Positive Solutions for Nonlinear First-Order m-Point Boundary Value Problem on Time Scales

Abstract

By means of fixed-point theorems, we investigate the existence of positive solutions for nonlinear first-order 𝑚-point boundary value problem 𝑥^Δ(𝑡)+𝑎(𝑡)𝑥(𝜎(𝑡))=𝑓(𝑡,𝑥(𝜎(𝑡))),𝑡∈[𝑡₁,𝑡_𝑚]⊂𝕋, 𝑥(𝑡₁)=∑^𝑚−1_𝑘=2𝛼_𝑘𝑥(𝑡_𝑘)+𝛼₁𝑥(𝜎(𝑡_𝑚)), where 𝕋 is a time scale, 0≤𝑡₁<𝑡₂<⋯<𝑡_𝑚−1<𝑡_𝑚, 𝛼₁,𝛼₂,…,𝛼_𝑚−1≥0 are given constants.

1. Introduction

The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his Ph.D. thesis in 1988 (see [1]). The time scales calculus has a tremendous potential for applications in some mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics, neural networks, and social sciences; see the monographs of Aulbach and Hilger [2], Bohner and Peterson [3, 4], and Lakshmikantham et al. [5] and the references therein.

A time scale 𝕋 is an arbitrary nonempty closed subset of real numbers ℝ. A book on the subject of time scales by Bohner and Peterson [3] also summarizes and organizes much of the time scale calculus. The closed interval in 𝕋 is defined as [𝑎,𝑏]={𝑡∈𝕋∶𝑎≤𝑡≤𝑏}, (1.1) where 𝑎,𝑏∈𝕋 with 𝑎<𝜌(𝑏).

In this study, we consider the nonlinear first-order 𝑚-point boundary value problem 𝑥^Δ(𝑡)+𝑎(𝑡)𝑥(𝜎(𝑡))=𝑓(𝑡,𝑥(𝜎(𝑡))),𝑡∈𝑡₁,𝑡_𝑚𝑥𝑡⊂𝕋,₁=^𝑚−1_𝑘=2𝛼_𝑘𝑥𝑡_𝑘+𝛼₁𝑥𝜎𝑡_𝑚, (1.2) where 𝕋 is a time scale, 0≤𝑡₁<𝑡₂<⋯<𝑡_𝑚−1<𝑡_𝑚,𝛼₁,𝛼₂,…,𝛼_𝑚−1≥0 are given constants. 𝑎 is regressive and rd-continuous, and 𝑓∶[𝑡₁,𝜎(𝑡_𝑚)]×ℝ⁺→ℝ⁺ is continuous.

In [6], Cabada studied the following first-order periodic boundary value problem on time scales: 𝑢^Δ(𝑡)((𝑡[]𝑢(𝑎)(𝜎(𝑏.=𝑓𝑡,𝑢)),𝑡∈𝑎,𝑏⊂𝕋,=𝑢)) (1.3) He developed the monotone method in the presence of lower and upper solutions to obtain the existence of extremal solutions. When 𝛼₁=1,𝛼₂=⋯=𝛼_𝑚−1=0, and 𝑎(𝑡)≡0, BVP (1.2) is reduced to (1.3).

In [7], Sun studied the first-order boundary value problem 𝑥^Δ(𝑡)(𝑥(𝜎(𝑡[]𝑥(0)(𝜎(𝑇,=𝑓))),𝑡∈0,𝑇⊂𝕋,=𝛽𝑥)) (1.4) where 0<𝛽<1. Some existence results for at least two positive solutions were established, by using Avery-Henderson fixed-point theorem. When 𝛼₂=⋯=𝛼_𝑚−1=0 and 𝑎(𝑡)≡0, BVP (1.2) is reduced to (1.5).

In [8], Shu and Chunhua are concerned with the existence of three positive solutions for the following nonlinear first-order boundary value problem on time scale: 𝑥^Δ(𝑡)(𝑥(𝜎(𝑡[]𝑥(0)(𝜎(𝑇,=𝑓))),𝑡∈0,𝑇⊂𝕋,=𝜂𝑥)) (1.5) where 𝑇>0 is fixed, 0,𝑇∈𝕋, and 𝑓∶[0,∞)→[0,∞) is continuous. When 𝛼₂=⋯=𝛼_𝑚−1=0 and 𝑎(𝑡)≡0, BVP (1.2) is reduced to (1.5).

Sun and Li [9] studied the following first-order periodic boundary value problem on time scales: 𝑥^Δ(𝑡)(𝑡)𝑥(𝜎(𝑡((𝜎(𝑡[]𝑥(0)(𝜎(𝑇.+𝑝))=𝑔𝑡,𝑥))),𝑡∈0,𝑇⊂𝕋,=𝑥)) (1.6) Conditions for the existence of at least one solution were obtained by using novel inequalities and the Schaefer fixed-point theorem. When 𝛼₁=1 and 𝛼₂=⋯=𝛼_𝑚−1=0, BVP (1.2) is reduced to (1.6).

In [10], Tian and Ge studied the existence and uniqueness results for first-order three-point boundary value problem 𝑥^Δ(𝑡)(𝑡)𝑥(𝜎(𝑡((𝜎(𝑡[]𝑥(0)(𝜉)(𝜎(𝑇,+𝑝))=𝑓𝑡,𝑥))),𝑡∈0,𝑇⊂𝕋,−𝛼𝑥=𝛽𝑥)) (1.7) by using several well-known fixed-point theorems. When 𝛼₃=⋯=𝛼_𝑚−1=0, BVP (1.2) is reduced to (1.7).

Motivated by [6–10], we establish some new and more general results for the existence of positive solutions for the problem (1.2) by applying fixed-point theorems in cones.

We have arranged the paper as follows. In Section 2, we give some lemmas which are needed later. In Section 3, we apply the Krasnosel'skii fixed-point theorem, Avery-Henderson fixed-point theorem, and Leggett-Williams fixed-point theorem to prove the existence of at least one, two, and three positive solutions to BVP (1.2). In Section 4, as an application, the examples are included to illustrate our results.

2. Preliminaries

Let ℬ denote the Banach space 𝐶[𝑡₁,𝜎(𝑡_𝑚)] with the norm ‖𝑥‖=sup_{𝑡∈[𝑡1,𝜎(𝑡𝑚)]}|𝑥(𝑡)|. For ℎ∈ℬ, we consider the following linear boundary value problem:𝑥^Δ(𝑡)+𝑝(𝑡)𝑥(𝜎(𝑡))=ℎ(𝑡),𝑡∈𝑡₁,𝑡_𝑚𝑥𝑡⊂𝕋,₁=^𝑚−1_𝑘=2𝛼_𝑘𝑥𝑡_𝑘+𝛼₁𝑥𝜎𝑡_𝑚. (2.1)

Lemma 2.1. For ℎ∈ℬ, BVP (2.1) has the unique solution 𝑥(𝑡)=1𝑒_𝑎𝑡,𝑡₁Γ𝛼₁∫^{𝜎(𝑡𝑚)}_𝑡1𝑒_𝑎𝑠,𝑡₁ℎ(𝑠)Δ𝑠𝑒_𝑎𝜎𝑡_𝑚,𝑡₁+^𝑚−1_𝑘=2𝛼_𝑘∫^𝑡𝑘_𝑡1𝑒_𝑎𝑠,𝑡₁ℎ(𝑠)Δ𝑠𝑒_𝑎𝑡_𝑘,𝑡₁+^𝑡_𝑡1𝑒_𝑎𝑠,𝑡₁ℎ(𝑠)Δ𝑠,𝑡∈𝑡₁,𝜎𝑡_𝑚, (2.2) where Γ=[1−∑^𝑚−1_𝑘=2(𝛼_𝑘/𝑒_𝑎(𝑡_𝑘,𝑡₁))−(𝛼₁/𝑒_𝑎(𝜎(𝑡_𝑚),𝑡₁))]⁻¹.

Proof. From 𝑥^Δ(𝑡)+𝑎(𝑡)𝑥(𝜎(𝑡))=ℎ(𝑡), we have 𝑥(𝑡)=1𝑒_𝑎𝑡,𝑡₁𝑥𝑡₁+^𝑡_𝑡1𝑒_𝑎𝑠,𝑡₁ℎ(𝑠)Δ𝑠. (2.3) By using the boundary condition, we get 𝛼1−¹𝑒_𝑎𝜎𝑡_𝑚,𝑡₁−^𝑚−1_𝑘=2𝛼_𝑘𝑒_𝑎𝑡_𝑘,𝑡₁𝑥𝑡₁=𝛼₁∫^{𝜎(𝑡𝑚)}_𝑡1𝑒_𝑎𝑠,𝑡₁ℎ(𝑠)Δ𝑠𝑒_𝑎𝜎𝑡_𝑚,𝑡₁+^𝑚−1_𝑘=2𝛼_𝑘∫^𝑡𝑘_𝑡1𝑒_𝑎𝑠,𝑡₁ℎ(𝑠)Δ𝑠𝑒_𝑎𝑡_𝑘,𝑡₁. (2.4) Thus, 𝑥 satisfies (2.2).

Let 𝐺(𝑡,𝑠) be Green's function for the boundary value problem 𝑥^Δ(𝑡)+𝑎(𝑡)𝑥(𝜎(𝑡))=ℎ(𝑡),𝑡∈𝑡₁,𝑡_𝑚𝑥𝑡⊂𝕋,₁=^𝑚−1_𝑘=2𝛼_𝑘𝑥𝑡_𝑘+𝛼₁𝑥𝜎𝑡_𝑚. (2.5) By Lemma 2.1, we obtain 𝐺()=𝐺𝑡,𝑠¹(𝑡,𝑠),𝑡₁≤𝑠≤𝜎(𝑠)≤𝑡₂,𝐺₂(𝑡,𝑠),𝑡₂≤𝑠≤𝜎(𝑠)≤𝑡₃,⋮𝐺_𝑚−2(𝑡,𝑠),𝑡_𝑚−2≤𝑠≤𝜎(𝑠)≤𝑡_𝑚−1,𝐺_𝑚−1(𝑡,𝑠),𝑡_𝑚−1≤𝑠≤𝑡_𝑚, (2.6) where 𝐺_𝑗()=𝑒𝑡,𝑠^𝑎𝑠,𝑡₁𝑒_𝑎𝑡,𝑡₁Γ𝛼₁𝑒_𝑎𝜎𝑡_𝑚,𝑡₁+∑^𝑚−1_𝑘=𝑗+1𝛼_𝑘𝑒_𝑎𝑡_𝑘,𝑡₁+1,𝜎(𝑠)≤𝑡,Γ𝑒_𝑎𝑠,𝑡₁𝑒_𝑎𝑡,𝑡₁𝛼₁𝑒_𝑎𝜎𝑡_𝑚,𝑡₁+∑^𝑚−1_𝑘=𝑗+1𝛼_𝑘𝑒_𝑎𝑡_𝑘,𝑡₁,𝑡≤𝑠, (2.7) for all 𝑗=1,2,…,𝑚−1.

Lemma 2.2. Green's function 𝐺(𝑡,𝑠) in (2.6) has the following properties:(i)𝐺(𝑡,𝑠)≥0 for (𝑡,𝑠)∈[𝑡₁,𝜎(𝑡_𝑚)]×[𝑡₁,𝑡_𝑚].(ii)𝑚≤𝐺(𝑡,𝑠)≤𝑀, where 𝑚=Γ𝛼₁/(𝑒_𝑎(𝜎(𝑡_𝑚),𝑡₁))² and 𝑀=Γ∑^𝑚−1_𝑘=1𝛼_𝑘+𝑒_𝑎(𝑡_𝑚−1,𝑡₁).(iii)𝐺(𝑡,𝑠)≥(𝑚/𝑀)sup_{(𝑡,𝑠)∈[𝑡1,𝜎(𝑡𝑚)]×[𝑡1,𝑡𝑚]}𝐺(𝑡,𝑠) for (𝑡,𝑠)∈[𝑡₁,𝜎(𝑡_𝑚)]×[𝑡₁,𝑡_𝑚].

Let ℬ denote the Banach space 𝐶[𝑡₁,𝜎(𝑡_𝑚)] with the norm ‖𝑥‖=max_{𝑡∈[𝑡1,𝜎(𝑡𝑚)]}|𝑥(𝑡)|. Define the cone 𝑃⊂ℬ by (𝑡)(𝑡)≥𝑚𝑃=𝑥∈ℬ∶𝑥≥0,𝑥𝑀‖𝑥‖on𝑡₁,𝜎𝑡_𝑚. (2.8) Equation (1.2) is equivalent to the nonlinear integral equation 𝑥(𝑡)=^{𝜎(𝑡𝑚)}_𝑡1𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝜎(𝑠)))Δ𝑠. (2.9)

We can define the operator 𝐴∶𝑃→ℬ by 𝐴𝑥(𝑡)=^{𝜎(𝑡𝑚)}_𝑡1𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝜎(𝑠)))Δ𝑠. (2.10) Therefore solving (2.9) in 𝑃 is equivalent to finding fixed-points of the operator 𝐴.

From Lemma 2.2, 𝐴𝑥(𝑡)≥0 for 𝑡∈[𝑡₁,𝜎(𝑡_𝑚)]. In addition, by using Lemma 2.2 we get 𝐴𝑥(𝑡)=^{𝜎(𝑡𝑚)}_𝑡1𝐺()𝑓((𝜎(𝑠≥𝑚𝑡,𝑠𝑠,𝑥)))Δ𝑠𝑀sup_{()∈𝑡𝑡,𝑠1𝑡,𝜎𝑚×𝑡1,𝑡𝑚}𝐺(𝑡,𝑠)^{𝜎(𝑡𝑚)}_𝑡1𝑓((𝜎(𝑠≥𝑚𝑠,𝑥)))Δ𝑠𝑀sup_{𝑡𝑡∈1𝑡,𝜎𝑚}^{𝜎(𝑡𝑚)}_𝑡1𝐺()𝑓((𝜎(𝑠=𝑚𝑡,𝑠𝑠,𝑥)))Δ𝑠𝑀‖𝐴𝑥‖. (2.11)

So, we have 𝐴∶𝑃→𝑃.

3. Main Results

To prove the existence of at least one positive solution for the BVP (1.2), we will need the following (Krasnosel'skii) fixed-point theorem.

Theorem 3.1 (Krasnosel'skii fixed-point theorem [11]). Let 𝐸 be a Banach space, and let 𝐾⊂𝐸 be a cone. Assume Ω₁ and Ω₂ are open bounded subsets of 𝐸 with 0∈Ω₁, Ω₁⊂Ω₂, and let 𝐴∶𝐾∩Ω₂⧵Ω₁→𝐾 (3.1) be a completely continuous operator such that either(i)‖𝐴𝑢‖≤‖𝑢‖ for 𝑢∈𝐾∩𝜕Ω₁,‖𝐴𝑢‖≥‖𝑢‖ for 𝑢∈𝐾∩𝜕Ω₂, or(ii)‖𝐴𝑢‖≥‖𝑢‖ for 𝑢∈𝐾∩𝜕Ω₁,‖𝐴𝑢‖≤‖𝑢‖ for 𝑢∈𝐾∩𝜕Ω₂ hold. Then A has a fixed-point in 𝐾∩(Ω₂⧵Ω₁).

Theorem 3.2. Let there exist numbers 𝑟,  𝑅 satisfying 0<𝑟<𝑅<∞ such that for 𝑡∈[𝑡₁,𝜎(𝑡_𝑚)]𝑓()<𝑥𝑡,𝑥𝑀𝜎𝑡_𝑚for𝑥∈[0,𝑟],𝑓(𝑡,𝑥)≥𝑀𝑥𝑚²𝜎𝑡_𝑚for𝑥∈[𝑅,∞). (3.2) Then BVP (1.2) has at least one positive solution 𝑥 satisfying 𝑟≤𝑥(𝑡)≤𝑅𝑀/𝑚, 𝑡∈[𝑡₁,𝜎(𝑡_𝑚)].

Proof. It is easy to check by the Arzela-Ascoli theorem that the operator 𝐴∶𝑃→𝑃 is completely continuous. Let us now define two bounded open sets as follows: Ω₁={𝑥∈ℬ∶‖𝑥‖<𝑟},Ω₂=𝑥∈ℬ∶‖𝑥‖<𝑅𝑀𝑚. (3.3) Then Ω₁⊂Ω₂. For 𝑥∈𝑃∩𝜕Ω₁, we obtain 𝐴𝑥(𝑡)=^{𝜎(𝑡𝑚)}_𝑡1𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝜎(𝑠)))Δ𝑠≤𝑀^{𝜎(𝑡𝑚)}_𝑡1𝑓((𝜎(𝑠∫𝑠,𝑥)))Δ𝑠≤𝑀^{𝜎(𝑡𝑚)𝑡1}𝑥(𝜎(𝑠))Δ𝑠𝑀𝜎𝑡_𝑚≤𝑟=‖𝑥‖. (3.4) Hence ‖𝐴𝑥‖≤‖𝑥‖ for 𝑥∈𝑃∩𝜕Ω₁.
If 𝑥∈𝑃∩𝜕Ω₂, then ‖𝑥‖=𝑅𝑀/𝑚 and 𝑥(𝑡)≥(𝑚/𝑀)‖𝑥‖=𝑅 for 𝑡∈[𝑡₁,𝜎(𝑡_𝑚)]. We have 𝐴𝑥(𝑡)=^{𝜎(𝑡𝑚)}_𝑡1𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝜎(𝑠)))Δ𝑠≥𝑚^{𝜎(𝑡𝑚)}_𝑡1𝑓((𝜎(𝑠𝑀∫𝑠,𝑥)))Δ𝑠≥𝑚^{𝜎(𝑡𝑚)𝑡1}𝑥(𝜎(𝑠))Δ𝑠𝑚²𝜎𝑡_𝑚≥𝑅𝑀𝑚=‖𝑥‖. (3.5)
Thus ‖𝐴𝑥‖≥‖𝑥‖ for 𝑥∈𝑃∩𝜕Ω₂. By the first part of Theorem 3.1, 𝐴 has a fixed-point in 𝑃∩(Ω₂⧵Ω₁). Therefore, the BVP (1.2) has at least one positive solution satisfying 𝑟≤𝑥(𝑡)≤𝑅𝑀/𝑚, 𝑡∈[𝑡₁,𝜎(𝑡_𝑚)].

Now, we will apply the following (Avery-Henderson) fixed-point theorem to prove the existence of at least two positive solutions to BVP (1.2).

Theorem 3.3 (see [12]). Let 𝑃 be a cone in a real Banach space 𝐸. Set 𝑃𝜙,𝑟=𝑢∈𝑃∶𝜙(𝑢)<𝑟. (3.6) If 𝜂 and 𝜙 are increasing, nonnegative continuous functionals on 𝑃, let 𝜃 be a nonnegative continuous functional on 𝑃 with 𝜃(0)=0 such that, for some positive constants 𝑟 and 𝑀, 𝜙(𝑢)≤𝜃(𝑢)≤𝜂(𝑢),‖𝑢‖≤𝑀𝜙(𝑢), (3.7) for all 𝑢∈𝑃(𝜙,𝑟). Suppose that there exist positive numbers 𝑝<𝑞<𝑟 such that 𝜃(𝜆𝑢)≤𝜆𝜃(𝑢),∀0≤𝜆≤1,𝑢∈𝜕𝑃𝜃,𝑞. (3.8) If 𝐴∶𝑃(𝜙,𝑟)→𝑃 is a completely continuous operator satisfying(i)𝜙(𝐴𝑢)>𝑟 for all 𝑢∈𝜕𝑃(𝜙,𝑟),(ii)𝜃(𝐴𝑢)<𝑞 for all 𝑢∈𝜕𝑃(𝜃,𝑞),(iii)𝑃(𝜂,𝑝)≠∅ and 𝜂(𝐴𝑢)>𝑝 for all 𝑢∈𝜕𝑃(𝜂,𝑝),
then 𝐴 has at least two fixed-points 𝑢₁ and 𝑢₂ such that 𝑝<𝜂𝑢₁with𝜃𝑢₁<𝑞,𝑞<𝜃𝑢₂with𝜙𝑢₂<𝑟. (3.9)

Theorem 3.4. Suppose there exist numbers 𝑝, 𝑞, and 𝑟 satisfying 0<𝑝<𝑞<𝑟 such that the function 𝑓 satisfies the following conditions:(i)𝑓(𝑡,𝑥)>𝑟/𝑚 for 𝑡∈[𝑡_𝑚−1,𝜎(𝑡_𝑚)] and 𝑥∈[𝑟,𝑟𝑀/𝑚(𝜎(𝑡_𝑚)−𝑡_𝑚−1)];(ii)𝑓(𝑡,𝑥)<𝑞/𝑀𝜎(𝑡_𝑚) for 𝑡∈[𝑡₁,𝜎(𝑡_𝑚)] and 𝑥∈[0,𝑞𝑀/𝑚];(iii)𝑓(𝑡,𝑥)>𝑝/𝑚(𝜎(𝑡_𝑚)−𝑡_𝑚−1) for 𝑡∈[𝑡_𝑚−1,𝜎(𝑡_𝑚)] and 𝑥∈[𝑝𝑚/𝑀,𝑝].
Then the BVP (1.2) has at least two positive solutions 𝑥₁ and 𝑥₂ such that 𝑝<sup_{𝑡𝑡∈1𝑡,𝜎𝑚}𝑥₁(𝑡)withsup_{𝑡𝑡∈𝑚−1𝑡,𝜎𝑚}𝑥₁(𝑡)<𝑞,𝑞<sup_{𝑡𝑡∈𝑚−1𝑡,𝜎𝑚}𝑥₂(𝑡)withinf_{𝑡𝑡∈𝑚−1𝑡,𝜎𝑚}𝑥₂(𝑡)<𝑟. (3.10)

Proof . Let the nonnegative increasing continuous functionals 𝜙, 𝜃, and 𝜂 be defined on the cone 𝑃 by 𝜙(𝑥)=inf_{𝑡𝑡∈𝑚−1𝑡,𝜎𝑚}𝑥(𝑡),𝜃(𝑥)=sup_{𝑡𝑡∈𝑚−1𝑡,𝜎𝑚}𝑥(𝑡),𝜂(𝑥)=sup_{𝑡𝑡∈1𝑡,𝜎𝑚}𝑥(𝑡). (3.11) For each 𝑥∈𝑃, we have 𝜙(𝑥)≤𝜃(𝑥)≤𝜂(𝑥) and 𝑥(𝑡)=^{𝜎(𝑡𝑚)}_𝑡1𝐺()𝑓((𝜎(𝑠≤𝑀𝑡,𝑠𝑠,𝑥)))Δ𝑠𝑚𝑚^{𝜎(𝑡𝑚)}_𝑡1𝑓((𝜎(𝑠≤𝑀𝑠,𝑥)))Δ𝑠𝑚inf_{(𝑡,𝑠)∈[𝑡1,𝜎(𝑡𝑚)]×[𝑡1,𝑡𝑚]}𝐺(𝑡,𝑠)^{𝜎(𝑡𝑚)}_𝑡1𝑓((𝜎(𝑠≤𝑀𝑠,𝑥)))Δ𝑠𝑚inf_{𝑡∈[𝑡1,𝜎(𝑡𝑚)]}^{𝜎(𝑡𝑚)}_𝑡1𝐺()𝑓((𝜎(𝑠=𝑀𝑡,𝑠𝑠,𝑥)))Δ𝑠𝑚𝜙(𝑥). (3.12) Then ‖𝑥‖≤(𝑀/𝑚)𝜙(𝑥). In addition, 𝜃(0)=0 and for all 𝑥∈𝑃, 𝜆∈[0,1] we obtain 𝜃(𝜆𝑥)=𝜆𝜃(𝑥).
Now we will verify the remaining conditions of Theorem 3.3.
Claim 1. If 𝑥∈𝜕𝑃(𝜙,𝑟), then 𝜙(𝐴𝑥)>𝑟. Since 𝑥∈𝜕𝑃(𝜙,𝑟), we have 𝑟=inf_{𝑡∈[𝑡𝑚−1,𝜎(𝑡𝑚)]}𝑥(𝑡)≤‖𝑥‖≤𝑟𝑀/𝑚 for 𝑡∈[𝑡_𝑚−1,𝜎(𝑡_𝑚)]. Then, we get 𝜙(𝐴𝑥)=^{𝜎(𝑡𝑚)}_𝑡1min_{𝑡∈[𝑡𝑚−1,𝜎(𝑡𝑚)]}𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝜎(𝑠)))Δ𝑠≥𝑚^{𝜎(𝑡𝑚)}_𝑡𝑚−1𝑓(𝑠,𝑥(𝜎(𝑠)))Δ𝑠>𝑟 (3.13) by hypothesis (i).
Claim 2. If 𝑥∈𝜕𝑃(𝜃,𝑞), then 𝜃(𝐴𝑥)<𝑞. Since 𝑥∈𝜕𝑃(𝜃,𝑞), 0≤𝑥(𝑡)≤‖𝑥‖≤(𝑀/𝑚)𝜙(𝑥)≤(𝑀/𝑚)𝜃(𝑥)=𝑞𝑀/𝑚 for 𝑡∈[𝑡₁,𝜎(𝑡_𝑚)]. Thus, by hypothesis (ii) we have 𝜃(𝐴𝑥)=^{𝜎(𝑡𝑚)}_𝑡1max_{𝑡𝑡∈𝑚−1𝑡,𝜎𝑚}𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝜎(𝑠)))Δ𝑠≤𝑀^{𝜎(𝑡𝑚)}_𝑡1𝑓(𝑠,𝑥(𝜎(𝑠)))Δ𝑠<𝑞. (3.14)
Claim 3. 𝑃(𝜂,𝑝)≠∅ and 𝜂(𝐴𝑥)>𝑝 for all 𝑥∈𝜕𝑃(𝜂,𝑝). Since 0∈𝑃 and 𝑝>0, 𝑃(𝜂,𝑝)≠∅. If 𝑥∈𝜕𝑃(𝜂,𝑝), we get (𝑚/𝑀)𝑝≤𝜙(𝑥)≤𝑥(𝑡)≤‖𝑥‖=𝑝 for 𝑡∈[𝑡_𝑚−1,𝜎(𝑡_𝑚)]. Hence, we obtain 𝜂(𝐴𝑥)≥^{𝜎(𝑡𝑚)}_𝑡1𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝜎(𝑠)))Δ𝑠≥𝑚^{𝜎(𝑡𝑚)}_𝑡𝑚−1𝑓(𝑠,𝑥(𝜎(𝑠)))Δ𝑠>𝑝 (3.15) by hypothesis (iii). This completes the proof.

To prove the existence of at least three positive solutions for the BVP (1.2), we will apply the following (Leggett-Williams) fixed-point theorem.

Theorem 3.5 (see [13]). Let P be a cone in the real Banach space 𝐸. Set 𝑃_𝑟{‖𝑥‖},𝑃(𝑥),‖𝑥‖.∶=𝑥∈𝑃∶<𝑟𝜓,𝑎,𝑏∶=𝑥∈𝑃∶𝑎≤𝜓≤𝑏 (3.16) Suppose 𝐴∶𝑃_𝑟→𝑃_𝑟 is a completely continuous operator and 𝜓 is a nonnegative continuous concave functional on 𝑃 with 𝜓(𝑢)≤‖𝑢‖ for all 𝑢∈𝑃_𝑟. If there exists 0<𝑝<𝑞<𝑙≤𝑟 such that the following condition hold:(i){𝑢∈𝑃(𝜓,𝑞,𝑙)∶𝜓(𝑢)>𝑞}≠∅ and 𝜓(𝐴𝑢)>𝑞 for all 𝑢∈𝑃(𝜓,𝑞,𝑙);(ii)‖𝐴𝑢‖<𝑝 for ‖𝑢‖≤𝑝;(iii)𝜓(𝐴𝑢)>𝑞 for 𝑢∈𝑃(𝜓,𝑞,𝑟) with ‖𝐴𝑢‖>𝑙,
then 𝐴 has at least three fixed-points 𝑢₁,𝑢₂, and 𝑢₃ in 𝑃_𝑟 satisfying 𝑢₁<𝑝,𝜓𝑢₂>𝑞,𝑝<𝑢₃with𝜓𝑢₃<𝑞. (3.17)

Theorem 3.6. Suppose that there exist numbers 𝑝, 𝑞, and 𝑟 satisfying 0<𝑝<𝑞<𝑞𝑀/𝑚≤𝑟 such that for 𝑡∈[𝑡₁,𝜎(𝑡_𝑚)] the function 𝑓 satisfies the following conditions:(i)𝑓(𝑡,𝑥)≤𝑟/𝑀𝜎(𝑡_𝑚), 𝑥∈[0,𝑟],(ii)𝑓(𝑡,𝑥)>𝑞/𝑚𝜎(𝑡_𝑚), 𝑥∈[𝑞,𝑞𝑀/𝑚],(iii)𝑓(𝑡,𝑥)<𝑝/𝑀𝜎(𝑡_𝑚), 𝑥∈[0,𝑝]. Then (1.2) has at least three positive solutions 𝑥₁,𝑥₂, and 𝑥₃ satisfying sup_{𝑡𝑡∈1𝑡,𝜎𝑚}𝑥₁(𝑡)<𝑝,𝑞<inf_{𝑡𝑡∈1𝑡,𝜎𝑚}𝑥₂(𝑡),𝑝<sup_{𝑡𝑡∈1𝑡,𝜎𝑚}𝑥₃(𝑡)withinf_{𝑡𝑡∈1𝑡,𝜎𝑚}𝑥₃(𝑡)<𝑞. (3.18)

Proof. Define the nonnegative continuous concave functional 𝜓∶𝑃→[0,∞) to be 𝜓(𝑥)∶=inf_{𝑡∈[𝑡1,𝜎(𝑡𝑚)]}𝑥(𝑡) and the cone 𝑃 as in (2.8). For all 𝑥∈𝑃, we have 𝜓(𝑥)≤‖𝑥‖. If 𝑥∈𝑃_𝑟, then 0≤𝑥≤𝑟 and 𝑓(𝑡,𝑥)≤𝑟/𝑀𝜎(𝑡_𝑚) from the hypothesis (i). Then we get ‖𝐴𝑥‖=sup_{𝑡𝑡∈1𝑡,𝜎𝑚}^{𝜎(𝑡𝑚)}_𝑡1𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝜎(𝑠)))Δ𝑠≤𝑀^{𝜎(𝑡𝑚)}_𝑡1𝑓(𝑠,𝑥(𝜎(𝑠)))Δ𝑠≤𝑟 (3.19) by Lemma 2.2. This proves that 𝐴∶𝑃_𝑟→𝑃_𝑟. Similarly, by the hypothesis (iii), the condition (ii) of Theorem 3.5 is satisfied.
Since 𝑞𝑀/𝑚∈𝑃(𝜓,𝑞,𝑞𝑀/𝑚) and 𝜓(𝑞𝑀/𝑚)>𝑞, {𝑦∈𝑃(𝜓,𝑞,𝑞𝑀/𝑚)∶𝜓(𝑥)>𝑞}≠∅. For all 𝑥∈𝑃(𝜓,𝑞,𝑞𝑀/𝑚), we have 𝑞≤inf_{𝑡∈[𝑡1,𝜎(𝑡𝑚)]}𝑥(𝑡)≤‖𝑥‖≤𝑞𝑀/𝑚 for 𝑡∈[𝑡₁,𝜎(𝑡_𝑚)]. Using the hypothesis (ii) and Lemma 2.2, we find 𝜓(𝐴𝑥)=^{𝜎(𝑡𝑚)}_𝑡1inf_{𝑡𝑡∈1𝑡,𝜎𝑚}𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝜎(𝑠)))Δ𝑠≥𝑚^{𝜎(𝑡𝑚)}_𝑡1𝑓(𝑠,𝑥(𝜎(𝑠)))Δ𝑠>𝑞. (3.20) Hence, the condition (i) of Theorem 3.5 holds.
For the condition (iii) of Theorem 3.5, we suppose that 𝑥∈𝑃(𝜓,𝑞,𝑟) with ‖𝐴𝑥‖>𝑞𝑀/𝑚. Then, from Lemma 2.2 we obtain 𝜓(𝐴𝑥)=inf_{𝑡𝑡∈1𝑡,𝜎𝑚}(𝑡)≥𝑚𝐴𝑥𝑀‖𝐴𝑥‖>𝑞. (3.21) This completes the proof.

4. Examples

Example 4.1. Let 𝕋=ℤ. We consider the first-order four-point BVP as follows: 𝑥^Δ(𝑡)+𝑥(𝜎(𝑡))=𝑥+5𝑥⁴[]𝑥(0)(1)(2)(6).+1,𝑡∈0,5⊂𝕋,=𝑥+𝑥+𝑥 (4.1) Taking 𝑎(𝑡)≡1,𝑡₁=0,𝑡₂=1,𝑡₃=2,𝑡₄=5,and𝛼₁=𝛼₂=𝛼₃=1, we have Γ=64/15,𝑚=1/960, and 𝑀=384/5. If we take 𝑝=0.001,𝑞=0.01, and 𝑟=0.02; then all the assumptions in Theorem 3.4 are satisfied. Finally, BVP (4.1) has at least two positive solutions 𝑥₁ and 𝑥₂ such that 0.001<sup_[]𝑡∈0,6𝑥₁(𝑡)withsup_[]𝑡∈2,6𝑥₁(𝑡)<0.01,0.01<sup_[]𝑡∈2,6𝑥₂(𝑡)withinf_[]𝑡∈2,6𝑥₂(𝑡)<0.02. (4.2)

Example 4.2. Let 𝕋=ℕ²₀. We consider the first-order four-point BVP as follows: 𝑥^Δ(𝑡)(𝜎(𝑡((𝜎(𝑡[]𝑥(0)(1)(4)(),+𝑥))=𝑓𝑡,𝑥))),𝑡∈0,9⊂𝕋,=𝑥+𝑥+𝑥16 (4.3) where 𝑎(𝑡)≡1,𝑡₁=0,𝑡₂=1,𝑡₃=4,𝑡₄=9,𝛼₁=𝛼₂=𝛼₃=1, and 𝑓()=𝑥𝑡,𝑥,()∈[]×[],1400𝑡,𝑥0,160,16870−2400𝑥+()∈[]×[],400−6870,𝑡,𝑥0,161,215𝑥882434+6870−15,()∈[]×[).441217𝑡,𝑥0,162,∞ (4.4) Hence, we obtain Γ=384/143,𝑚=1/54912, and 𝑀=2296/143. If we take 𝑝=1,𝑞=2, and 𝑟=1764870; then all the assumptions in Theorem 3.6 are satisfied. Finally, BVP (4.3) has at least three positive solutions 𝑥₁,𝑥₂, and 𝑥₃ such that sup_{[]𝑡∈0,16}𝑥₁(𝑡)<1,2<inf_{[]𝑡∈0,16}𝑥₂(𝑡),1<sup_{[]𝑡∈0,16}𝑥₃(𝑡)withinf_{[]𝑡∈0,16}𝑥₃(𝑡)<2. (4.5)