_𝑖(𝑢^Δ_𝑛0(𝜉_𝑖))−𝜌_𝑛0≤^𝑚_𝑖=1𝜓_𝑖(𝛼^Δ(𝜉_𝑖))−^𝑚_𝑖=1𝜓^∗_𝑖(𝑢^Δ_𝑛0(𝜉_𝑖))−𝜌_𝑛0<0, (2.32) this is a contradiction.
(c) If there exist sequences {𝑖_𝑙}={1,2,…,𝑙} and {𝑖_𝑘}={1,2,…,𝑘} such that 𝛼^Δ(𝜉_𝑖𝑙)<𝑢^Δ_𝑛0(𝜉_𝑖𝑙) and 𝑢^Δ_𝑛0(𝜉_𝑖𝑘)≤𝛼^Δ(𝜉_𝑖𝑘), here 𝑙+𝑘=𝑚. Essentially the same reasoning as before we have 0<𝛼_𝑛0(𝑇)−𝑢_𝑛0(𝑇)=𝛼(𝑇)−∑^𝑚_𝑖=1𝜓^∗_𝑖(𝑢^Δ_𝑛0(𝜉_𝑖))−𝜌_𝑛0<0, this is a contradiction.
Thus, Cases 1–2 imply (2.19) is established. In particular, since 𝛼(𝑡)≤𝛼_𝑛0(𝑡) for 𝑡∈[0,𝑇]_𝕋, we obtain 𝛼(𝑡)≤𝛼_𝑛0(𝑡)≤𝑢_𝑛0(𝑡)for𝑡∈[0,𝑇]_𝕋.
Essentially the same reasoning as the proof of inequality (2.19) we obtain 𝑢_𝑛0(𝑡)≤𝛽for𝑡∈[0,𝑇]_𝕋.
Hence 𝛼(𝑡)≤𝛼_𝑛0(𝑡)≤𝑢_𝑛0(𝑡)≤𝛽(𝑡)for𝑡∈[0,𝑇]_𝕋. (2.33)
Now, we discuss the boundary value problem (𝜑_𝑝(𝑢^Δ(𝑡)))^∇+𝑞(𝑡)ℎ^∗_𝑛0+1(𝑡,𝑢(𝑡))=0,𝑡∈(0,𝑇)_𝕋,𝑢(0)=𝜌_𝑛0+1,𝑢(𝑇)−^𝑚_𝑖=1𝜓^∗_𝑖(𝑢^Δ(𝜉_𝑖))=𝜌_𝑛0+1, (2.34) where ℎ^∗_𝑛0+1ℎ(𝑡,𝑢(𝑡))=^𝑛0+1(𝑡,𝛼_𝑛0+1(𝑡))+𝑟(𝛼_𝑛0+1(𝑡)−𝑢(𝑡)),𝑢(𝑡)≤𝛼_𝑛0+1ℎ(𝑡),_𝑛0+1(𝑡,𝑢(𝑡)),𝛼_𝑛0+1(𝑡)≤𝑢(𝑡)≤𝑢_𝑛0ℎ(𝑡),_𝑛0+1(𝑡,𝑢_𝑛0(𝑡))+𝑟(𝑢_𝑛0(𝑡)−𝑢(𝑡)),𝑢(𝑡)≥𝑢_𝑛0𝜓(𝑡),_∗𝑖(𝑧_𝑖𝜓)=^𝑖(𝛼^Δ(𝜉_𝑖)),𝑧_𝑖≤𝛼^Δ_𝑛0+1(𝜉_𝑖)=𝛼^Δ(𝜉_𝑖𝜓),_𝑖(𝑧_𝑖),𝛼^Δ_𝑛0+1(𝜉_𝑖)≤𝑧_𝑖≤𝑢^Δ_𝑛0(𝜉_𝑖𝜓),_𝑖(𝑢^Δ_𝑛0(𝜉_𝑖)),𝑧_𝑖≥𝑢^Δ_𝑛0(𝜉_𝑖),𝑖=1,…,𝑚. (2.35) Schauder's fixed point theorem guarantees that the boundary value problem (2.34) has a solution 𝑢_𝑛0+1(𝑡)∈𝐶^Δ[0,𝑇]_𝕋 with 𝜑_𝑝(𝑢^Δ_𝑛0+1(𝑡))∈𝐶^∇(0,𝑇)_𝕋.
Essentially the same reasoning as the proof of inequality (2.33), we have 𝛼(𝑡)≤𝛼_𝑛0+1(𝑡)≤𝑢_𝑛0+1(𝑡)≤𝑢_𝑛0(𝑡)for𝑡∈[0,𝑇]_𝕋. (2.36)
If there exists 𝑢_𝑘(𝑡) for some 𝑘∈{𝑛₀+1,𝑛₀+2,…} satisfying 𝛼_𝑘(𝑡)≤𝑢_𝑘(𝑡)≤𝑢_𝑘−1(𝑡) for 𝑡∈[0,𝑇]_𝕋. Then we investigate the boundary value problem (𝜑_𝑝(𝑢^Δ(𝑡)))^∇+𝑞(𝑡)ℎ^∗_𝑘+1(𝑡,𝑢(𝑡))=0,𝑡∈(0,𝑇)_𝕋,𝑢(0)=𝜌_𝑘+1,𝑢(𝑇)−∑^𝑚_𝑖=1𝜓^∗_𝑖(𝑢^Δ(𝜉_𝑖))=𝜌_𝑘+1, (2.37) where ℎ^∗_𝑘+1ℎ(𝑡,𝑢(𝑡))=^𝑘+1(𝑡,𝛼_𝑘+1(𝑡))+𝑟(𝛼_𝑘+1(𝑡)−𝑢(𝑡)),𝑢(𝑡)≤𝛼_𝑘+1ℎ(𝑡),_𝑘+1(𝑡,𝑢(𝑡)),𝛼_𝑘+1(𝑡)≤𝑢(𝑡)≤𝑢_𝑘ℎ(𝑡),_𝑘+1(𝑡,𝑢_𝑘(𝑡))+𝑟(𝑢_𝑘(𝑡)−𝑢(𝑡)),𝑢(𝑡)≥𝑢_𝑘𝜓(𝑡),_∗𝑖(𝑧_𝑖𝜓)=^𝑖(𝛼^Δ(𝜉_𝑖)),𝑧_𝑖≤𝛼^Δ_𝑘+1(𝜉_𝑖)=𝛼^Δ(𝜉_𝑖𝜓),_𝑖(𝑧_𝑖),𝛼^Δ_𝑘+1(𝜉_𝑖)≤𝑧_𝑖≤𝑢^Δ_𝑘(𝜉_𝑖𝜓),_Δ𝑖(𝑢_𝑘(𝜉_𝑖)),𝑧_𝑖≥𝑢_𝑘(𝜉_𝑖),𝑖=1,…,𝑚. (2.38) It follows from Schauder's fixed point theorem that the boundary value problem (2.37) has a solution 𝑢_𝑘+1(𝑡)∈𝐶^Δ[0,𝑇]_𝕋 with 𝜑_𝑝(𝑢^Δ_𝑘+1(𝑡))∈𝐶^∇(0,𝑇)_𝕋.
By using the similar arguments as above, we have 𝛼(𝑡)≤𝛼_𝑘+1(𝑡)≤𝑢_𝑘+1(𝑡)≤𝑢_𝑘(𝑡)for𝑡∈[0,𝑇]_𝕋. (2.39) Hence, for each 𝑛∈{𝑛₀,𝑛₀+1,…}, the mathematical induction implies that 𝛼(𝑡)≤𝛼_𝑛(𝑡)≤𝑢_𝑛(𝑡)≤𝑢_𝑛−1(𝑡)≤⋯≤𝑢_𝑛0(𝑡)≤𝛽(𝑡)for𝑡∈[0,𝑇]_𝕋. (2.40) Denote 𝑅_𝑛0=sup1{|𝑓(𝑡,𝑦)|∶𝑡∈[2^𝑛0+1,𝑇]_𝕋,𝛼(𝑡)≤𝑦≤𝑢_𝑛0(𝑡)}. (2.41) It follows from Lemma 1.6 that there exist 𝜏^∗₁,𝜏^∗₂∈[1/2^𝑛0+1,𝑇)_𝕋 satisfy 𝑢^Δ_𝑛(𝜏^∗₁𝑢)≤^𝑛(𝑇)−𝑢_𝑛(1/2^𝑛0+1)𝑇−1/2^𝑛0+1≤𝑢^Δ_𝑛(𝜏^∗₂). (2.42) From (2.42), we have 𝜑_𝑝(𝑢^Δ_𝑛(𝑡))=𝜑_𝑝(𝑢^Δ_𝑛(𝜏^∗₁))−^𝑡_𝜏∗1𝑞(𝑠)𝑓(𝑠,𝑢(𝑠))∇𝑠≤𝜑_𝑝(𝑢_𝑛(𝑇)−𝑢_𝑛(1/2^𝑛0+1)𝑇−1/2^𝑛0+1)+𝑅_𝑛0^𝑇_1/2𝑛0+1𝑞(𝑠)∇𝑠for1𝑡∈[2^𝑛0+1,𝑇]_𝕋,𝜑_𝑝(𝑢^Δ_𝑛(𝑡))=𝜑_𝑝(𝑢^Δ_𝑛(𝜏^∗₂))−^𝑡_𝜏∗2𝑞(𝑠)𝑓(𝑠,𝑢(𝑠))∇𝑠≥𝜑_𝑝(𝑢_𝑛(𝑇)−𝑢_𝑛(1/2^𝑛0+1)𝑇−1/2^𝑛0+1)−𝑅_𝑛0^𝑇_1/2𝑛0+1𝑞(𝑠)∇𝑠for1𝑡∈[2^𝑛0+1,𝑇]_𝕋. (2.43)
So there exists a positive number 𝐾₀ such that |𝑢^Δ_𝑛(𝑡)|≤𝐾₀. By Lemma 1.8, we have {𝑢_𝑛(𝑡)}^∞_𝑛=𝑛0+1isabounded,equicontinuousfamilyon1𝑡∈[2^𝑛0+1,𝑇]_𝕋. (2.44) The Arzela-Ascoli theorem on time scales [2] guarantees the existence of a subsequence ℕ_𝑛0 of integers and a function 𝑧_𝑛0(𝑡)∈𝐶[1/2^𝑛0+1,𝑇]_𝕋 with 𝑢_𝑛(𝑡) converging uniformly to 𝑧_𝑛0(𝑡) on [1/2^𝑛0+1,𝑇]_𝕋 as 𝑛→∞ through ℕ_𝑛0. Similarly {𝑢_𝑛(𝑡)}^∞_𝑛=𝑛0+1isabounded,equicontinuousfamilyon1𝑡∈[2^𝑛0+2,𝑇]_𝕋. (2.45) Thus there is a subsequence ℕ_𝑛0+1 of ℕ_𝑛0 and a function 𝑧_𝑛0+1(𝑡)∈𝐶[1/2^𝑛0+2,𝑇]_𝕋 with 𝑢_𝑛(𝑡) converging uniformly to 𝑧_𝑛0+1(𝑡) on [1/2^𝑛0+2,𝑇]_𝕋 as 𝑛→∞ through ℕ_𝑛0+1. Since ℕ_𝑛0+1⊆ℕ_𝑛0, we have 𝑧_𝑛0+1(𝑡)=𝑧_𝑛0(𝑡) on [1/2^𝑛0+1,𝑇]_𝕋. Proceed inductively to obtain subsequence of integers ℕ_𝑛0⊇ℕ_𝑛0+1⊇⋯⊇ℕ_𝑛⊇⋯ and functions 𝑧_𝑛(𝑡)∈𝐶[1/2^𝑛+1,𝑇]_𝕋 with 𝑢_𝑛(𝑡)converginguniformlyto𝑧_𝑛(𝑡)on[1/2^𝑛+1,𝑇]_𝕋 as 𝑛→∞throughℕ_𝑛 and 𝑧_𝑛(𝑡)=𝑧_𝑛−1(𝑡)on[1/2^𝑛,𝑇]_𝕋.
Now, we define a function 𝑢∶[0,𝑇]→[0,∞) with 𝑢(𝑡)=𝑧_𝑛(𝑡) on [1/2^𝑛+1,𝑇]_𝕋 and 𝑢(0)=0. Notice, 𝑢(𝑡) is well defined and 𝛼(𝑡)≤𝑢(𝑡)≤𝑢_𝑛0(𝑡)≤𝛽 for 𝑡∈(0,𝑇)_𝕋. Nextly fix 𝑡∈(0,𝑇)_𝕋 and let 𝑙∈{𝑛₀,𝑛₀+1,…} be such that 𝑡∈(1/2^𝑙+1,𝑇)_𝕋, let ℕ^∗_𝑙={𝑛∈ℕ_𝑙∶𝑛≥𝑙}, we have 𝜓^∗_𝑖(𝑢_𝑛(𝜉_𝑖))=𝜓_𝑖(𝑢_𝑛(𝜉_𝑖)),ℎ^∗_𝑛(𝑡,𝑢_𝑛(𝑡))=ℎ_𝑛(𝑡,𝑢_𝑛(𝑡))=𝑓(𝑡,𝑢_𝑛(𝑡))for𝑛∈ℕ^∗_𝑙. (2.46) Hence, for 𝑛∈ℕ^∗_𝑙, we have 𝑢_𝑛which is the positive solution of the following boundary value problem (𝜑_𝑝(𝑢^Δ_𝑛(𝑡)))^∇+𝑞(𝑡)𝑓(𝑡,𝑢_𝑛1(𝑡))=0,𝑡∈(2^𝑛,𝑇)_𝕋,𝑢_𝑛(0)=𝜌_𝑛,𝑢_𝑛(𝑇)−^𝑚_𝑖=1𝜓_𝑖(𝑢^Δ_𝑛(𝜉_𝑖))=𝜌_𝑛. (2.47) Let 𝑛→∞ through ℕ^∗_𝑙, we have 𝑢(𝑡)that satisfies (𝜑_𝑝(𝑢^Δ(𝑡)))^∇+𝑞(𝑡)𝑓(𝑡,𝑢(𝑡))=0,𝑡∈(0,𝑇)_𝕋,𝑢(0)=0,𝑢(𝑇)−^𝑚_𝑖=1𝜓_𝑖(𝑢^Δ(𝜉_𝑖))=0. (2.48) It remains to show that 𝑢(𝑡) is continuous at 0. Now by lim_𝑛→∞𝑢_𝑛(0)=0, there exists 𝑛₁∈{𝑛₀,𝑛₀+1,…} with 𝑢_𝑛1(0)<𝜀/2. Since 𝑢_𝑛1(𝑡)∈𝐶[0,𝑇]_𝕋 there exists 𝛿_𝑛1∈(0,𝑇)_𝕋 with 𝑢_𝑛1(𝑡)<𝜀/2 for 𝑡∈[0,𝛿_𝑛1)_𝕋. By the monotonicity of {𝑢_𝑛(𝑡)}_𝑛∈ℕ0 for each 𝑡∈[0,𝑇]_𝕋, we have 𝛼(𝑡)≤𝑢_𝑛(𝑡)≤𝑢_𝑛1(𝑡)<𝜀/2for𝑡∈[0,𝛿_𝑛1)_𝕋and𝑛≥𝑛₁, which means 𝛼(𝑡)≤𝑢(𝑡)<𝜀/2 for 𝑡∈[0,𝛿_𝑛1)_𝕋. So 𝑢(𝑡) is continuous at 0.

If we replace 𝑡∈[1/2^𝑛+1,𝑇]_𝕋 with 𝑡∈[1/2^𝑛+1,𝑇−1/2^𝑛+1]_𝕋, the singularity occurs at 𝑢=0,𝑡=0 and 𝑡=𝑇.

If we replace 𝑡∈[1/2^𝑛+1,𝑇]_𝕋 with 𝑡∈[0,𝑇−1/2^𝑛+1]_𝕋, the singularity occurs at 𝑢=0 and 𝑡=𝑇.

If we replace 𝑡∈[1/2^𝑛+1,𝑇]_𝕋 with 𝑡∈[0,𝑇]_𝕋, the singularity occurs at 𝑢=0.

So it is easily obtain the analogue of Theorem 2.1 in this section. See the following remark.

Remark 2.2. If (A3) is appropriately adjusted, we can replace 𝑡∈[1/2^𝑛+1,𝑇]_𝕋 in (A1) by 1𝑡∈[2^𝑛+11,𝑇−2^𝑛+1]_𝕋,1𝑡∈[0,𝑇−2^𝑛+1]_𝕋, (2.49)(2.50) or 𝑡∈[0,𝑇]_𝕋. (2.51) For example, if (2.49) occurs, (A3) is replaced by
(A3^′) There exists a function 𝛽∈𝐶[0,𝑇]_𝕋∩𝐶^Δ(0,𝑇]_𝕋,𝜑_𝑝(𝛽^Δ)∈𝐶^∇(0,𝑇)_𝕋 such that 𝛽≥𝛼and𝛽≥𝜌_𝑛0 for 𝑡∈[0,𝑇]_𝕋,𝛽(𝑇)−∑^𝑚_𝑖=1𝜓_𝑖(𝛽^Δ(𝜉_𝑖))>0,−(𝜑_𝑝(𝛽^Δ))^∇≥𝑞(𝑡)𝑓(𝑡,𝛽)for𝑡∈(0,𝑇)_𝕋,−(𝜑_𝑝(𝛽^Δ))^∇≥𝑞(𝑡)𝑓(1/2^𝑛0+1,𝛽) for 𝑡∈(0,1/2^𝑛0+1)_𝕋 and −(𝜑_𝑝(𝛽^Δ))^∇≥𝑞(𝑡)𝑓(𝑇−1/2^𝑛0+1,𝛽)for𝑡∈(𝑇−1/2^𝑛0+1,𝑇)_𝕋.
Assume that (H1)–(H3), (A1) and (A2) hold, and in addition suppose the following conditions are satisfied:
(A4) −(𝜑_𝑝(𝛼^Δ))^∇<𝑞(𝑡)𝑓(𝑡,𝑢) for (𝑡,𝑢)∈(0,𝑇]_𝕋×{𝑢∈𝐶[0,𝑇]_𝕋∩𝐶^Δ(0,𝑇]_𝕋∶0<𝑢≤𝛼}; (2.52)
(A5) There exists a function 𝛽∈𝐶[0,𝑇]_𝕋∩𝐶^Δ(0,𝑇]_𝕋,𝜑_𝑝(𝛽^Δ)∈𝐶^∇(0,𝑇)_𝕋 such that 𝛽≥𝜌_𝑛0 for 𝑡∈[0,𝑇]_𝕋,𝛽(𝑇)−∑^𝑚_𝑖=1𝜓_𝑖(𝛽^Δ(𝜉_𝑖))>0,−(𝜑_𝑝(𝛽^Δ))^∇≥𝑞(𝑡)𝑓(𝑡,𝛽) for 𝑡∈(0,𝑇)_𝕋 and −(𝜑_𝑝(𝛽^Δ))^∇≥𝑞(𝑡)𝑓(1/2^𝑛0+1,𝛽) for 𝑡∈(0,1/2^𝑛0+1)_𝕋;
(A6) 𝛽(𝑇)≥𝛼(𝑇).

Then the result in Theorem 2.1 is also true. This follows immediately from Theorem 2.1 if we show (A3) holds. That is to say, if we show 𝛽≥𝛼 for 𝑡∈[0,𝑇]_𝕋, then the result holds. Assume it is not true, in view of (A6) we obtain 𝛽−𝛼 has a negative minimum for some 𝜏₆∈(0,𝑇)_𝕋, so (𝛽−𝛼)^Δ(𝜏₆)≥0 and essentially the same reasoning as the proof of inequality (2.20), we have (𝜑_𝑝(𝛼^Δ))^∇(𝜏₆)≤(𝜑_𝑝(𝛽^Δ))^∇(𝜏₆). However, by (A4), (A5) and 𝛼(𝜏₆)>𝛽(𝜏₆)>0, we obtain −(𝜑_𝑝(𝛼^Δ))^∇(𝜏₆)<𝑞(𝜏₆)𝑓(𝜏₆,𝛽(𝜏₆)). Hence (𝜑_𝑝(𝛼^Δ))^∇(𝜏₆)−(𝜑_𝑝(𝛽^Δ))^∇(𝜏₆)≥(𝜑_𝑝(𝛼^Δ))^∇(𝜏₆)+𝑞(𝜏₆)𝑓(𝜏₆,𝛽(𝜏₆))>0, which implies a contradiction.

Corollary 2.3. Let 𝑛₀∈{1,2,…} be fixed, suppose (H1)–(H3), (A1), (A2) and (A4)–(A6) hold, then the boundary value problem (1.1) and (1.2) has a solution 𝑢∈𝐶[0,𝑇]_𝕋∩𝐶^Δ(0,𝑇]_𝕋,𝜑_𝑝(𝑢^Δ)∈𝐶^∇(0,𝑇)_𝕋 with 𝑢≥𝛼 for 𝑡∈[0,𝑇]_𝕋.

3. Construction of 𝜶 and 𝜷

In this section, we consider how to construct a lower solution 𝛼 and an upper solution 𝛽 in certain circumstances. In this section, we assume that ^𝑚_𝑖=1𝜓_𝑖(𝑥_𝑖)≥0for𝑥_𝑖∈ℝ. (3.1)

Lemma 3.1. Assume that there exists a nonincreasing positive sequence {𝜀_𝑛} with lim_𝑛→∞𝜀_𝑛=0, then there exist a function 𝜆(𝑡)∈𝐶^Δ[0,𝑇]_𝕋 satisfying(i)𝜑_𝑝(𝜆^Δ(𝑡))∈𝐶^∇[0,𝑇]_𝕋,𝜆(𝑡)>0 for 𝑡∈(0,𝑇]_𝕋 and max_{𝑡∈[0,𝑇]𝕋}|(𝜑_𝑝(𝜆^Δ(𝑡)))^∇|>0;(ii)𝜆(0)=0,𝜆(𝑇)−∑^𝑚_𝑖=1𝜓_𝑖(𝜆^Δ(𝜉_𝑖))<0 and 0<𝜆(𝑡)≤𝜀_𝑛 for 𝑡∈(0,𝑇]_𝕋.

Proof. Let 𝑒_𝑛=[1/2^𝑛+1,𝑇]_𝕋(𝑛≥𝑛₀). Assume that 𝑟∶[0,𝑇]_𝕋→[0,∞) be such that 𝑟(0)=0,𝑟(𝑡)=𝜀_𝑛^𝑝−1/(2𝑇)^𝑝+1 for 𝑡∈𝑒_𝑛⧵𝑒_𝑛−1,𝑛≥𝑛₀ and 𝑟(𝑡)=𝜀_𝑛^𝑝−1₀/(2𝑇)^𝑝+1 for 𝑡∈[1/2^𝑛0,𝑇]_𝕋. Let 𝑢(𝑡)=∫^𝑡₀𝑟(𝑠)Δ𝑠,𝑣(𝑡)=[∫^𝑡₀𝑢(𝑠)∇𝑠]^1/(𝑝−1),𝑤(𝑡)=∫^𝑡₀𝑣(𝑠)Δ𝑠. Suppose 𝜏₇∈𝑒_𝑛⧵𝑒_𝑛−1for𝑛≥𝑛₀,𝜏₈∈(0,𝑇]_𝕋 satisfy 𝜏₇<𝜏₈ and 2𝜏₈−𝑇≥𝜏₇. It is easy to show that 𝑢,𝑣,𝑤∶[0,𝜏₇]_𝕋→[0,∞) are continuous and increasing. Denote 𝑎(𝑡)=[𝑐₀(𝜏₈−𝑡)+𝑐₁𝑡]^1/(𝑝−1)for𝑡∈[𝜏₇,𝑇]_𝕋, (3.2) here 𝑐₀𝜏=−⁷𝜏₈𝑢(𝜏₇1)+𝜏₈(𝑣(𝜏₇))^𝑝−1,𝑐₁=𝜏₈−𝜏₇𝜏₈𝑢(𝜏₇1)+𝜏₈(𝑣(𝜏₇))^𝑝−1. (3.3) Hence, 𝑎(𝑡)>0 for 𝑡∈[𝜏₇,𝑇]_𝕋 and is nondecreasing. Define 𝑏(𝑡)=^𝑡_𝜏7𝑎(𝑠)Δ𝑠+𝑤(𝜏₇)for𝑡∈[𝜏₇,𝜏₈]_𝕋,𝑃(𝑡)=𝑏(𝑡),𝑡∈[𝜏₇,𝜏₈]_𝕋,𝑏(2𝜏₈−𝑡),𝑡∈[𝜏₈,𝑇]_𝕋,𝜆(𝑡)=𝑤(𝑡),𝑡∈[0,𝜏₇]_𝕋,𝑃(𝑡),𝑡∈[𝜏₇,𝑇]_𝕋. (3.4) We can easily prove 𝑤(𝜏₇)=𝑃(𝜏₇),𝑤^Δ(𝜏₇)=𝑃^Δ(𝜏₇),(𝜑_𝑝(𝑤^Δ))^∇(𝜏₇)=(𝜑_𝑝(𝑃^Δ))^∇(𝜏₇) and 𝑤∈𝐶^Δ[0,𝜏₇]_𝕋,𝑃∈𝐶^Δ[𝜏₇,𝑇]_𝕋,𝜑_𝑝(𝑤^Δ)∈𝐶^∇[0,𝜏₇]_𝕋,𝜑_𝑝(𝑃^Δ)∈𝐶^∇[𝜏₇,𝑇]_𝕋. Thus, we have 𝜆∈𝐶^Δ[0,𝑇]_𝕋 and 𝜑_𝑝(𝜆^Δ)∈𝐶^∇[0,𝑇]_𝕋 with max_{0≤𝑡≤𝑇}|(𝜑_𝑝(𝜆^Δ))^∇(𝑡)|>0. Now since 𝑤(𝑡)>0 for 𝑡∈(0,𝜏₇]_𝕋 and 𝑃(𝑡)>0 for 𝑡∈[𝜏₇,𝑇]_𝕋, we have 𝜆(𝑡)>0 for 𝑡∈(0,𝑇]_𝕋. On the other hand, 𝑢(𝜏₇)=∫^𝜏7₀𝑟(𝑠)Δ𝑠≤𝜏₇𝜀_𝑛^𝑝−1(2𝑇)^𝑝+1<𝜀_𝑛^𝑝−1(2𝑇)^𝑝,𝑣(𝜏₇)=[^𝜏7₀𝑢(𝑠)∇𝑠]^1/(𝑝−1)<(𝜏₇𝜀_𝑛^𝑝−1(2𝑇)^𝑝)^1/(𝑝−1)<𝜀_𝑛2^{𝑝/(𝑝−1)}𝑇,𝑤(𝜏₇)<𝜏₇×𝜀_𝑛2^{𝑝/(𝑝−1)}𝑇<𝜀_𝑛2, (3.5) by the monotonicity of 𝑃(𝑡) on [𝜏₇,𝜏₈]_𝕋,[𝜏₈,𝑇]_𝕋, respectively, we have 𝜆(𝜏₈)=max_{𝑡∈[𝜏7,𝑇]𝕋}=𝜆(𝑡)_𝜏8𝜏7𝑎(𝑠)Δ𝑠+𝑤(𝜏₇)≤(𝜏₈−𝜏₇)max_{𝑡∈[𝜏7,𝜏8]𝕋}[𝑐₀(𝜏₈−𝑡)+𝑐₁𝑡]^1/(𝑝−1)+𝑤(𝜏₇)≤(𝜏₈−𝜏₇)[(𝜏₈−𝜏₇)𝑢(𝜏₇)+(𝑣(𝜏₇))^𝑝−1]^1/(𝑝−1)+𝑤(𝜏₇)𝜀<𝑇[𝑇^𝑛𝑝−1(2𝑇)^𝑝+𝜀_𝑛^𝑝−12(2𝑇)^𝑝−1]^1/(𝑝−1)+𝜀_𝑛2<𝜀_𝑛2+𝜀_𝑛2=𝜀_𝑛. (3.6) Consequently, 0<𝜆(𝑡)≤𝜀_𝑛,𝑡∈(0,𝑇]_𝕋.
Without loss of generality, ∑^𝑚_𝑖=1𝜓_𝑖(𝜆^Δ(𝜉_𝑖))≥𝜀_𝑛>𝜆(𝑇). We have 𝜆(𝑇)−^𝑚_𝑖=1𝜓_𝑖(𝜆^Δ(𝜉_𝑖))<0. (3.7)

Now we discuss how to construct a lower solution 𝛼(𝑡) in (A2) and (A4).

(A7) For each 𝑛∈{1,2,…}, there exist a constant 𝑘₀ and a strictly monotone decreasing sequence {𝜌_𝑛} with lim_𝑛→∞𝜌_𝑛=0, and 𝑞(𝑡)𝑓(𝑡,𝑢)≥𝑘₀ for (𝑡,𝑢)∈[1/2^𝑛+1,𝑇]_𝕋×{𝑢∈𝐶[0,𝑇]_𝕋∩𝐶^Δ(0,𝑇]_𝕋∶0<𝑢≤𝜌_𝑛};

(A8) There exists a function 𝛽∈𝐶[0,𝑇]_𝕋∩𝐶^Δ(0,𝑇]_𝕋,𝜑_𝑝(𝛽^Δ)∈𝐶^∇(0,𝑇)_𝕋 such that 𝛽≥0 for 𝑡∈[0,𝑇]_𝕋,𝛽(𝑇)−∑^𝑚_𝑖=1𝜓_𝑖(𝛽^Δ(𝜉_𝑖))>0, −(𝜑_𝑝(𝛽^Δ))^∇≥𝑞(𝑡)𝑓(𝑡,𝛽)for𝑡∈(0,𝑇)_𝕋 and −(𝜑_𝑝(𝛽^Δ))^∇≥𝑞(𝑡)𝑓(1/2^𝑛0+1,𝛽) for 𝑡∈(0,1/2^𝑛0+1)_𝕋.

Theorem 3.2. Let 𝑛₀∈{3,4,…} be fixed. If (H1)–(H3), (3.1) and (A7)-(A8) hold, then boundary value problem (1.1) and (1.2) has a solution 𝑢∈𝐶[0,𝑇]_𝕋∩𝐶^Δ(0,𝑇]_𝕋 with 𝜑_𝑝(𝑢^Δ)∈𝐶^∇(0,𝑇)_𝕋 and 𝑢(𝑡)>0 for 𝑡∈(0,𝑇]_𝕋.

Proof. By Corollary 2.3, we need only show that conditions (A1), (A2), (A4)–(A6) are satisfied. Without loss of generality, suppose 𝛽(𝑡)>𝜌_𝑛0for𝑡∈[0,𝑇]_𝕋,𝛽(𝑇)−^𝑚_𝑖=1𝜓_𝑖(𝛽^Δ(𝜉_𝑖))>𝜌_𝑛0, (3.8) by (A7), (A8) and (3.8), we obtain that (A1) and ( A5) hold.
From Lemma 3.1 there exists a function 𝜆(𝑡)∈𝐶^Δ[0,𝑇]_𝕋 satisfying(i)𝜑_𝑝(𝜆^Δ(𝑡))∈𝐶^∇[0,𝑇]_𝕋,𝜆(𝑡)>0 for 𝑡∈(0,𝑇]_𝕋 and 𝑅₁=max_{𝑡∈[0,𝑇]𝕋}|(𝜑_𝑝(𝜆^Δ(𝑡)))^∇|>0.(ii)𝜆(0)=0,𝜆(𝑇)−∑^𝑚_𝑖=1𝜓_𝑖(𝜆^Δ(𝜉_𝑖))<0 and 0<𝜆(𝑡)≤𝜌_𝑛 for 𝑡∈(0,𝑇]_𝕋.
Assume 𝑚=min{1,(𝑘₀/2𝑅₁)^1/(𝑝−1),𝜌_𝑛0/|𝜆|}. Let 𝛼(𝑡)=𝑚𝜆(𝑡) for 𝑡∈[0,𝑇]_𝕋. Then 𝛼(𝑡)∈𝐶[0,𝑇]_𝕋∩𝐶^Δ(0,𝑇]_𝕋,𝜑_𝑝(𝛼^Δ(𝑡))∈𝐶^∇(0,𝑇)_𝕋,𝛼(0)=0 with 0<𝛼(𝑡)≤𝜆(𝑡)≤𝜌_𝑛 for 𝑡∈(1/2^𝑛+1,𝑇]_𝕋. Without loss of generality, we have 𝛼(𝑇)−∑^𝑚_𝑖=1𝜓_𝑖(𝛼^Δ(𝜉_𝑖))<0. For arbitrary (𝑡,𝑢)∈(0,𝑇]_𝕋×{𝑢∈𝐶[0,𝑇]_𝕋∩𝐶^Δ(0,𝑇]_𝕋∶0<𝑢≤𝛼(𝑡)}, there exists 𝑛∈{𝑛₀,𝑛₀+1,…} such that (𝑡,𝑢)∈[1/2^𝑛+1,𝑇]_𝕋×{𝑢∈𝐶[0,𝑇]_𝕋∩𝐶^Δ(0,𝑇]_𝕋∶0<𝑢≤𝛼(𝑡)}. We have 𝑞(𝑡)𝑓(𝑡,𝑢)+(𝜑_𝑝(𝛼^Δ(𝑡)))^∇≥𝑘₀+(𝜑_𝑝(𝑚𝜆^Δ(𝑡)))^∇=𝑘₀+𝑚^𝑝−1(𝜑_𝑝(𝜆^Δ(𝑡)))^∇≥𝑘₀−𝑚^𝑝−1|(𝜑_𝑝(𝜆^Δ(𝑡)))^∇|≥𝑘₀−𝑘₀2𝑅₁|(𝜑_𝑝(𝜆^Δ(𝑡)))^∇|≥𝑘₀−𝑘₀2𝑅₁max_{𝑡∈[0,𝑇]}|(𝜑_𝑝(𝜆^Δ(𝑡)))^∇𝑘|=⁰2>0. (3.9) Thus (A4) holds and (A2) is also true if 𝑢(𝑡)=𝛼(𝑡). Also since 𝛼(𝑇)≤sup_{𝑡∈[0,𝑇]𝕋}|𝛼(𝑡)|=𝑚sup_{𝑡∈[0,𝑇]𝕋}|𝜆|≤𝜌_𝑛0, we have 𝛽(𝑇)≥𝜌_𝑛0≥𝛼(𝑇), then (A6) is fulfilled. By Corollary 2.3, the boundary value problem (1.1) and (1.2) has a solution 𝑢(𝑡)∈𝐶[0,𝑇]_𝕋∩𝐶^Δ(0,𝑇]_𝕋,𝜑_𝑝(𝑢^Δ(𝑡))∈𝐶^∇(0,𝑇)_𝕋 with 𝑢(𝑡)≥0 for 𝑡∈(0,𝑇]_𝕋.

We can replace 𝑡∈[1/2^𝑛+1,𝑇]_𝕋 with 𝑡∈[0,𝑇−1/2^𝑛+1]_𝕋 or 𝑡∈[1/2^𝑛+1,𝑇−1/2^𝑛+1]_𝕋. So it is easily obtain (see Remark 2.2) the analogue of Theorem 3.2 in this section.

Looking at Theorem 3.2, it is difficulty for us to discuss examples in constructing 𝛽(𝑡) in (A8). The following theorem removes (A8) and replaces it with an easy verified condition.

Theorem 3.3. Let 𝑛₀∈{1,2,…} be fixed. If (H1)–(H3), (A1) and (A2) hold, in addition suppose that the following conditions are satisfied: 𝑀₁>0,𝑀₂>max{sup_{𝑡∈[0,𝑇]}𝛼(𝑡),𝜌_𝑛0},here𝑀₁,𝑀₂areconstants,𝑞(𝑡)𝑓(𝑡,𝑀₁𝑡+𝑀₂)≤0for𝑡∈(0,𝑇)_𝕋,1𝑞(𝑡)𝑓(2^𝑛0+1,𝑀₁𝑡+𝑀₂)≤0for1𝑡∈(0,2^𝑛0+1)_𝕋,𝑀₁𝑇+𝑀₂−∑^𝑚_𝑖=1𝜓_𝑖(𝑀₁)>0. (3.10)(3.11) Then boundary value problem (1.1) and (1.2) has a solution 𝑢∈𝐶[0,𝑇]_𝕋∩𝐶^Δ(0,𝑇]_𝕋 with 𝜑_𝑝(𝑢^Δ)∈𝐶^∇(0,𝑇)_𝕋 and 𝑢>0 for 𝑡∈(0,𝑇]_𝕋.

Proof. Denote 𝛽(𝑡)=𝑀₁𝑡+𝑀₂ for 𝑡∈[0,𝑇]_𝕋, then 𝛽(𝑡)∈𝐶[0,𝑇]_𝕋∩𝐶^Δ(0,𝑇]_𝕋,𝜑_𝑝(𝛽^Δ(𝑡))∈𝐶^∇(0,𝑇)_𝕋 with 𝛽(𝑡)≥𝛼(𝑡) and 𝛽(𝑡)≥𝜌_𝑛0 for 𝑡∈[0,𝑇]_𝕋,𝛽(𝑇)−^𝑚_𝑖=1𝜓_𝑖(𝛽^Δ(𝜉_𝑖))>0, (3.12) with −(𝜑_𝑝(𝛽^Δ(𝑡)))^∇≥𝑞(𝑡)𝑓(𝑡,𝛽)for𝑡∈(0,𝑇)_𝕋,−(𝜑_𝑝(𝛽(𝑡)^Δ(𝑡)))^∇1≥𝑞(𝑡)𝑓(2^𝑛0+1,𝛽(𝑡))for1𝑡∈(0,2^𝑛0+1)_𝕋, (3.13) then (A3) holds. By Theorem 2.1 the result holds.

From Theorems 3.2 and 3.3 we have the following theorem.

Theorem 3.4. Let 𝑛₀∈{1,2,…} be fixed. If (H1)–(H3), (3.1) and (A7) hold, in addition suppose there exist constants 𝑀₁,𝑀₂>0 such that (3.11) and (3.5) are true. Then the problem (1.1) and (1.2) has a solution 𝑢∈𝐶[0,𝑇]_𝕋∩𝐶^Δ(0,𝑇]_𝕋 with 𝜑_𝑝(𝑢^Δ)∈𝐶^∇(0,𝑇)_𝕋 and 𝑢>0 for 𝑡∈(0,𝑇]_𝕋.

Proof. Without loss of generality suppose 𝜌_𝑛0<𝑀₂, by (A7) we have (A1) which holds and 𝑀₂>𝜌_𝑛0>𝜌_𝑛0+1>…,lim_𝑛→∞𝜌_𝑛=0. (3.14) By the similar way as the proof of the Theorem 3.2, there exists a function 𝛼∈𝐶[0,𝑇]_𝕋∩𝐶^Δ(0,𝑇]_𝕋,𝜑_𝑝(𝛼^Δ)∈𝐶^∇(0,𝑇)_𝕋 with 𝛼(0)=0,𝛼(𝑇)−∑^𝑚_𝑖=1𝜓_𝑖(𝛼^Δ(𝜉_𝑖))<0,𝛼(𝑡)>0 for 𝑡∈(0,𝑇]_𝕋, such that −(𝜑_𝑝(𝛼^Δ(𝑡)))^∇≤𝑞(𝑡)𝑓(𝑡,𝛼(𝑡)) for 𝑡∈(0,𝑇)_𝕋 and 𝛼(𝑡)≤sup_{𝑡∈[0,𝑇]𝕋}|𝛼|≤𝜌_𝑛0. This together with (3.14) we have 𝑀₂>max{sup_{𝑡∈[0,𝑇]}𝛼(𝑡),𝜌_𝑛0}. Thus all the conditions of the Theorem 3.3 are fulfilled.

4. An Example

In this section, we present an example to illustrate our results. Let 1𝕋={0}∪{(2)^ℕ1}∪[2,1]. (4.1) Consider the following boundary value problem −(|𝑢^Δ(𝑡)|²𝑢^Δ(𝑡))^∇=𝑞(𝑡)𝑓(𝑡,𝑢(𝑡))for𝑡∈(0,1)_𝕋,1𝑢(0)=0,𝑢(1)−5𝑢^Δ(181)−𝑢10^Δ(141)−5𝑢^Δ(341)−𝑢10^Δ(1)=0. (4.2) It is obvious that 𝑇=1,𝑝=4,𝑚=4, 𝜓₁(𝑥)=𝜓₃(𝑥)=(1/5)𝑥,𝜓₂(𝑥)=𝜓₄(𝑥)=(1/10)𝑥. Denote 𝑞(𝑡)=𝑡⁸+5and𝑓(𝑡,𝑢(𝑡))=𝑡/𝑢⁷(𝑡)+𝑢⁷(𝑡)−𝜆², here 𝜆²≥2 is constant. Let 𝑛₀∈{1,2,…},𝜌_𝑛=(1/2^𝑛+1(𝜆²+𝑎₁))^1/7 and 𝑘₀=𝑎₁>0isaconstant. We have 𝜌_𝑛0≤1. Note that (H1)–(H3) and (3.1) hold. For 𝑛∈{1,2,…},𝑡∈[1/2^𝑛+1,1]_𝕋 and 0<𝑢≤𝜌_𝑛, we have 𝑞(𝑡)𝑓(𝑡,𝑢)≥(𝑡⁸1+5)(2^𝑛+1𝜌⁷_𝑛−𝜆²)≥(𝑡⁸+5)(𝜆²+𝑎₁−𝜆²)>𝑎₁, (4.3) which implies (A7) is satisfied.

Now we show that (A8) holds with 𝛽=𝑡^1/7.

Notice that if 𝑡∈(1/2,1]_𝕋, then 𝛽^Δ(𝑡)=𝛽^′(𝑡)=(1/7)𝑡^−6/7,|𝛽^Δ(𝑡)|²𝛽^Δ1(𝑡)=𝑡343^−18/7,(|𝛽^Δ(𝑡)|²𝛽^Δ(𝑡))^∇=−18𝑡2401^−25/7≤0. (4.4)

If 𝑡=1/2, then 𝛽^Δ(𝑡)=(1/7)𝑡^−6/7 and |𝛽^Δ(𝑡)|²𝛽^Δ(𝑡)=(1/343)𝑡^−18/7, (|𝛽^Δ(𝑡)|²𝛽^Δ(𝑡))^∇=2(13432)^−18/7−2(13434)^−18/7≈−0.1714≤0. (4.5)

If 𝑡=1/2^𝑛(𝑛=2,3,…), then 𝜎(𝑡)=2𝑡,𝜌(𝑡)=𝑡/2, 𝜇(𝑡)=𝑡,𝜈(𝑡)=𝑡/2, we have 𝛽^Δ1(𝑡)=𝑡[(2𝑡)^1/7−𝑡^1/7],|𝛽^Δ(𝑡)|²𝛽^Δ1(𝑡)=𝑡³[(2𝑡)^1/7−𝑡^1/7]³, (4.6) by induction, one gets (|𝛽^Δ(𝑡)|²𝛽^Δ(𝑡))^∇=2^4𝑛+11[(2^𝑛−1)^1/71−(2^𝑛)^1/7]³−2^4𝑛+41[(2^𝑛)^1/71−(2^𝑛+1)^1/7]³≤0. (4.7) Thus, for 𝑡∈(0,1)_𝕋, we have (|𝛽^Δ(𝑡)|²𝛽^Δ(𝑡))^∇+𝑞(𝑡)𝑓(𝑡,𝛽)≤(𝑡⁸𝑡+5)(𝑡+(𝑡^1/7)⁷−𝜆²)≤(𝑡⁸+5)(2−𝜆²)≤0,(|𝛽^Δ(𝑡)|²𝛽^Δ(𝑡))^∇1+𝑞(𝑡)𝑓(2^𝑛0+1,𝛽)≤(𝑡⁸1+5)(2^𝑛0+1(𝑡^1/7)+(𝑡^1/7)⁷−𝜆²)≤(𝑡⁸+5)(2−𝜆²)≤0for1𝑡∈(0,2^𝑛0+1)_𝕋. (4.8) Now 𝛽(1)−𝜓₁(𝛽^Δ(18))−𝜓₂1(𝛽(4))−𝜓₃(𝛽^Δ(34))−𝜓₄(𝛽(1))>0. (4.9) Hence, all conditions of the Theorem 3.2 are satisfied. As a result, the problem (4.2) has a positive solution.

Acknowledgments

This paper is supported by XZIT under Grant XKY2008311 and DEGP under Grant 0709-03.