Solvability of a Higher-Order Three-Point Boundary Value Problem on Time Scales

Abstract

We consider a higher-order three-point boundary value problem on time scales. A new existence result is first obtained by using a fixed point theorem due to Krasnoselskii and Zabreiko. Later, under certain growth conditions imposed on the nonlinearity, several sufficient conditions for the existence of a nonnegative and nontrivial solution are obtained by using Leray-Schauder nonlinear alternative. Our conditions imposed on nonlinearity are all very easy to verify; as an application, some examples to demonstrate our results are given.

1. Introduction

We are concerned with the following even-order three-point boundary value problem on time scales 𝕋: (−1)^𝑛𝑦^Δ2𝑛(𝑡)=𝑓(𝑦^𝜎𝛼(𝑡)),𝑡∈[𝑎,𝑏]⊂𝕋,_𝑖+1𝑦^Δ2𝑖(𝜂)+𝛽_𝑖+1𝑦^Δ2𝑖+1(𝑎)=𝑦^Δ2𝑖(𝑎),𝛾_𝑖+1𝑦^Δ2𝑖(𝜂)=𝑦^Δ2𝑖(𝜎(𝑏)),0≤𝑖≤𝑛−1,(−1)^𝑛𝑦^Δ2𝑛(𝑡)=𝑓(𝑡,𝑦^𝜎𝛼(𝑡)),𝑡∈[𝑎,𝑏]⊂𝕋,_𝑖+1𝑦^Δ2𝑖(𝜂)+𝛽_𝑖+1𝑦^Δ2𝑖+1(𝑎)=𝑦^Δ2𝑖(𝑎),𝛾_𝑖+1𝑦^Δ2𝑖(𝜂)=𝑦^Δ2𝑖(𝜎(𝑏)),0≤𝑖≤𝑛−1, (1.1)(1.2)𝑛≥1,𝑎<𝜂<𝜎(𝑏); we assume that 𝜎(𝑏) is right dense so that 𝜎^𝑗(𝑏)=𝜎(𝑏) for 𝑗≥1 and that for each 1≤𝑖≤𝑛, 𝛼_𝑖,𝛽_𝑖,𝛾_𝑖 coefficients satisfy the following condition:

(𝐻)0≤𝛼_𝑖<𝜎(𝑏)−𝛾_𝑖𝜂+(𝛾_𝑖−1)(𝑎−𝛽_𝑖)𝜎(𝑏)−𝜂,𝛽_𝑖≥0,0<𝛾_𝑖<𝜎(𝑏)−𝑎+𝛽_𝑖𝜂−𝑎+𝛽_𝑖. (1.3)

Throughout this paper, we let 𝕋 be any time scale and let [𝑎,𝑏] be a subset of 𝕋 such that [𝑎,𝑏]={𝑡∈𝕋∶𝑎≤𝑡≤𝑏}. Some preliminary definitions and theorems on time scales can be found in [1–5] which are excellent references for the calculus of time scales.

In recent years, there is much attention paid to the existence of positive solution for second-order multipoint and higher-order two-point boundary value problems on time scales; for details, see [6–16] and references therein. However, to the best of our knowledge, there are not many results concerning multipoint boundary value problems of higher-order on time scales; we refer the readers to [17–20] for some recent results.

We would like to mention some results of Anderson and Avery [17], Anderson and Karaca [18], Han and Liu [19], and Yaslan [20]. In [17], Anderson and Avery studied the following even-order three-point BVP:

(−1)^𝑛𝑥^(Δ∇)𝑛𝑥(𝑡)=𝜆ℎ(𝑡)𝑓(𝑥(𝑡)),𝑡∈[𝑎,𝑐]⊂𝕋,_(Δ∇)𝑖(𝑎)=0,𝑥^(Δ∇)𝑖(𝑐)=𝛽𝑥^(Δ∇)𝑖(𝑏),0≤𝑖≤𝑛−1. (1.4) They have studied the existence of at least one positive solution to the BVP (1.4) using the functional-type cone expansion-compression fixed point theorem.

In [18], Anderson and Karaca were concerned with the dynamic three-point boundary value problem (1.2) and the eigenvalue problem (−1)^𝑛𝑦^Δ2𝑛(𝑡)=𝜆𝑓(𝑡,𝑦^𝜎(𝑡)) with the same boundary conditions where 𝜆 is a positive parameter. Existence results of bounded solutions of a noneigenvalue problem were first established as a result of the Schauder fixed point theorem. Second, the monotone method was discussed to ensure the existence of solutions of the BVP (1.2). Third, they established criteria for the existence of at least one positive solution of the eigenvalue problem by using the Krasnosel’skii fixed point theorem. Later, they investigated the existence of at least two positive solutions of the BVP (1.2) by using the Avery-Henderson fixed point theorem.

In [19], Han and Liu studied the existence and uniqueness of nontrivial solution for the following third-order 𝑝-Laplacian 𝑚-point eigenvalue problems on time scales:

(𝜙_𝑝(𝑢^Δ∇))^∇+𝜆𝑓(𝑡,𝑢(𝑡),𝑢^Δ(𝑡))=0,𝑡∈(0,𝑇),𝛼𝑢(0)−𝛽𝑢^Δ(0)=0,𝑢(𝑇)=^𝑚−2_𝑖=1𝑎_𝑖𝑢(𝜉_𝑖),𝑢^Δ∇(0)=0, (1.5) where 𝜙_𝑝(𝑠) is 𝑝-Laplacian operator, that is, 𝜙_𝑝(𝑠)=|𝑠|^𝑝−2𝑠,𝑝>1,𝜆>0 is a parameter, and 0<𝜉₁<⋯<𝜉_𝑚−2<𝜌(𝑇). They obtained several sufficient conditions of the existence and uniqueness of nontrivial solution of the BVP (1.5) when 𝜆 is in some interval. Their approach was based on the Leray-Schauder nonlinear alternative.

Very recently, Yaslan [20] investigated the existence of solutions to the nonlinear even-order three-point boundary value problem on time scales 𝕋:

(−1)^𝑛𝑦^Δ2𝑛(𝑡)=𝑓(𝑡,𝑦(𝜎(𝑡))),𝑡∈[𝑡₁,𝑡₃𝑦]⊂𝕋,_Δ2𝑖+1(𝑡₁)=0,𝛼𝑦^Δ2𝑖(𝜎(𝑡₃))+𝛽𝑦^Δ2𝑖+1(𝜎(𝑡₃))=𝑦^Δ2𝑖+1(𝑡₂) (1.6) for 0≤𝑖≤𝑛−1, where 𝛼>0 and 𝛽>1 are given constants. On the one hand, the author established criteria for the existence of at least one solution and of at least one positive solution for the BVP (1.6) by using the Schauder fixed point theorem and Krasnosel’skii fixed point theorem, respectively. On the other hand, the author investigated the existence of multiple positive solutions to the BVP (1.6) by using Avery-Henderson fixed point theorem and Leggett-Williams fixed point theorem.

In this paper, motivated by [21], firstly, a new existence result for (1.1) is obtained by using a fixed point theorem, which is due to KrasnoseÍskĭ and Zabreĭko [22]. Particularly, 𝑓 may not be sublinear. Secondly, some simple criteria for the existence of a nonnegative solution of the BVP (1.2) are established by using Leray-Schauder nonlinear alternative. Thirdly, we investigate the existence of a nontrivial solution of the BVP (1.2); our approach is also based on the application of Leray-Schauder nonlinear alternative. Particularly, we do not require any monotonicity and nonnegativity on 𝑓. Our conditions imposed on 𝑓 are all very easy to verify; our method is motivated by [1, 21, 23, 24].

2. Preliminaries

To state and prove the main results of this paper, we need the following lemmas.

Lemma 2.1 (see [18]). For 1≤𝑖≤𝑛, let 𝐺_𝑖(𝑡,𝑠) be Green's function for the following boundary value problem: −𝑦^Δ2𝛼(𝑡)=0,𝑡∈[𝑎,𝑏]⊂𝕋,_𝑖𝑦(𝜂)+𝛽_𝑖𝑦^Δ(𝑎)=𝑦(𝑎),𝛾_𝑖𝑦(𝜂)=𝑦(𝜎(𝑏)), (2.1) and let 𝑑_𝑖=(𝛾_𝑖−1)(𝑎−𝛽_𝑖)+(1−𝛼_𝑖)𝜎(𝑏)+𝜂(𝛼_𝑖−𝛾_𝑖). Then, for 1≤𝑖≤𝑛, 𝐺_𝑖𝐺(𝑡,𝑠)=^𝑖1𝐺(𝑡,𝑠),𝑎≤𝑠≤𝜂,_𝑖2(𝑡,𝑠),𝜂<𝑠≤𝑏, (2.2) where 𝐺_𝑖1=1(𝑡,𝑠)𝑑_𝑖[𝛾_𝑖(𝑡−𝜂)+𝜎(𝑏)−𝑡](𝜎(𝑠)+𝛽_𝑖−𝑎),𝜎(𝑠)≤𝑡,[𝛾_𝑖(𝜎(𝑠)−𝜂)+𝜎(𝑏)−𝜎(𝑠)](𝑡+𝛽_𝑖−𝑎)+𝛼_𝑖𝐺(𝜂−𝜎(𝑏))(𝑡−𝜎(𝑠)),𝑡≤𝑠,_𝑖21(𝑡,𝑠)=𝑑_𝑖[𝜎(𝑠)(1−𝛼_𝑖)+𝛼_𝑖𝜂+𝛽_𝑖−𝑎](𝜎(𝑏)−𝑡)+𝛾_𝑖(𝜂−𝑎+𝛽_𝑖)(𝑡−𝜎(𝑠)),𝜎(𝑠)≤𝑡,[𝑡(1−𝛼_𝑖)+𝛼_𝑖𝜂+𝛽_𝑖−𝑎](𝜎(𝑏)−𝜎(𝑠)),𝑡≤𝑠. (2.3)

Lemma 2.2 (see [18]). Under condition (𝐻), for 1≤𝑖≤𝑛, Green’s function 𝐺_𝑖(𝑡,𝑠) in (2.2) possesses the following property: 𝐺_𝑖(𝑡,𝑠)>0,(𝑡,𝑠)∈(𝑎,𝜎(𝑏))×(𝑎,𝑏). (2.4)

Lemma 2.3 (see [18]). Assume that (𝐻) holds. Then, for 1≤𝑖≤𝑛, Green’s function 𝐺_𝑖(𝑡,𝑠) in (2.2) satisfies 𝐺_𝑖(𝑡,𝑠)≤max{𝐺_𝑖(𝑎,𝑠),𝐺_𝑖1(𝜎(𝑠),𝑠),𝑑_𝑖(𝜂−𝑎+𝛽_𝑖)(𝜎(𝑏)−𝜎(𝑠))},𝑡,𝑠∈[𝑎,𝜎(𝑏)]×[𝑎,𝑏],0<𝛾_𝑖𝐺≤1,_𝑖(𝑡,𝑠)≤max{𝐺_𝑖(𝜎(𝑏),𝑠),𝐺_𝑖(𝜎(𝑠),𝑠)},𝑡,𝑠∈[𝑎,𝜎(𝑏)]×[𝑎,𝑏],1<𝛾_𝑖<𝜎(𝑏)−𝑎+𝛽_𝑖𝜂−𝑎+𝛽_𝑖. (2.5)

Lemma 2.4 (see [18]). Assume that condition (𝐻) is satisfied. For 𝐺 as in (2.2), take 𝐻₁(𝑡,𝑠)∶=𝐺₁(𝑡,𝑠) and recursively define 𝐻_𝑗(𝑡,𝑠)=_𝑎^𝜎(𝑏)𝐻_𝑗−1(𝑡,𝑟)𝐺_𝑗(𝑟,𝑠)Δ𝑟 (2.6) for 2≤𝑗≤𝑛. Then 𝐻_𝑛(𝑡,𝑠) is Green’s function for the homogeneous problem: (−1)^𝑛𝑦^Δ2𝑛𝛼(𝑡)=0,𝑡∈[𝑎,𝑏]⊂𝕋,_𝑖+1𝑦^Δ2𝑖(𝜂)+𝛽_𝑖+1𝑦^Δ2𝑖+1(𝑎)=𝑦^Δ2𝑖(𝑎),𝛾_𝑖+1𝑦^Δ2𝑖(𝜂)=𝑦^Δ2𝑖(𝜎(𝑏)),0≤𝑖≤𝑛−1. (2.7)

Lemma 2.5 (see [18]). Assume that (𝐻) holds. If one defines 𝐾=∏^𝑛−1_𝑗=1𝐾_𝑗, then the Green function 𝐻_𝑛(𝑡,𝑠) in Lemma 2.4 satisfies the following inequalities: 0≤𝐻_𝑛(𝑡,𝑠)≤𝐾‖𝐺_𝑛(⋅,𝑠)‖,(𝑡,𝑠)∈[𝑎,𝜎(𝑏)]×[𝑎,𝑏], (2.8) where 𝐾_𝑗=_𝑎^𝜎(𝑏)‖𝐺_𝑗(⋅,𝑠)‖Δ𝑠>0,1≤𝑗≤𝑛. (2.9)

Lemma 2.6 (see [22]). Let𝑋 be a Banach space and let 𝐹∶𝑋→𝑋 be completely continuous. If there exists a bounded and linear operator 𝐴∶𝑋→𝑋 such that 1 is not an eigenvalue of 𝐴 and lim_{‖𝑢‖→∞}‖𝐹(𝑢)−𝐴(𝑢)‖‖𝑢‖=0, (2.10) then 𝐹 has a fixed point in 𝑋.

Lemma 2.7 (see [25]). Let 𝑋 be a real Banach space, let Ω be a bounded open subset of 𝑋, 0∈Ω, and let 𝐹∶Ω→𝑋 be a completely continuous operator. Then either there exist 𝑥∈𝜕Ω, 𝜆>1 such that 𝐹(𝑥)=𝜆𝑥 or there exists a fixed point 𝑥^∗∈Ω.

Suppose that ℬ denotes the Banach space 𝐶[𝑎,𝜎(𝑏)] with the norm ‖𝑦‖=sup_{𝑡∈[𝑎,𝜎(𝑏)]}|𝑦(𝑡)|.

3. Existence Results

In this section, we apply Lemmas 2.6 and 2.7 to establish some existence criteria for (1.1) and (1.2).

Theorem 3.1. Suppose that condition (𝐻) holds, and 𝑓∶ℝ→ℝ is continuous with lim_𝑠→∞(𝑓(𝑠)/𝑠)=𝑚. If |𝑚|<𝑑=[^𝑛_𝑗=1𝐾_𝑗]⁻¹, (3.1) then the BVP (1.1) has a solution 𝑦^∗, and 𝑦^∗≠0 when 𝑓(0)≠0.

Proof. Define the integral operator 𝐹∶ℬ→ℬ by 𝐹𝑦(𝑡)=_𝑎^𝜎(𝑏)𝐻_𝑛(𝑡,𝑠)𝑓(𝑦^𝜎(𝑠))Δ𝑠 (3.2) for 𝑡∈[𝑎,𝜎(𝑏)]. Obviously, the solutions of the BVP (1.1) are the fixed points of operator 𝐹. From the proof of Theorem 3.1 of [18], we can know that 𝐹∶ℬ→ℬ is completely continuous.
In order to apply Lemma 2.6, we consider the following BVP:
(−1)^𝑛𝑦^Δ2𝑛(𝑡)=𝑚𝑦^𝜎𝛼(𝑡),𝑡∈[𝑎,𝑏]⊂𝕋,_𝑖+1𝑦^Δ2𝑖(𝜂)+𝛽_𝑖+1𝑦^Δ2𝑖+1(𝑎)=𝑦^Δ2𝑖(𝑎),𝛾_𝑖+1𝑦^Δ2𝑖(𝜂)=𝑦^Δ2𝑖(𝜎(𝑏)),0≤𝑖≤𝑛−1. (3.3) Define the integral operator 𝐴∶ℬ→ℬ by 𝐴𝑦(𝑡)=𝑚_𝑎^𝜎(𝑏)𝐻_𝑛(𝑡,𝑠)𝑦^𝜎(𝑠)Δ𝑠 (3.4) for 𝑡∈[𝑎,𝜎(𝑏)]. Then it is easy to check that 𝐴∶ℬ→ℬ is completely continuous (so bounded) linear operator and that solutions of the BVP (3.3) are the fixed points of operator 𝐴 and conversely.
First, we claim that 1 is not an eigenvalue of 𝐴.
In fact, if 𝑚=0, then it is obvious that the BVP (3.3) has no nontrivial solution.
If 𝑚≠0 and the BVP (3.3) has a nontrivial solution 𝑦, then ‖𝑦‖>0, and so
‖𝑦‖=‖𝐴𝑦‖=sup_{𝑡∈[𝑎,𝜎(𝑏)]}|𝑚_𝑎^𝜎(𝑏)𝐻_𝑛(𝑡,𝑠)𝑦^𝜎(𝑠)Δ𝑠|=|𝑚|sup_{𝑡∈[𝑎,𝜎(𝑏)]}|_𝑎^𝜎(𝑏)𝐻_𝑛(𝑡,𝑠)𝑦^𝜎(𝑠)Δ𝑠|≤|𝑚|_𝑎^𝜎(𝑏)𝐾‖𝐺_𝑛(⋅,𝑠)‖|𝑦^𝜎(𝑠)|Δ𝑠≤|𝑚|(^𝑛_𝑗=1𝐾_𝑗1)‖𝑦‖<𝑑⋅𝑑‖𝑦‖=‖𝑦‖, (3.5) which is impossible. So, 1 is not an eigenvalue of 𝐴.
Next, we will show that
lim_{‖𝑦‖→∞}‖𝐹(𝑦)−𝐴(𝑦)‖‖𝑦‖=0. (3.6) In fact, for any 𝜖>0, since lim_𝑠→∞(𝑓(𝑠)/𝑠)=𝑚, there must exist a number 𝑌₁>0 such that |𝑓(𝑠)−𝑚𝑠|<𝜖|𝑠|,|𝑠|>𝑌₁. (3.7) Let 𝑀=max_{|𝑠|≤𝑌1}|𝑓(𝑠)|. (3.8) Then for any 𝑦∈ℬ and ‖𝑦‖>𝑌 (𝑌>0), we distinguish the following two cases.Case 1 (𝑌<𝑌₁). In this case, choose 𝑌 such that 𝑀+|𝑚|𝑌₁𝑌<𝜖. (3.9) Thus, when 𝑠∈[𝑎,𝜎(𝑏)] and |𝑦^𝜎(𝑠)|≤𝑌₁, we have |𝑓(𝑦^𝜎(𝑠))−𝑚𝑦^𝜎(𝑠)|≤|𝑓(𝑦^𝜎(𝑠))|+|𝑚||𝑦^𝜎(𝑠)|≤𝑀+|𝑚|𝑌₁<𝜖𝑌<𝜖‖𝑦‖, (3.10) which together with (3.7) implies that |𝑓(𝑦^𝜎(𝑠))−𝑚𝑦^𝜎(𝑠)|<𝜖‖𝑦‖,‖𝑦‖>𝑌. (3.11)Case 2 (𝑌≥𝑌₁). In this case, when 𝑠∈[𝑎,𝜎(𝑏)], from (3.7), we see that |𝑓(𝑦^𝜎(𝑠))−𝑚𝑦^𝜎(𝑠)|<𝜖|𝑦^𝜎(𝑠)|≤𝜖‖𝑦‖. (3.12) Thus, we can deduce from (3.11) and (3.12) that for any 𝑦∈ℬ and ‖𝑦‖>𝑌|𝑓(𝑦^𝜎(𝑠))−𝑚𝑦^𝜎(𝑠)|<𝜖‖𝑦‖,∀𝑠∈[𝑎,𝜎(𝑏)]. (3.13) From (3.13), we have ‖𝐹(𝑦)−𝐴(𝑦)‖=sup_{𝑡∈[𝑎,𝜎(𝑏)]}|_𝑎^𝜎(𝑏)𝐻_𝑛(𝑡,𝑠)[𝑓(𝑦^𝜎(𝑠))−𝑚𝑦^𝜎(𝑠)]Δ𝑠|≤sup_{𝑡∈[𝑎,𝜎(𝑏)]𝑎}^𝜎(𝑏)𝐻_𝑛(𝑡,𝑠)|𝑓(𝑦^𝜎(𝑠))−𝑚𝑦^𝜎(𝑠)|Δ𝑠≤𝜖‖𝑦‖(^𝑛_𝑗=1𝐾_𝑗)=𝜖𝑑‖𝑦‖. (3.14) that is to say, lim_{‖𝑦‖→∞}‖𝐹(𝑦)−𝐴(𝑦)‖‖𝑦‖=0. (3.15) Then, it follows from Lemma 2.6 that 𝐹 has a fixed point 𝑦^∗∈ℬ. In other words, 𝑦^∗ is a solution of the BVP (1.1). Moreover, we can assert that 𝑦^∗ is nontrivial when 𝑓(0)≠0. In fact, if 𝑓(0)≠0, then (−1)^𝑛(0)^Δ2𝑛=0≠𝑓(0), (3.16) that is, 0 is not a solution of the BVP (1.1).

Corollary 3.2. Assume that condition (𝐻) holds, and 𝑓∶[0,+∞)→[0,+∞) is continuous with lim_𝑠→+∞(𝑓(𝑠)/𝑠)=0. Then the BVP (1.1) has a nonnegative solution.

Proof. Let 𝑓^∗(𝑠)=𝑓(𝑠),𝑠≥0,𝑓(−𝑠),𝑠<0, (3.17) then 𝑓^∗∶ℝ→[0,+∞) is continuous, and from lim_𝑠→+∞(𝑓(𝑠)/𝑠)=0, we know that lim_𝑠→+∞𝑓^∗(𝑠)𝑠=0. (3.18) Consider the following BVP: (−1)^𝑛𝑦^Δ2𝑛(𝑡)=𝑓^∗(𝑦^𝜎𝛼(𝑡)),𝑡∈[𝑎,𝑏]⊂𝕋,_𝑖+1𝑦^Δ2𝑖(𝜂)+𝛽_𝑖+1𝑦^Δ2𝑖+1(𝑎)=𝑦^Δ2𝑖(𝑎),𝛾_𝑖+1𝑦^Δ2𝑖(𝜂)=𝑦^Δ2𝑖(𝜎(𝑏)),0≤𝑖≤𝑛−1. (3.19) It follows from Theorem 3.1 that the BVP (3.19) has a solution 𝑦^∗, that is, (−1)^𝑛(𝑦^∗)^Δ2𝑛(𝑡)=𝑓^∗((𝑦^∗)^𝜎𝛼(𝑡)),𝑡∈[𝑎,𝑏]⊂𝕋,_𝑖+1(𝑦^∗)^Δ2𝑖(𝜂)+𝛽_𝑖+1(𝑦^∗)^Δ2𝑖+1(𝑎)=(𝑦^∗)^Δ2𝑖(𝑎),𝛾_𝑖+1(𝑦^∗)^Δ2𝑖(𝜂)=(𝑦^∗)^Δ2𝑖(𝜎(𝑏)),0≤𝑖≤𝑛−1. (3.20) Since 𝐻_𝑛(𝑡,𝑠) and 𝑓^∗ are nonnegative, we can get that 𝑦^∗≥0 on [𝑎,𝜎(𝑏)]. Consequently, from the definition of 𝑓^∗, we have (−1)^𝑛(𝑦^∗)^Δ2𝑛(𝑡)=𝑓((𝑦^∗)^𝜎(𝑡)),𝑡∈[𝑎,𝑏]⊂𝕋. (3.21) It follows from the boundary conditions of (3.20) and (3.21) that 𝑦^∗ is a nonnegative solution of the BVP (1.1).

Remark 3.3. In Corollary 3.2, we only need that lim_𝑠→+∞(𝑓(𝑠)/𝑠)=0. Thus, 𝑓 may not be sublinear.

Theorem 3.4. Assume that condition (𝐻) holds, and 𝑓∶[𝑎,𝜎(𝑏)]×[0,+∞)→[0,+∞)iscontinuouswith𝑓(𝑡,𝑢)>0for(𝑡,𝑢)∈[𝑎,𝜎(𝑏)]×(0,+∞),𝑓(𝑡,𝑢)≤𝜑(𝑡)𝑔(𝑢)on[𝑎,𝜎(𝑏)]×[0,+∞)with𝑔≥0continuous,andnondecreasingon[0,+∞)and𝜑∶[𝑎,𝜎(𝑏)]→(0,∞)continuous∃𝑟>0with𝑟>𝑔(‖𝑟‖)_𝑎^𝜎(𝑏)𝜑(𝑠)𝐾‖𝐺_𝑛(⋅,𝑠)‖Δ𝑠, (3.22)(3.23)(3.24) where 𝐾 and 𝐺_𝑛(⋅,𝑠) are defined in Lemmas 2.5 and 2.1, respectively. Then the BVP (1.2) has a nonnegative solution y₁ with‖𝑦₁‖<𝑟.

Proof. We consider the following boundary value problem: (−1)^𝑛𝑦^Δ2𝑛(𝑡)=𝜆𝑓^∗(𝑡,𝑦^𝜎𝛼(𝑡)),𝑡∈[𝑎,𝑏]⊂𝕋,_𝑖+1𝑦^Δ2𝑖(𝜂)+𝛽_𝑖+1𝑦^Δ2𝑖+1(𝑎)=𝑦^Δ2𝑖(𝑎),𝛾_𝑖+1𝑦^Δ2𝑖(𝜂)=𝑦^Δ2𝑖(𝜎(𝑏)),0≤𝑖≤𝑛−1. (3.25) where 0<𝜆<1, and 𝑓^∗(𝑡,𝑢)=𝑓(𝑡,𝑢),𝑢≥0,𝑓(𝑡,0),𝑢<0. (3.26) Let 𝑦 be any solution of (3.25). Then 𝑦(𝑡)=𝜆_𝑎^𝜎(𝑏)𝐻_𝑛(𝑡,𝑠)𝑓^∗(𝑠,𝑦^𝜎(𝑠))Δ𝑠 (3.27) for 𝑡∈[𝑎,𝜎(𝑏)]. We note that 𝑦(𝑡)≥0 for 𝑡∈[𝑎,𝜎(𝑏)]. It follows from condition (3.23) in Theorem 3.4 that for 𝑡∈[𝑎,𝜎(𝑏)]𝑦(𝑡)≤_𝑎^𝜎(𝑏)𝐻_𝑛(𝑡,𝑠)𝜑(𝑠)𝑔(|𝑦^𝜎(𝑠)|)Δ𝑠≤𝑔(‖𝑦‖)_𝑎^𝜎(𝑏)𝜑(𝑠)𝐾‖𝐺_𝑛(⋅,𝑠)‖Δ𝑠. (3.28) Consequently, ‖𝑦‖≤𝑔(‖𝑦‖)_𝑎^𝜎(𝑏)𝜑(𝑠)𝐾‖𝐺_𝑛(⋅,𝑠)‖Δ𝑠, (3.29) which together with the condition (3.24) in Theorem 3.4 implies that ‖𝑦‖≠𝑟.
Let 𝑁∶ℬ→ℬ be given by
𝑁𝑦(𝑡)=_𝑎^𝜎(𝑏)𝐻_𝑛(𝑡,𝑠)𝑓^∗(𝑠,𝑦^𝜎(𝑠))Δ𝑠. (3.30) It is easy to show that 𝑁∶ℬ→ℬ is completely continuous.
Let
𝑈={𝑢∈ℬ∶‖𝑢‖<𝑟}. (3.31) Since ‖𝑦‖≠𝑟, any solution 𝑦∈𝜕𝑈 of 𝑦=𝜆𝑁𝑦 with 0<𝜆<1 cannot occur. Lemma 2.7 guarantees that 𝑁 has a fixed point 𝑦₁ in 𝑈. In other words, the BVP (1.2) has a solution 𝑦₁∈ℬ with ‖𝑦₁‖<𝑟.

Theorem 3.5. Assume that condition (𝐻) is satisfied. Suppose that 𝑓(𝑡,0)≢0, 𝑡∈[𝑎,𝜎(𝑏)], 𝑓∶[𝑎,𝜎(𝑏)]×ℝ→ℝ is continuous, and there exist nonnegative integrable functions 𝑘, ℎ such that 𝐾|𝑓(𝑡,𝑦)|≤𝑘(𝑡)|𝑦|+ℎ(𝑡),(𝑡,𝑦)∈[𝑎,𝜎(𝑏)]×ℝ,_{𝑎𝜎(𝑏)}‖𝐺_𝑛(⋅,𝑠)‖𝑘(𝑠)Δ𝑠<1. (3.32) Then, the BVP (1.2) has at least one nontrivial solution 𝑦^∗∈ℬ.

Proof. Let 𝐴=𝐾_𝑎^𝜎(𝑏)‖𝐺_𝑛(⋅,𝑠)‖ℎ(𝑠)Δ𝑠,𝐵=𝐾_𝑎^𝜎(𝑏)‖𝐺_𝑛(⋅,𝑠)‖𝑘(𝑠)Δ𝑠. (3.33) By hypothesis 𝐵<1. Since 𝑓(𝑡,0)≢0, there exists [𝑚,𝑛]⊂[𝑎,𝜎(𝑏)] such that min_{𝑡∈[𝑚,𝑛]}|𝑓(𝑡,0)|>0. On the other hand, from the condition ℎ(𝑡)≥|𝑓(𝑡,0)|,a.e.𝑡∈[𝑎,𝜎(𝑏)], we know that 𝐴>0.
Let 𝑑=𝐴(1−𝐵)⁻¹, Ω_𝑑={𝑦∈ℬ∶‖𝑦‖<𝑑}. For 𝑡∈[𝑎,𝜎(𝑏)], the operator 𝑇 is defined by
𝑇𝑦(𝑡)=_𝑎^𝜎(𝑏)𝐻_𝑛(𝑡,𝑠)𝑓(𝑠,𝑦^𝜎(𝑠))Δ𝑠, (3.34) from the proof of Theorems 3.1 and 3.4, we have known that 𝑇∶ℬ→ℬ is a completely continuous operator, and the BVP (1.2) has at least one nontrivial solution 𝑦^∗∈ℬ if and only if 𝑦^∗ is a fixed point of 𝑇 in ℬ.
Suppose 𝑦∈𝜕Ω_𝑑, 𝜆>1 such that 𝑇𝑦=𝜆𝑦, then
𝜆𝑑=𝜆‖𝑦‖=‖𝑇𝑦‖=sup_{𝑡∈[𝑎,𝜎(𝑏)]}|𝑇𝑦(𝑡)|=sup_{𝑡∈[𝑎,𝜎(𝑏)]}|_𝑎^𝜎(𝑏)𝐻_𝑛(𝑡,𝑠)𝑓(𝑠,𝑦^𝜎(𝑠))Δ𝑠|≤𝐾_𝑎^𝜎(𝑏)‖𝐺_𝑛(⋅,𝑠)‖|𝑓(𝑠,𝑦^𝜎(𝑠))|Δ𝑠≤𝐾_𝑎^𝜎(𝑏)‖𝐺_𝑛(⋅,𝑠)‖[𝑘(𝑠)|𝑦^𝜎(𝑠)|+ℎ(𝑠)]Δ𝑠=𝐾_𝑎^𝜎(𝑏)‖𝐺_𝑛(⋅,𝑠)‖𝑘(𝑠)|𝑦^𝜎(𝑠)|Δ𝑠+𝐾_𝑎^𝜎(𝑏)‖𝐺_𝑛(⋅,𝑠)‖ℎ(𝑠)Δ𝑠≤𝐵‖𝑦‖+𝐴=𝐵𝑑+𝐴. (3.35) Therefore (𝜆−1)𝑑≤𝐴−(1−𝐵)𝑑=𝐴−𝐴=0, (3.36) which contradicts 𝜆>1. By Lemma 2.7, 𝑇 has a fixed point 𝑦^∗∈Ω_𝑑. Noting 𝑓(𝑡,0)≢0, the BVP (1.2) has at least one nontrivial solution 𝑦^∗∈ℬ. This completes the proof.

Corollary 3.6. Assume that condition (𝐻) is satisfied. Suppose that 𝑓(𝑡,0)≢0, 𝑡∈[𝑎,𝜎(𝑏)], 𝑓∶[𝑎,𝜎(𝑏)]×ℝ→ℝ is continuous, and there exist nonnegative integrable functions 𝑘,ℎ such that |𝑓(𝑡,𝑦)|≤𝑘(𝑡)|𝑦|+ℎ(𝑡),(𝑡,𝑦)∈[𝑎,𝜎(𝑏)]×ℝ,𝑘(𝑡)<𝑒=[^𝑛_𝑗=1𝐾_𝑗]⁻¹,𝑡∈[𝑎,𝜎(𝑏)]. (3.37) Then, the BVP (1.2) has at least one nontrivial solution 𝑦^∗∈ℬ.

Proof. In this case, we have 𝐾_𝑎^𝜎(𝑏)‖𝐺_𝑛(⋅,𝑠)‖𝑘(𝑠)Δ𝑠<𝐾𝑒_𝑎^𝜎(𝑏)‖𝐺_𝑛(⋅,𝑠)‖Δ𝑠=𝐾[^𝑛_𝑗=1𝐾_𝑗]⁻¹_𝑎^𝜎(𝑏)‖𝐺_𝑛(⋅,𝑠)‖Δ𝑠=1. (3.38) By Theorem 3.5, this completes the proof.

Corollary 3.7. Assume that condition (𝐻) is satisfied. Suppose that 𝑓(𝑡,0)≢0, 𝑡∈[𝑎,𝜎(𝑏)], 𝑓∶[𝑎,𝜎(𝑏)]×ℝ→ℝ is continuous, and there exist nonnegative integrable functions k, h such that |𝑓(𝑡,𝑦)|≤𝑘(𝑡)|𝑦|+ℎ(𝑡),(𝑡,𝑦)∈[𝑎,𝜎(𝑏)]×ℝ,lim_|𝑙|→∞max_{𝑡∈[𝑎,𝜎(𝑏)]}|𝑓(𝑡,𝑙)𝑙|<[∏^𝑛_𝑗=1𝐾_𝑗]⁻¹. (3.39) Then, the BVP (1.2) has at least one nontrivial solution 𝑦^∗∈ℬ.

Proof. Let 𝜖₁=12[(^𝑛_𝑗=1𝐾_𝑗)⁻¹−lim_|𝑙|→∞max_{𝑡∈[𝑎,𝜎(𝑏)]}|𝑓(𝑡,𝑙)𝑙|], (3.40) then, there exists 𝑐>0such that|𝑓(𝑡,𝑙)|≤[(^𝑛_𝑗=1𝐾_𝑗)⁻¹−𝜖₁]|𝑙|,(𝑡,𝑙)∈[𝑎,𝜎(𝑏)]×ℝ⧵(−𝑐,𝑐). (3.41) Set 𝑀=max{𝑓(𝑡,𝑙)∣(𝑡,𝑙)∈[𝑎,𝜎(𝑏)]×[−𝑐,𝑐]}, (3.42) and it follows from (3.41) and (3.42) that |𝑓(𝑡,𝑙)|≤[(^𝑛_𝑗=1𝐾_𝑗)⁻¹−𝜖₁]|𝑙|+𝑀,(𝑡,𝑙)∈[𝑎,𝜎(𝑏)]×ℝ. (3.43) By Corollary 3.6, we can deduce that Corollary 3.7 is true.

4. Two Examples

In the section, we present two examples to explain our results.

Example 4.1. Let 𝕋=ℝ[0,1)∪{1,2,3,4,5}∪[6,7], and consider the following BVP: −𝑦^Δ2(𝑡)=𝑓(𝑦^𝜎3(𝑡)),𝑡∈[0,5]⊂𝕋,52𝑦(32)+3𝑦^Δ1(0)=𝑦(0),62𝑦(3)=𝑦(𝜎(5)), (4.1) where 𝑓(𝑦)=(17737/89902)𝑦+1/101. It is easy to check that 30<5=𝛼₁<𝜎(𝑏)−𝛾₁𝜂+(𝛾₁−1)(𝑎−𝛽₁)=𝜎(𝑏)−𝜂6−(1/6)⋅(2/3)+(1/6−1)(0−2/3)=6−2/329,𝛽24₁=231>0,0<6=𝛾₁<𝜎(𝑏)−𝑎+𝛽₁𝜂−𝑎+𝛽₁=6−1+(2/3)2/3−0+2/3=5, (4.2) and therefore, the condition (𝐻) is satisfied. By computation, we can get that 𝑑₁=(𝛾₁−1)(𝑎−𝛽₁)+(1−𝛼₁)𝜎(𝑏)+𝜂(𝛼₁−𝛾₁)=146,𝐺45₁𝐺(𝑡,𝑠)=¹¹2(𝑡,𝑠),0≤𝑠≤3,𝐺₁₂2(𝑡,𝑠),3<𝑠≤5, (4.3) where 𝐺₁₁(𝑡,𝑠)=45[114662(𝑡−32)+𝜎(5)−𝑡](𝜎(𝑠)+3[1),𝜎(𝑠)≤𝑡,62(𝜎(𝑠)−32)+6−𝜎(𝑠)](𝑡+33)+5(23𝐺−6)(𝑡−𝜎(𝑠)),𝑡≤𝑠,₁₂(𝑡,𝑠)=453146[𝜎(𝑠)(1−53)+5×23+231](6−𝑡)+6(23+233)(𝑡−𝜎(𝑠)),𝜎(𝑠)≤𝑡,[𝑡(1−53)+5×23+23](6−𝜎(𝑠)),𝑡≤𝑠. (4.4)From the proof of Lemma 2.5 in [18], we can get that ‖𝐺₁(⋅,𝑠)‖=max{𝐺₁(0,𝑠),𝐺₁1(𝜎(𝑠),𝑠),𝑑₁(𝜂−𝑎+𝛽₁)(𝜎(𝑏)−𝜎(𝑠))}=𝐺₁(𝜎(𝑠),𝑠), (4.5) and for 𝛾₁=1/6∈(0,1] and 𝛼₁=3/5∈(0,1), we have 𝐾₁=45146₀^2/3[162(𝑠−32)+6−𝑠](𝑠+3−0)𝑑𝑠+45146¹_2/33[𝑠(1−53)+5×23+23+−0](6−𝑠)𝑑𝑠_5𝑠=1453146[(𝑠+1)(1−53)+5×23+23=][6−(𝑠+1)]89902.17739 (4.6) Thus, we can obtain that lim_𝑦→∞𝑓(𝑦)𝑦=lim_𝑦→∞(17737/89902)𝑦+1/101𝑦=1773789902<𝑑=𝐾₁⁻¹=1773989902. (4.7) By Theorem 3.1, it is easy to get that the BVP (4.1) has a solution 𝑦^∗.

Example 4.2. Let us introduce an example to illustrate the usage of Theorem 3.5. Let 𝑛=2,𝕋={(2/3)^𝑛∶𝑛∈ℕ₀}∪{0}∪[1,2],𝑎=8/27,𝜂=4/9,𝑏=2/3,𝛼₁=1/2,𝛽₁=1/9,𝛼₂=1/10,𝛽₂=7/27,𝛾₁=3/2, and 𝛾₂=2. Then condition (𝐻) is satisfied. Green’s function 𝐺₁(𝑡,𝑠) in Lemma 2.1 is 𝐺₁𝐺(𝑡,𝑠)=¹¹8(𝑡,𝑠),427≤𝑠≤9,𝐺₁₂4(𝑡,𝑠),92<𝑠≤3, (4.8) where 𝐺₁₁(𝑡,𝑠)=274[324(𝑡−93)+1−𝑡][21𝑠+9−8327],2[3𝑠≤𝑡,2(324𝑠−93)+1−21𝑠](𝑡+9−8127)+2(493−1)(𝑡−2𝐺𝑠),𝑡≤𝑠,₁₂(𝑡,𝑠)=274[321𝑠(1−21)+2×49+19−8327](1−𝑡)+2(49−8+12793)(𝑡−23𝑠),21𝑠≤𝑡,[𝑡(1−21)+2×49+19−8327](1−2𝑠),𝑡≤𝑠. (4.9) Green’s function 𝐺₂(𝑡,𝑠) in Lemma 2.1 is 𝐺₂𝐺(𝑡,𝑠)=²¹8(𝑡,𝑠),427≤𝑠≤9,𝐺₂₂4(𝑡,𝑠),92<𝑠≤3, (4.10) where 𝐺₂₁(𝑡,𝑠)=5454[2(𝑡−93)+1−𝑡][27𝑠+−827327],23𝑠≤𝑡,[2(24𝑠−93)+1−27𝑠](𝑡+−827127)+(41093−1)(𝑡−2𝐺𝑠),𝑡≤𝑠,₂₂(𝑡,𝑠)=544[321𝑠(1−110)+×4109+7−827427](1−𝑡)+2(9−8+727327)(𝑡−23𝑠),21𝑠≤𝑡,[𝑡(1−110)+×4109+7−827327](1−2𝑠),𝑡≤𝑠. (4.11) Since 𝑠∈[𝑎,(𝛾₁(𝜂−𝑎+𝛽₁)−𝛼₁𝜂−𝛽₁+𝑎)/(1−𝛼₁))=[8/27,19/27), by using the cases in the proof of Lemma 2.5 of [18], we can know that ‖𝐺₁(⋅,𝑠)‖=𝐺₁(𝜎(2/3),𝑠). Therefore, we have 𝐾₁=133/324.
Set 𝑓(𝑡,𝑦)=𝑡|𝑦|sin𝑦+𝑡³−2sin𝑡,𝑘(𝑡)=𝑡, and ℎ(𝑡)=𝑡³+2sin𝑡. Then it is easy to prove that
𝑓(𝑡,𝑦)≤𝑘(𝑡)|𝑦|+ℎ(𝑡),(𝑡,𝑦)∈[𝑎,𝜎(𝑏)]×ℝ. (4.12) On the other hand, since 𝑠∈[𝑎,(𝛾₂(𝜂−𝑎+𝛽₂)−𝛼₂𝜂−𝛽₂+𝑎)/(1−𝛼₂))=[8/27,218/243), we can know that ‖𝐺₂(⋅,𝑠)‖=𝐺₂(1,𝑠) by using the cases in the proof of Lemma 2.5 of [18]. Therefore, we have _𝑎^𝜎(𝑏)‖𝐺₂(⋅,𝑠)‖𝑘(𝑠)Δ𝑠=6⋅11⋅16⋅475⋅27³. (4.13) Thus 𝐾_𝑎^𝜎(𝑏)‖𝐺₂(⋅,𝑠)‖𝑘(𝑠)Δ𝑠=(²⁻¹_𝑗=1𝐾_𝑗)_𝑎^𝜎(𝑏)‖𝐺₂(⋅,𝑠)‖𝑘(𝑠)Δ𝑠=133⋅3246⋅11⋅16⋅475⋅27³<1. (4.14) Hence, by Theorem 3.5, the BVP (1.2) has at least one nontrivial solution 𝑦^∗.

Acknowledgment

Project supported by the Youth Science Foundation of Shanxi Province (2009021001-2).