First-Order Boundary Value Problem with Nonlinear Boundary Condition on Time Scales

Abstract

This work is concerned with the following first-order dynamic equation on time scale 𝕋,  𝑥^Δ(𝑡)+𝑝(𝑡)𝑥(𝜎(𝑡))=𝑓(𝑡,𝑥(𝑡)),𝑡∈[0,𝑇]_𝕋 with the nonlinear boundary condition 𝑥(0)=𝑔(𝑥(𝜎(𝑇))). By applying monotone iteration method, we not only obtain the existence of positive solutions, but also establish iterative schemes for approximating the solutions.

1. Introduction

The theory of time scales was introduced by Hilger in his Ph.D. thesis [1] in 1988 in order to unify continuous and discrete analysis. The study of dynamic equations on time scales is a fairly new subject and research in this area is rapidly growing. For some basic definitions and relevant results on time scales, see [2, 3].

Recently, first-order boundary value problems (BVPs for short) on time scales have attracted much attention from many authors. For example, for first-order periodic boundary value problem (PBVP for short) on time scales𝑢^Δ(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡∈[𝑎,𝑏]_𝕋,𝑢(𝑎)=𝑢(𝜎(𝑏)), (1.1) Cabada [4] developed the method of lower and upper solutions coupled with the monotone iterative techniques to derive the existence of extremal solutions. In [5], Cabada and Vivero studied the existence of solutions for the first-order dynamic equation with nonlinear functional boundary value conditions𝑢^Δ(𝑡)=𝑓𝑡,𝑢^𝜎(𝑡),forΔ−a.e.𝑡∈[𝑎,𝑏]_𝕋,𝐵(𝑢(𝑎),𝑢)=0. (1.2) They proved the uniqueness of solutions and developed the monotone iterative technique when 𝐵(𝑥,𝑦)=𝑥−𝑔(𝑦^𝜎(𝑏)) and 𝑔 was a continuous and nonincreasing function. In 2007, Sun and Li [6] considered the first-order PBVP on time scales𝑥^Δ(𝑡)+𝑝(𝑡)𝑥(𝜎(𝑡))=𝑔(𝑡,𝑥(𝜎(𝑡))),𝑡∈[0,𝑇]_𝕋,𝑥(0)=𝑥(𝜎(𝑇)). (1.3) Some existence criteria of at least one solution were established by using novel inequalities and the Schaefer fixed point theorem. In [7], by applying several well-known fixed point theorems, Sun and Li obtained some existence and multiplicity criteria of positive solutions for the first-order PBVP on time scales𝑥^Δ(𝑡)+𝑝(𝑡)𝑥(𝜎(𝑡))=𝑓(𝑥(𝑡)),𝑡∈[0,𝑇]_𝕋,𝑥(0)=𝑥(𝜎(𝑇)). (1.4) In 2010, Zhao and Sun [8] investigated the first-order PBVP on time scales𝑥^Δ(𝑡)+𝑝(𝑡)𝑥(𝜎(𝑡))=𝑓(𝑡,𝑥(𝑡)),𝑡∈[0,𝑇]_𝕋,𝑥(0)=𝑥(𝜎(𝑇)). (1.5) Some existence criteria of positive solutions were established, and the method used was the monotone iterative technique. For other related results, one can refer to [9–14] and the references therein.

Motivated greatly by the above-mentioned works, in this paper, we are interested in the existence and iteration of positive solutions for the following first-order dynamic equation with nonlinear boundary condition on time scales:𝑥^Δ(𝑡)+𝑝(𝑡)𝑥(𝜎(𝑡))=𝑓(𝑡,𝑥(𝑡)),𝑡∈[0,𝑇]_𝕋,𝑥(0)=𝑔(𝑥(𝜎(𝑇))), (1.6) where 𝕋 is an arbitrary time scale, 𝑇>0 is fixed, and 0,𝑇∈𝕋. For each interval 𝐈 of ℝ, we denote by 𝐈_𝕋=𝐈∩𝕋. Throughout this paper, we always assume that 𝑝∶[0,𝑇]_𝕋→(0,+∞) is right-dense continuous. Here, a solution 𝑥 of the BVP (1.6) is said to be positive if 𝑥 is nonnegative and nontrivial. By applying monotone iteration method, we not only obtain the existence of positive solutions for the BVP (1.6), but also establish iterative schemes for approximating the solutions. It is worth mentioning that the initial terms of our iterative schemes are constant functions, which implies that the iterative schemes are significant and feasible. In our arguments, the following monotone iteration method [15] is very crucial.

Theorem 1.1. Let 𝐾 be a normal cone of a Banach space 𝐸 and 𝑣₀≤𝑤₀. Suppose that(a₁)𝑇∶[𝑣₀,𝑤₀]→𝐸 is completely continuous,(a₂)𝑇 is monotone increasing on [𝑣₀,𝑤₀],(a₃)𝑣₀ is a lower solution of 𝑇, that is, 𝑣₀≤𝑇𝑣₀,(a₄)𝑤₀ is an upper solution of 𝑇; that is, 𝑇𝑤₀≤𝑤₀.Then, the iterative sequences 𝑣_𝑛=𝑇𝑣_𝑛−1,𝑤_𝑛=𝑇𝑤_𝑛−1(𝑛=1,2,3,…) (1.7) satisfy 𝑣₀≤𝑣₁≤⋯≤𝑣_𝑛≤⋯≤𝑤_𝑛≤⋯≤𝑤₁≤𝑤₀, (1.8) and converge to, respectively, 𝑣 and 𝑤∈[𝑣₀,𝑤₀], which are fixed points of 𝑇.

2. Main Results

Theorem 2.1. Assume that 𝑓∶[0,𝑇]_𝕋×[0,+∞)→[0,+∞) and 𝑔∶[0,+∞)→[0,+∞) are continuous and ∫₀^𝜎(𝑇)𝑓(𝑠,0)Δ𝑠>0. If there exists a constant 𝑅>0 such that the following conditions are satisfied: (H₁)𝑓(𝑡,𝑢)≤𝑓(𝑡,𝑣)≤𝑅/2𝜎(𝑇), 𝑡∈[0,𝑇]_𝕋, 0≤𝑢≤𝑣≤𝑅,(H₂)𝑔(𝑢)≤𝑔(𝑣)≤𝑅/2, 0≤𝑢≤𝑣≤𝑅,then the BVP (1.6) has positive solutions.

Proof. Let 𝐸=𝑥∣𝑥∶[0,𝜎(𝑇)]_𝕋→ℝiscontinuous (2.1) be equipped with the norm ‖𝑥‖=max_{[(𝑇𝑡∈0,𝜎)]𝕋}|𝑥(𝑡)|. (2.2) Then, 𝐸 is a Banach space. Denote 𝐾=𝑥∈𝐸∣𝑥(𝑡)≥0,𝑡∈[0,𝜎(𝑇)]_𝕋. (2.3) Then, 𝐾 is a normal cone of 𝐸. Now, if we define an operator Φ∶𝐾→𝐾 by (𝑡)=1Φ𝑥)(𝑒_𝑝(𝑡,0)^𝑡₀𝑒_𝑝(𝑠,0)𝑓(𝑠,𝑥(𝑠))Δ𝑠+𝑔(𝑥(𝜎(𝑇))),𝑡∈[0,𝜎(𝑇)]_𝕋, (2.4) then it is easy to know that fixed points of Φ are nonnegative solutions of the BVP (1.6).
Let 𝑣₀(𝑡)≡0 and 𝑤₀(𝑡)≡𝑅, 𝑡∈[0,𝜎(𝑇)]_𝕋. Now, we divide our proof into the following steps.
Step 1. We verify that Φ∶[𝑣₀,𝑤₀]→𝐾 is completely continuous.
First, we will show that Φ∶[𝑣₀,𝑤₀]→𝐾 is continuous.
Let 𝑥_𝑛(𝑛=1,2,…),𝑥∈[𝑣₀,𝑤₀] and lim_𝑛→∞𝑥_𝑛=𝑥. Then,0≤𝑥_𝑛(𝑡)≤𝑅,0≤𝑥(𝑡)≤𝑅,for𝑡∈[0,𝜎(𝑇)]_𝕋. (2.5)
For any given 𝜖>0, since 𝑓 is uniformly continuous on [0,𝑇]_𝕋×[0,𝑅], there exists 𝛿₁>0 such that for any 𝑢₁,𝑢₂∈[0,𝑅] with |𝑢₁−𝑢₂|<𝛿₁,𝑓𝑠,𝑢₁−𝑓𝑠,𝑢₂<𝜖2𝜎(𝑇)𝑒_𝑝(𝜎(𝑇))[],0,𝑠∈0,𝑇_𝕋. (2.6)
On the other hand, since 𝑔 is continuous at 𝑥(𝜎(𝑇)), there exists 𝛿₂>0 such that for any 𝑢∈[0,+∞) with |𝑢−𝑥(𝜎(𝑇))|<𝛿₂,𝑔(𝑢)(𝑥(𝜎(𝑇<𝜖−𝑔)))2. (2.7)
Let 𝛿=min{𝛿₁,𝛿₂}. Then, it follows from lim_𝑛→∞𝑥_𝑛=𝑥 that there exists a positive integer 𝑁 such that for any 𝑛>𝑁,𝑥_𝑛(𝑠)−𝑥(𝑠)<𝛿,𝑠∈[0,𝜎(𝑇)]_𝕋. (2.8)
In view of (2.6), (2.7), and (2.8), we know that for any 𝑛>𝑁,Φ𝑥_𝑛(𝑡)−(𝑡=1Φ𝑥)()𝑒_𝑝(𝑡,0)^𝑡₀𝑒_𝑝(𝑠,0)𝑓𝑠,𝑥_𝑛(𝑠)−𝑓(𝑠,𝑥(𝑠))Δ𝑠+𝑔𝑥_𝑛(𝜎(𝑇))−𝑔(𝑥(𝜎(𝑇)))≤𝑒_𝑝(𝜎(𝑇),0)₀^𝜎(𝑇)𝑓𝑠,𝑥_𝑛(𝑠)−𝑓(𝑠,𝑥(𝑠))Δ𝑠+𝑔𝑥_𝑛(𝜎(𝑇(𝑥(𝜎(𝑇[(𝑇))−𝑔)))<𝜖,𝑡∈0,𝜎)]_𝕋, (2.9) which indicates that lim_𝑛→∞Φ𝑥_𝑛=Φ𝑥. So, Φ∶[𝑣₀,𝑤₀]→𝐾 is continuous.
Next, we will show that Φ∶[𝑣₀,𝑤₀]→𝐾 is compact.
Let 𝑋⊂[𝑣₀,𝑤₀] be a bounded set. Then, there exists a constant 𝐿>0 such that for any 𝑥∈𝑋 and 𝑠∈[0,𝜎(𝑇)]_𝕋, 0≤𝑥(𝑠)≤𝐿. Define 𝑄=max_{𝑠∈[0,𝑇]𝕋,𝑢∈[0,𝐿]}𝑓(𝑠,𝑢) and 𝑀=max_{𝑢∈[0,𝐿]}𝑔(𝑢).
On the one hand, for any 𝑥∈𝑋, we have(𝑡)=1Φ𝑥)(𝑒_𝑝(𝑡,0)^𝑡₀𝑒_𝑝(𝑠,0)𝑓(𝑠,𝑥(𝑠))Δ𝑠+𝑔(𝑥(𝜎(𝑇)))≤𝑒_𝑝(𝜎(𝑇),0)₀^𝜎(𝑇)𝑓((𝑠(𝑥(𝜎(𝑇(𝑇)𝑒𝑠,𝑥))Δ𝑠+𝑔)))≤𝑄𝜎_𝑝(𝜎(𝑇),0)+𝑀,𝑡∈[0,𝜎(𝑇)]_𝕋, (2.10) which shows that Φ(𝑋) is uniformly bounded.
On the other hand, for any 𝑥∈𝑋 and 𝑡₁,𝑡₂∈[0,𝜎(𝑇)]_𝕋 with 𝑡₁≤𝑡₂, we have(Φ𝑥)𝑡₂−(Φ𝑥)𝑡₁=1𝑒_𝑝𝑡₂−1,0𝑒_𝑝𝑡₁,0^𝑡1₀𝑒_𝑝()𝑓((𝑠(𝑥(𝜎(𝑇+1𝑠,0𝑠,𝑥))Δ𝑠+𝑔)))𝑒_𝑝𝑡₂,0^𝑡2_𝑡1𝑒_𝑝()𝑓((𝑠≤1𝑠,0𝑠,𝑥))Δ𝑠𝑒_𝑝𝑡₁−1,0𝑒_𝑝𝑡₂,0₀^𝜎(𝑇)𝑒_𝑝()𝑓((𝑠(𝑥(𝜎(𝑇+𝑠,0𝑠,𝑥))Δ𝑠+𝑔)))_𝑡2𝑡1𝑒_𝑝()𝑓((𝑠≤1𝑠,0𝑠,𝑥))Δ𝑠𝑒_𝑝𝑡₁−1,0𝑒_𝑝𝑡₂(𝑇)𝑒,0𝑄𝜎_𝑝(𝜎(𝑇),0)+𝑀+𝑄𝑒_𝑝(𝜎(𝑇),0)𝑡₂−𝑡₁, (2.11) which implies that Φ(𝑋) is equicontinuous. Consequently, Φ∶[𝑣₀,𝑤₀]→𝐾 is compact.Step 2. We assert that Φ is monotone increasing on [𝑣₀,𝑤₀].
Suppose that 𝑢,𝑣∈[𝑣₀,𝑤₀] and 𝑢≤𝑣. Then, 0≤𝑢(𝑡)≤𝑣(𝑡)≤𝑅, 𝑡∈[0,𝜎(𝑇)]_𝕋. By (H₁) and (H₂), we have(𝑡)=1Φ𝑢)(𝑒_𝑝(𝑡,0)^𝑡₀𝑒_𝑝()𝑓((𝑠(𝑢(𝜎(𝑇≤1𝑠,0𝑠,𝑢))Δ𝑠+𝑔)))𝑒_𝑝(𝑡,0)^𝑡₀𝑒_𝑝()𝑓((𝑠(𝑣(𝜎(𝑇=(𝑡)[(𝑇𝑠,0𝑠,𝑣))Δ𝑠+𝑔)))Φ𝑣)(,𝑡∈0,𝜎)]_𝕋, (2.12) which shows that Φ𝑢≤Φ𝑣.Step 3. We prove that 𝑣₀ is a lower solution of Φ.
For any 𝑡∈[0,𝜎(𝑇)]_𝕋, it is obvious thatΦ𝑣₀𝑡)=1(𝑒_𝑝(𝑡,0)^𝑡₀𝑒_𝑝(𝑠,0)𝑓(𝑠,0)Δ𝑠+𝑔(0)≥0=𝑣₀(𝑡), (2.13) which implies that 𝑣₀≤Φ𝑣₀.Step 4. We show that 𝑤₀ is an upper solution of Φ.
In view of (H₁) and (H₂), we haveΦ𝑤₀(𝑡)=^𝑡₀𝑒_𝑝(𝑠,0)𝑒_𝑝()𝑓()𝑔(𝑅)𝑡,0𝑠,𝑅Δ𝑠+𝑒_𝑝()≤𝑡,0_𝑡0𝑓()(𝑅)≤𝑠,𝑅Δ𝑠+𝑔_0𝜎(𝑇)𝑓(𝑠,𝑅)Δ𝑠+𝑔(𝑅)≤𝑅=𝑤₀(𝑡),𝑡∈[0,𝜎(𝑇)]_𝕋, (2.14) which indicates that Φ𝑤₀≤𝑤₀.Step 5. We claim that the BVP (1.6) has positive solutions.
In fact, if we construct sequences {𝑣_𝑛}^∞_𝑛=1 and {𝑤_𝑛}^∞_𝑛=1 as follows:𝑣_𝑛=Φ𝑣_𝑛−1,𝑤_𝑛=Φ𝑤_𝑛−1,𝑛=1,2,3,…, (2.15) then it follows from Theorem 1.1 that 𝑣₀≤𝑣₁≤⋯≤𝑣_𝑛≤⋯≤𝑤_𝑛≤⋯≤𝑤₁≤𝑤₀, (2.16) and {𝑣_𝑛}^∞_𝑛=0 and {𝑤_𝑛}^∞_𝑛=0 converge to, respectively, 𝑣 and 𝑤∈[𝑣₀,𝑤₀], which are solutions of the BVP (1.6). Moreover, it follows from Φ𝑣₀(𝜎(𝑇=1))𝑒_𝑝(𝜎(𝑇),0)₀^𝜎(𝑇)𝑒_𝑝()𝑓()(0≥1𝑠,0𝑠,0Δ𝑠+𝑔)𝑒_𝑝(𝜎(𝑇),0)₀^𝜎(𝑇)𝑓(𝑠,0)Δ𝑠>0 (2.17) that 0<Φ𝑣₀(𝜎(𝑇))≤(Φ𝑣)(𝜎(𝑇))=𝑣(𝜎(𝑇))≤𝑤(𝜎(𝑇)), (2.18) which shows that 𝑣 and 𝑤 are positive solutions of the BVP (1.6).

Example 2.2. Let 𝕋=[0,1]∪[2,3]. We consider the following BVP on 𝕋: 𝑥^Δ(𝑡)(𝑡)𝑥(𝜎(𝑡=𝑡+𝑝))𝑥(𝑡)+39,𝑡∈[0,3]_𝕋,𝑥(0)=𝑥²(3).12 (2.19)
Since 𝑓(𝑡,𝑥)=𝑡√𝑥+3/9 and 𝑔(𝑥)=𝑥²/12, if we choose 𝑅=6, then all the conditions of Theorem 2.1 are fulfilled. So, it follows from Theorem 2.1 that the BVP (2.19) has positive solutions 𝑣 and 𝑤. Furthermore, if we construct sequences {𝑣_𝑛}^∞_𝑛=1 and {𝑤_𝑛}^∞_𝑛=1 as follows:𝑣_𝑛=Φ𝑣_𝑛−1,𝑤_𝑛=Φ𝑤_𝑛−1,𝑛=1,2,3,…, (2.20) where 𝑣₀(𝑡)≡0 and 𝑤₀(𝑡)≡6, then lim_𝑛→∞𝑣_𝑛=𝑣,lim_𝑛→∞𝑤_𝑛=𝑤. (2.21)

Acknowledgment

This paper is supported by the National Natural Science Foundation of China (10801068).