Existence of Positive Solutions for Neumann Boundary Value Problem with a Variable Coefficient

Abstract

We consider the existence of positive solutions for the Neumann boundary value problem 𝑥^′′(𝑡)+𝑚²(𝑡)𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡))+𝑒(𝑡),𝑡∈(0,1),𝑥^′(0)=0,𝑥^′(1)=0, where 𝑚∈𝐶([0,1],(0,+∞)),𝑒∈𝐶[0,1], and 𝑓∶[0,1]×(0,+∞)→[0,+∞) is continuous. The theorem obtained is very general and complements previous known results.

1. Introduction

The existence of solutions of Neumann boundary value problem of second-order ordinary differential equations has been studied by many authors; see Sun et al. [1], Cabada and Pouso [2], Cabada et al. [3], Canada et al. [4], Chu et al. [5], Jiang, and Liu [6], Yazidi [7], Sun and Li [8] and the references therein.

Recently, Chu et al. [5] have studied the existence of positive solution to the Neumann boundary value problem 𝑥^′′(𝑡)+𝑚²𝑥(𝑡)((𝑡(𝑡)(),𝑥=𝑓𝑡,𝑥))+𝑒,𝑡∈0,1_′(0)=0,𝑥^′(1)=0, (1.1) where 𝑚∈(0,𝜋/2) is a constant, 𝑒∈𝐶[0,1] and nonlinearity 𝑓(𝑡,𝑥) may be singular at 𝑥=0. Their approach was based upon the nonlinear alternative principle of Leray-Schauder and Green's function, 𝐺₁(𝑡,𝑠), of the associated linear problem 𝑥^′′(𝑡)+𝑚²𝑥(𝑡)(),𝑥=0,𝑡∈0,1_′(0)=0,𝑥^′(1)=0. (1.2) Notice that Green's function 𝐺₁(𝑡,𝑠) can be explicitly expressed by 𝐺₁()=()𝑡,𝑠cos𝑚1−𝑠cos𝑚𝑡()𝑚sin𝑚,0≤𝑡≤𝑠≤1,cos𝑚1−𝑡cos𝑚𝑠𝑚sin𝑚,0≤𝑠≤𝑡≤1. (1.3)

In this paper, we will consider the more general problem 𝑥^′′(𝑡)+𝑚²(𝑡)𝑥(𝑡)((𝑡(𝑡)(),𝑥=𝑓𝑡,𝑥))+𝑒,𝑡∈0,1_′(0)=0,𝑥^′(1)=0, (1.4) where 𝑚∈𝐶([0,1],(0,+∞)),𝑒∈𝐶[0,1], and 𝑓∶[0,1]×(0,+∞)→[0,+∞) is continuous.

Of course, the natural question is what would happen when the constant 𝑚 in (1.1) is replaced with a function 𝑚(𝑡)? Obviously, Green's function of the associated linear problem 𝑥^′′(𝑡)+𝑚²(𝑡)𝑥(𝑡)(),𝑥=0,𝑡∈0,1_′(0)=0,𝑥^′(1)=0 (1.5) cannot be explicitly expressed by elementary functions! The primary contribution of this paper is to construct Green's function associated with the Neumann boundary value problem with a variable coefficient (1.5) and study the properties of the Green's function. We apply the Krasnoselskii and Guo fixed point theorem as an application. This application was first made by Erbe and Wang [9] to ordinary differential equations. Since that time, there has been a tremendous amount of work to study the existence of positive solutions to BVPs for ordinary differential equations. Once we obtain Theorem 2.2, many of those applications would work here as well.

The rest of the paper is organized as follows: Section 2 is devoted to constructing Green's function and proves some preliminary results. In Section 3, we state and prove our main results. In Section 4, an example illustrates the applicability of the main existence result.

2. Preliminaries and Lemmas

Let us fix some notation to be used. Given 𝜑∈𝐿¹[0,1], we write 𝜑≻0 if 𝜑≥0 for a.e. 𝑡∈[0,1], and it is positive in a set of positive measure. Let us denote by 𝑝^∗ and 𝑝_∗ the essential supremum and infimum of a given function 𝑝∈𝐿¹[0,1] if they exist. To study the boundary value problem (1.4), we need restriction on 𝑚(𝑡)(𝐻₀)𝑚∈𝐶[0,1],and0<𝑚∶=inf_𝑡∈[0,1]𝑚(𝑡),𝑚∶=sup_𝑡∈[0,1]𝑚(𝑡)<𝜋/2.

To rewrite (1.4) to an equivalent integral equation, we need to construct Green's function of the corresponding linear problem. To do this, we need the following.

Lemma 2.1. Let (𝐻₀) hold. Suppose 𝜑 and 𝜓 be the solution of the linear problems 𝜑^′′(𝑡)+𝑚²(𝑡)𝜑(𝑡)[],𝜑=0,𝑡∈0,1_′(0)(0)𝜓=0,𝜑=1,_′′(𝑡)+𝑚²(𝑡)𝜓(𝑡)[],𝜓=0,𝑡∈0,1_′(1)=0,𝜓(1)=1, (2.1) respectively. Then(i)𝜑(𝑡)>0 on [0,1], and 𝜑^′(𝑡)<0 on (0,1];(ii)(ii) 𝜓(𝑡)>0 on [0,1], and 𝜓^′(𝑡)>0 on [0,1).

Proof. We will give a proof for (i) only. The proof of (ii) follows in a similar manner.
It is easy to see that the problem 𝑥^′′(𝑡)+𝑚²𝑥(𝑡)[],𝑥=0,𝑡∈0,1_′(0)=0,𝑥(0)=1 (2.2) has the unique solution 𝑥(𝑡)=cos𝑚𝑡 and 𝑡∈[0,1]. From (𝐻₀), we know that cos𝑚𝑡>0,𝑡∈[0,1]. (2.3) On the other hand, for all 𝑡∈[0,1], we have (cos𝑚𝑡)^′′+𝑚²(𝑡)cos𝑚𝑡=−𝑚²cos𝑚𝑡+𝑚²(𝑡)cos𝑚𝑡=𝑚²(𝑡)−𝑚²cos𝑚𝑡≤0. (2.4) By using comparison theorem (see [10]), we obtain cos𝑚𝑡≤𝜑(𝑡),𝑡∈[0,1]. (2.5) Therefore, we have from (2.3) and (2.5) that 0<cos𝑚𝑡≤𝜑(𝑡),𝑡∈[0,1]. (2.6) Thus 𝜑^′′(𝑡)=−𝑚²(𝑡)𝜑(𝑡)<0,𝑡∈[0,1]. (2.7) From the fact 𝜑^′(0)=0 and (2.7), we obtain 𝜑^′(𝑡)<0 on (0,1].
Now, let 𝐺()=1𝑡,𝑠𝜓^′(0)𝜓(𝑡)𝜑(𝑠)𝜓(𝑠)𝜑(𝑡),0≤𝑠≤𝑡≤1,,0≤𝑡≤𝑠≤1. (2.8)

Theorem 2.2. Let (𝐻₀) hold. Then for any 𝑦∈𝐶[0,1], the problem 𝑥^′′(𝑡)+𝑚²(𝑡)𝑥(𝑡)(𝑡)(),𝑥=𝑦,𝑡∈0,1_′(0)=0,𝑥^′(1)=0 (2.9) is equivalent to the integral equation 𝑥(𝑡)=¹₀𝐺(𝑡,𝑠)𝑦(𝑠)d𝑠. (2.10)

Proof. First we show that the unique solution of (2.9) can be represented by (2.10).
In fact, we know that the equation𝑥^′′(𝑡)+𝑚²(𝑡)𝑥(𝑡)=0,𝑡∈[0,1] (2.11) has known two linear independent solutions 𝜑 and 𝜓 since _𝜑^{𝜑(0)𝜓(0)}_{′(0)𝜓′(0)}=𝜓^′(0)≠0.
Now by the method of variation of constants, we can obtain that the unique solution of the problem (2.9) can be represented by𝑥(𝑡)=¹₀𝐺(𝑡,𝑠)𝑦(𝑠)d𝑠, (2.12) where 𝐺(𝑡,𝑠) is as (2.8).
Next we check that the function defined by (2.10) is a solution of (2.9).
From (2.10), we know that𝑥(𝑡)=^𝑡₀𝜓(𝑡)𝜑(𝑠)𝜓^′(0)𝑦(𝑠)d𝑠+¹_𝑡𝜓(𝑠)𝜑(𝑡)𝜓^′(0)𝑦(𝑠)𝑥d𝑠,_′(𝑡)=𝜓^′(𝑡)^𝑡₀𝜑(𝑠)𝜓^′(0)𝑦(𝑠)d𝑠+𝜑^′(𝑡)¹_𝑡𝜓(𝑠)𝜓^′(0)𝑦(𝑠)𝑥d𝑠,_′′(𝑡)=𝜓^′′(𝑡)^𝑡₀𝜑(𝑠)𝜓^′(0)𝑦(𝑠)d𝑠+𝜑^′′(𝑡)¹_𝑡𝜓(𝑠)𝜓^′(0)𝑦(𝑠)𝜓d𝑠+^′(𝑡)𝜑(𝑡)𝜓^′(0)𝑦(𝑡)−𝜑^′(𝑡)𝜓(𝑡)𝜓^′(0)𝑦(𝑡). (2.13) So that 𝑥^′′(𝑡)+𝑚²(𝑡)𝑥(𝑡)=1𝜓^′(0)𝜑(𝑡)𝜓(𝑡)𝜑^′(𝑡)𝜓^′(𝑡)𝑦(𝑡)=1𝜓^′(0)𝜑(0)𝜓(0)𝜑^′(0)𝜓^′(0)𝑦(𝑡)=𝑦(𝑡). (2.14) Finally, it is easy to see that 𝑥^′(0)=𝜑^′(0)¹₀𝜓(𝑠)𝜓^′(0)𝑦(𝑠)d𝑠=0,𝑥^′(1)=𝜓^′(1)¹₀𝜑(𝑠)𝜓^′(0)𝑦(𝑠)d𝑠=0. (2.15)

From Lemma 2.1, we know that 𝐺(𝑡,𝑠)>0,∀𝑡,𝑠∈[0,1]. (2.16) Let 𝐴=min_{0≤𝑡,𝑠≤1}𝐺(𝑡,𝑠),𝐵=max_{0≤𝑡,𝑠≤1}𝐺(𝑡,𝑠),𝜎=𝐴/𝐵. Then 𝐵>𝐴>0 and 0<𝜎<1.

In order to prove the main result of this paper, we need the following fixed-point theorem of cone expansion-compression type due to Krasnoselskii's (see [11]).

Theorem 2.3. Let 𝐸 be a Banach space, and 𝐾⊂𝐸 is a cone in 𝐸. Assume that Ω₁ and Ω₂ are open subsets of 𝐸 with 𝜃∈Ω₁ and Ω₁⊂Ω₂. Let 𝑇∶𝐾∩(Ω₂⧵Ω₁)→𝐾 be a completely continuous operator. In addition, suppose that either (i)‖𝑇𝑢‖≤‖𝑢‖,∀𝑢∈𝐾∩𝜕Ω₁ and ‖𝑇𝑢‖≥‖𝑢‖,∀𝑢∈𝐾∩𝜕Ω₂ or(ii)‖𝑇𝑢‖≥‖𝑢‖,∀𝑢∈𝐾∩𝜕Ω₁ and ‖𝑇𝑢‖≤‖𝑢‖,∀𝑢∈𝐾∩𝜕Ω₂ holds.Then 𝑇 has a fixed point in 𝐾∩(Ω₂⧵Ω₁).

3. Main Results

In this section, we state and prove the main results of this paper.

Let us define the function 𝛾(𝑡)=¹₀𝐺(𝑡,𝑠)𝑒(𝑠)d𝑠, (3.1) which is just the unique solution of the linear problem (2.9) with 𝑦(𝑡)=𝑒(𝑡). For our constructions, let 𝐸=𝐶[0,1], with norm ‖𝑥‖=sup_0≤𝑡≤1|𝑥(𝑡)|. Define a cone 𝑃, by 𝑃=𝑥∈𝐸∣𝑥(𝑡)≥0on[0,1],andmin_0≤𝑡≤1𝑥(𝑡)≥𝜎‖𝑥‖. (3.2)

Theorem 3.1. Let (𝐻₀) hold. Suppose that there exist a constant 𝑟>0 such that (𝐻₁)there exist continuous, nonnegative functions 𝑔,ℎ, and 𝑘, such that 0≤𝑓(𝑡,𝑥)≤𝑘(𝑡)𝑔(𝑥)+ℎ(𝑥)∀(𝑡,𝑥)∈[0,1]×(0,𝑟], (3.3)𝑔(𝑥)>0 is nonincreasing, and ℎ(𝑥)/𝑔(𝑥)is nondecreasing in 𝑥∈(0,𝑟];(𝐻₂)𝑟−𝛾^∗/(𝑔(𝜎𝑟)(1+ℎ(𝑟)/𝑔(𝑟)))>𝐾^∗, here 𝐾(𝑡)=∫¹₀𝐺(𝑡,𝑠)𝑘(𝑠)d𝑠;(𝐻₃)there exists a continuous function 𝜙_𝑟≻0 such that 𝑓(𝑡,𝑥)≥𝜙_𝑟(𝑡)∀(𝑡,𝑥)∈[0,1]×(0,𝑟]; (3.4)(𝐻₄)𝜙_𝑟(𝑡)+𝑒(𝑡)≻0forall𝑡∈[0,1]. Then problem (1.4) has at least one positive solution 𝑥 with 0<‖𝑥‖<𝑟.

Remark 3.2. When 𝑚(𝑡)≡𝑚,𝑡∈[0,1], then (1.4) reduces to (1.1), (𝐻₀) reduce to 𝑚∈(0,𝜋/2). So Theorem 3.1 is more extensive than [5, Theorem  3.1].

Proof of Theorem 3.1. Let 𝛿=min_0≤𝑡≤1∫¹₀𝐺(𝑡,𝑠)𝜙_𝑟(𝑠)d𝑠+𝛾_∗. Choose 𝑛₀∈{1,2,…} such that 1/𝑛₀<𝜎𝑟₁, where 0<𝑟₁<𝑚𝑖𝑛{𝛿,𝑟} is a constant. Let 𝑁₀={𝑛₀+1,𝑛₀+2,…}. Fix 𝑛∈𝑁₀. Consider the boundary value problem 𝑥^′′(𝑡)+𝑚²(𝑡)𝑥(𝑡)=𝑓_𝑛((𝑡(𝑡)(),𝑥𝑡,𝑥))+𝑒,𝑡∈0,1_′(0)=0,𝑥^′(1)=0, (3.1_𝑛) where 𝑓_𝑛()=𝑓()1𝑡,𝑥𝑡,𝑥,if𝑥≥𝑛,𝑓1𝑡,𝑛1,if0≤𝑥≤𝑛. (3.5) We note that 𝑥 is a solution of (3.1_𝑛) if and only if 𝑥(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓_𝑛(𝑠,𝑥(𝑠))+𝑒(𝑠)d𝑠,0≤𝑡≤1. (3.6) Define an integral operator 𝑇_𝑛∶𝑃→𝐸 by 𝑇_𝑛𝑥(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓_𝑛(𝑠,𝑥(𝑠))+𝑒(𝑠)d𝑠,0≤𝑡≤1,𝑥∈𝑃. (3.7) Then, (3.1_𝑛) is equivalent to the fixed point equation 𝑥=𝑇_𝑛𝑥. We seek a fixed point of 𝑇_𝑛 in the cone 𝑃.
Set Ω₁={𝑥∈𝐸∣‖𝑥‖<𝑟₁},Ω₂={𝑥∈𝐸∣‖𝑥‖<𝑟}. If 𝑥∈𝑃∩(Ω₂⧵Ω₁), then𝑟≥𝑥(𝑡)≥𝜎‖𝑥‖≥𝜎𝑟₁>1𝑛₀>1𝑛>0on[0,1]. (3.8) Notice from (2.16), (𝐻₃), and (𝐻₄) that, for 𝑥∈𝑃∩(Ω₂⧵Ω₁),(𝑇_𝑛𝑥)(𝑡)≥0 on [0,1]. Also, for 𝑥∈𝑃∩(Ω₂⧵Ω₁), we have 𝑇_𝑛𝑥(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓_𝑛(𝑠,𝑥(𝑠))+𝑒(𝑠)d𝑠≤max_{0≤𝑡,𝑠≤1}𝐺(𝑡,𝑠)¹₀𝑓_𝑛(𝑠,𝑥(𝑠))+𝑒(𝑠)d𝑠,𝑡∈[0,1], (3.9) so that 𝑇_𝑛𝑥≤max_{0≤𝑡,𝑠≤1}𝐺(𝑡,𝑠)¹₀𝑓_𝑛(𝑠,𝑥(𝑠))+𝑒(𝑠)d𝑠. (3.10) And next, if 𝑥∈𝑃∩(Ω₂⧵Ω₁), we have by (3.10), min_0≤𝑡≤1𝑇_𝑛𝑥(𝑡)=min_0≤𝑡≤1¹₀𝐺(𝑡,𝑠)𝑓_𝑛(𝑠,𝑥(𝑠))+𝑒(𝑠)d𝑠≥min_{0≤𝑡,𝑠≤1}𝐺(𝑡,𝑠)¹₀𝑓_𝑛(𝑠,𝑥(𝑠))+𝑒(𝑠)d𝑠=𝜎max_{0≤𝑡,𝑠≤1}𝐺(𝑡,𝑠)¹₀𝑓_𝑛((𝑠(𝑠𝑇𝑠,𝑥))+𝑒)d𝑠≥𝜎_𝑛𝑥. (3.11) As a consequence, 𝑇_𝑛∶𝑃∩(Ω₂⧵Ω₁)→𝑃. In addition, standard arguments show that 𝑇_𝑛 is completely continuous.
If 𝑥∈𝑃 with ‖𝑥‖=𝑟, then𝑟≥𝑥(𝑡)≥𝜎‖𝑥‖=𝜎𝑟>𝜎𝑟₁>1𝑛₀>1𝑛>0on[0,1], (3.12) and we have by (𝐻₁),(𝐻₂), and (𝐻₃)𝑇_𝑛𝑥(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓_𝑛((𝑠(𝑠≤𝑠,𝑥))+𝑒)d𝑠₁₀𝐺(𝑡,𝑠)𝑘(𝑠)𝑔(𝑥(𝑠))+ℎ(𝑥(𝑠))d𝑠+𝛾^∗()ℎ(𝑟)≤𝑔𝜎𝑟1+𝑔(𝑟)𝐾^∗+𝛾^∗<𝑟=‖𝑥‖,𝑡∈[0,1]. (3.13) Thus, ‖𝑇_𝑛𝑥‖≤‖𝑥‖. Hence, 𝑇_𝑛𝑥≤‖𝑥‖,for𝑥∈𝑃∩𝜕Ω₂. (3.14)
If 𝑥∈𝑃 with ‖𝑥‖=𝑟₁, then𝑟>𝑟₁≥𝑥(𝑡)≥𝜎‖𝑥‖=𝜎𝑟₁>1𝑛₀>1𝑛>0on[0,1], (3.15) and we have by (𝐻₃) and (𝐻₄)𝑇_𝑛𝑥12=¹₀𝐺12,𝑠𝑓_𝑛((𝑠(𝑠≥𝑠,𝑥))+𝑒)d𝑠₁₀𝐺12,𝑠𝜙_𝑟(𝑠)+𝑒(𝑠)d𝑠≥min_0≤𝑡≤1¹₀𝐺(𝑡,𝑠)𝜙_𝑟(𝑠)+𝑒(𝑠)d𝑠≥min_0≤𝑡≤1¹₀𝐺(𝑡,𝑠)𝜙_𝑟(𝑠)d𝑠+𝛾_∗=𝛿>𝑟₁=‖𝑥‖. (3.16) Thus, ‖𝑇_𝑛𝑥‖≤‖𝑥‖. Hence, 𝑇_𝑛𝑥≤‖𝑥‖,for𝑥∈𝑃∩𝜕Ω₁. (3.17)
Applying (ii) of Theorem 2.3 to (3.14) and (3.17) yields that 𝑇_𝑛 has a fixed point 𝑥_𝑛∈𝑃∩(Ω₂⧵Ω₁), and 𝑟₁≤‖𝑥_𝑛‖≤𝑟. As such, 𝑥_𝑛 is a solution of (3.1_𝑛), and𝑟≥𝑥_𝑛(𝑡)≥𝜎𝑥_𝑛≥𝜎𝑟₁>1𝑛₀>1𝑛>0,𝑡∈[0,1]. (3.18)
Next we prove the fact𝑥^′_𝑛≤𝐻 (3.19) for some constant 𝐻>0 and for all 𝑛>𝑛₀. To this end, integrating the first equation of (3.1_𝑛) from 0 to 1, we obtain ¹₀𝑚²(𝑡)𝑥_𝑛(𝑡)d𝑡=¹₀𝑓_𝑛𝑡,𝑥_𝑛(𝑡)+𝑒(𝑡)d𝑡. (3.20) Then 𝑥^′_𝑛=max_0≤𝑡≤1𝑥^′_𝑛(𝑡)=max_0≤𝑡≤1^𝑡₀𝑥_𝑛^′′(𝑠)d𝑠=max_0≤𝑡≤1^𝑡₀𝑓_𝑛𝑠,𝑥_𝑛(𝑠)+𝑒(𝑠)−𝑚²(𝑠)𝑥_𝑛(𝑠≤)d𝑠₁₀𝑓_𝑛𝑠,𝑥_𝑛(𝑠)+𝑒(𝑠)d𝑠+¹₀𝑚²(𝑠)𝑥_𝑛(𝑠)d𝑠=2¹₀𝑚²(𝑠)𝑥_𝑛(𝑠)d𝑠≤2𝑚²𝑟=∶𝐻. (3.21)
The fact ‖𝑥_𝑛‖≤𝑟 and (3.19) show that {𝑥_𝑛}_𝑛∈𝑁0 is a bounded and equicontinuous family on [0,1]. Now the Arzela-Ascoli Theorem guarantees that {𝑥_𝑛}_𝑛∈𝑁0 has a subsequence, {𝑥_𝑛𝑘}_𝑘∈ℕ, converging uniformly on [0,1] to a function 𝑥∈𝐶[0,1]. From the fact ‖𝑥_𝑛‖≤𝑟 and (3.18), 𝑥 satisfies 𝜎𝑟₁≤𝑥(𝑡)≤𝑟 for all 𝑡∈[0,1]. Moreover, 𝑥_𝑛𝑘 satisfies the integral equation𝑥_𝑛𝑘(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓_𝑛𝑠,𝑥_𝑛𝑘(𝑠)+𝑒(𝑠)d𝑠. (3.22) Let 𝑘→∞, and we arrive at 𝑥(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠))+𝑒(𝑠)d𝑠, (3.23) where the uniform continuity of 𝑓(𝑡,𝑥) on [0,1]×[𝜎𝑟₁,𝑟] is used. Therefore, 𝑥 is a positive solution of boundary value problem (1.4). Finally, it is not difficult to show that, ‖𝑥‖<𝑟.

By Theorem 3.1, we have the following Corollary.

Corollary 3.3. Let (𝐻₀) hold. Assume that there exist continuous functions 𝑏,𝑏≻0 and 𝜆>0 such that (𝐹)0≤𝑏(𝑡)/𝑥^𝜆≤𝑓(𝑡,𝑥)≤𝑏(𝑡)/𝑥^𝜆, for all 𝑥>0 and 𝑡∈[0,1].Then problem (1.4) has at least one positive solution if one of the following two conditions holds:(i)𝑒_∗≥0;(ii)𝑒^∗<0,𝑏_∗+(𝐵^∗/𝜎^𝜆)^𝜆/𝜆+1𝑒_∗>0, where 𝐵(𝑡)=∫¹₀𝐺(𝑡,𝑠)𝑏(𝑠)d𝑠.

Remark 3.4. When 𝑚(𝑡)≡𝑚,𝑡∈[0,1], then (1.4) reduces to (1.1), (𝐻₀) reduce to 𝑚∈(0,𝜋/2). So Corollary 3.3 is more extensive than [5, Corollary  3.1].

4. Example

Consider second-order Neumann boundary value problem 𝑥^′′(𝑡)+𝜋(123−𝑡)²𝑥(𝑡)=√2𝑡¹⁵𝑥⁻¹(𝑡)(),𝑥+1,𝑡∈0,1_′(0)=0,𝑥^′(1)=0. (4.1) Here ,𝑓(𝑡,𝑥)=√2𝑡¹⁵[𝑥⁻¹+1],(𝑡,𝑥)∈[0,1]×(0,+∞),𝑒(𝑡)≡0,𝑚(𝑡)=(𝜋/12)(3−𝑡),𝑡∈[0,1]. Obviously, (𝐻₀) is satisfied. Let 𝑘(𝑡)=√2𝑡¹⁵,𝑔(𝑥)=1/𝑥,ℎ(𝑥)≡1,𝜙_𝑟(𝑡)=√2𝑡¹⁵/2,𝑟=2, then we can check that (𝐻₁),(𝐻₃), and (𝐻₄) are satisfied. In addition, for 𝑟=2, we have 𝑟−𝛾^∗𝑔((𝑟)(𝑟=𝜎𝑟)1+ℎ/𝑔)4𝜎3=43min_{0≤𝑡,𝑠≤1}𝐺(𝑡,𝑠)max_{0≤𝑡,𝑠≤1}𝐺()≥𝑡,𝑠4cos²𝑚3=23. (4.2) On the other hand, by Lemma 2.1, we have 𝜓^′(0)=¹₀𝑚²(𝑡)𝜓(𝑡)d𝑡≥𝑚²cos√𝑚=2𝜋²72. (4.3) By (4.3), we have 𝐾^∗=max_0≤𝑡≤1¹₀𝐺(𝑡,𝑠)√2𝑠¹⁵√d𝑠≤2(0)≤916𝜓′2𝜋²<23. (4.4) Hence, 𝑟−𝛾^∗/(𝑔(𝜎𝑟)(1+ℎ(𝑟)/𝑔(𝑟)))>𝐾^∗. So that (𝐻₂) is satisfied. According to Theorem 3.1, the boundary value problem (4.1) has at least one positive solution 𝑥 with 0<‖𝑥‖<2.

For boundary value problem (4.1), however, we cannot obtain the above conclusion by Theorem 3.1 of paper [5] since 𝑚(𝑡)=(𝜋/12)(3−𝑡),𝑡∈[0,1] is not a constant. These imply that Theorem 3.1 in this paper complement and improve those obtained in [5].