A New Solution Operator of One-Dimensional 𝑝-Laplacian with a Sign-Changing Weight and Its Application

Abstract

We establish a new solution operator for the following problem −𝜑_𝑝(𝑢′)′=𝑔, 𝑡∈(0,1), 𝑢(0)=0=𝑢(1), where 𝜑_𝑝(𝑥)=|𝑥|^𝑝−2𝑥, 𝑝>1. 𝑔 may be singular at the boundary or change signs or may not be in 𝐿¹(0,1) so that this solution operator can cover larger class of 𝑔 than previously known ones. As an application, by checking complete continuity of the solution operator, we show the existence of nontrivial solutions for 𝑝-Laplacian 𝜑_𝑝(𝑢′)′+𝜆ℎ(𝑡)𝑓(𝑢(𝑡))=0, 𝑡∈(0,1), 𝑢(0)=0=𝑢(1), where 𝜆>0 a parameter and 𝑓∈𝐶(ℝ,ℝ) and 𝑓(0)>0 and ℎ may change signs or may be beyond of 𝐿¹(0,1).

1. Introduction

The solution operator approach has been used to show the existence of solutions for wide class of nonlinear differential equations. As it is wellknown, a solution operator plays a key role when we employ nonlinear analytic methods such as degree theory, fixed point index theory, and bifurcation theory. In this paper, we are concerned with a solution operator for one-dimensional 𝑝-Laplacian with a sign-changing singular weight.

Let us summarize a brief history about solution operators for one-dimensional 𝑝-Laplacian with Dirichlet boundary condition −𝜑_𝑝𝑢^′′(),𝑢(0)(1),=𝑔,𝑡∈0,1=0=𝑢 (𝐴) where 𝜑_𝑝(𝑥)=|𝑥|^𝑝−2𝑥, 𝑝>1, and 𝑔∈𝐿¹_loc(0,1).

First, when 𝑔∈𝐿¹(0,1), a solution operator was introduced by Manásevich and Mawhin [1, 2]. In this case, we know that all solutions of (𝐴) are in 𝐶¹[0,1] so that by a solution of problem (𝐴), we understand a function 𝑢∈𝐶¹[0,1] with 𝜑_𝑝(𝑢′) absolutely continuous which satisfies (𝐴). Key step of establishing a solution operator for this case is to show that, for each 𝑔∈𝐿¹(0,1), the equation ¹₀𝜑_𝑝⁻¹𝑎𝑔+^𝑠₀𝑔(𝜏)𝑑𝜏𝑑𝑡=0 (1.1) has a unique solution 𝑎=𝑎(𝑔) where 𝑎∶𝐿¹(0,1)→ℝ is a continuous function. Defining 𝐺𝑔(𝑡)≜^𝑡₀𝜑_𝑝⁻¹𝑎𝑔+^𝑠₀𝑔(𝜏)𝑑𝜏𝑑𝑠, (1.2) problem (𝐴) can be equivalently written as 𝑢=𝐺(𝑔).

Secondly, when 𝑔≥0 and satisfies ₀^1/2𝜑_𝑝⁻¹_𝑠^1/2𝑔(𝜏)𝑑𝜏𝑑𝑠+¹_1/2𝜑_𝑝^−1𝑠_1/2𝑔(𝜏)𝑑𝜏𝑑𝑠<∞, (1.3) a solution operator was studied by Agarwal et al. [3]. We note that a function satisfying (1.3) needs not be in 𝐿¹(0,1). In this case, we know that solutions of (𝐴) may not be in 𝐶¹[0,1] so that, by a solution of problem (𝐴), we understand a function 𝑢∈𝐶[0,1]∩𝐶¹(0,1) with 𝜑_𝑝(𝑢′) absolutely continuous which satisfies (𝐴). Define a function 𝑥 by 𝑥(𝑡)=^𝑡₀𝜑_𝑝^−1𝑡_𝑠𝑔(𝜏)𝑑𝜏𝑑𝑠−¹_𝑡𝜑_𝑝^−1𝑠_𝑡𝑔(𝜏)𝑑𝜏𝑑𝑠 (1.4) for 0<𝑡<1. It is easy to see that 𝑥 is continuous, strictly increasing in (0,1) and 𝑥(0⁺)<0<𝑥(1⁻). Thus, 𝑥 has a unique zero in (0,1), say 𝐴_𝑔. Then, we see that ^𝐴𝑔₀𝜑_𝑝^−1𝐴𝑔_𝑠𝑔(𝜏)𝑑𝜏𝑑𝑠=¹_𝐴𝑔𝜑_𝑝^−1𝑠_𝐴𝑔𝑔(𝜏)𝑑𝜏𝑑s, (1.5) and defining 𝐺𝑔(𝑡)≜∫^𝑡₀𝜑_𝑝⁻¹∫^𝐴𝑔_𝑠𝑔(𝜏)𝑑𝜏𝑑𝑠,0≤𝑡≤𝐴_𝑔,∫¹_𝑡𝜑_𝑝⁻¹∫^𝑠_𝐴𝑔𝑔(𝜏)𝑑𝜏𝑑𝑠,𝐴_𝑔≤𝑡≤1, (1.6) problem (𝐴) can be equivalently written as 𝑢=𝐺(𝑔).

Now let us consider the case that 𝑔 may change signs and belongs to ℋ⊂𝐿¹_loc(0,1) which is defined as ℋ=𝑔∶₀^1/2𝜑_𝑝⁻¹_𝑠^1/2𝑔(𝜏)𝑑𝜏𝑑𝑠+¹_1/2𝜑_𝑝^−1𝑠_1/2𝑔(𝜏)𝑑𝜏𝑑𝑠<∞. (1.7) Defining 𝑔∶(0,1)→ℝ by 𝑔(𝑡)=𝑡^−𝛼1,0<𝑡≤2,1−1,2<𝑡<1, (1.8) where 1<𝛼<𝑝, it is not hard to check that 𝑔∈ℋ, but 𝑔∉𝐿¹(0,1).

For 𝑔∈ℋ but 𝑔∉𝐿¹(0,1), existence of solution for (1.1) is not known so that Manásevich-Mawhin's approach cannot be applied directly. For 𝑔 sign changing, the uniqueness of zeros of function 𝑥 in (1.4) is not guaranteed so that Agarwal et al.’s approach cannot be applied directly either. The aim of this paper is to introduce a new solution operator for (𝐴) under this situation.

As an application, we will make use of newly established solution operator to show the existence of nontrivial solutions with respect to a parameter for the following type of nonlinear problem 𝜑_𝑝𝑢^′′(𝑡)𝑓(𝑢(𝑡(),𝑢(0)(1),+𝜆ℎ))=0,𝑡∈0,1=0=𝑢 𝑃_𝜆 where 𝜆>0 a parameter, ℎ∈ℋ, 𝑓∈𝐶(ℝ,ℝ), and 𝑓(0)>0.

This kind of problems can be classified as cases 𝑓(0)>0, 𝑓(0)=0, or 𝑓(0)<0, and, under additional nonlinear growth conditions, it is easy to see that bifurcation phenomena usually happen for the case of 𝑓(0)>0. In this paper, we will employ the global continuation theorem to get an existence result. One of the major steps for proof is to show the complete continuity of corresponding solution operator for 𝑃_𝜆. There have been many studies for 𝑓 nonnegative with 𝑓(0)>0 and ℎ≥0. Many authors are interested in finding positive solutions mainly employing the fixed-point index theory on a cone. Since ℎ may change signs in our case, this method is not suitable. Hai [4] investigated semilinear boundary value problems with a sign-changing weight; however, his method is also restricted to fit our quasilinear case.

The paper is organized as follow. In Section 2, we establish a solution operator for (𝐴), when 𝑔∈ℋ. In Section 3, we prove the continuity and compactness of the solution operator for problem 𝑃_𝜆. In Section 4, we obtain the existence of non-trivial solutions for 𝑃_𝜆.

2. A Solution Operator

In this section, we construct a new solution operator for (𝐴).

Theorem 2.1. For each 𝑔∈ℋ, there is a unique solution 𝑢∈𝐶[0,1]∩𝐶¹(0,1) for (𝐴).

The proof follows the next two lemmas. We first show the existence of a solution for (𝐴).

Lemma 2.2. For each 𝑔∈ℋ, there is a unique constant 𝛼=𝛼(𝑔)∈ℝ such that a function 𝑢 defined by 𝑢(𝑡)=∫^𝑡₀𝜑_𝑝⁻¹𝛼+∫_𝑠^1/2𝑔(𝑟)1𝑑𝑟𝑑𝑠,0≤𝑡≤2,∫¹_𝑡𝜑_𝑝⁻¹−𝛼+∫^𝑠_1/2𝑔(𝑟)1𝑑𝑟𝑑𝑠,2≤𝑡≤1, (2.1) satisfies 𝑢∈𝐶[0,1]∩𝐶¹(0,1) and 𝑢 solves (𝐴). Furthermore, we have ₀^1/2𝜑_𝑝⁻¹𝛼+_𝑠^1/2𝑔(𝑟)𝑑𝑟𝑑𝑠=¹_1/2𝜑_𝑝⁻¹−𝛼+^𝑠_1/2𝑔(𝑟)𝑑𝑟𝑑𝑠. (2.2)

Remark 2.3. For 𝑎,𝑏>0, 𝜑_𝑝⁻¹(𝑎+𝑏)=(𝑎+𝑏)^1/(𝑝−1)≤𝑎^1/(𝑝−1)+𝑏^1/(𝑝−1)2,𝑝>2,_{(2−𝑝)/(𝑝−1)}𝑎^1/(𝑝−1)+𝑏^1/(𝑝−1),1<𝑝≤2. (2.3) Thus for 0<𝑡≤1/2, |𝑢(𝑡)|≤𝐶_𝑝^𝑡₀|𝛼|^1/(𝑝−1)𝑑𝑡+^𝑡₀𝜑_𝑝⁻¹_𝑠^1/2𝑔(𝑟)𝑑𝑟𝑑𝑠<∞, (2.4) where 𝐶_𝑝=21,𝑝>2,_{(2−𝑝)/(𝑝−1)},1<𝑝≤2. (2.5) Similarly, we get an upper bound of |𝑢| on (1/2,1). It is clear that 𝑢(0)=0=𝑢(1).

Proof of Lemma 2.2. Define 𝑣_𝛼(𝑡)≜^𝑡₀𝜑_𝑝⁻¹𝛼+_𝑠^1/2𝑔(𝑟)1𝑑𝑟𝑑𝑠,0≤𝑡≤2,𝑤_𝛼(𝑡)≜¹_𝑡𝜑_𝑝⁻¹−𝛼+^𝑠_1/2𝑔(𝑟)1𝑑𝑟𝑑𝑠,2≤𝑡≤1. (2.6) Then, by Lebesgue dominated convergence theorem, we see that 𝑣_𝛼(1/2) and 𝑤_𝛼(1/2) are continuous on 𝛼. Because of the strict monotonicity of 𝜑_𝑝, it is easy to show that 𝑣_𝛼(1/2) is strictly increasing on 𝛼 and 𝑤_𝛼(1/2) is strictly decreasing on 𝛼.
We claim that 𝑣_𝛼(1/2)→∞, 𝑤_𝛼(1/2)→−∞ as 𝛼→∞, and 𝑣_𝛼(1/2)→−∞, 𝑤_𝛼(1/2)→∞ as 𝛼→−∞. Since (𝛼/2)+∫_𝑠^1/2𝑔(𝑟)𝑑𝑟>0 for 1/3≤𝑠≤1/2, for sufficiently large 𝛼>0, we have 𝑣_𝛼12=₀^1/2𝜑_𝑝⁻¹𝛼+_𝑠^1/2𝑔(𝑟)=𝑑𝑟𝑑𝑠_01/3𝜑_𝑝⁻¹𝛼+_𝑠^1/2𝑔(𝑟)𝑑𝑟𝑑𝑠+^1/2_1/3𝜑_𝑝⁻¹𝛼+_𝑠^1/2𝑔(𝑟)≥1𝑑𝑟𝑑𝑠6𝛼2^1/(𝑝−1)+₀^1/3𝜑_𝑝⁻¹_𝑠^1/2𝑔(𝑟)≥1𝑑𝑟𝑑𝑠6𝛼2^1/(𝑝−1)−₀^1/2𝜑_𝑝⁻¹_𝑠^1/2𝑔(𝑟)𝑑𝑟𝑑𝑠⟶∞, (2.7) as 𝛼→∞. Since −𝛼/2+∫^𝑠_1/2𝑔(𝑟)𝑑𝑟<0 for 1/2≤𝑠≤2/3, for sufficiently large 𝛼>0, we have 𝑤_𝛼12=¹_1/2𝜑_𝑝⁻¹−𝛼+^𝑠_1/2𝑔(𝑟)𝑑𝑟𝑑𝑠=^2/3_1/2𝜑_𝑝⁻¹−𝛼+^𝑠_1/2𝑔(𝑟)𝑑𝑟𝑑𝑠+¹_2/3𝜑_𝑝⁻¹−𝛼+^𝑠_1/2𝑔(𝑟)1𝑑𝑟𝑑𝑠≤−6𝛼2^1/(𝑝−1)+¹_2/3𝜑_𝑝^−1𝑠_1/2𝑔(𝑟)1𝑑𝑟𝑑𝑠≤−6𝛼2^1/(𝑝−1)+¹_1/2𝜑_𝑝^−1𝑠_1/2𝑔(𝑟)𝑑𝑟𝑑𝑠⟶−∞, (2.8) as 𝛼→∞. Similarly, since 𝛼/2+∫_𝑠^1/2𝑔(𝑟)𝑑𝑟<0 for 1/3≤𝑠≤1/2, for negatively large 𝛼<0, we have 𝑣_𝛼12=₀^1/2𝜑_𝑝⁻¹𝛼+_𝑠^1/2𝑔(𝑟)=𝑑𝑟𝑑𝑠_01/3𝜑_𝑝⁻¹𝛼+_𝑠^1/2𝑔(𝑟)𝑑𝑟𝑑𝑠+^1/2_1/3𝜑_𝑝⁻¹𝛼+_𝑠^1/2𝑔(𝑟)1𝑑𝑟𝑑𝑠≤−6−𝛼2^1/(𝑝−1)+₀^1/3𝜑_𝑝⁻¹_𝑠^1/2𝑔(𝑟)1𝑑𝑟𝑑𝑠≤−6−𝛼2^1/(𝑝−1)+₀^1/2𝜑_𝑝⁻¹_𝑠^1/2𝑔(𝑟)𝑑𝑟𝑑𝑠⟶−∞, (2.9) as 𝛼→−∞. Since −𝛼/2+∫^𝑠_1/2𝑔(𝑟)𝑑𝑟>0 for 1/2≤𝑠≤2/3, for negatively large 𝛼<0, we have 𝑤_𝛼12=¹_1/2𝜑_𝑝⁻¹−𝛼+^𝑠_1/2𝑔(𝑟)𝑑𝑟𝑑𝑠=^2/3_1/2𝜑_𝑝⁻¹−𝛼+^𝑠_1/2𝑔(𝑟)𝑑𝑟𝑑𝑠+¹_2/3𝜑_𝑝⁻¹−𝛼+^𝑠_1/2𝑔(𝑟)≥1𝑑𝑟𝑑𝑠6−𝛼2^1/(𝑝−1)+¹_2/3𝜑_𝑝^−1𝑠_1/2𝑔(𝑟)≥1𝑑𝑟𝑑𝑠6−𝛼2^1/(𝑝−1)−¹_1/2𝜑_𝑝^−1𝑠_1/2𝑔(𝑟)𝑑𝑟𝑑𝑠⟶∞, (2.10) as 𝛼→−∞. Thus, by intermediate value theorem, there is a unique 𝛼₀∈ℝ such that 𝑣_𝛼0(1/2)=𝑤_𝛼0(1/2). Put 𝑢(𝑡)=𝑣_𝛼0(𝑡)1,0≤𝑡≤2,𝑤_𝛼0(𝑡),12≤𝑡≤1. (2.11) Then, 𝑢∈𝐶[0,1]∩𝐶¹(0,1) and 𝑢 solves (𝐴).

From the uniqueness of 𝛼, the following is obvious.

Corollary 2.4. Suppose that 𝛼 is a function defined in Lemma 2.2. Then, 𝛼 has the following property: 𝛼𝜆𝑔=𝜆𝛼𝑔, (2.12) for all 𝜆∈ℝ.

The following lemma guarantees the uniqueness of solutions for (𝐴).

Lemma 2.5. Assume that, for 𝑖=1,2, 𝑢_𝑖∈𝐶[0,1]∩𝐶¹(0,1) satisfies −𝜑_𝑝𝑢^′′=𝑔_𝑖(),𝑢(0)(1).,𝑡∈0,1=0=𝑢 𝐴_𝑖 If 𝑔₁≤𝑔₂ or 𝑔₂≤𝑔₁ on [0,1], then 𝑢₁≤𝑢₂ or 𝑢₂≤𝑢₁ on [0,1], respectively.

Proof. We only prove the first case. We may use similar argument to show the second case. Let 𝑢_𝑖 be a solution of 𝐴_𝑖, 𝑖=1,2, and assume 𝑔₁≤𝑔₂ for 0≤𝑡≤1. We want to show 𝑢₁(𝑡)≤𝑢₂(𝑡), 0≤𝑡≤1. If this is false, then there exists 𝑡₀∈(0,1) with 𝑢₁(𝑡₀)>𝑢₂(𝑡₀). Hence, there is an interval (𝑎,𝑏)⊂(0,1) with 𝑡₀∈(𝑎,𝑏) such that 𝑢₁(𝑡)>𝑢₂(𝑡), 𝑎<𝑡<𝑏 and 𝑢₁(𝑎)−𝑢₂(𝑎)=𝑢₁(𝑏)−𝑢₂(𝑏)=0. Let 𝑚=𝑢₁(𝐵)−𝑢₂(𝐵) be the positive maximum of 𝑢₁(𝑡)−𝑢₂(𝑡) on [𝑎,𝑏]. Then, 𝐵∈[𝑎,𝑏] and 𝑢^′₁(𝐵)=𝑢^′₂(𝐵). Integrating both sides (𝐴₁) and (𝐴₂) over [𝑠,𝐵], 𝑎<𝑠<𝐵, we have 𝑢^′₁(𝑠)=𝜑_𝑝⁻¹𝜑_𝑝𝑢^′₁(𝐵)+^𝐵_𝑠𝑔₁(𝑟),𝑢𝑑𝑟_′2(𝑠)=𝜑_𝑝⁻¹𝜑_𝑝𝑢^′₂(𝐵)+^𝐵_𝑠𝑔₂(𝑟)𝑑𝑟. (2.13) Integrating both sides of the above equalities over [𝑎,𝐵], we get 𝑢₁(𝐵)−𝑢₁(𝑎)=^𝐵_𝑎𝜑_𝑝⁻¹𝜑_𝑝𝑢^′₁(𝐵)+^𝐵_𝑠𝑔₁(𝑟)𝑢𝑑𝑟𝑑𝑠,₂(𝐵)−𝑢₂(𝑎)=^𝐵_𝑎𝜑_𝑝⁻¹𝜑_𝑝𝑢^′₂(𝐵)+^𝐵_𝑠𝑔₂(𝑟)𝑑𝑟𝑑𝑠. (2.14) Thus, we obtain 0<𝑚=𝑢₁(𝐵)−𝑢₂(𝐵)≤0, (2.15) which is a contradiction. This completes the proof.

Let us define 𝐺𝑔(𝑡)=∫^𝑡₀𝜑_𝑝⁻¹𝛼+∫_𝑠^1/2𝑔(𝑟)1𝑑𝑟𝑑𝑠,0≤𝑡≤2,∫¹_𝑡𝜑_𝑝⁻¹−𝛼+∫^𝑠_1/2𝑔(𝑟)1𝑑𝑟𝑑𝑠,2≤𝑡≤1. (2.16) Then, problem (𝐴) can be equivalently written as 𝑢=𝐺(𝑔).

3. Application: Compactness

In this and the following sections, as an application of new solution operator, we consider a problem of one-dimensional 𝑝-Laplacian with a singular weight as follows: 𝜑_𝑝𝑢^′′(𝑡)𝑓(𝑢(𝑡(),𝑢(0)(1),+𝜆ℎ))=0,𝑡∈0,1=0=𝑢 𝑃_𝜆 where 𝜆>0 a parameter, ℎ∈ℋ, and 𝑓∈𝐶(ℝ,ℝ). Note that we do not need any other assumptions on 𝑓 except the continuity in this section. Employing the solution operator 𝐺 established in Section 2, we may rewrite 𝑃_𝜆 equivalently as 𝑢=𝐺(𝜆,𝑢), (3.1) where 𝐺∶(0,∞)×𝐶[0,1]→𝐶[0,1] is defined by 𝐺(𝑡)≜∫𝜆,𝑢)(^𝑡0𝜑_𝑝⁻¹𝛼𝜆ℎ𝑓(𝑢)+∫_𝑠^1/2(𝑟)𝑓(𝑢(𝑟1𝜆ℎ))𝑑𝑟𝑑𝑠,0≤𝑡≤2,∫¹_𝑡𝜑_𝑝⁻¹−𝛼𝜆ℎ𝑓(𝑢)+∫^𝑠_1/2(𝑟)𝑓(𝑢(𝑟1𝜆ℎ))𝑑𝑟𝑑𝑠,2≤𝑡≤1. (3.2) The purpose of this section is to show that 𝐺 is completely continuous on (0,∞)×𝐶[0,1]. For this, we need to know the properties of function 𝛼∶(0,∞)×𝐶[0,1]→ℝ.

Lemma 3.1. 𝛼 sends bounded sets in (0,∞)×𝐶[0,1] into bounded sets in ℝ.

Proof. Assume that {𝜆_𝑛} and {𝑢_𝑛} are bounded on (0,∞) and 𝐶[0,1], respectively. For convenience, we denote 𝛼_𝑛=𝛼(𝜆_𝑛ℎ𝑓(𝑢_𝑛)). Suppose that {𝛼_𝑛} is unbounded in ℝ. Then, there is a subsequence {𝛼_𝑛𝑘} such that 𝛼_𝑛𝑘→∞ or 𝛼_𝑛𝑘→−∞ as 𝑘→∞. Here, we consider the first case 𝛼_𝑛𝑘→∞. We can prove the second case by the same argument. As in (2.2) of Lemma 2.2, we have ₀^1/2𝜑_𝑝⁻¹𝛼_𝑛𝑘+_𝑠^1/2𝜆_𝑛𝑘ℎ(𝑟)𝑓𝑢_𝑛𝑘(𝑟)=𝑑𝑟𝑑𝑠_11/2𝜑_𝑝⁻¹−𝛼_𝑛𝑘+^𝑠_1/2𝜆_𝑛𝑘ℎ(𝑟)𝑓𝑢_𝑛𝑘(𝑟)𝑑𝑟𝑑𝑠. (3.3) As in the proof of Lemma 2.2, we get the following inequality for the left-hand side 0≤₀^1/2𝜑_𝑝⁻¹𝛼_𝑛𝑘−𝑀_𝑠^1/2|ℎ(𝑟≤)|𝑑𝑟𝑑𝑠_01/2𝜑_𝑝⁻¹𝛼_𝑛𝑘+_𝑠^1/2𝜆_𝑛𝑘ℎ(𝑟)𝑓𝑢_𝑛𝑘(𝑟)𝑑𝑟𝑑𝑠, (3.4) for sufficiently large 𝑘, where 𝑀=sup{|𝜆_𝑛𝑘|‖𝑓(𝑢_𝑛𝑘)‖_∞}. On other hand, as in the proof of Lemma 2.2, we get the following inequality for the right-hand side ¹_1/2𝜑_𝑝⁻¹−𝛼_𝑛𝑘+^𝑠_1/2𝜆_𝑛𝑘ℎ(𝑟)𝑓𝑢_𝑛𝑘(𝑟)≤𝑑𝑟𝑑𝑠_11/2𝜑_𝑝⁻¹−𝛼_𝑛𝑘+𝑀^𝑠_1/2|ℎ(𝑟)|𝑑𝑟𝑑𝑠<0, (3.5) for sufficiently large 𝑘, and this is a contradiction.

Lemma 3.2. 𝛼 is continuous on (0,∞)×𝐶[0,1].

Proof. Assume that 𝜆_𝑛→𝜆 in ℝ and 𝑢_𝑛→𝑢 in 𝐶[0,1]. Then, there exists a constant 𝑀>0 such that 𝜆_𝑛+𝑢_𝑛∞≤𝑀,∀𝑛. (3.6) Lemma 3.1 implies that {𝛼(𝜆_𝑛ℎ𝑓(𝑢_𝑛))} is bounded in ℝ. Hence, it has a convergent subsequence denoted by {𝛼(𝜆_𝑛𝑘ℎ𝑓(𝑢_𝑛𝑘))}. Let 𝛼𝜆_𝑛𝑘ℎ𝑓𝑢_𝑛𝑘⟶𝛼 (3.7) as 𝑘→∞. As in (2.2) of Lemma 2.2, we have ₀^1/2𝜑_𝑝⁻¹𝛼𝜆_𝑛𝑘ℎ𝑓𝑢_𝑛𝑘+_𝑠^1/2𝜆_𝑛𝑘ℎ(𝑟)𝑓𝑢_𝑛𝑘(𝑟)=𝑑𝑟𝑑𝑠_11/2𝜑_𝑝⁻¹−𝛼𝜆_𝑛𝑘ℎ𝑓𝑢_𝑛𝑘+^𝑠_1/2𝜆_𝑛𝑘ℎ(𝑟)𝑓𝑢_𝑛𝑘(𝑟)𝑑𝑟𝑑𝑠. (3.8) By Lebesgue dominated convergence theorem, as 𝑘→∞, we get ₀^1/2𝜑_𝑝⁻¹𝛼+_𝑠^1/2𝜆ℎ(𝑟)𝑓(𝑢(𝑟))𝑑𝑟𝑑𝑠=¹_1/2𝜑_𝑝⁻¹−𝛼+^𝑠_1/2𝜆ℎ(𝑟)𝑓(𝑢(𝑟))𝑑𝑟𝑑𝑠. (3.9) By the uniqueness of 𝛼, we obtain 𝛼=𝛼(𝜆ℎ𝑓(𝑢)), and this proves the continuity of 𝛼.

We have the following corollary using similar argument in the proof of Lemma 3.2.

Corollary 3.3. Let 𝜆_𝑛∈(0,∞) satisfy 𝜆_𝑛→0 as 𝑛→∞. Then, 𝛼(𝜆_𝑛)→0 as 𝑛→∞.

Theorem 3.4. 𝐺 is continuous and compact on (0,∞)×𝐶[0,1].

Proof. First, we show the continuity of 𝐺. Assume that 𝜆_𝑛→𝜆 on (0,∞) and 𝑢_𝑛→𝑢 in 𝐶[0,1]. Then, from the continuity of 𝛼 and Lebesgue dominated convergence theorem, we have𝐺𝜆_𝑛,𝑢_𝑛−𝐺(𝜆,𝑢)_∞=sup_0≤𝑡≤1𝐺𝜆_𝑛,𝑢_𝑛(𝑡)−𝐺(𝜆,𝑢)(𝑡)≤sup_{0≤𝑡≤1/2}^𝑡₀𝜑_𝑝⁻¹𝛼𝜆_𝑛ℎ𝑓𝑢_𝑛+_𝑠^1/2𝜆_𝑛ℎ(𝑟)𝑓𝑢_𝑛(𝑟)𝑑𝑟−𝜑_𝑝⁻¹𝛼𝜆ℎ𝑓(𝑢)+_𝑠^1/2𝜆ℎ(𝑟)𝑓(𝑢(𝑟))𝑑𝑟𝑑𝑠+sup_{1/2≤𝑡≤1}¹_𝑡𝜑_𝑝⁻¹𝛼𝜆_𝑛ℎ𝑓𝑢_𝑛+^𝑠_1/2𝜆_𝑛ℎ(𝑟)𝑓𝑢_𝑛(𝑟)𝑑𝑟−𝜑_𝑝⁻¹𝛼𝜆ℎ𝑓(𝑢)+^𝑠_1/2𝜆ℎ(𝑟)𝑓(𝑢(𝑟))𝑑𝑟𝑑𝑠⟶0. (3.10)
Next, we show that 𝐺 is uniformly bounded. Let ℳ be a bounded subset of (0,∞)×𝐶[0,1], and let (𝜆,𝑢)∈ℳ, then using the fact that 𝛼 sends bounded sets on (0,∞)×𝐶[0,1] into bounded sets on ℝ and taking 𝑀=sup{|𝜆|‖𝑓(𝑢)‖_∞∶(𝜆,𝑢)∈ℳ}, we obtain |𝐺(𝑡≤∫𝜆,𝑢)()|^𝑡0𝜑_𝑝⁻¹𝛼𝜆ℎ𝑓(𝑢)+𝑀∫_𝑠^1/2|ℎ(𝑟1)|𝑑𝑟𝑑𝑠,0≤𝑡≤2,∫¹_𝑡𝜑_𝑝⁻¹𝛼𝜆ℎ𝑓(𝑢)+𝑀∫^𝑠_1/2|ℎ(𝑟1)|𝑑𝑟𝑑𝑠,2≤∫≤𝑡≤1,_01/2𝜑_𝑝⁻¹|𝐾(𝑀,ℎ)|+𝑀∫_𝑠^1/2|ℎ(𝑟1)|𝑑𝑟𝑑𝑠<∞,0≤𝑡≤2,∫¹_1/2𝜑_𝑝⁻¹|𝐾(𝑀,ℎ)|+𝑀∫^𝑠_1/2|ℎ(𝑟1)|𝑑𝑟𝑑𝑠<∞,2≤𝑡≤1, (3.11) where 𝐾(𝑀,ℎ) is a constant and the above upper bound is independent of (𝜆,𝑢)∈ℳ.
Finally, we show that 𝐺(ℳ) is equicontinuous. By Lemma 3.1, {𝛼(𝜆ℎ𝑓(𝑢))∶(𝜆,𝑢)∈ℳ} is bounded so by taking 𝑁=sup{𝜑_𝑝⁻¹(|𝛼(𝜆ℎ𝑓(𝑢))|)∶(𝜆,𝑢)∈ℳ}, we obtain for 0≤𝑡₁,𝑡₂≤1/2, 𝐺(𝜆,𝑢)𝑡₁−𝐺(𝜆,𝑢)𝑡₂≤^𝑡2_𝑡1𝜑_𝑝⁻¹𝛼𝜆ℎ𝑓(𝑢)+_𝑠^1/2𝜆ℎ(𝑟)𝑓(𝑢(𝑟))𝑑𝑟𝑑𝑠≤𝐶_𝑝𝑁𝑡₂−𝑡₁+𝐶_𝑝𝑀^𝑡2_𝑡1𝜑_𝑝⁻¹_𝑠^1/2|ℎ(𝑟)|𝑑𝑟𝑑𝑠. (3.12) Since ℎ∈ℋ, the above upper bound tends to 0 as |𝑡₁−𝑡₂| does. For case 1/2≤𝑡₁, 𝑡₂≤1, we may handle similar way. For case 0≤𝑡₁≤1/2, 1/2≤𝑡₂≤1 (or case 0≤𝑡₂≤1/2,1/2≤𝑡₁≤1), we can compute the following: 𝐺(𝜆,𝑢)𝑡₁−𝐺(𝜆,𝑢)𝑡₂=^𝑡1₀𝜑_𝑝⁻¹𝛼𝜆ℎ𝑓(𝑢)+_𝑠^1/2(𝑟)𝑓(𝑢(𝑟−𝜆ℎ))𝑑𝑟𝑑𝑠_1𝑡2𝜑_𝑝⁻¹−𝛼𝜆ℎ𝑓(𝑢)+^𝑠_1/2(𝑟)𝑓(𝑢(𝑟=𝜆ℎ))𝑑𝑟𝑑𝑠_01/2𝜑_𝑝⁻¹𝛼𝜆ℎ𝑓(𝑢)+_𝑠^1/2(𝑟)𝑓(𝑢(𝑟−𝜆ℎ))𝑑𝑟𝑑𝑠_𝑡1/21𝜑_𝑝⁻¹𝛼𝜆ℎ𝑓(𝑢)+_𝑠^1/2(𝑟)𝑓(𝑢(𝑟−𝜆ℎ))𝑑𝑟𝑑𝑠_11/2𝜑_𝑝⁻¹−𝛼𝜆ℎ𝑓(𝑢)+^𝑠_1/2(𝑟)𝑓(𝑢(𝑟−𝜆ℎ))𝑑𝑟𝑑𝑠_𝑡21/2𝜑_𝑝⁻¹−𝛼𝜆ℎ𝑓(𝑢)+^𝑠_1/2(𝑟)𝑓(𝑢(𝑟=𝜆ℎ))𝑑𝑟𝑑𝑠_𝑡21/2𝜑_𝑝⁻¹−𝛼𝜆ℎ𝑓(𝑢)+^𝑠_1/2(𝑟)𝑓(𝑢(𝑟−𝜆ℎ))𝑑𝑟𝑑𝑠_𝑡1/21𝜑_𝑝⁻¹𝛼𝜆ℎ𝑓(𝑢)+_𝑠^1/2𝜆ℎ(𝑟)𝑓(𝑢(𝑟))𝑑𝑟𝑑𝑠≤2𝐶_𝑝^𝑡2_𝑡1𝜑_𝑝⁻¹𝛼𝜆ℎ𝑓(𝑢)+𝑀₁𝜑_𝑝^−1𝑠_1/2|ℎ(𝑟)|𝑑𝑟+𝜑_𝑝⁻¹_𝑠^1/2|ℎ(𝑟)|𝑑𝑟𝑑𝑠≤2𝐶_𝑝𝑁𝑡₂−𝑡₁+2𝐶_𝑝𝑀₁^𝑡2_𝑡1𝜑_𝑝^−1𝑠_1/2|ℎ(𝑟)|𝑑𝑟+𝜑_𝑝⁻¹_𝑠^1/2|ℎ(𝑟)|𝑑𝑟𝑑𝑠, (3.13) where 𝑀₁=𝜑_𝑝⁻¹(𝑀). Thus, by Arzela-Ascoli Theorem, 𝐺 is compact, and the proof is complete.

4. Application: Existence

In this section, we show the existence of nontrivial solutions for problem 𝑃_𝜆: 𝜑_𝑝𝑢^′′(𝑡)𝑓(𝑢(𝑡(),𝑢(0)(1),+𝜆ℎ))=0,𝑡∈0,1=0=𝑢 𝑃_𝜆 where 𝜆>0 a parameter, ℎ∈ℋ, and 𝑓∈𝐶(ℝ,ℝ). In addition, we assume that (F)𝑓(0)>0 and 𝑓_∞≡lim_𝑠→∞𝑓(𝑠)/𝑠^𝑝−1=0.

Based on the solution operator and its complete continuity established in Section 3, we show the existence of nontrivial solutions of 𝑃_𝜆 for all 𝜆>0. We will employ the following theorem the so-called global continuation theorem for the proof of our existence theorem.

Let 𝑋 be a Banach space and 𝐺∶ℝ×𝑋→ℝ completely continuous with 𝐺(0,𝑢)=0, and consider 𝑢=𝐺(𝜆,𝑢). (4.1) We denote 𝒮 as the set of solutions for (4.1), ℝ⁺=[0,∞) and ℝ⁻=(−∞,0].

Theorem 4.1 (see [5, Theorem 3.2]). Let 𝑋 be a Banach space and 𝐺∶ℝ×𝑋→𝑋 continuous and compact and 𝐺(0,𝑢)=0. Then, 𝒮 contains a pair of unbounded components 𝒞⁺,𝒞⁻ in ℝ⁺×𝑋 and ℝ⁻×𝑋, respectively, and 𝒞⁺∩𝒞⁻={(0,0)}.

We now state our main existence theorem.

Theorem 4.2. Assume that ℎ∈ℋ and 𝑓 holds (𝐹). Then 𝑃_𝜆 has non-trivial solutions for all 𝜆>0.

Let us take 𝑋=𝐶[0,1] with the norm ‖⋅‖_∞. We know that, to solve 𝑃_𝜆, it is enough to solve 𝑢=𝐺(𝜆,𝑢), (4.2) where 𝐺 is defined in Section 3. And we have shown that 𝐺 is continuous and compact in Section 3. Since 𝐺(0,𝑢)=0 is obvious, employing Theorem 4.1, we obtain an unbounded continuum 𝒞⁺. If we provide a priori boundedness of solutions for 𝑃_𝜆, then the unbounded continuum guarantees the existence of solutions for all 𝜆>0. Therefore, to complete the proof of Theorem 4.2, we need to give a priori estimate of solutions for 𝑃_𝜆.

Lemma 4.3. Assume that ℎ∈ℋ and 𝑓 holds (𝐹). Let (𝜆,𝑢) be a solution for 𝑃_𝜆 with 𝜆∈(0,Λ]. Then, there exists a constant 𝐶(Λ) which depends only on Λ such that ‖𝑢‖_∞≤𝐶(Λ).

Proof. Let us define a function 𝐾∶(0,∞)×ℝ→ℝ which satisfies the following property: 𝐾(𝜆_𝑛,ℎ)→0 as 𝜆_𝑛→0. Since ℎ∈ℋ, for given Λ>0, there exists 𝜀>0 such that Λ^𝑝−1max₀^1/2𝜑_𝑝⁻¹|𝐾(𝜀,ℎ)|+𝜀_𝑠^1/2|ℎ(𝑟)|𝑑𝑟𝑑𝑠,_11/2𝜑_𝑝⁻¹|𝐾(𝜀,ℎ)|+𝜀^𝑠_1/2|ℎ(𝑟<1)|𝑑𝑟𝑑𝑠2. (4.3) Then, since 𝑓_∞=0, for 𝜀 given above, there exists 𝑁>0 such that for all 𝑠 with |𝑠|≥𝑁, 𝑓(𝑠)≤𝜀|𝑠|^𝑝−1. (4.4) Since 𝑓 is continuous on [−𝑁,𝑁], for 𝑁 given above, there exists a maximum value 𝑀>0 for |𝑓| on [−𝑁,𝑁], that is, for all 𝑠 with |𝑠|≤𝑁, we have |𝑓(𝑠)|≤𝑀. Let us divide the interval [0,1] into 𝐼 and 𝐽 as {[]∶|𝑢(𝑡},{[]∶|𝑢(𝑡},𝐼∶=𝑡∈0,1)|≥𝑁𝐽∶=𝑡∈0,1)|≤𝑁 (4.5) respectively. Then, we can estimate solution 𝑢 as follows: |𝑢(𝑡≤∫)|^𝑡0𝜑_𝑝⁻¹𝛼𝜆ℎ𝑓(𝑢)+∫_𝑠^1/2𝜆ℎ(𝑟)𝑓(𝑢(𝑟1))𝑑𝑟𝑑𝑠,0≤𝑡≤2,∫¹_𝑡𝜑_𝑝⁻¹𝛼𝜆ℎ𝑓(𝑢)+∫^𝑠_1/2𝜆ℎ(𝑟)𝑓(𝑢(𝑟1))𝑑𝑟𝑑𝑠,2=≤𝑡≤1,_𝐼+_𝐽, (4.6) where ∫_Γ with Γ=𝐼 or 𝐽 can be given as _Γ=∫_Γ𝜑_𝑝⁻¹𝛼𝜆ℎ𝑓(𝑢)+∫_𝑠^1/2𝜆ℎ(𝑟)𝑓(𝑢(𝑟1))𝑑𝑟𝑑𝑠,0≤𝑡≤2,∫_Γ𝜑_𝑝⁻¹𝛼𝜆ℎ𝑓(𝑢)+∫^𝑠_1/2𝜆ℎ(𝑟)𝑓(𝑢(𝑟1))𝑑𝑟𝑑𝑠,2≤𝑡≤1. (4.7) By Lemma 3.1, if |𝜆𝑓(𝑢)|≤𝑀 for some 𝑀>0, then there exists a constant 𝐾(𝑀,ℎ)>0 such that |𝛼(𝜆ℎ𝑓(𝑢))|≤𝐾(𝑀,ℎ). Thus, on 𝐽, we have _𝐽=∫_𝐽𝜑_𝑝⁻¹𝛼𝜆ℎ𝑓(𝑢)+∫_𝑠^1/2𝜆ℎ(𝑟)𝑓(𝑢(𝑟1))𝑑𝑟𝑑𝑠,0≤𝑡≤2,∫_𝐽𝜑_𝑝⁻¹𝛼𝜆ℎ𝑓(𝑢)+∫^𝑠_1/2𝜆ℎ(𝑟)𝑓(𝑢(𝑟1))𝑑𝑟𝑑𝑠,2≤∫≤𝑡≤1,_𝐽𝜑_𝑝⁻¹|𝐾(Λ𝑀,ℎ)|+Λ𝑀∫_𝑠^1/2|ℎ(𝑟1)|𝑑𝑟𝑑𝑠,0≤𝑡≤2,∫_𝐽𝜑_𝑝⁻¹|𝐾(Λ𝑀,ℎ)|+Λ𝑀∫^𝑠_1/2|ℎ(𝑟1)|𝑑𝑟𝑑𝑠,2≤∫≤𝑡≤1,_01/2𝜑_𝑝⁻¹|𝐾(Λ𝑀,ℎ)|+Λ𝑀∫_𝑠^1/2|ℎ(𝑟1)|𝑑𝑟𝑑𝑠,0≤𝑡≤2,∫₀^1/2𝜑_𝑝⁻¹|𝐾(Λ𝑀,ℎ)|+Λ𝑀∫^𝑠_1/2|ℎ(𝑟1)|𝑑𝑟𝑑𝑠,2≤𝑡≤1,≤𝐾𝐶_Λ, (4.8) where 𝐶_Λ=max{|𝐾(Λ𝑀,ℎ)|,Λ𝑀} and 𝐾=max₀^1/2𝜑_𝑝⁻¹1+_𝑠^1/2|ℎ(𝑟)|𝑑𝑟𝑑𝑠,¹_1/2𝜑_𝑝⁻¹1+^𝑠_1/2|ℎ(𝑟)|𝑑𝑟𝑑𝑠. (4.9) On 𝐼, using Corollaries 2.4 and 3.3, and 𝜆/Λ<1, we obtain _𝐼≤∫_𝐼𝜑_𝑝⁻¹𝛼𝜆Λℎ𝑓(𝑢)‖𝑢‖_∞^𝑝−1+∫_𝑠^1/2𝜆Λ|ℎ(𝑟𝑓(𝑢))|‖𝑢‖_∞^𝑝−1𝑑𝑟𝑑𝑠Λ^𝑝−1‖𝑢‖_∞1,0≤𝑡≤2,∫_𝐼𝜑_𝑝⁻¹𝛼𝜆Λℎ𝑓(𝑢)‖𝑢‖_∞^𝑝−1+∫^𝑠_1/2𝜆Λ|ℎ(𝑟𝑓(𝑢))|‖𝑢‖_∞^𝑝−1𝑑𝑟𝑑𝑠Λ^𝑝−1‖𝑢‖_∞,12≤∫≤𝑡≤1,_𝐼𝜑_𝑝⁻¹𝐾(𝜀,ℎ)+𝜀∫_𝑠^1/2|ℎ(𝑟)|𝑑𝑟𝑑𝑠Λ^𝑝−1‖𝑢‖_∞1,0≤𝑡≤2,∫_𝐼𝜑_𝑝⁻¹𝐾(𝜀,ℎ)+𝜀∫^𝑠_1/2|ℎ(𝑟)|𝑑𝑟𝑑𝑠Λ^𝑝−1‖𝑢‖_∞,12≤∫≤𝑡≤1,_01/2𝜑_𝑝⁻¹𝐾(𝜀,ℎ)+𝜀∫_𝑠^1/2|ℎ(𝑟)|𝑑𝑟𝑑𝑠Λ^𝑝−1‖𝑢‖_∞1,0≤𝑡≤2,∫¹_1/2𝜑_𝑝⁻¹𝐾(𝜀,ℎ)+𝜀∫^𝑠_1/2|ℎ(𝑟)|𝑑𝑟𝑑𝑠Λ^𝑝−1‖𝑢‖_∞,12≤1≤𝑡≤1,2‖𝑢‖_∞. (4.10) Therefore, we have |𝑢(𝑡)|≤𝐾𝐶_Λ+12‖𝑢‖_∞, (4.11) so that ‖𝑢‖_∞≤2𝐾𝐶_Λ, (4.12) and this completes the proof.

Acknowledgment

The authors express their thanks to Professor Yuki Naito for valuable suggestions. The first author was supported by the 2010 Research Fund of the University of Ulsan.