Existence of Three Positive Solutions to Some p-Laplacian Boundary Value Problems

Abstract

We obtain, by using the Leggett-Williams fixed point theorem, sufficient conditions that ensure the existence of at least three positive solutions to some p-Laplacian boundary value problems on time scales.

1. Introduction

The study of dynamic equations on time scales goes back to the 1989 Ph.D. thesis of Hilger [1, 2] and is currently an area of mathematics receiving considerable attention [3–7]. Although the basic aim of the theory of time scales is to unify the study of differential and difference equations in one and the same subject, it also extends these classical domains to hybrid and in-between cases. A great deal of work has been done since the eighties of the XX century in unifying the theories of differential and difference equations by establishing more general results in the time scale setting [8–12].

Boundary value 𝑝-Laplacian problems for differential equations and finite difference equations have been studied extensively (see, e.g., [13] and references therein). Although many existence results for dynamic equations on time scales are available [14, 15], there are not many results concerning 𝑝-Laplacian problems on time scales [16–19]. In this paper we prove new existence results for three classes of 𝑝-Laplacian boundary value problems on time scales. In contrast with our previous works [17, 18], which make use of the Krasnoselskii fixed point theorem and the fixed point index theory, respectively, here we use the Leggett-Williams fixed point theorem [20, 21] obtaining multiplicity of positive solutions. The application of the Leggett-Williams fixed point theorem for proving multiplicity of solutions for boundary value problems on time scales was first introduced by Agarwal and O’Regan [22] and is now recognized as an important tool to prove existence of positive solutions for boundary value problems on time scales [23–28].

The paper is organized as follows. In Section 2 we present some necessary results from the theory of time scales (Section  2.1) and the theory of cones in Banach spaces (Section  2.2). We end Section  2.2 with the Leggett-Williams fixed point theorem for a cone-preserving operator, which is our main tool in proving existence of positive solutions to the boundary value problems on time scales we consider in Section 3. The contribution of the paper is Section 3, which is divided into three parts. The purpose of the first part (Section  3.1) is to prove existence of positive solutions to the nonlocal 𝑝-Laplacian dynamic equation on time scales −𝜙_𝑝𝑢^Δ(𝑡)^∇=𝜆𝑓(𝑢(𝑡))∫^𝑇₀𝑓(𝑢(𝜏))∇𝜏²,𝑡∈(0,𝑇)_𝕋, (1) satisfying the boundary conditions 𝜙_𝑝𝑢^Δ(0)−𝛽𝜙_𝑝𝑢^Δ𝜂𝑢(𝑇)𝜂=0,−𝛽𝑢=0, (2) where 𝜂∈(0,𝑇)_𝕋, 𝜙_𝑝(⋅) is the 𝑝-Laplacian operator defined by 𝜙_𝑝(𝑠)=|𝑠|^𝑝−2𝑠, 𝑝>1, and (𝜙_𝑝)⁻¹=𝜙_𝑞 with 𝑞 the Holder conjugate of 𝑝, that is, 1/𝑝+1/𝑞=1. The concrete value of 𝑝 is connected with the application at hands. For 𝑝=2, problem (1)-(2) describes the operation of a device flowed by an electric current, for example, thermistors [29], which are devices made from materials whose electrical conductivity is highly dependent on the temperature. Thermistors have the advantage of being temperature measurement devices of low cost, high resolution, and flexibility in size and shape. Constant 𝜆 in (1) is a dimensionless parameter that can be identified with the square of the applied potential difference at the ends of a conductor, 𝑓(𝑢) is the temperature-dependent resistivity of the conductor, and 𝛽 in (2) is a transfer coefficient supposed to verify 0<𝛽<1. For a more detailed discussion about the physical justification of (1)-(2) the reader is referred to [17]. Theoretical analysis (existence, uniqueness, regularity, and asymptotic results) for thermistor problems with various types of boundary and initial conditions has received significant attention in the last few years for the particular case 𝕋=ℝ [30–34]. The second part of our results (Section  3.2) is concerned with the following quasilinear elliptic problem: −𝜙_𝑝𝑢^Δ(𝑡)^∇=𝑓(𝑢(𝑡))+ℎ(𝑡),𝑡∈(0,𝑇)_𝕋,𝑢^Δ(0)=0,𝑢(𝑇)−𝑢𝜂=0, (3) where 𝜂∈(0,𝑇)_𝕋. Results on existence of infinitely many radial solutions to (3) are proved in the literature using (i) variational methods, where solutions are obtained as critical points of some energy functional on a Sobolev space, with 𝑓 satisfying appropriate conditions [35, 36]; (ii) methods based on phase-plane analysis and the shooting method [37]; (iii) the technique of time maps [38]. For 𝑝=2, ℎ≡0, and 𝕋=ℝ, problem (3) becomes a well-known boundary value problem of differential equations. Our results generalize earlier works to the case of a generic time scale 𝕋, 𝑝≠2, and ℎ not identically zero. Finally, the third part of our contribution (Section  3.3) is devoted to the existence of positive solutions to the 𝑝-Laplacian dynamic equation 𝜙_𝑝𝑢^Δ(𝑡)^∇(𝑡)𝑓(𝑢(𝑡)(𝜔(𝑡()+𝜆𝑎,𝑢)))=0,𝑡∈0,𝑇_𝕋,𝑢(𝑡)=𝜓(𝑡),𝑡∈[−𝑟,0]_𝕋,𝑢(0)−𝐵₀𝑢^Δ(0)=0,𝑢^Δ(𝑇)=0 (4) on a time scale 𝕋 such that 0,𝑇∈𝕋^𝜅_𝜅, −𝑟∈𝕋 with −𝑟≤0<𝑇, where 𝜆>0. This problem is considered in [39] where the author applies the Krasnoselskii fixed point theorem to obtain one positive solution to (4). Here we use the same conditions as in [39], but applying Leggett-Williams’ theorem we are able to obtain more: we prove existence of at least three positive solutions and we are able to localize them.

2. Preliminaries

Here we just recall the basic concepts and results needed in the sequel. For an introduction to time scales the reader is referred to [3, 8–10, 40, 41] and references therein; for a good introduction to the theory of cones in Banach spaces we refer the reader to the book [42].

2.1. Time Scales

A time scale 𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ. The operators 𝜎 and 𝜌 from 𝕋 to 𝕋 are defined in [1, 2] as 𝜎(𝑡){}𝜌(𝑡){}=inf𝜏∈𝕋∣𝜏>𝑡∈𝕋,=sup𝜏∈𝕋∣𝜏<𝑡∈𝕋 (5) and are called the forward jump operator and the backward jump operator, respectively. A point 𝑡∈𝕋 is left-dense, left-scattered, right-dense, and right-scattered if 𝜌(𝑡)=𝑡,𝜌(𝑡)<𝑡, and 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. Following [43], we also introduce the set 𝕋^𝜅_𝜅=𝕋^𝜅∩𝕋_𝜅.

Let 𝑓∶𝕋→ℝ and 𝑡∈𝕋^𝜅 (assume 𝑡 is not left-scattered if 𝑡=sup𝕋), then the delta derivative of 𝑓 at the point 𝑡 is defined to be the number 𝑓^Δ(𝑡) (provided it exists) with the property that for each 𝜖>0 there is a neighborhood 𝑈 of 𝑡 such that 𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡)(𝜎(𝑡)−𝑠)≤|𝜎(𝑡)−𝑠|∀𝑠∈𝑈. (6) Similarly, for 𝑡∈𝕋_𝜅 (assume 𝑡 is not right-scattered if 𝑡=inf𝕋), the nabla derivative of 𝑓 at the point 𝑡 is defined in [44] to be the number 𝑓^∇(𝑡) (provided it exists) with the property that for each 𝜖>0 there is a neighborhood 𝑈 of 𝑡 such that 𝑓𝜌(𝑡)−𝑓(𝑠)−𝑓^∇(𝑡)𝜌(𝑡)−𝑠≤𝜌(𝑡)−𝑠∀𝑠∈𝑈. (7) If 𝕋=ℝ, then 𝑓^Δ(𝑡)=𝑓^∇(𝑡)=𝑓^′(𝑡). If 𝕋=ℤ, then 𝑓^Δ(𝑡)=𝑓(𝑡+1)−𝑓(𝑡) is the forward difference operator while 𝑓^∇(𝑡)=𝑓(𝑡)−𝑓(𝑡−1) is the backward difference operator.

A function 𝑓 is left-dense continuous (i.e., 𝑙𝑑-continuous), if 𝑓 is continuous at each left-dense point in 𝕋 and its right-sided limit exists at each right-dense point in 𝕋. If 𝑓 is 𝑙𝑑-continuous, then there exists 𝐹 such that 𝐹^∇(𝑡)=𝑓(𝑡) for any 𝑡∈𝕋_𝜅. We then introduce the nabla integral by ^𝑏_𝑎𝑓(𝑡)∇𝑡=𝐹(𝑏)−𝐹(𝑎). (8) We define right-dense continuous (𝑟𝑑-continuous) functions in a similar way. If 𝑓 is 𝑟𝑑-continuous, then there exists 𝐹 such that 𝐹^Δ(𝑡)=𝑓(𝑡) for any 𝑡∈𝕋^𝜅, and we define the delta integral by ^𝑏_𝑎𝑓(𝑡)Δ𝑡=𝐹(𝑏)−𝐹(𝑎). (9)

2.2. Cones in Banach Spaces

In this paper 𝕋 is a time scale with 0∈𝕋_𝜅 and 𝑇∈𝕋^𝜅. We use ℝ⁺ and ℝ⁺₀ to denote, respectively, the set of positive and nonnegative real numbers. By [0,𝑇]_𝕋 we denote the set [0,𝑇]∩𝕋. Similarly, (0,𝑇)_𝕋=(0,𝑇)∩𝕋. Let 𝐸=ℂ_𝑙𝑑([0,𝑇]_𝕋,ℝ). It follows that 𝐸 is a Banach space with the norm ‖𝑢‖=max_[0,𝑇]𝕋|𝑢(𝑡)|.

Definition 1. Let 𝐸 be a real Banach space. A nonempty, closed, and convex set 𝑃⊂𝐸 is called a cone if it satisfies the following two conditions:(i)𝑢∈𝑃, 𝜆≥0, implies 𝜆𝑢∈𝑃; (ii)𝑢∈𝑃, −𝑢∈𝑃, implies 𝑢=0.

Every cone 𝑃⊂𝐸 induces an ordering in 𝐸 given by 𝑢≤𝑣iff𝑣−𝑢∈𝑃. (10)

Definition 2. Let 𝐸 be a real Banach space and 𝑃⊂𝐸 a cone. A function 𝛼∶𝑃→ℝ⁺₀ is called a nonnegative continuous concave functional if 𝛼 is continuous and 𝛼𝑡𝑥+(1−𝑡)𝑦≥𝑡𝛼(𝑥)+(1−𝑡)𝛼𝑦 (11) for all 𝑥,𝑦∈𝑃 and 0≤𝑡≤1.

Let 𝑎,𝑏, and 𝑟 be positive constants, 𝑃_𝑟={𝑢∈𝑃|‖𝑢‖<𝑟}, 𝑃(𝛼,𝑎,𝑏)={𝑢∈𝑃|𝑎≤𝛼(𝑢),‖𝑢‖<𝑏}. The following fixed point theorem provides the existence of at least three positive solutions. The origin in 𝐸 is denoted by ∅. The proof of the Leggett-Williams fixed point theorem can be found in Guo and Lakshmikantham [42] or Leggett and Williams [45].

Theorem 3 (Leggett-Williams’ Theorem). Let 𝑃 be a cone in a real Banach space 𝐸. Let 𝐺∶𝑃_𝑐→𝑃_𝑐 be a completely continuous map and 𝛼 a nonnegative continuous concave functional on 𝑃 such that 𝛼(𝑢)≤‖𝑢‖ for all 𝑢∈𝑃_𝑐. Suppose there exist positive constants 𝑎,𝑏, and 𝑑 with 0<𝑎<𝑏<𝑑≤𝑐 such that(i){𝑢∈𝑃(𝛼,𝑏,𝑑)|𝛼(𝑢)>𝑏}≠∅ and 𝛼(𝐺𝑢)>𝑏 for all 𝑢∈𝑃(𝛼,𝑏,𝑑);(ii)‖𝐺𝑢‖<𝑎 for all 𝑢∈𝑃_𝑎;(iii)𝛼(𝐺𝑢)>𝑏 for all 𝑢∈𝑃(𝛼,𝑏,𝑐) with ‖𝐺𝑢‖>𝑑. Then 𝐺 has at least three fixed points 𝑢₁, 𝑢₂, and 𝑢₃ satisfying 𝑢₁<𝑎,𝑏<𝛼𝑢₂,𝑢₃>𝑎,𝛼𝑢₃<𝑏. (12)

3. Main Results

We prove existence of three positive solutions to different 𝑝-Laplacian problems on time scales: in Section  3.1 we study problem (1)-(2), in Section  3.2 problem (3), and finally (4) in Section  3.3.

3.1. Nonlocal Thermistor Problem

By a solution 𝑢∶𝕋→ℝ of (1)-(2) we mean a delta differentiable function such that 𝑢^Δ and (|𝑢^Δ|^𝑝−2𝑢^Δ)^∇ are both continuous on 𝕋^𝜅_𝜅 and 𝑢 satisfies (1)-(2). We consider the following hypothesis:(H1)𝑓∶ℝ→ℝ⁺ is a continuous function.

Lemma 4 (see Lemma  3.1 of [17]). Assume that hypothesis (H1) on function 𝑓 is satisfied. Then 𝑢 is a solution to (1)-(2) if and only if 𝑢∈𝐸 is a solution to the integral equation 𝑢(𝑡)=−^𝑡₀𝜙_𝑞𝑔(𝑠)Δ𝑠+𝐵, (13) where 𝑔(𝑠)=^𝑠₀𝜆ℎ(𝑢(𝑟))∇𝑟−𝐴,𝐴=𝜙_𝑝𝑢^Δ(0)=−𝜆𝛽1−𝛽^𝜂₀ℎ(𝑢(𝑟ℎ(𝑢(𝑡=(𝑢(𝑡))∇𝑟,))𝜆𝑓))∫^𝑇₀𝑓(𝑢(𝜏))∇𝜏²,(0)=1𝐵=𝑢1−𝛽^𝑇₀𝜙_𝑞𝑔(𝑠)Δ𝑠−𝛽^𝜂₀𝜙_𝑞𝑔(𝑠)Δ𝑠. (14)

Lemma 5. Suppose (H1) holds. Then a solution 𝑢 to (1)-(2) satisfies 𝑢(𝑡)≥0 for 𝑡∈(0,𝑇)_𝕋.

Proof. We have 𝐴=−𝜆𝛽/(1−𝛽)∫^𝜂₀ℎ(𝑢(𝑟))∇𝑟≤0. Then, 𝑔(𝑠)=𝜆∫^𝑠₀ℎ(𝑢(𝑟))∇𝑟−𝐴≥0. It follows that 𝜙_𝑝(𝑔(𝑠))≥0. Since 0<𝛽<1, we also have 𝑢(0)=1=𝐵1−𝛽^𝑇₀𝜙_𝑞𝑔(𝑠)Δ𝑠−𝛽^𝜂₀𝜙_𝑞𝑔(𝑠)≥1Δ𝑠𝛽1−𝛽^𝑇₀𝜙_𝑞𝑔(𝑠)Δ𝑠−𝛽^𝜂₀𝜙_𝑞𝑔(𝑠)𝑢(𝑇)(0)−Δ𝑠≥0,=𝑢^𝑇₀𝜙_𝑞𝑔(𝑠)=Δ𝑠−𝛽1−𝛽^𝜂₀𝜙_𝑞𝑔(𝑠)+1Δ𝑠1−𝛽^𝑇₀𝜙_𝑞𝑔(𝑠)Δ𝑠−^𝑇₀𝜙_𝑞𝑔(𝑠)=Δ𝑠−𝛽1−𝛽^𝜂₀𝜙_𝑞𝑔(𝑠)𝛽Δ𝑠+1−𝛽^𝑇₀𝜙_𝑞𝑔(𝑠)=𝛽Δ𝑠1−𝛽^𝑇₀𝜙_𝑞𝑔(𝑠)Δ𝑠−^𝜂₀𝜙_𝑞𝑔(𝑠)Δ𝑠≥0. (15) If 𝑡∈(0,𝑇)_𝕋, then 𝑢(𝑡)=𝑢(0)−^𝑡₀𝜙_𝑞𝑔(𝑠)Δ𝑠≥−^𝑇₀𝜙_𝑞𝑔(𝑠)Δ𝑠+𝑢(0)=𝑢(𝑇)≥0. (16) Consequently, 𝑢(𝑡)≥0 for 𝑡∈(0,𝑇)_𝕋.

On the other hand, we have 𝜙_𝑝(𝑢^Δ(𝑠))=𝜙_𝑝(𝑢^Δ(0))−∫^𝑠₀𝜆ℎ(𝑢(𝑟))∇𝑟≤0. Since 𝐴=𝜙_𝑝(𝑢^Δ(0))≤0, then 𝑢^Δ≤0. This means that ||𝑢||=𝑢(0), inf_{𝑡∈(0,𝑇)𝕋}𝑢(𝑡)=𝑢(𝑇). Moreover, 𝜙_𝑝(𝑢^Δ(𝑠)) is nonincreasing, which implies with the monotonicity of 𝜙_𝑝 that 𝑢^Δ is a nonincreasing function on (0,𝑇)_𝕋. Hence, 𝑢 is concave. In order to apply Theorem 3, let us define the cone 𝑃⊂𝐸 by []𝑃=𝑢∈𝐸∣𝑢isnonnegative,decreasingon0,𝑇_𝕋andconcaveon𝐸. (17) We also define the nonnegative continuous concave functional 𝛼∶𝑃→ℝ⁺₀ by 𝛼(𝑢)=min_{𝑡∈𝜉,𝑇−𝜉𝕋}𝑢(𝑡)𝑇,𝜉∈0,2,∀𝑢∈𝑃. (18) It is easy to see that problem (1)-(2) has a solution 𝑢=𝑢(𝑡) if and only if 𝑢 is a fixed point of the operator 𝐺∶𝑃→𝐸 defined by 𝐺𝑢(𝑡)=−^𝑡₀𝜙_𝑞𝑔(𝑠)Δ𝑠+𝐵, (19) where 𝑔 and 𝐵 are as in Lemma 4.

Lemma 6. Let 𝐺 be defined by (19). Then,(i)𝐺(𝑃)⊆𝑃; (ii)𝐺∶𝑃→𝑃 is completely continuous.

Proof. (i)holds clearly from above. (ii)Suppose that 𝐷⊆𝑃 is a bounded set and let 𝑢∈𝐷. Then, |𝐺𝑢(𝑡)|=−^𝑡₀𝜙_𝑞𝑔(𝑠)≤Δ𝑠+𝐵−^𝑡₀𝜙_𝑞^𝑠₀𝜆𝑓(𝑢(𝑟))∫^𝑇₀𝑓(𝑢(𝜏))∇𝜏²+|𝐵|≤∇𝑟−𝐴Δ𝑠_𝑇0𝜙_𝑞^𝑠₀𝜆sup_𝑢∈𝐷𝑓(𝑢)𝑇inf_𝑢∈𝐷𝑓(𝑢)²|𝐵|,|𝐴|=∇𝑟−𝐴Δ𝑠+𝜆𝛽1−𝛽^𝜂₀ℎ(𝑢(𝑟=))∇𝑟𝜆𝛽1−𝛽^𝜂₀𝑓(𝑢(𝑟))∫^𝑇₀𝑓(𝑢(𝜏))∇𝜏²≤∇𝑟𝜆𝛽1−𝛽sup_𝑢∈𝐷𝑓(𝑢)𝑇inf_𝑢∈𝐷𝑓(𝑢)²𝜂. (20)
In the same way, we have |𝐵|≤11−𝛽^𝑇₀𝜙_𝑞𝑔(𝑠)≤1Δ𝑠1−𝛽^𝑇₀𝜙_𝑞𝜆sup_𝑢∈𝐷𝑓(𝑢)𝑇inf_𝑢∈𝐷𝑓(𝑢)²𝛽𝑠+𝜂1−𝛽Δ𝑠. (21) It follows that |𝐺𝑢(𝑡)|≤^𝑇₀𝜙_𝑞𝜆sup_𝑢∈𝐷𝑓(𝑢)𝑇inf_𝑢∈𝐷𝑓(𝑢)²𝑠+𝛽𝜂|𝐵|.1−𝛽Δ𝑠+ (22) As a consequence, we get ‖𝐺𝑢‖≤2−𝛽1−𝛽^𝑇₀𝜙_𝑞𝜆sup_𝑢∈𝐷𝑓(𝑢)𝑇inf_𝑢∈𝐷𝑓(𝑢)²𝑠+𝛽𝜂≤21−𝛽Δ𝑠𝜙1−𝛽^𝑞𝜆sup_𝑢∈𝐷𝑓(𝑢)𝑇inf_𝑢∈𝐷𝑓(𝑢)^2𝑇0𝜙_𝑞𝑠+𝛽𝜂1−𝛽Δ𝑠. (23) Then 𝐺(𝐷) is bounded on the whole bounded set 𝐷. Moreover, if 𝑡₁,𝑡₂∈[0,𝑇]_𝕋 and 𝑢∈𝐷, then we have for a positive constant 𝑐𝐺𝑢𝑡₂−𝐺𝑢𝑡₁≤^𝑡2_𝑡1𝜙_𝑞𝑔(𝑠)Δ𝑠≤𝑐𝑡₂−𝑡₁. (24) We see that the right-hand side of the above inequality goes uniformly to zero when |𝑡₂−𝑡₁|→0. Then by a standard application of the Arzela-Ascoli theorem we have that 𝐺∶𝑃→𝑃 is completely continuous.

We can also easily obtain the following properties.

Lemma 7. (i) 𝛼(𝑢)=𝑢(𝑇−𝜉)≤‖𝑢‖ for all 𝑢∈𝑃;
(ii) 𝛼(𝐺𝑢)=(𝐺𝑢)(𝑇−𝜉);
(iii) ‖𝐺𝑢‖=𝐺𝑢(0).

We now state the main result of Section  3.1.

Theorem 8. Suppose that (𝐻1) is verified and there exist positive constants 𝑎, 𝑏, 𝑐, and 𝑑 such that 0<𝜁𝑎=𝑎₁<𝑏<𝑑−𝑇²𝑐𝜙_𝑞(1/T)<𝑑<𝜁𝑐=𝑐₁, with 𝜁=(𝑇²/(1−𝛽)) 𝜙_𝑞(1/𝑇). One further imposes 𝑓 to satisfy the following hypotheses:(H2)min_{0≤𝑢≤𝑎1}𝑓(𝑢)≥𝜆²/(𝑇(1−𝛽)𝜙_𝑝(𝑎)) uniformly for all 𝑡∈[0,𝑇]_𝕋;(H3)min_{0≤𝑢≤𝑐1}𝑓(𝑢)≥𝜆²/(𝑇(1−𝛽)𝜙_𝑝(𝑐)) uniformly for all 𝑡∈[0,𝑇]_𝕋;(H4)min_{𝑏≤𝑢≤𝑑}𝑓(𝑢)≥𝜙_𝑝(𝑏𝐵₁) uniformly for all 𝑡∈[0,𝑇]_𝕋, where 𝐵₁=1−𝛽𝜙𝛽𝜉^𝑝(𝑇−𝜉)𝜙_𝑝𝜆𝑇sup_{𝑏≤𝑢≤𝑑}𝑓(𝑢)². (25)Then the boundary value problem (1)-(2) has at least three positive solutions 𝑢₁, 𝑢₂, and 𝑢₃, verifying 𝑢₁<𝑎,𝑏<𝛼𝑢₂,𝑢₃>𝑎,𝛼𝑢₃<𝑏. (26)

Proof. The proof passes by several lemmas. We have already seen in Lemma 6 that the operator 𝐺 is completely continuous. We now show the following Lemma.

Lemma 9. The following relations hold: 𝐺𝑃_𝑐1⊂𝑃_𝑐1,𝐺𝑃_𝑎1⊂𝑃_𝑎1. (27)

Proof. Obviously, 𝐺𝑃_𝑎1⊂𝑃. Moreover, for all 𝑢∈𝑃_𝑎1, we have 0≤𝑢(𝑡)≤𝑎₁. On the other hand we have 𝐺𝑢(𝑡)=−^𝑡₀𝜙_𝑞𝑔(𝑠)1Δ𝑠+1−𝛽^𝑇₀𝜙_𝑞𝑔(𝑠)−𝛽Δ𝑠1−𝛽^𝜂₀𝜙_𝑞𝑔(𝑠)Δ𝑠, (28) and for all 𝑢∈𝐺𝑃_𝑎1 we have 0≤𝑢(𝑡)≤𝑎₁. Then, ||≤1𝐺𝑢1−𝛽^𝑇₀𝜙_𝑞𝑔(𝑠)Δ𝑠. (29) We have ℎ(𝑢(𝑡=(𝑢(𝑡))𝜆𝑓))∫^𝑇₀𝑓(𝑢(𝜏))∇𝜏²,𝑔(𝑠)=^𝑠₀𝜆ℎ(𝑢(𝑟))∇𝑟+𝜆𝛽1−𝛽^𝜂₀ℎ(𝑢(𝑟≤))∇𝑟𝜆+𝜆𝛽1−𝛽^𝑇₀ℎ(𝑢(𝑟≤𝜆))∇𝑟1−𝛽^𝑇₀ℎ(𝑢(𝑟))∇𝑟. (30) Using (H2) it follows that 𝜙_𝑞𝑔(𝑠)≤𝑎𝑇𝜙_𝑞1𝑇. (31) Then we get |𝐺𝑢|≤𝑎₁,𝐺𝑃_𝑎1⊂𝑃_𝑎1. (32) Similarly, using (H3) we get 𝐺𝑃_𝑐1⊂𝑃_𝑐1.

Lemma 10. The set {𝑢∈𝑃(𝛼,𝑏,𝑑)∣𝛼(𝑢)>𝑏} (33) is nonempty, and 𝛼(𝐺𝑢)>𝑏,if𝑢∈𝑃(𝛼,𝑏,𝑑). (34)

Proof. Let 𝑢=(𝑏+𝑑)/2. Then, 𝑢∈𝑃,‖𝑢‖=(𝑏+𝑑)/2≤𝑑, and 𝛼(𝑢)≥(𝑏+𝑑)/2>𝑏. The first part of the lemma is proved. For 𝑢∈𝑃(𝛼,𝑏,𝑑) we have 𝑏≤𝑢≤𝑑. If 𝑡∈[𝜉,𝑇]_𝕋, then 𝛼(𝐺𝑢)=(𝐺𝑢)(𝑇−𝜉)=−₀^𝑇−𝜉𝜙_𝑞𝑔(𝑠)≥𝛽Δ𝑠+𝐵1−𝛽^𝑇_𝑇−𝜉𝜙_𝑞𝑔(𝑠)Δ𝑠. (35) Since 𝐴≤0, we have by using (H4) 𝑔(𝑠)=𝜆^𝑠₀ℎ(𝑢(𝑟))∇𝑟−𝐴≥𝜆^𝑠₀ℎ(𝑢(𝑟))∇𝑟≥𝜆^𝑠₀𝑓(𝑢)𝑇sup_{𝑏≤𝑢≤𝑑}𝑓(𝑢)²∇𝑢≥𝜆𝑏𝐵₁^𝑝−1𝑇sup_{𝑏≤𝑢≤𝑑}𝑓(𝑢)²𝑠. (36) Using the fact that 𝜙_𝑞 is nondecreasing we get 𝜙_𝑞𝑔(𝑠)≥𝜙_𝑞𝜆𝑏𝐵₁^𝑝−1𝑇sup_{𝑏≤𝑢≤𝑑}𝑓(𝑢)²𝑠≥𝑏𝐵₁𝜙_𝑞𝜆𝑇sup_{𝑏≤𝑢≤𝑑}𝑓(𝑢)²𝜙_𝑞(𝑠). (37) Using the expression of 𝐵₁𝛼()≥𝛽𝐺𝑢1−𝛽𝑏𝐵₁𝜙_𝑞𝜆𝑇sup_{𝑏≤𝑢≤𝑑}𝑓(𝑢)^{2𝑇𝑇−𝜉}𝜙_𝑞(𝑠)Δ𝑠≥𝑏𝐵₁𝛽𝜙1−𝛽^𝑞𝜆𝑇sup_{𝑏≤𝑢≤𝑑}𝑓(𝑢)²𝜙_𝑞(𝑇−𝜉)𝜉≥𝑏. (38)

Lemma 11. For all 𝑢∈𝑃(𝛼,𝑏,𝑐₁) with ‖𝐺𝑢‖>𝑑 one has 𝛼(𝐺𝑢)>𝑏. (39)

Proof. If 𝑢∈𝑃(𝛼,𝑏,𝑐₁) and ‖𝐺𝑢‖>𝑑, then 0≤𝑢(𝑡)≤𝑐₁. Using hypothesis (H3) and the fact that 0<𝛽<1, it follows that 𝛼(𝐺𝑢)=𝐺𝑢(𝑇−𝜉)=−₀^𝑇−𝜉𝜙_𝑞𝑔(𝑠)Δ𝑠+𝐵≥−^𝑇₀𝜙_𝑞𝑔(𝑠)(0)≥‖‖Δ𝑠+𝐺𝑢𝐺𝑢−𝑇²𝑐𝜙_𝑞1𝑇≥𝑑−𝑇²𝑐𝜙_𝑞1𝑇>𝑏. (40) Gathering Lemmas 4 to 11 and applying Theorem 3, there exist at least three positive solutions 𝑢₁, 𝑢₂, and 𝑢₃ to (1)-(2) verifying 𝑢₁<𝑎,𝑏<𝛼𝑢₂,𝑢₃>𝑎,𝛼𝑢₃<𝑏. (41)

Example 12. Let 𝕋={1−(1/2)^ℕ0}∪{1}, where ℕ₀ denotes the set of all nonnegative integers. Consider the 𝑝-Laplacian dynamic equation −𝜙_𝑝𝑢^Δ(𝑡)^∇=𝜆𝑓(𝑢(𝑡))∫^𝑇₀𝑓(𝑢(𝜏))∇𝜏²,𝑡∈(0,𝑇)_𝕋, (42) satisfying the boundary conditions 𝜙_𝑝𝑢^Δ(0)−𝛽𝜙_𝑝𝑢^Δ14𝑢(1)1=0,−𝛽𝑢4=0, (43) where 𝑝=3/2, 𝑞=3, 𝜂=1/4, 𝛽=1/2, 𝜆=1, 𝑇=1, and 𝑓(𝑢)=2√4()√2,0≤𝑢≤1,𝑢−1+232,1≤𝑢≤2,2+2√2,32√≤𝑢≤10,2𝑢+22−18,10≤𝑢≤16. (44) Choose 𝑎₁=1=2𝑎, 𝑏=3/2, 𝑐₁=16=2𝑐, and 𝑑=10. It is easy to see that 𝜁=2, 𝐵₁=1/(2(2+√2)) and min𝑓(𝑢)∶𝑢∈0,𝑎₁=2√2≥𝜆²𝑇1−𝛽𝜙_𝑝(𝑎)=2√𝑎=2√𝑓(𝑢)2,min∶𝑢∈0,𝑐₁=2√2≥𝜆²𝑇1−𝛽𝜙_𝑝(𝑐)=2√𝑐=1√2,min𝑓(𝑢)∶𝑢∈[𝑏,𝑑]=2+2√2≥𝜙_𝑝𝑏𝐵₁=𝑏𝐵₁=342+2√2. (45) Then, hypotheses (H1)–(H4) are satisfied. Therefore, by Theorem 8, problem (42)-(43) has at least three positive solutions.

3.2. Quasilinear Elliptic Problem

We are interested in this section in the study of the following quasilinear elliptic problem: −𝜙_𝑝𝑢^Δ(𝑡)^∇=𝑓(𝑢(𝑡))+ℎ(𝑡),𝑡∈(0,𝑇)_𝕋,𝑢^Δ(0)=0,𝑢(𝑇)−𝑢𝜂=0, (46) where 𝜂∈(0,𝑇)_𝕋. We assume the following hypotheses: (A1) function 𝑓∶ℝ→ℝ⁺₀ is continuous;(A2) function ℎ∶(0,𝑇)_𝕋→ℝ⁺₀ is left dense continuous, that is, ℎ∈ℂ_𝑙𝑑(0,𝑇)_𝕋,ℝ⁺₀,ℎ∈𝐿^∞. (47)Similarly as in Section  3.1, we prove existence of solutions by constructing an operator whose fixed points are solutions to (46). The main ingredient is, again, the Leggett-Williams fixed point theorem (Theorem 3). We can easily see that (46) is equivalent to the integral equation 𝑢(𝑡)=𝜙_𝑞^𝑇_𝜂𝑓(𝑢(𝑟(𝑟)+))+ℎ∇𝑟_𝑡0𝜙_𝑞^𝑇_𝑠𝑓(𝑢(𝑟))+ℎ(𝑟)∇𝑟Δ𝑠. (48) On the other hand, we have −(𝜙_𝑝(𝑢^Δ))^∇=𝑓(𝑢(𝑡))+ℎ(𝑡). Since 𝑓, ℎ≥0, we have (𝜙_𝑝(𝑢^Δ))^∇≤0 and 𝜙_𝑝(𝑢^Δ(𝑡₂))≤𝜙_𝑝(𝑢^Δ(𝑡₁)) for any 𝑡₁,𝑡₂∈[0,𝑇]_𝕋 with 𝑡₁≤𝑡₂. It follows that 𝑢^Δ(𝑡₂)≤𝑢^Δ(𝑡₁) for 𝑡₁≤𝑡₂. Hence, 𝑢^Δ(𝑡) is a decreasing function on [0,𝑇]_𝕋. Then, 𝑢 is concave. In order to apply Theorem 3 we define the cone []𝑃=𝑢∈𝐸∣𝑢isnonnegative,increasingon0,𝑇_𝕋,andconcaveon𝐸. (49) For 𝜉∈(0,𝑇/2) we also define the nonnegative continuous concave functional 𝛼∶𝑃→ℝ⁺₀ by 𝛼(𝑢)=min_{𝑡∈𝜉,𝑇−𝜉𝕋}𝑢(𝑡),𝑢∈𝑃 (50) and the operator 𝐹∶𝑃→𝐸 by 𝐹𝑢(𝑡)=𝜙_𝑞^𝑇_𝜂𝑓(𝑢(𝑟(𝑟)+))+ℎ∇𝑟_𝑡0𝜙_𝑞^𝑇_𝑠𝑓(𝑢(𝑟))+ℎ(𝑟)∇𝑟Δ𝑠. (51) It is easy to see that (46) has a solution 𝑢=𝑢(𝑡) if and only if 𝑢 is a fixed point of the operator 𝐹. For convenience, we introduce the following notation: 𝛾=(1+𝑇)𝜙_𝑞(𝑇),‖ℎ‖𝐴=𝑎−𝛼_∞^1/(𝑝−1)𝛼𝑎,where𝛼=𝜙_𝑞2^𝑝−2𝜙_𝑞(𝑇)(𝑇+1),𝐵=𝜙_𝑝𝑇−𝜂. (52)

Theorem 13. Suppose that hypotheses (A1) and (A2) are satisfied; there exist positive constants 𝑎, 𝑏, 𝑐, and 𝑑 with 0<𝛾𝑎=𝑎₁<𝑏<𝑑−2^𝑝−2(𝑇−𝜉)𝜙_𝑞(𝑇−𝜉)‖ℎ‖^1/(𝑝−1)+𝑏𝐵<𝑑<𝛾𝑐=𝑐₁ (53) and, in addition to (A1) and (A2), that 𝑓 satisfies (A3)max_{0≤𝑢≤𝑎}𝑓(𝑢)≤𝜙_𝑝(𝑎𝐴);(A4)max_{0≤𝑢≤𝑐}𝑓(𝑢)≤𝜙_𝑝(𝑐𝐴);(A5)min_{𝑏≤𝑢≤𝑑}𝑓(𝑢)≥𝜙_𝑝(𝑏𝐵). Then problem (46) has at least three positive solutions 𝑢₁, 𝑢₂, and 𝑢₃, verifying 𝑢₁<𝑎,𝑏<𝛼𝑢₂,𝑢₃>𝑎,𝛼𝑢₃<𝑏. (54)

Proof. As done for Theorem 8, the proof is divided in several steps. We first show that 𝐹∶𝑃→𝑃 is completely continuous. Indeed, 𝐹 is obviously continuous. Let 𝑈_𝛿={𝑢∈𝑃∣‖𝑢‖≤𝛿}. (55) It is easy to see that for 𝑢∈𝑈_𝛿 there exists a constant 𝑐>0 such that |𝐹𝑢(𝑡)|≤𝑐. On the other hand, let 𝑡₁,𝑡₂∈(0,𝑇)_𝕋, 𝑢∈𝑈_𝛿. Then there exists a positive constant 𝑐 such that 𝐹𝑢𝑡₂−𝐹𝑢𝑡₁≤𝑐𝑡₂−𝑡₁, (56) which converges uniformly to zero when |𝑡₂−𝑡₁| tends to zero. Using the Arzela-Ascoli theorem we conclude that 𝐹∶𝑃→𝑃 is completely continuous.
We now show that 𝐹𝑃_𝑐1⊂𝑃_𝑐1,𝐹𝑃_𝑎1⊂𝑃_𝑎1. (57) For all 𝑢∈𝑃_𝑎1 we have 0≤𝑢≤𝑎₁ and ‖𝐹(𝑢)‖≤𝜙_𝑞^𝑇_𝜂(𝑎𝐴)^𝑝−1+‖ℎ‖_∞+∇𝑟_𝑇0𝜙_𝑞^𝑇_𝑠(𝑎𝐴)^𝑝−1+‖ℎ‖_∞∇𝑟Δ𝑠≤𝜙_𝑞(𝑎𝐴)^𝑝−1+‖ℎ‖_∞+𝑇−𝜂_𝑇0𝜙_𝑞(𝑎𝐴)^𝑝−1+‖ℎ‖_∞(𝑇−𝑠)Δ𝑠≤𝜙_𝑞(𝑎𝐴)^𝑝−1+‖ℎ‖_∞𝜙_𝑞(𝑇)+𝜙_𝑞(𝑎𝐴)^𝑝−1+‖ℎ‖_∞^𝑇₀𝜙_𝑞(𝑇−𝑠)Δ𝑠≤𝜙_𝑞(𝑎𝐴)^𝑝−1+‖ℎ‖_∞^{1/(𝑝−1)𝑝−1}𝜙_𝑞(𝑇)+𝜙_𝑞(𝑎𝐴)^𝑝−1+‖ℎ‖_∞^{1/(𝑝−1)𝑝−1}𝜙_𝑞(𝑇)𝑇≤𝜙_𝑞(𝑎𝐴)^𝑝−1+‖ℎ‖_∞^{1/(𝑝−1)𝑝−1}𝜙_𝑞(𝑇)(𝑇+1). (58) Using the elementary inequality 𝑥^𝑝+𝑦^𝑝≤2^𝑝−1𝑥+𝑦^𝑝 (59) and the form of 𝐴, it follows that ‖𝐹(𝑢)‖≤𝜙_𝑞(𝑇+1)2^𝑝−2𝑎𝐴+‖ℎ‖_∞^1/(𝑝−1)≤𝜙_𝑞(𝑇+1)2^𝑝−2𝑎=𝛾𝑎=𝑎₁. (60) Then 𝐹𝑃_𝑎1⊂𝑃_𝑎1. In a similar way we prove that 𝐹𝑃_𝑐1⊂𝑃_𝑐1.
Our following step is to show that {()(𝑢)}𝛼()().𝑢∈𝑃𝛼,𝑏,𝑑∣𝛼>𝑏≠∅,𝐹𝑢>𝑏,if𝑢∈𝑃𝛼,𝑏,𝑑 (61) The first point is obvious. Let us prove the second part of (61). For 𝑢∈𝑃(𝛼,𝑏,𝑑) we have 𝑏≤𝑢≤𝑑, if 𝑡∈[𝜉,𝑇]_𝕋. Then, using (A2) we have 𝛼(𝐹𝑢)=𝐹𝑢(𝜉)≥𝜙_𝑞^𝑇_𝜂𝑓(𝑢(𝑟))∇𝑟+^𝜉₀𝜙_𝑞^𝑇_𝑠𝑓(𝑢(𝑟))∇𝑟Δ𝑠≥𝜙_𝑞^𝑇_𝜂𝑓(𝑢(𝑟))∇𝑟≥𝑏𝐵𝜙_𝑞(𝑇−𝜉)≥𝑏. (62) Finally we prove that 𝛼(𝐹𝑢)>𝑏 for all 𝑢∈𝑃(𝛼,𝑏,𝑐₁) and ‖𝐹𝑢‖>𝑑: 𝛼(𝐹𝑢)=𝐹𝑢(𝜉)=𝜙_𝑞^𝑇_𝜂𝑓(𝑢(𝑟))+ℎ(𝑟)∇𝑟+^𝜉₀𝜙_𝑞^𝑇_𝑠𝑓(𝑢(𝑟))+ℎ(𝑟)∇𝑟Δ𝑠=𝜙_𝑞^𝑇_𝜂𝑓(𝑢(𝑟(𝑟)+))+ℎ∇𝑟_𝑇0𝜙_𝑞^𝑇_𝑠𝑓(𝑢(𝑟(𝑟)−))+ℎ∇𝑟Δ𝑠_𝑇𝜉𝜙_𝑞^𝑇_𝑠𝑓(𝑢(𝑟(𝑟)≥‖‖−))+ℎ∇𝑟Δ𝑠𝐹𝑢^𝑇_𝜉𝜙_𝑞^𝑇_𝑠𝑓(𝑢(𝑟(𝑟)≥‖‖))+ℎ∇𝑟Δ𝑠𝐹𝑢−^𝑇_𝜉𝜙_𝑞^𝑇_𝜉𝑓(𝑢(𝑟(𝑟)≥‖‖−))+ℎ∇𝑟Δ𝑠𝐹𝑢^𝑇_𝜉𝜙_𝑞(𝑇−𝜉)‖ℎ‖+𝜙_𝑝(≥‖‖−()𝜙𝑏𝐵)Δ𝑠𝐹𝑢𝑇−𝜉_𝑞(𝑇−𝜉)𝜙_𝑞‖ℎ‖+𝜙_𝑝(𝑏𝐵). (63) Using again the elementary inequality 𝑥^𝑝+𝑦^𝑝≤2^𝑝−1(𝑥+𝑦)^𝑝 we get that 𝛼(𝐹𝑢)≥‖𝐹𝑢‖−(𝑇−𝜉)𝜙_𝑞(𝑇−𝜉)‖ℎ‖^1/(𝑝−1)+𝑏𝐵≥𝑑−2^𝑝−2(𝑇−𝜉)𝜙_𝑞(𝑇−𝜉)‖ℎ‖^1/(𝑝−1)+𝑏𝐵≥𝑏. (64) By Theorem 3 there exist at least three positive solutions 𝑢₁, 𝑢₂, and 𝑢₃ to (46) satisfying ‖𝑢₁‖<𝑎, 𝑏<𝛼(𝑢₂), ‖𝑢₃‖>𝑎, and 𝛼(𝑢₃)<𝑏.

Example 14. Let 𝕋={1−(1/2)^ℕ0}∪{1}, where ℕ₀ denotes the set of all nonnegative integers. Consider the 𝑝-Laplacian dynamic equation 𝜙_𝑝𝑢^Δ(𝑡)^∇+𝑓(𝑢(𝑡))=0,𝑡∈[0,1]_𝕋, (65) satisfying the boundary conditions 𝑢(1)1−𝑢2=0,𝑢^▵(0)=0, (66) where 𝑝=3/2, 𝑞=3, 𝑎(𝑡)≡1, ℎ≡0, 𝑇=1, and 𝑓(𝑢)=√221,0≤𝑢≤2,41𝑢−2+√22,123≤𝑢≤2,√4+22,32≤𝑢≤25. (67) Choose 𝑎=1/2, 𝑏=3/2, 𝑐=25, and 𝑑=3. It is easy to see that 𝛾=2, 𝐴=1, 𝐵=√2/2, 𝛼=1, and 𝑓(𝑢)1max∶𝑢∈0,2=√22≤(𝑎𝐴)^𝑝−1=√√𝑎=22,𝑓(𝑢)[√max∶𝑢∈0,25]=4+22≤()≃4,707𝑐𝐴^𝑝−1=𝑐^𝑝−1=√𝑓(𝑢)3𝑐=5,min∶𝑢∈2√,3=4+22≥()≃4,707𝑏𝐵^𝑝−1=√𝑏𝐵≃1,02. (68) Therefore, by Theorem 13, problem (65)-(66) has at least three positive solutions.

3.3. A p-Laplacian Functional Dynamic Equation on Time Scales with Delay

Let 𝕋 be a time scale with 0,𝑇∈𝕋^𝜅_𝜅, −𝑟∈𝕋 with −𝑟≤0<𝑇. We are concerned in this section with the existence of positive solutions to the 𝑝-Laplacian dynamic equation 𝜙_𝑝𝑢^Δ(𝑡)^∇+𝜆𝑎(𝑡)𝑓(𝑢(𝑡),𝑢(𝜔(𝑡)))=0,𝑡∈(0,𝑇)_𝕋,𝑢(𝑡)=𝜓(𝑡),𝑡∈[−𝑟,0]_𝕋,𝑢(0)−𝐵₀𝑢^Δ(0)=0,𝑢^Δ(𝑇)=0, (69) where 𝜆>0. We define 𝑋=ℂ_𝑙𝑑([0,𝑇]_𝕋,ℝ), which is a Banach space with the maximum norm ‖𝑢‖=max_[0,𝑇]𝕋|𝑢(𝑡)|. We note that 𝑢 is a solution to (69) if and only if 𝑢(𝑡)=𝐵₀𝜙_𝑞∫^𝑇₀(𝑟)(𝑢(𝑟)(𝜔(𝑟𝜆𝑎×𝑓,𝑢)))∇𝑟+∫^𝑡₀𝜙_𝑞∫^𝑇_𝑠(𝑟)(𝑢(𝑟)(𝜔(𝑟[]𝜆𝑎×𝑓,𝑢)))∇𝑟Δ𝑠if𝑡∈0,𝑇_𝕋,𝜓(𝑡)if𝑡∈[−𝑟,0]_𝕋. (70) Let 𝐾=𝑢∈𝑋∣𝑢isnonnegativeandconcaveon𝐸. (71) Clearly 𝐾 is a cone in the Banach space 𝑋. For each 𝑢∈𝑋, we extend 𝑢 to [−𝑟,0]_𝕋 with 𝑢(𝑡)=𝜓(𝑡) for 𝑡∈[−𝑟,0]_𝕋. We also define the nonnegative continuous concave functional 𝛼∶𝑃→ℝ⁺₀ by 𝛼(𝑢)=min_{𝑡∈𝜉,𝑇−𝜉𝕋}𝑢(𝑡)𝑇,𝜉∈0,2,∀𝑢∈𝐾. (72) For 𝑡∈[0,𝑇]_𝕋, define 𝑄∶𝐾→𝑋 as 𝑄𝑢(𝑡)=𝐵₀𝜙_𝑞^𝑇₀(𝑟)𝑓(𝑢(𝑟)(𝜔(𝑟+𝜆𝑎,𝑢)))∇𝑟_𝑡0𝜙_𝑞^𝑇_𝑠𝜆𝑎(𝑟)𝑓(𝑢(𝑟),𝑢(𝜔(𝑟)))∇𝑟Δ𝑠. (73)

Lemma 15. Let 𝑢₁ be a fixed point of 𝑄 in the cone 𝐾. Define 𝑢(𝑡)=𝑢₁,𝑡∈[0,𝑇]_𝕋,𝜓(𝑡),𝑡∈[−𝑟,0]_𝕋. (74) It follows that (74) is a positive solution to (69) satisfying ‖𝑄𝑢‖≤𝑇+𝛾𝜆^𝑞−1×𝜙_𝑞^𝑇₀𝑎(𝑟)𝑓(𝑢(𝑟),𝑢(𝜔(𝑟)))∇𝑟for𝑡∈[0,𝑇]_𝕋. (75)

Proof. ‖𝑄𝑢‖=(𝑄𝑢)(𝑇)=𝐵₀𝜙_𝑞^𝑇₀𝜆𝑎(𝑟)𝑓(𝑢(𝑟),𝑢(𝜔(𝑟)))∇𝑟+^𝑇₀𝜙_𝑞^𝑇_𝑠(𝑟)𝑓(𝑢(𝑟)(𝜔(𝑟≤𝜆𝜆𝑎,𝑢)))∇𝑟Δ𝑠𝑇+𝛾^𝑞−1𝜙_𝑞^𝑇₀𝑎(𝑟)𝑓(𝑢(𝑟),𝑢(𝜔(𝑟)))∇𝑟. (76)

From (73) and (75) it follows that (i)𝑄(𝐾)⊂𝐾; (ii)𝑄∶𝐾→𝐾 is completely continuous; (iii)𝑢(𝑡)≥𝛿/(𝑇+𝛾)‖𝑢‖, 𝑡∈[0,𝑇]_𝕋.

Depending on the signature of the delay 𝜔, we set the following two subsets of [0,𝑇]_𝕋: 𝑌₁∶=𝑡∈[0,𝑇]_𝕋(𝑡);𝑌∣𝜔<0₂∶=𝑡∈[0,𝑇]_𝕋∣𝜔(𝑡)≥0. (77)

In the remainder of this section, we suppose that 𝑌₁ is nonempty and ∫_𝑌1𝑎(𝑟)∇𝑟>0. For convenience we also denote 𝜙𝑙∶=^𝑝∫^𝑇₀𝑎(𝑟)∇𝑟𝜆^𝑞−1𝜙𝑇+𝛾,𝑚∶=^𝑝∫^𝑇₀𝑎(𝑟)∇𝑟𝛿𝜆^𝑞−1. (78)

Theorem 16. Suppose that there exist positive constants 𝑎, 𝑏, 𝑐, and 𝑑 such that 0<𝑎<𝑏<𝛿𝑑/(𝑇+𝛾)<𝑑<𝑐. Assume that the following hypotheses (C1)–(C8) hold: (C1)𝑓∶ℝ⁺₀×ℝ⁺₀→ℝ⁺₀ is continuous;(C2) function 𝑎∶(0,𝑇)_𝕋→ℝ⁺₀ is left-dense continuous;(C3) 𝜓∶[−𝑟,0]_𝕋→ℝ⁺₀ is continuous;(C4) 𝜔∶[0,𝑇]_𝕋→[−𝑟,𝑇]_𝕋 is continuous, 𝜔(𝑡)≤𝑡 for all 𝑡;(C5)𝐵₀∶ℝ→ℝ is continuous and there are 0<𝛿≤𝛾 such that 𝛿𝑠≤𝐵₀(𝑠)≤𝛾𝑠for𝑠∈ℝ⁺₀; (79)(C6)lim_𝑥→0⁺𝑓(𝑥,𝜓(𝑠))/𝑥^𝑝−1<𝑙^𝑝−1, uniformly in 𝑠∈[−𝑟,0]_𝕋;(C7) lim_𝑥1→0⁺_;𝑥2→0⁺𝑓(𝑥₁,𝑥₂)/max{𝑥₁^𝑝−1,𝑥₂^𝑝−1}<𝑙^𝑝−1;(C8)lim_𝑥→∞𝑓(𝑥,𝜓(𝑠))/𝑥^𝑝−1>𝑚^𝑝−1, uniformly in 𝑠∈[−𝑟,0]_𝕋. Then, for each 0<𝜆<∞ the boundary value problem (69) has at least three positive solutions 𝑢₁, 𝑢₂, and 𝑢₃ verifying 𝑢₁<𝑎,𝑏<𝛼𝑢₂,𝑢₃>𝑎,𝛼𝑢₃<𝑏. (80)
Proof. The proof passes by three lemmas.

Lemma 17. The following relations hold: 𝑄𝑃_𝑎⊂𝑃_𝑎,𝑄𝑃_𝑐⊂𝑃_𝑐. (81)

Proof. Using condition (C6) for 𝜀₁>0 such that 0<𝑥≤𝜀₁, we have 𝑓𝑥,𝜓(𝑠)<(𝑙𝑥)^𝑝−1foreach𝑠∈[−𝑟,0]_𝕋. (82) Applying condition (C7) we get 𝑓𝑥₁,𝑥₂<max𝑥₁^𝑝−1,𝑥₂^𝑝−1𝑙^𝑝−1 (83) for 𝜀₂>0 such that 0<𝑥₁≤𝜀₂, 0<𝑥₂≤𝜀₂. Put 𝜀=min{𝜀₁,𝜀₂}. Then for ‖𝑢‖≤𝑎 and from (75) we have ||≤‖‖≤𝜆𝑄𝑢𝑄𝑢𝑇+𝛾^𝑞−1𝜙_𝑞^𝑇₀𝑎(𝑟)𝑓(𝑢(𝑟)(𝜔(𝑟=𝜆,𝑢)))∇𝑟𝑇+𝛾^𝑞−1𝜙_𝑞𝑌1𝑎(𝑟)𝑓𝑢(𝑟)(𝜔(𝑟+,𝜓))∇𝑟_𝑌2𝑎(𝑟)𝑓(𝑢(𝑟)(𝜔(𝑟𝜆,𝑢)))∇𝑟≤𝑙𝑇+𝛾^𝑞−1‖𝑢‖𝜙_𝑞^𝑇₀𝑎(𝑟)𝜆∇𝑟≤𝑙𝑇+𝛾^𝑞−1𝑎𝜙_𝑞^𝑇₀𝑎(𝑟)∇𝑟=𝑎. (84) Then 𝑄𝑃_𝑎⊂𝑃_𝑎. Similarly one can show that 𝑄𝑃_𝑐⊂𝑃_𝑐.

Lemma 18. The set {𝑢∈𝑃(𝛼,𝑏,𝑑)∣𝛼(𝑢)>𝑏} (85) is nonempty, and 𝛼(𝑄𝑢)>𝑏,if𝑢∈𝑃(𝛼,𝑏,𝑑). (86)

Proof. Applying hypothesis (𝐶8) we have 𝑓𝑢,𝜓(𝑠)>𝜙_𝑝(𝑚𝑢)foreach𝑠∈[−𝑟,0]_𝕋. (87) We also have 𝑢(𝑡)≥𝛿/(𝑇+𝛾)‖𝑢‖. Let 𝑢∈𝑃(𝛼,𝑏,𝑑). Then, 𝑏≤𝑢(𝑡)≤𝑑. Hence, 𝛼()=()≥𝛿𝑄𝑢𝑄𝑢)(𝑇−𝜉‖‖𝑇+𝛾𝑄𝑢≥𝛿𝜙_𝑞^𝑇₀𝜆𝑎(𝑟)𝑓(𝑢(𝑟),𝑢(𝜔(𝑟)))∇𝑟≥𝛿𝜆^𝑞−1𝜙_𝑞𝑌1𝑎(𝑟)𝑓𝑢(𝑟)(𝜔(𝑟+,𝜓))∇𝑟_𝑌2𝑎(𝑟)𝑓𝑢(𝑟),𝜓(𝜔(𝑟))∇𝑟≥𝛿𝜆^𝑞−1𝜙_𝑞𝑌1𝑎(𝑟)𝑓𝑢(𝑟),𝜓(𝜔(𝑟))∇𝑟≥𝑚𝛿𝜆^𝑞−1min_𝑡∈𝑌1{𝑢(𝑡)}𝜙_𝑞𝑌1𝑎(𝑟)∇𝑟≥𝑏𝑚𝛿𝜆^𝑞−1𝜙_𝑞𝑌1𝑎(𝑟)∇𝑟≥𝑏. (88)

Lemma 19. For all 𝑢∈𝑃(𝛼,𝑏,𝑐) and ‖𝑄𝑢‖>𝑑 one has 𝛼(𝑄𝑢)>𝑏.

Proof. Using the fact that 𝛿≤𝑇+𝛾, we have 𝛼()=()≥𝛿𝑄𝑢𝑄𝑢)(𝑇−𝜉‖‖>𝑇+𝛾𝑄𝑢𝛿𝑑𝑇+𝛾≥𝑏. (89)

Applying the Leggett-Williams theorem (Theorem 3), the proof of Theorem 16 is complete.

Example 20. Let 𝕋=[−3/4,−1/4]∪{0,3/4}∪{(1/2)^ℕ0}, where ℕ₀ denotes the set of all nonnegative integers. Consider the following 𝑝-Laplacian functional dynamic equation on the time scale 𝕋: Φ_𝑝𝑢^Δ(𝑡)^∇+𝑢₁+𝑢₂²=0,𝑡∈(0,1)_𝕋,𝜓(𝑡)−3≡0,𝑡∈4,0_𝕋,𝑢(0)−𝐵₀𝑢^Δ14=0,𝑢^Δ(1)=0, (90) where 𝑇=1, 𝑝=3/2, 𝑞=3, 𝑎(𝑡)≡1,𝐵₀(𝑠)=𝑠, 𝑤(𝑡)∶[0,1]_𝕋→[−3/4,1]_𝕋 with 𝑤(𝑡)=𝑡−3/4, 𝑟=3/4, 𝜂=1/4, 𝑙=1/2, 𝑚=1, and 𝑓(𝑢,𝜓(𝑡))=𝑢², 𝑓(𝑢₁,𝑢₂)=(𝑢₁+𝑢₂)². We deduce that 𝑌₁=[0,3/4)_𝕋, 𝑌₂=[3/4,1]_𝕋. It is easy to see that hypotheses (C1)–(C5) are verified. On the other hand, notice that lim_𝑥→0⁺𝑓(𝑥,𝜓(𝑠))/𝑥^𝑝−1=0<𝑙^𝑝−1 and lim_𝑥→∞𝑓(𝑥,𝜓(𝑠))/(𝑥^𝑝−1)=+∞>𝑚^𝑝−1. Thus, hypotheses (C6)–(C8) are obviously satisfied. Then, by Theorem 16, problem (90) has at least three positive solutions of the form 𝑢(𝑡)=𝑢_𝑖(𝑡),𝑡∈[0,1]_𝕋𝜓(𝑡)−3,𝑖=1,2,3,,𝑡∈4,0_𝕋. (91)

Acknowledgments

This work was supported by FEDER funds through COMPETE—Operational Programme Factors of Competitiveness (Programa Operacional Factores de Competitividade)—by Portuguese funds through the Center for Research and Development in Mathematics and Applications (University of Aveiro), and the Portuguese Foundation for Science and Technology (FCT—Fundação para a Ciência e a Tecnologia), within project PEst-C/MAT/UI4106/2011 with COMPETE no. FCOMP-01-0124-FEDER-022690. The authors were also supported by the project New Explorations in Control Theory Through Advanced Research (NECTAR) cofinanced by FCT, Portugal and the Centre National de la Recherche Scientifique et Technique (CNRST), Morocco.