Existence of Positive Solutions for Third-Order 𝑚-Point Boundary Value Problems with Sign-Changing Nonlinearity on Time Scales

Abstract

A four-functional fixed point theorem and a generalization of Leggett-Williams fixed point theorem are used, respectively, to investigate the existence of at least one positive solution and at least three positive solutions for third-order 𝑚-point boundary value problem on time scales with an increasing homeomorphism and homomorphism, which generalizes the usual 𝑝-Laplacian operator. In particular, the nonlinear term 𝑓(𝑡,𝑢) is allowed to change sign. As an application, we also give some examples to demonstrate our results.

1. Introduction

The theory of time scales was introduced by Hilger in his Ph.D. thesis in 1990 [1]. Theoretically, this new theory not only unifies continuous and discrete equations, but has also exhibited much more complicated dynamics on time scales. Moreover, the study of dynamic equations on time scales has led to several important applications, for example, insect population models, biology, neural networks, heat transfer, and epidemic models (see [2, 3] and references therein). Some preliminary definitions and theorems on time scales also can be found in books [2, 3], which are excellent references for calculus on time scales.

Multipoint boundary value problem (BVP) arises in a variety of different areas of applied mathematics and physics [4]. The study of multipoint BVPs for linear second-order ordinary differential equations was initiated by Ilin and Moiseev [5], since then many authors studied more general nonlinear multipoint boundary value problems ([6, 7] and references therein). Recently, when 𝜑_𝑝 is 𝑝-Laplacian operator, that is, 𝜑_𝑝(𝑢)=|𝑢|^𝑝−2𝑢(𝑝>1) and the nonlinear term does not depend on the first-order derivative, the existence problems of positive solutions of boundary value problems have attracted much attention. One can notice that the oddness of the 𝑝-Laplacian operator is key to the proof. However, in the operator 𝜙 is not necessary odd, so it improves and generalizes the 𝑝-Laplacian operator.

There are few works that investigate the existence of positive solutions by using four functionals fixed point theorem [8] or a generalization of Leggett-Williams fixed point theorem [9]. Avery et al. [8] present a four-functional fixed point theorem, which is a major generalization of the original Leggett-Williams fixed point theorem [10]. To our knowledge, this fixed point theorem so far has only been used by Sun et al. [11]. By using the four-functional fixed point theorem and five-functional fixed point theorem, they obtain the existence criteria of at least one positive solution and three positive solutions for the nonlocal BVP with 𝑝-Laplacian. Liu and Sun [12] discussed the existence of at least three and arbitrary odd number positive solutions for 𝑚-point boundary value problems on time scale. They used a generalization of Leggett-Williams fixed point theorem [9].

There have been extensive studies on BVP with sign-changing nonlinearity [13–17]. Most of these results are obtained by using the fixed point theorem on cone [18, 19]. To the best of our knowledge, positive solutions of third-order 𝑚-point BVP for an increasing homeomorphism and homomorphism with sign-changing nonlinearity on time scales by using four functionals fixed point theorem [8] or a generalization of Leggett-Williams fixed point theorem [9] have not been considered till now. Our aim in this paper is to fill the gap.

In this paper, motivated by above results, we consider the existence of at least one or three positive solutions of the following third-order 𝑚-point boundary value problem (BVP) on time scales𝜙𝑢^Δ∇(𝑡)^∇+𝑎(𝑡)𝑓(𝑡,𝑢(𝑡))=0,𝑡∈[0,𝑇]_𝕋,𝑢(0)=^𝑚−2_𝑖=1𝑏_𝑖𝑢𝜉_𝑖,𝑢^Δ(𝑇)=0,𝜙𝑢^Δ∇(0)=^𝑚−2_𝑖=1𝑐_𝑖𝜙𝑢^Δ∇𝜉_𝑖, (1.1) where 𝜙∶ℝ→ℝ is an increasing homeomorphism and homomorphism with 𝜙(0)=0. A projection 𝜙∶ℝ→ℝ is called an increasing homeomorphism and homomorphism if the following conditions are satisfied.(i)If 𝑥≤𝑦, then 𝜙(𝑥)≤𝜙(𝑦), for all 𝑥,𝑦∈ℝ;(ii)𝜙 is continuous bijection and its inverse mapping is also continuous;(iii)𝜙(𝑥𝑦)=𝜙(𝑥)𝜙(𝑦), for all 𝑥,𝑦∈ℝ.

We will assume that the following conditions are satisfied throughout this paper:(H1)𝑏_𝑖,𝑐_𝑖∈[0,+∞),0<𝜉₁<𝜉₂<⋯<𝜉_𝑚−2<𝜌(𝑇),0<∑^𝑚−2_𝑖=1𝑏_𝑖<1,0<∑^𝑚−2_𝑖=1𝑐_𝑖<1; (H2)𝑓∶[0,𝑇]_𝕋×[0,+∞)→(−∞,+∞) is continuous, 𝑎∈𝒞_ld([0,𝑇]_𝕋,[0,+∞)), and there exist 𝑡₀∈[0,𝑇]_𝕋 such that 𝑎(𝑡₀)>0; (H3)𝜁=min{𝑡∈𝕋∶𝑡>𝑇/2} exist.

By using a four-functional fixed point theorem [8], we establish the existence of at least one positive solution, and by using a generalization of Leggett-Williams fixed point theorem [9], we establish the existence of at least three positive solutions for the above boundary value problem. In particular, the nonlinear term 𝑓(𝑡,𝑢(𝑡)) is allowed to change sign. The remainder of this paper is organized as follows. Section 2 is devoted to some preliminary lemmas. We give and prove our main results in Section 3. Finally, in Section 4, we give two examples to demonstrate our results.

2. Preliminary Lemmas

In this section, we present some definitions and lemmas, which will be needed in the proof of the main results.

Lemma 2.1 (see [13]). If condition (H1) holds, then for ℎ∈𝒞_ld[0,𝑇]_𝕋, the boundary value problem (BVP), 𝑢^Δ∇+ℎ(𝑡)=0,𝑡∈[0,𝑇]_𝕋,𝑢(0)=^𝑚−2_𝑖=1𝑏_𝑖𝑢𝜉_𝑖,𝑢^Δ(𝑇)=0, (2.1) has a unique solution 𝑢(𝑡)=^𝑡₀()ℎ(𝑠)∑𝑇−𝑠∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)ℎ(𝑠)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖. (2.2)

Lemma 2.2. If condition (H1) holds, then for ℎ∈𝒞_ld[0,𝑇]_𝕋, the boundary value problem (BVP), 𝜙𝑢^Δ∇(𝑡)^∇+ℎ(𝑡)=0,𝑡∈[0,𝑇]_𝕋,𝑢(0)=^𝑚−2_𝑖=1𝑏_𝑖𝑢𝜉_𝑖,𝑢^Δ(𝑇)=0,𝜙𝑢^Δ∇(0)=^𝑚−2_𝑖=1𝑐_𝑖𝜙𝑢^Δ∇𝜉_𝑖, (2.3) has a unique solution 𝑢(𝑡)=^𝑡₀(𝑠−𝑇)𝜙⁻¹−^𝑠₀ℎ(𝑟)∑∇𝑟+𝐷∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑠−𝑇)𝜙⁻¹−∫^𝑠₀ℎ(𝑟)∇𝑟+𝐷∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖, (2.4) where 𝐷=−(∑^𝑚−2_𝑖=1𝑐_𝑖∫^𝜉𝑖₀ℎ(𝑟)∇𝑟)/(1−∑^𝑚−2_𝑖=1𝑐_𝑖).

Proof. Integrating both sides of equation in (2.3) on [0,𝑡], we have 𝜙𝑢^Δ∇(𝑡)=𝜙𝑢^Δ∇(0)−^𝑡₀ℎ(𝑟)∇𝑟. (2.5) So, 𝜙𝑢^Δ∇𝜉_𝑖=𝜙𝑢^Δ∇(0)−^𝜉𝑖₀ℎ(𝑟)∇𝑟. (2.6) By boundary value condition 𝜙(𝑢^Δ∇(0))=∑^𝑚−2_𝑖=1𝑐_𝑖𝜙(𝑢^Δ∇(𝜉_𝑖)), we have 𝜙𝑢^Δ∇(0)∑=−^{𝑚−2𝑖=1}𝑐_𝑖∫^𝜉𝑖₀ℎ(𝑟)∇𝑟1−∑^𝑚−2_𝑖=1𝑐_𝑖. (2.7) By (2.5) and (2.7), we know 𝑢^Δ∇(𝑡)=𝜙⁻¹−∑^𝑚−2_𝑖=1𝑐_𝑖∫^𝜉𝑖₀ℎ(𝑟)∇𝑟1−∑^𝑚−2_𝑖=1𝑐_𝑖−^𝑡₀ℎ(𝑟)∇𝑟. (2.8) This together with Lemma 2.1 implies that 𝑢(𝑡)=^𝑡₀(𝑠−𝑇)𝜙⁻¹−^𝑠₀ℎ(𝑟)∑∇𝑟+𝐷∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑠−𝑇)𝜙⁻¹−∫^𝑠₀ℎ(𝑟)∇𝑟+𝐷∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖, (2.9) where 𝐷=−(∑^𝑚−2_𝑖=1𝑐_𝑖∫^𝜉𝑖₀ℎ(𝑟)∇𝑟)/(1−∑^𝑚−2_𝑖=1𝑐_𝑖). The proof is complete.

Lemma 2.3. Let condition (H1) hold. If ℎ∈𝒞_ld[0,𝑇]_𝕋 and ℎ(𝑡)≥0, then the unique solution 𝑢(𝑡) of (2.3) satisfies (i)𝑢(𝑡)≥0,𝑡∈[0,𝑇]_𝕋, (ii)inf_{𝑡∈[0,𝑇]𝕋}𝑢(𝑡)≥𝜎‖𝑢‖, where 𝜎=∑^𝑚−2_𝑖=1𝑏_𝑖𝜉_𝑖/(𝑇−∑^𝑚−2_𝑖=1𝑏_𝑖(𝑇−𝜉_𝑖)),‖𝑢‖=sup_{𝑡∈[0,𝑇]𝕋}|𝑢(𝑡)|.

Proof. (i) Let 𝜓₀(𝑠)∶=𝜙⁻¹(−∫^𝑠₀ℎ(𝑟)∇𝑟+𝐷). Consider the following equation: −^𝑠₀ℎ(𝑟)∇𝑟+𝐷=−^𝑠₀ℎ(𝑟)∑∇𝑟−^{𝑚−2𝑖=1}𝑐_𝑖∫^𝜉𝑖₀ℎ(𝑟)∇𝑟1−∑^𝑚−2_𝑖=1𝑐_𝑖≤0. (2.10) So, we have 𝜓₀(𝑠)≤0. Hence 𝑢(𝑡)≥0,𝑡∈[0,𝑇]_𝕋.
(ii) By 𝑢^Δ∇(𝑡)=𝜙⁻¹(−∫^𝑡₀ℎ(𝑟)∇𝑟+𝐷)≤0, we can know that the graph of 𝑢(𝑡) is concave down on (0,𝑇)_𝕋 and 𝑢^Δ(𝑡) is nonincreasing on [0,𝑇]_𝕋. This together with the assumption that the boundary condition 𝑢^Δ(𝑇)=0 implies that 𝑢^Δ(𝑡)≥0 for 𝑡∈[0,𝑇]_𝕋. This implies that‖𝑢‖=𝑢(𝑇),min_{𝑡∈[0,𝑇]𝕋}𝑢(𝑡)=𝑢(0). (2.11) For all 𝑖∈{1,2,…,𝑚−2}, we have from the concavity of 𝑢 that 𝑢𝜉_𝑖−𝑢(0)𝜉_𝑖≥𝑢(𝑇)−𝑢(0)𝑇, (2.12) that is, 𝑢𝜉_𝑖(0)+𝜉−𝑢^𝑖𝑇𝑢(0)≥𝜉_𝑖𝑇𝑢(𝑇). (2.13) This together with the boundary condition 𝑢(0)=∑^𝑚−2_𝑖=1𝑏_𝑖𝑢(𝜉_𝑖) implies that min_{𝑡∈[0,𝑇]𝕋}𝑢(𝑡)≥∑^𝑚−2_𝑖=1𝑏_𝑖𝜉_𝑖𝑇−∑^𝑚−2_𝑖=1𝑏_𝑖𝑇−𝜉_𝑖𝑢(𝑇). (2.14) This completes the proof.

Let 𝐸=𝒞_ld[0,𝑇]_𝕋 be equipped with the norm ‖𝑢‖=max_{𝑡∈[0,𝑇]𝕋}|𝑢(𝑡)|. Clearly, it follows that (𝐸,‖𝑢‖) is a Banach space. For the convenience, let 𝜓(𝑠)=𝜙⁻¹−∑^𝑚−2_𝑖=1𝑐_𝑖∫^𝜉𝑖₀𝑎(𝑟)∇𝑟1−∑^𝑚−2_𝑖=1𝑐_𝑖−^𝑠₀𝑎(𝑟)∇𝑟. (2.15) We define two cones by 𝑃=𝑢∶𝑢∈𝐸,𝑢(𝑡)≥0,𝑡∈[0,𝑇]_𝕋,𝐾=𝑢∶𝑢∈𝐸,𝑢(𝑡)isconcave,nondecreasing,nonnegativeon[0,𝑇]_𝕋,min_{[]𝑡∈0,𝑇𝕋}𝑢(𝑡)≥𝜎‖𝑢‖, (2.16) where 𝜎 is as in Lemma 2.3. Define the operators 𝐹∶𝑃→𝐸 and 𝑆∶𝐾→𝐸 by setting (𝐹𝑢)(𝑡)=^𝑡₀(𝑠−𝑇)𝜙⁻¹−^𝑠₀𝑎(𝑟)𝑓((𝑟+∑𝑟,𝑢))∇𝑟+𝐴∇𝑠_{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑠−𝑇)𝜙⁻¹−∫^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢(𝑟))∇𝑟+𝐴∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖, (2.17) where 𝐴=−(∑^𝑚−2_𝑖=1𝑐_𝑖∫^𝜉𝑖₀𝑎(𝑟)𝑓(𝑟,𝑢(𝑟))∇𝑟)/(1−∑^𝑚−2_𝑖=1𝑐_𝑖), (𝑆𝑢)(𝑡)=^𝑡₀()𝜑(𝑠)∑𝑠−𝑇∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑠−𝑇)𝜑(𝑠)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖, (2.18) where 𝜑(𝑠)=𝜙⁻¹(−∫^𝑠₀𝑎(𝑟)𝑓⁺(𝑟,𝑢(𝑟))∇𝑟+𝐴),𝐴=−(∑^𝑚−2_𝑖=1𝑐_𝑖∫^𝜉𝑖₀𝑎(𝑟)𝑓⁺(𝑟,𝑢(𝑟))∇𝑟)/(1−∑^𝑚−2_𝑖=1𝑐_𝑖), and 𝑓⁺(𝑡,𝑢(𝑡))=max{𝑓(𝑡,𝑢(𝑡)),0}. Obviously, 𝑢 is a solution of the BVP (1.1) if and only if 𝑢 is a fixed point of operator 𝐹.

Lemma 2.4. 𝑆∶𝐾→𝐾 is completely continuous.

Proof. It is easy to see that 𝑆𝐾⊂𝐾 by 𝑓⁺≥0 and Lemma 2.3. By Arzela-Ascoli theorem and Lebesgue dominated convergence theorem, we can easily prove that operator 𝑆 is completely continuous.

3. Main Results

In this section, we prove the existence of at least one positive solution to the BVP (1.1) by applying a four-functional fixed point theorem [8]. Also, by using the generalization of Leggett-Williams fixed point theorem [9], we prove the existence of at least three positive solutions for the BVP (1.1).

Let 𝛼 and Ψ be nonnegative continuous concave functionals on 𝑃, and let 𝛽 and 𝜃 be nonnegative continuous convex functionals on 𝑃, then for positive numbers 𝑟, 𝑗, 𝑣, and 𝑅, we define the following sets: 𝑄=(𝑥)(𝑥),𝑈=(𝑥,𝑉()=(𝑥).𝛼,𝛽,𝑟,𝑅𝑥∈𝑃∶𝑟≤𝛼,𝛽≤𝑅Ψ,𝑗𝑥∈𝑄𝛼,𝛽,𝑟,𝑅∶𝑗≤Ψ)𝜃,𝑣𝑥∈𝑄𝛼,𝛽,𝑟,𝑅∶𝜃≤𝑣 (3.1)

Theorem 3.1 (see [8]). If 𝑃 is a cone in a real Banach space 𝐸, 𝛼 and Ψ are nonnegative continuous concave functionals on 𝑃, 𝛽, and 𝜃 are nonnegative continuous convex functionals on 𝑃 and there exist positive numbers 𝑟, 𝑗, 𝑣, and 𝑅, such that 𝐴∶𝑄𝛼,𝛽,𝑟,𝑅→𝑃 (3.2) is a completely continuous operator, and 𝑄(𝛼,𝛽,𝑟,𝑅) is a bounded set. If(i){𝑥∈𝑈(Ψ,𝑗)∶𝛽(𝑥)<𝑅}∩{𝑥∈𝑉(𝜃,𝑣)∶𝑟<𝛼(𝑥)}≠∅, (ii)𝛼(𝐴𝑥)≥𝑟, for all 𝑥∈𝑄(𝛼,𝛽,𝑟,𝑅), with 𝛼(𝑥)=𝑟 and 𝑣<𝜃(𝐴𝑥),(iii)𝛼(𝐴𝑥)≥𝑟, for all 𝑥∈𝑉(𝜃,𝑣), with 𝛼(𝑥)=𝑟,(iv)𝛽(𝐴𝑥)≤𝑅, for all 𝑥∈𝑄(𝛼,𝛽,𝑟,𝑅), with 𝛽(𝑥)=𝑅 and Ψ(𝐴𝑥)<𝑗,(v)𝛽(𝐴𝑥)≤𝑅, for all 𝑥∈𝑈(Ψ,𝑗), with 𝛽(𝑥)=𝑅, then 𝐴 has a fixed point 𝑥 in 𝑄(𝛼,𝛽,𝑟,𝑅).

Suppose 𝜏,𝜂∈𝕋 with 0<𝜏<𝜂<𝑇. For convenience, we take the notations ∑𝑤=^{𝑚−2𝑖=1}𝑏_𝑖1−∑^𝑚−2_𝑖=1𝑏_𝑖,𝑁_𝜂=^𝜂₀(𝑠−𝑇)𝜓(𝑠)∇𝑠,𝑁_𝑚−2=^𝜉𝑚−2₀()𝜓(𝑠)𝑁𝑠−𝑇∇𝑠,_𝜏=^𝜏₀(𝑠−𝑇)𝜓(𝑠)∇𝑠,𝑁_𝑇=^𝑇₀()𝜓(𝑠)𝑁𝑠−𝑇∇𝑠,₀=^𝜏₀(𝑠−𝑇)𝜙⁻¹−^𝑠₀𝑎(𝑟)∇𝑟∇𝑠. (3.3)

Theorem 3.2. Assume that (H1), (H2) hold if there exist constants 𝑟, 𝑗, 𝑣, 𝑅 with 𝑅>max{(𝑁_𝑇+𝑁_𝑚−2𝑤)/𝑁_𝜏,𝑇/𝜏}𝑗,𝑣≥max{((𝑁_𝜂+𝑁_𝑚−2𝑤)/𝑁_𝜏)𝑗,(𝜂/𝜏)𝑟}, 𝑟<𝑗, and suppose that 𝑓(𝑡,𝑢) satisfies the following conditions: (B1)𝑓(𝑡,𝑢)≤𝜙(𝑅/(𝑁_𝑇+𝑁_𝑚−2𝑤)) for all (𝑡,𝑢)∈[0,𝑇]_𝕋×[0,𝑅];(B2)𝑓(𝑡,𝑢)≥𝜙(𝑟/𝑁₀) for all (𝑡,𝑢)∈[0,𝜏]_𝕋×[𝑟,𝑣];(B3)𝑓(𝑡,𝑢)≥0 for all (𝑡,𝑢)∈[0,𝑇]_𝕋×[𝜎𝑟,𝑅]. Then the BVP (1.1) has a fixed point 𝑢∈𝑃 such that min_{𝑡∈𝜏,𝜂𝕋}𝑢(𝑡)≥𝑟,max_{[]𝑡∈0,𝑇𝕋}𝑢(𝑡)≤𝑅. (3.4)

Define maps 𝛼(𝑢)=Ψ(𝑢)=min_{𝑡∈𝜏,𝜂𝕋}𝑢(𝑡),𝜃(𝑢)=max_{𝑡∈𝜏,𝜂𝕋}𝑢(𝑡),𝛽(𝑢)=max_{[]𝑡∈0,𝑇𝕋}𝑢(𝑡). (3.5) Let 𝑄(𝛼,𝛽,𝑟,𝑅), 𝑈(Ψ,𝑗), and 𝑉(𝜃,𝑣) be defined by (3.1).

Proof. We first show that 𝑄(𝛼,𝛽,𝑟,𝑅) is bounded and 𝑆∶𝑄(𝛼,𝛽,𝑟,𝑅)→𝐾 is completely continuous. For all 𝑢∈𝑄(𝛼,𝛽,𝑟,𝑅), we have ‖𝑢‖=max_{𝑡∈[0,𝑇]𝕋}𝑢(𝑡)=𝛽(𝑢)≤𝑅, which means that 𝑄(𝛼,𝛽,𝑟,𝑅) is a bounded set. According to Lemma 2.4, it is clear that 𝑆∶𝑄(𝛼,𝛽,𝑟,𝑅)→𝐾 is completely continuous.
Let 𝑢₀=𝑗𝑁_𝜏^𝑡₀()𝜓(𝑠)∑𝑠−𝑇∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑠−𝑇)𝜓(𝑠)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖. (3.6)
Clearly, 𝑢₀∈𝐾. By direct calculation, Ψ𝑢₀=𝑢₀(𝜏)=𝑗𝑁_𝜏^𝜏₀()𝜓(𝑠)∑𝑠−𝑇∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑠−𝑇)𝜓(𝑠)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖≥𝑗𝑁_𝜏^𝜏₀()𝜓(𝑠)𝛽𝑢𝑠−𝑇∇𝑠=𝑗,₀=𝑗𝑁_𝜏^𝑇₀()𝜓(𝑠)∑𝑠−𝑇∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑠−𝑇)𝜓(𝑠)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖≤𝑗𝑁_𝜏^𝑇₀()𝜓(𝑠)∑𝑠−𝑇∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑚−2₀(𝑠−𝑇)𝜓(𝑠)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖=𝑗𝑁_𝜏𝑁_𝑇+𝜔𝑁_𝑚−2𝜃𝑢<𝑅,₀=𝑗𝑁_𝜏^𝜂₀()𝜓(𝑠)∑𝑠−𝑇∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑠−𝑇)𝜓(𝑠)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖≤𝑗𝑁_𝜏^𝜂₀()𝜓(𝑠)∑𝑠−𝑇∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑚−2₀(𝑠−𝑇)𝜓(𝑠)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖=𝑗𝑁_𝜏𝑁_𝜂+𝜔𝑁_𝑚−2𝛼𝑢≤𝑣,₀=Ψ𝑢₀≥𝑗>𝑟. (3.7) So, 𝑢₀∈{𝑢∈𝑈(Ψ,𝑗)∶𝛽(𝑢)<𝑅}∩{𝑢∈𝑉(𝜃,𝑣)∶𝑟<𝛼(𝑢)}, which means that (i) in Theorem 3.1 is satisfied.
For all 𝑢∈𝑄(𝛼,𝛽,𝑟,𝑅), with 𝛼(𝑢)=𝑟 and 𝑣<𝜃(𝑆𝑢), we have 𝛼()=(𝜏)≥𝜏𝑆𝑢𝑆𝑢)(𝜂()𝜂=𝜏𝑆𝑢𝜂𝜃()>𝜏𝑆𝑢𝜂𝑣≥𝑟, (3.8) and for all 𝑢∈𝑄(𝛼,𝛽,𝑟,𝑅), with 𝛽(𝑢)=𝑅 and Ψ(𝑆𝑢)<𝑗, we have 𝛽()=(𝑇)≤𝑇𝑆𝑢𝑆𝑢)(𝜏(𝜏)=𝑇𝑆𝑢)(𝜏Ψ()<𝑇𝑆𝑢𝜏𝑗<𝑅. (3.9) Hence (ii) and (iv) in Theorem 3.1 are fulfilled.
Lastly, we consider Theorem 3.1 (iii) and (v). For all 𝑉(𝜃,𝑣), with 𝛼(𝑢)=𝑟, 𝛼(𝑆𝑢)=^𝜏₀()𝜑(𝑠)∑𝑠−𝑇∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑠−𝑇)𝜑(𝑠)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖≥^𝜏₀(𝑠−𝑇)𝜙⁻¹−^𝑠₀𝑎(𝑟)𝑓⁺((𝑟∑𝑟,𝑢))∇𝑟−^{𝑚−2𝑖=1}𝑐_𝑖∫^𝜉𝑖₀𝑎(𝑟)𝑓⁺(𝑟,𝑢(𝑟))∇𝑟1−∑^𝑚−2_𝑖=1𝑐_𝑖≥∇𝑠_𝜏0(𝑠−𝑇)𝜙⁻¹−^𝑠₀𝑎(𝑟)𝑓⁺((𝑟𝑟𝑟,𝑢))∇𝑟∇𝑠≥𝑁₀^𝜏₀(𝑠−𝑇)𝜙⁻¹−^𝑠₀𝑎(𝑟)∇𝑟∇𝑠=𝑟, (3.10) and for all 𝑢∈𝑈(Ψ,𝑗), with 𝛽(𝑢)=𝑅, 𝛽(𝑆𝑢)=^𝑇₀()𝜑(𝑠)∑𝑠−𝑇∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑠−𝑇)𝜑(𝑠)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖≤^𝑇₀()𝜑(𝑠)∑𝑠−𝑇∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑚−2₀(𝑠−𝑇)𝜑(𝑠)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖≤𝑅𝑁_𝑇+𝑁_𝑚−2𝑤^𝑇₀(𝑠−𝑇)𝜓(𝑠)∇𝑠+𝑤^𝜉𝑚−2₀(𝑠−𝑇)𝜓(𝑠)∇𝑠=𝑅. (3.11) Thus, all conditions of Theorem 3.1 are satisfied. 𝑆 has a fixed point 𝑢 in 𝑄(𝛼,𝛽,𝑟,𝑅). Clearly, 𝜎𝑟≤𝑢(𝑡)≤𝑅,𝑡∈[0,𝑇]_𝕋. By condition (B3), we have 𝑓(𝑡,𝑢(𝑡))≥0,𝑡∈[0,𝑇]_𝕋, that is, 𝑓⁺(𝑡,𝑢(𝑡))=𝑓(𝑡,𝑢(𝑡)). Hence, 𝐹𝑢=𝑆𝑢. This means that 𝑢 is a fixed point of operator 𝐹. Therefore, the BVP (1.1) has at least one positive solution.

Suppose 𝛼 and 𝛽 are two nonnegative continuous convex functionals satisfying ‖𝑥‖≤𝐿max𝛼(𝑥),𝛽(𝑥),𝑥∈𝑃, (3.12) where 𝐿 is a positive constant, and Ω=𝑥∈𝑃∶𝛼(𝑥)<𝑟,𝛽(𝑥)<𝑙≠∅,𝑟>0,𝑙>0. (3.13)

Let 𝑟>𝑎>0,𝑙>0 be given, 𝛼,𝛽 nonnegative continuous convex functionals on 𝑃 satisfying the relation (3.12) and (3.13), and 𝛾 a nonnegative continuous concave functional on 𝑃. We define the following convex sets: 𝑃𝛼,𝑟;𝛽,𝑙=𝑢∈𝑃∶𝛼(𝑢)<𝑟,𝛽(𝑢)<𝑙,𝑃=(𝑢)(𝑢),𝑃=(𝑢)(𝑢)(𝑢),𝛼,𝑟;𝛽,𝑙𝑢∈𝑃∶𝛼≤𝑟,𝛽≤𝑙𝛼,𝑟;𝛽,𝑙;𝛾,𝑎𝑢∈𝑃∶𝛼<𝑟,𝛽<𝑙,𝛾>𝑎𝑃𝛼,𝑟;𝛽,𝑙;𝛾,𝑎=𝑢∈𝑃∶𝛼(𝑢)≤𝑟,𝛽(𝑢)≤𝑙,𝛾(𝑢)≥𝑎. (3.14)

Theorem 3.3 (see [9]). Let 𝔹 be a Banach space, 𝑃⊂𝔹 a cone, and 𝑟₂≥𝑑>𝑏>𝑟₁>0,𝑙₂≥𝑙₁>0. Assume that 𝛼 and 𝛽 are nonnegative continuous convex functionals satisfying (3.12) and (3.13), 𝛾 is a nonnegative continuous concave functional on 𝑃 such that 𝛾(𝑢)≤𝛼(𝑢) for all 𝑢∈𝑃(𝛼,𝑟₂;𝛽,𝑙₂), and 𝐴∶𝑃(𝛼,𝑟₂;𝛽,𝑙₂)→𝑃(𝛼,𝑟₂;𝛽,𝑙₂) is a completely continuous operator. Suppose(i){𝑢∈𝑃(𝛼,𝑑;𝛽,𝑙₂;𝛾,𝑏)∶𝛾(𝑢)>𝑏}≠∅,𝛾(𝐴𝑢)>𝑏 for 𝑢∈𝑃(𝛼,𝑑;𝛽,𝑙₂;𝛾,𝑏);(ii)𝛼(𝐴𝑢)<𝑟₁,𝛽(𝐴𝑢)<𝑙₁ for 𝑢∈𝑃(𝛼,𝑟₁;𝛽,𝑙₁);(iii)𝛾(𝐴𝑢)>𝑏 for 𝑢∈𝑃(𝛼,𝑟₂;𝛽,𝑙₂;𝛾,𝑏) with 𝛼(𝐴𝑢)>𝑑. Then 𝐴 has at least three fixed points 𝑢₁,𝑢₂,𝑢₃∈𝑃(𝛼,𝑟₂;𝛽,𝑙₂) with 𝑢₁∈𝑃𝛼,𝑟₁;𝛽,𝑙₁,𝑢₂∈𝑃𝛼,𝑟₂;𝛽,𝑙₂(𝑢),𝑢;𝛾,𝑏∶𝛾>𝑏₃∈𝑃𝛼,𝑟₂;𝛽,𝑙₂⧵𝑃𝛼,𝑟₂;𝛽,𝑙₂;𝛾,𝑏∪𝑃𝛼,𝑟₁;𝛽,𝑙₁. (3.15)

It is easy to see that for 𝑢∈𝐾,𝑢(𝑡)≥(𝑡/𝑇)‖𝑢‖,𝑡∈[0,𝑇]_𝕋. So we get𝑢(𝑡)≥𝜁𝑇‖𝑢‖>12‖𝑢‖,𝑡∈𝜁,𝑇_𝕋, (3.16) where 𝜁 is as in (H3). Let 𝛼, 𝛽, and 𝛾 be defined on the cone 𝐾 by 𝛼(𝑢)=max_{[]𝑡∈0,𝑇𝕋}𝑢(𝑡),𝛽(𝑢)=min_{[]𝑡∈0,𝑇𝕋}𝑢(𝑡),𝛾(𝑢)=min_{𝑡∈𝜁,𝑇𝕋}𝑢(𝑡), (3.17) respectively. Then it is easy to see that ‖𝑢‖=max{𝛼(𝑢),𝛽(𝑢)} and (3.12), (3.13) hold. Now, for convenience, let (3.3) and 𝑀=∫^𝜁₀(𝑠−𝑇)𝜓(𝜁)∇𝑠.

Theorem 3.4. Assume (H1)–(H3) hold, 𝑓(𝑡,0) is not equivalent to 0 for 𝑡∈[0,𝑇]_𝕋. If there are positive numbers 𝑟₂≥2𝑏>𝑏>𝑟₁>0, 𝑙₂≥𝑙₁>0 with 𝑏/𝑀≤min{𝑟₂/(𝑁_𝑇+𝑁_𝑚−2𝑤),𝑙₂/(𝑁_𝑚−2𝑤)}, such that the following conditions are satisfied. (C1)0≤𝑓(𝑡,𝑢)≤min{𝜙(𝑟₂/(𝑁_𝑇+𝑁_𝑚−2𝑤)),𝜙(𝑙₂/(𝑁_𝑚−2𝑤))} for (𝑡,𝑢)∈[0,𝑇]_𝕋×[0,𝑟₂],(C2)𝑓(𝑡,𝑢)>𝜙(𝑏/𝑀) for (𝑡,𝑢)∈[𝜁,𝑇]_𝕋×[𝑏,2𝑏],(C3)𝑓(𝑡,𝑢)<min{𝜙(𝑟₁/(𝑁_𝑇+𝑁_𝑚−2𝑤)),𝜙(𝑙₁/(𝑁_𝑚−2𝑤))} for (𝑡,𝑢)∈[0,𝑇]_𝕋×[0,𝑟₁]. Then the BVP (1.1) has at least three positive solutions 𝑢₁,𝑢₂,𝑢₃ satisfying max_{[]𝑡∈0,𝑇𝕋}𝑢₁(𝑡)<𝑟₁,min_{[]𝑡∈0,𝑇𝕋}𝑢₁(𝑡)<𝑙₁,𝑏<min_{𝑡∈𝜁,𝑇𝕋}𝑢₂(𝑡)≤max_{[]𝑡∈0,𝑇𝕋}𝑢₂(𝑡)≤𝑟₂,min_{[]𝑡∈0,𝑇𝕋}𝑢₂(𝑡)≤𝑙₂,max_{𝑡∈[0,𝑇]𝕋}𝑢₃(𝑡)<2𝑏,withmin_{𝑡∈[𝜁,𝑇]𝕋}𝑢₃(𝑡)<𝑏,min_{𝑡∈[0,𝑇]𝕋}𝑢₃(𝑡)≤𝑙₂. (3.18)

Proof. By the definition of 𝑆, it suffices to show that the conditions of Theorem 3.3 hold with respect to the operator 𝑆. Firstly, we show that if the condition (C1) is satisfied, then 𝑆∶𝐾𝛼,𝑟₂;𝛽,𝑙₂→𝐾𝛼,𝑟₂;𝛽,𝑙₂. (3.19)
If 𝑢∈𝐾(𝛼,𝑟₂;𝛽,𝑙₂), then assumption (C1) implies 𝑓()𝜙𝑟𝑡,𝑢≤min²𝑁_𝑇+𝑁_𝑚−2𝑤𝑙,𝜙²𝑁_𝑚−2𝑤,𝑡∈[0,𝑇]_𝕋,𝛼(𝑆𝑢)=^𝑇₀()𝜑(𝑠)∑𝑠−𝑇∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑠−𝑇)𝜑(𝑠)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖≤^𝑇₀()𝜑(𝑠)∑𝑠−𝑇∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑚−2₀(𝑠−𝑇)𝜑(𝑠)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖≤𝑟₂𝑁_𝑇+𝑁_𝑚−2𝑤^𝑇₀(𝑠−𝑇)𝜓(𝑠)∇𝑠+𝑤^𝜉𝑚−2₀(𝑠−𝑇)𝜓(𝑠)∇𝑠=𝑟₂,𝛽()=∑𝑆𝑢^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑠−𝑇)𝜑(𝑠)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖≤𝑙₂𝑁_𝑚−2𝑤𝑤^𝜉𝑚−2₀(𝑠−𝑇)𝜓(𝑠)∇𝑠=𝑙₂. (3.20) Therefore, (3.19) holds. In the same way, if 𝑢∈𝐾(𝛼,𝑟₁;𝛽,𝑙₁), then condition (C3) implies 𝑓()𝜙𝑟𝑡,𝑢<min¹𝑁_𝑇+𝑁_𝑚−2𝑤𝑙,𝜙¹𝑁_𝑚−2𝑤,𝑡∈[0,𝑇]_𝕋. (3.21) As the argument above, we have 𝑆∶𝐾(𝛼,𝑟₁;𝛽,𝑙₁)→𝐾(𝛼,𝑟₁;𝛽,𝑙₁). Thus, condition (ii) of Theorem 3.3 holds. To check the condition (i) of Theorem 3.3, we choose 𝑢₀(𝑡)=2𝑏𝑡𝑇,𝑡∈[0,𝑇]_𝕋. (3.22) It is easy to see that 𝑢₀(𝑡)∈𝐾(𝛼,2𝑏;𝛽,𝑙₂;𝛾,𝑏),𝛾(𝑢₀)>𝑏. Therefore, for 𝑢∈𝐾(𝛼,2𝑏;𝛽,𝑙₂;𝛾,𝑏), there are 𝑏≤𝑢(𝑡)≤2𝑏,𝑡∈[𝜁,𝑇]_𝕋. Hence, condition (C2) implies 𝑓(𝑡,𝑢)>𝜙(𝑏/𝑀),𝑡∈[𝜁,𝑇]_𝕋. So, by the definition of 𝛾, 𝛾(𝑆𝑢)=^𝜁₀()𝜑(𝑠)∑𝑠−𝑇∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑠−𝑇)𝜑(𝑠)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖≥^𝜁₀()𝜑(𝑠)=𝑠−𝑇∇𝑠_𝜁0(𝑠−𝑇)𝜙⁻¹−^𝑠₀𝑎(𝑟)𝑓⁺((𝑟∑𝑟,𝑢))∇𝑟−^{𝑚−2𝑖=1}𝑐_𝑖∫^𝜉𝑖₀𝑎(𝑟)𝑓⁺(𝑟,𝑢(𝑟))∇𝑟1−∑^𝑚−2_𝑖=1𝑐_𝑖≥∇𝑠_𝜁0(𝑠−𝑇)𝜙⁻¹−^𝜁₀𝑎(𝑟)𝑓⁺((𝑟∑𝑟,𝑢))∇𝑟−^{𝑚−2𝑖=1}𝑐_𝑖∫^𝜉𝑖₀𝑎(𝑟)𝑓⁺(𝑟,𝑢(𝑟))∇𝑟1−∑^𝑚−2_𝑖=1𝑐_𝑖>𝑏∇𝑠𝑀^𝜁₀(𝑠−𝑇)𝜙⁻¹−^𝜁₀𝑎(𝑟)∑∇𝑟−^{𝑚−2𝑖=1}𝑐_𝑖∫^𝜉𝑖₀𝑎(𝑟)∇𝑟1−∑^𝑚−2_𝑖=1𝑐_𝑖∇𝑠=𝑏. (3.23) Hence, condition (i) of Theorem 3.3 holds. We finally prove that (iii) in Theorem 3.3 holds. In fact, for 𝑢∈𝐾(𝛼,2𝑏;𝛽,𝑙₂;𝛾,𝑏) with 𝛼(𝑆𝑢)>2𝑏, we have 𝛾(𝑆𝑢)>𝑏. Thus from Theorem 3.3, the BVP (1.1) has at least three fixed points 𝑢₁,𝑢₂,𝑢₃∈𝐾(𝛼,𝑟₂;𝛽,𝑙₂). Clearly, 0≤𝑢₁(𝑡)≤𝑟₁, 𝜎𝑏<𝑢₂(𝑡)≤𝑟₂, 0≤𝑢₃(𝑡)≤2𝑏, 𝑡∈[0,𝑇]_𝕋. By condition (C1), we have 𝑓(𝑡,𝑢(𝑡))≥0, 𝑡∈[0,𝑇]_𝕋, that is, 𝑓⁺(𝑡,𝑢(𝑡))=𝑓(𝑡,𝑢(𝑡)). Hence, 𝐹𝑢=𝑆𝑢. This means that 𝑢₁,𝑢₂,𝑢₃ are fixed points of operator 𝐹. Therefore, the BVP (1.1) has at least three positive solutions.

4. Example

Example 4.1. Let 𝕋=[0,1/2]∪{0.6}∪[0.7,1]. If we choose 𝑇=1, 𝑎(𝑡)=1,𝑏₁=1/3, 𝑐₁=1/2, 𝜉₁=1/4 in the boundary value problem (1.1), then we have the following BVP on time scale 𝕋: 𝜙𝑢^Δ∇(𝑡)^∇+𝑓(𝑡,𝑢(𝑡))=0,𝑡∈[0,1]_𝕋,𝑢(0)=13𝑢14,𝑢^Δ(1)=0,𝜙𝑢^Δ∇(0)=12𝜙𝑢^Δ∇14, (4.1) where 𝜙(𝑢)=𝑢𝑢,𝑢≤0,₂𝑓()=()[],𝑢>0,𝑡,𝑢95𝑢−0.0001+0.0007,0,1_𝕋×[],[]0,0.00010.0007,0,1_𝕋×[0.0001,∞). (4.2)
Set 𝜏=1/3, 𝜂=0.6, by calculation, 1𝑤=2,𝑁_𝑇=17216000,𝑁_𝑚−2=31384,𝑁_𝜏=73648,𝑁₀=7,𝑁162_𝜂=2533112000,𝜎=9, (4.3) and let 𝑟=0.001,𝑗=𝑟+10⁻⁴, 𝑅=0.012, and 𝑣=0.0035 with 0.012>0.0033,0.0035≥0.002455. Clearly, we can verify that the conditions in Theorem 3.2 are fulfilled. Thus, by Theorem 3.2, the BVP (4.1) has a fixed point 𝑢 such that min_{𝑡∈[1/3,0.6]𝕋}𝑢(𝑡)≥0.001,max_{𝑡∈[0,1]𝕋}𝑢(𝑡)≤0.012. (4.4)

Example 4.2. Let 𝕋=[0,1]∪{1.2}∪{1.5}∪[1.8,2]. If we choose 𝑇=2, 𝑎(𝑡)=1, 𝑏₁=1/4, 𝑐₁=1/3, 𝜉₁=2/3 in the boundary value problem (1.1), then we have the following BVP on time scale 𝕋∶𝜙𝑢^Δ∇(𝑡)^∇+𝑓(𝑡,𝑢(𝑡))=0,𝑡∈[0,2]_𝕋,𝑢(0)=14𝑢23,𝑢^Δ(2)=0,𝜙𝑢^Δ∇(0)=13𝜙𝑢^Δ∇23, (4.5) where 𝜙(𝑢)=𝑢𝑢,𝑢≤0,₂𝑓()=1,𝑢>0,𝑡,𝑢180𝑡+𝑢10³,𝑡∈[0,2]_𝕋[],1,𝑢∈0,4880𝑡+5𝑢,𝑡∈[0,2]_𝕋(],1,𝑢∈4,200()[]80𝑡+520−𝑢,𝑡∈0,2_𝕋,𝑢∈(200,∞). (4.6) Obviously the hypotheses (H1), (H2) hold and 𝑓(𝑡,0) is not equivalent to 0 on [0,2]_𝕋. By simple calculations, we have 𝜁=1.2,1𝑤=3,𝑁_𝑇=1.859,𝑁_𝑚−2=0.716,𝑀=2.5398. (4.7) If we choose 𝑟₂=200, 𝑏=2, 𝑟₁=1/2, and 𝑙₂=80, 𝑙₁=20, then 𝑓(𝑡,𝑢) satisfies(1)0≤𝑓(𝑡,𝑢)≤320.025≤min{𝜙(𝑟₂/(𝑁_𝑇+𝑁_𝑚−2𝑤)),𝜙(𝑙₂/𝑁_𝑚−2𝑤)}=9090 for (𝑡,𝑢)∈[0,2]_𝕋×[0,200];(2)𝑓(𝑡,𝑢)≥0.815>𝜙(𝑏/𝑀)=0.62 for (𝑡,𝑢)∈[1.2,2]_𝕋×[2,4],(3)𝑓(𝑡,𝑢)≤0.0375<min{𝜙(𝑟₁/(𝑁_𝑇+𝑁_𝑚−2𝑤)),𝜙(𝑙₁/(𝑁_𝑚−2𝑤))}=0.0568. So, all conditions of Theorem 3.4 hold. Thus by Theorem 3.4, the BVP (4.5) has at least three positive solutions 𝑢₁, 𝑢₂, 𝑢₃ such that max_{𝑡∈[0,2]𝕋}𝑢₁(𝑡)<12,min_{𝑡∈[0,2]𝕋}𝑢₁(𝑡)<20,2<min_{𝑡∈[1.2,2]𝕋}𝑢₂(𝑡)≤max_{𝑡∈[0,2]𝕋}𝑢₂(𝑡)≤200,min_{𝑡∈[0,2]𝕋}𝑢₂(𝑡)≤80,max_{𝑡∈[0,2]𝕋}𝑢₃(𝑡)<4,withmin_{𝑡∈[1.2,2]𝕋}𝑢₃(𝑡)<2,min_{𝑡∈[0,2]𝕋}𝑢₃(𝑡)≤80. (4.8)