Eigenvalue Problem of Nonlinear Semipositone Higher Order Fractional Differential Equations

Abstract

We study the eigenvalue interval for the existence of positive solutions to a semipositone higher order fractional differential equation −𝒟_𝐭^𝜇𝑥(𝑡) = 𝜆𝑓(𝑡,𝑥(𝑡),𝒟_𝐭^𝜇1𝑥(𝑡),𝒟_𝐭^𝜇2𝑥(𝑡),…,𝒟_𝐭^𝜇𝑛−1𝑥(𝑡))…𝒟_𝐭^𝜇𝑖𝑥(0) = 0,1≤𝑖≤𝑛−1,𝒟_𝐭^{𝜇𝑛−1+1}𝑥(0)=0,𝒟_𝐭^𝜇𝑛−1𝑥(1)=∑^𝑚−2_𝑗=1𝑎_𝑗𝒟_𝐭^𝜇𝑛−1𝑥(𝜉_𝑗),  where 𝑛−1<𝜇≤𝑛,  𝑛≥3, 0<𝜇₁<𝜇₂<⋯<𝜇_𝑛−2<𝜇_𝑛−1, 𝑛−3<𝜇_𝑛−1<𝜇−2, 𝑎_𝑗∈ℝ,0<𝜉₁<𝜉₂<⋯<𝜉_𝑚−2<1 satisfying 0<∑^𝑚−2_𝑗=1𝑎_𝑗𝜉^{𝜇−𝜇𝑛−1}_𝑗⁻¹<1, 𝒟_𝐭^𝜇 is the standard Riemann-Liouville derivative, 𝑓∈𝐶((0,1)×ℝ^𝑛,(−∞,+∞)), and 𝑓 is allowed to be changing-sign. By using reducing order method, the eigenvalue interval of existence for positive solutions is obtained.

1. Introduction

In this paper, we consider the eigenvalue interval for existence of positive solutions to the following semipositone higher order fractional differential equation: −𝓓_𝐭^𝜇𝑥(𝑡)=𝜆𝑓𝑡,𝑥(𝑡),𝓓_𝐭^𝜇1𝑥(𝑡),𝓓_𝐭^𝜇2𝑥(𝑡),…,𝓓_𝐭^𝜇𝑛−1𝑥(𝑡,𝓓_𝐭𝜇𝑖)𝑥(0)=0,1≤𝑖≤𝑛−1,𝓓_𝐭^{𝜇𝑛−1+1}𝑥(0)𝓓=0,_{𝐭𝜇𝑛−1}𝑥(1)=^𝑚−2_𝑗=1𝑎_𝑗𝓓_𝐭^𝜇𝑛−1𝑥𝜉_𝑗, (1.1) where 𝑛−1<𝜇≤𝑛,   𝑛≥3,  𝑛∈ℕ,  0<𝜇₁<𝜇₂<⋯<𝜇_𝑛−2<𝜇_𝑛−1, and 𝑛−3<𝜇_𝑛−1<𝜇−2, 𝑎_𝑗∈ℝ,  0<𝜉₁<𝜉₂<⋯<𝜉_𝑚−2<1 satisfying 0<∑^𝑚−2_𝑗=1𝑎_𝑗𝜉^{𝜇−𝜇𝑛−1}_𝑗⁻¹<1, 𝓓_𝐭^𝜇 is the standard Riemann-Liouville derivative, 𝑓∶(0,1)×ℝ^𝑛→(−∞,+∞) is continuous.

Recently, one has found that fractional models can sufficiently describe the operation of variety of computational, economic mathematics, physical, and biological processes and systems, see [1–9]. Accordingly, considerable attention has been paid to the solution of fractional differential equations, integral equations, and fractional partial differential equations of physical phenomena [10–24]. One of the most frequently used tools in the theory of fractional calculus is furnished by the Riemann-Liouville operators. It possesses advantages of fast convergence, higher stability and higher accuracy to derive the solution of different types of fractional equations.

In this work, we will deal with the eigenvalue interval for existence of positive solutions to the higher order fractional differential equation when 𝑓 may be negative. This type of differential equation is called semipositone problem which arises in many interesting applications as pointed out by Lions in [25]. For example, the semipositone differential equation which can be derived from chemical reactor theory, design of suspension bridges, combustion, and management of natural resources, see [26–28]. To our knowledge, few results were established, especially for higher order multipoint boundary value problems with the fractional derivatives.

2. Preliminaries and Lemmas

We use the following assumptions in this paper:(𝐁𝟏) 𝑓∶(0,1)×ℝ^𝑛→(−∞,+∞) is continuous, and there exist functions 𝛼,𝛽∈𝐿¹[(0,1),(0,+∞)] and continuous function ℎ∶ℝ^𝑛→[0,+∞) such that −𝛼(𝑡)≤𝑓𝑡,𝑥₁,𝑥₂,…,𝑥_𝑛≤𝛽(𝑡)ℎ𝑥₁,𝑥₂,…,𝑥_𝑛,𝑡,𝑥₁,𝑥₂,…,𝑥_𝑛∈(0,1)×ℝ^𝑛. (2.1)

Now we begin this section with some preliminaries of fractional calculus. Let 𝜇> 0 and 𝑛=[𝜇]+1=𝑁+1, where 𝑁 is the smallest integer greater than or equal to 𝜇. For a function 𝑥∶(0,1)→ℝ, we define the fractional integral of order 𝜇 of 𝑥 as 𝐼^𝜇𝑥(𝑡)=1Γ𝜇^𝑡₀(𝑡−𝑠)^𝜇−1𝑥(𝑠)𝑑𝑠 (2.2) provided the integral exists. The fractional derivative of order 𝜇 of a continuous function 𝑥 is defined by 𝓓_𝐭^𝜇𝑥(𝑡)=1Γ𝑑𝑛−𝜇𝑑𝑡^𝑛𝑡₀(𝑡−𝑠)^{𝑛−𝜇−1}𝑥(𝑠)𝑑𝑠, (2.3) provided the right side is pointwise defined on (0,+∞). We recall the following properties [8, 9] which are useful for the sequel.

Lemma 2.1 (see [8, 9]). (1) If 𝑥∈𝐿¹(0,1),  𝜌>𝜎>0, and 𝑛∈ℕ, then 𝐼^𝜌𝐼^𝜎𝑥(𝑡)=𝐼^𝜌+𝜎𝑥(𝑡),𝓓_𝐭^𝜎𝐼^𝜌𝑥(𝑡)=𝐼^𝜌−𝜎𝑥(𝑡),𝓓_𝐭^𝜎𝐼^𝜎𝑥(𝑡)(𝑡),𝑑=𝑥𝑑𝑡^𝑛𝓓_𝐭^𝜎𝑥(𝑡)=𝓓_𝐭^𝑛+𝜎𝑥(𝑡). (2.4)(2.5)(2) If 𝜈>0,  𝜎>0, then 𝓓_𝐭^𝜈𝑡^𝜎−1=Γ(𝜎)Γ()𝑡𝜎−𝜈^{𝜎−𝜈−1}. (2.6)

Lemma 2.2 (see [8]). Assume that 𝑥∈𝐿¹(0,1) and 𝜇>0. Then 𝐼^𝜇𝓓_𝐭^𝜇𝑥(𝑡)=𝑥(𝑡)+𝑐₁𝑡^𝜇−1+𝑐₂𝑡^𝜇−2+⋯+𝑐_𝑛𝑡^𝜇−𝑛, (2.7) where 𝑐_𝑖∈ℝ(𝑖=1,2,…,𝑛), 𝑛 is the smallest integer greater than or equal to 𝜇.

Let 𝑥(𝑡)=𝐼^𝜇𝑛−1𝑣(𝑡), and consider the following modified integro-differential equation: −𝓓_𝐭^{𝜇−𝜇𝑛−1}𝑣(𝑡)=𝜆𝑓𝑡,𝐼^𝜇𝑛−1𝑣(𝑡),𝐼^{𝜇𝑛−1−𝜇1}𝑣(𝑡),…,𝐼^{𝜇𝑛−1−𝜇𝑛−2}𝑣(𝑡)(𝑡,𝑣(0),𝑣)=𝑣^′(0)=0𝑣(1)=^𝑚−2_𝑗=1𝑎_𝑗𝑣𝜉_𝑗. (2.8)

The following Lemmas 2.3–2.5 are obtained by Zhang et al. [10].

Lemma 2.3. The higher order multipoint boundary value problem (1.1) has a positive solution if and only if nonlinear integro-differential equation (2.8) has a positive solution. Moreover, if 𝑣 is a positive solution of (2.8), then 𝑥(𝑡)=𝐼^𝜇𝑛−1𝑣(𝑡) is positive solution of the higher order multipoint boundary value problem (1.1).

Lemma 2.4. If 2<𝜇−𝜇_𝑛−1<3 and 𝛼∈𝐿¹[0,1], then the boundary value problem 𝓓_𝐭^{𝜇−𝜇𝑛−1}𝑤(𝑡)(𝑡)𝑤(0)+𝜆𝛼=0,=𝑤^′(0)=0,𝑤(1)=^𝑚−2_𝑗=1𝑎_𝑗𝑤𝜉_𝑗 (2.9) has the unique solution 𝑤(𝑡)=𝜆¹₀𝐾(𝑡,𝑠)𝛼(𝑠)𝑑𝑠, (2.10) where 𝐾()()+𝑡𝑡,𝑠=𝑘𝑡,𝑠^{𝜇−𝜇𝑛−1−1}1−∑^𝑚−2_𝑗=1𝑎_𝑗𝜉^{𝜇−𝜇𝑛−1}_𝑗^−1𝑚−2_𝑗=1𝑎_𝑗𝑘𝜉_𝑗,𝑠, (2.11) is the Green function of the boundary value problem (2.9), and 𝑘()=[𝑡(𝑡,𝑠1−𝑠)]^{𝜇−𝜇𝑛−1−1}−(𝑡−𝑠)^{𝜇−𝜇𝑛−1−1}Γ𝜇−𝜇_𝑛−1[𝑡(,0≤𝑠≤𝑡≤1,1−𝑠)]^{𝜇−𝜇𝑛−1−1}Γ𝜇−𝜇_𝑛−1,0≤𝑡≤𝑠≤1. (2.12)

Lemma 2.5. The Green function of the boundary value problem (2.9) satisfies 𝐾(𝑡,𝑠)≤𝜏, (2.13) where ∑𝜏=1+^𝑚−2_𝑗=1𝑎_𝑗1−𝜉^{𝜇−𝜇𝑛−1}_𝑗⁻¹Γ𝜇−𝜇_𝑛−11−∑^𝑚−2_𝑗=1𝑎_𝑗𝜉^{𝜇−𝜇𝑛−1}_𝑗⁻¹. (2.14)
Define a modified function [⋅]^∗ for any 𝑧∈𝐶[0,1] by [𝑧(𝑡)]^∗=𝑧(𝑡)(𝑡)(𝑡),𝑧≥0,0,𝑧<0, (2.15) and consider the following boundary value problem −𝓓_𝐭^{𝜇−𝜇𝑛−1}𝑢(𝑡)=𝜆𝑓𝑡,𝐼^𝜇𝑛−1[𝑢(𝑡)−𝑤(𝑡)]^∗,𝐼^{𝜇𝑛−1−𝜇1}[𝑢(𝑡)−𝑤(𝑡)]^∗𝐼⋯,_{𝜇𝑛−1−𝜇𝑛−2}[𝑢(𝑡)−𝑤(𝑡)]^∗,[𝑢(𝑡)−𝑤(𝑡)]^∗(𝑡,𝑢(0)+𝛼)=𝑢^′(0)=0,𝑢(1)=^𝑚−2_𝑗=1𝑎_𝑗𝑢𝜉_𝑗. (2.16)

Lemma 2.6. Suppose 𝑢(𝑡)≥𝑤(𝑡),  𝑡∈[0,1] is a solution of the problem (2.16), then 𝑢−𝑤 is a positive solution of the problem (2.8), consequently, 𝐼^𝜇𝑛−1[𝑢(𝑡)−𝑤(𝑡)] is also a positive solution of the semipositone higher differential equation (1.1).

Proof. Since 𝑢 is a solution of the BVP (2.16) and 𝑢(𝑡)≥𝑤(𝑡) for any 𝑡∈[0,1], then we have −𝓓_𝐭^{𝜇−𝜇𝑛−1}𝑢(𝑡)=𝜆𝑓𝑡,𝐼^𝜇𝑛−1[𝑢(𝑡)−𝑤(𝑡)],𝐼^{𝜇𝑛−1−𝜇1}[𝑢(𝑡)(𝑡𝐼−𝑤)],…,_{𝜇𝑛−1−𝜇𝑛−2}[𝑢(𝑡)(𝑡,[𝑢(𝑡)(𝑡(𝑡,𝑢(0)−𝑤)]−𝑤)]+𝛼)=𝑢^′(0)=0,𝑢(1)=^𝑚−2_𝑗=1𝑎_𝑗𝑢𝜉_𝑗. (2.17) Let 𝑣=𝑢−𝑤, then we have 𝑤(0)=𝑤^′(0)=0,𝑤(1)=^𝑚−2_𝑗=1𝑎_𝑗𝑤𝜉_𝑗, (2.18) and 𝓓_𝐭^{𝜇−𝜇𝑛−1}𝑣(𝑡)=𝓓_𝐭^{𝜇−𝜇𝑛−1}𝑢(𝑡)−𝓓_𝐭^{𝜇−𝜇𝑛−1}𝑤(𝑡), which implies that −𝓓_𝐭^{𝜇−𝜇𝑛−1}𝑢(𝑡)=−𝓓_𝐭^{𝜇−𝜇𝑛−1}𝑣(𝑡)+𝜆𝛼(𝑡). (2.19) Substituting the above into (2.17), then 𝑣=𝑢−𝑤 solves the (2.8), that is, 𝑢−𝑤 is a positive solution of the semipositone differential equation (2.8). By Lemma 2.3, 𝐼^𝜇𝑛−1[𝑢(𝑡)−𝑤(𝑡)] is a positive solution of the singular semipositone differential equation (1.1). This completes the proof of Lemma 2.5.

Let 𝛾(𝑡)=1−^𝑚−2_𝑗=1𝑎_𝑗𝜉^{𝜇−𝜇𝑛−1}_𝑗⁻¹𝑡^{𝜇−𝜇𝑛−1−1}(1−𝑡)+^𝑚−2_𝑗=1𝑎_𝑗𝜉^{𝜇−𝜇𝑛−1}_𝑗⁻¹1−𝜉_𝑗𝑡^{𝜇−𝜇𝑛−1−1}. (2.20)

Lemma 2.7 (see [10]). The solution 𝑤(𝑡) of (2.9) satisfies 𝑤(𝑡)≤𝜆𝜂𝛾(𝑡),𝑡∈[0,1], (2.21) where ∫𝜂=¹⁰𝛼(𝑠)𝑑𝑠Γ𝜇−𝜇_𝑛−11−∑^𝑚−2_𝑗=1𝑎_𝑗𝜉^{𝜇−𝜇𝑛−1}_𝑗⁻¹. (2.22)

It is well known that the BVP (2.17) is equivalent to the fixed points for the mapping 𝑇 by (𝑇𝑢)(𝑡)=𝜆¹₀𝐾(𝑡,𝑠)𝑓𝑠,𝐼^𝜇𝑛−1[𝑢(𝑠)−𝑤(𝑠)],𝐼^{𝜇𝑛−1−𝜇1}[𝑢(𝑠)(𝑠𝐼−𝑤)],…,_{𝜇𝑛−1−𝜇𝑛−2}[𝑢(𝑠)−𝑤(𝑠)],[𝑢(𝑠)−𝑤(𝑠)]+𝛼(𝑠)𝑑𝑠. (2.23)

The basic space used in this paper is 𝐸=𝐶([0,1];ℝ), where ℝ is the set of real numbers. Obviously, the space 𝐸 is a Banach space if it is endowed with the norm as follows: ‖𝑢‖=max_𝑡∈[0,1]|𝑢(𝑡)|, (2.24) for any 𝑢∈𝐸. Let (𝑡)≥1𝑃=𝑢∈𝐸∶𝑢2𝛾(𝑡)‖𝑢‖, (2.25) then 𝑃 is a cone of 𝐸.

Lemma 2.8. Assume that (𝐁𝟏) holds. Then 𝑇∶𝑃→𝑃 is a completely continuous operator.

Proof. By using similar method to [10] and standard arguments, according to the Ascoli-Arzela Theorem, one can show that 𝑇∶𝑃→𝑃 is a completely continuous operator.

Lemma 2.9 (see [29]). Let 𝐸 be a real Banach space, 𝑃⊂𝐸 be a cone. Assume Ω₁,Ω₂ are two bounded open subsets of 𝐸 with 𝜃∈Ω₁,  Ω₁⊂Ω₂, and let 𝑇∶𝑃∩(Ω₂⧵Ω₁)→𝑃 be a completely continuous operator such that either(1)‖𝑇𝑥‖≤‖𝑥‖,𝑥∈𝑃∩𝜕Ω₁ and ‖𝑇𝑥‖≥‖𝑥‖,𝑥∈𝑃∩𝜕Ω₂, or (2)‖𝑇𝑥‖≥‖𝑥‖,𝑥∈𝑃∩𝜕Ω₁ and ‖𝑇𝑥‖≤‖𝑥‖,𝑥∈𝑃∩𝜕Ω₂.Then 𝑇 has a fixed point in 𝑃∩(Ω₂⧵Ω₁).

3. Main Results

Theorem 3.1. Suppose that (𝐁𝟏) holds, and lim_{∑𝑛𝑖=1|𝑥𝑖|→∞}min_{𝑡∈[1/4,3/4]}𝑓𝑡,𝑥₁,𝑥₂,…,𝑥_𝑛∑^𝑛_𝑖=1𝑥_𝑖=+∞. (3.1) Then there exists some constant 𝜆^∗>0 such that the higher order multipoint boundary value problem (1.1) has at least one positive solution for any 𝜆∈(0,𝜆^∗).

Proof. By Lemma 2.8, we know 𝑇 is a completely continuous operator. Take 𝑟=𝜏¹₀𝛽(𝑠)+𝛼(𝑠)𝑑𝑠, (3.2) where 𝜏 is defined by Lemma 2.5. Let Ω₁={𝑢∈𝑃∶‖𝑢‖<𝑟}. Then, for any 𝑢∈𝜕Ω₁,  𝑠∈[0,1], we have 0≤[𝑢(𝑠)−𝑤(𝑠)]^∗≤𝑢(𝑠)≤‖𝑢‖≤𝑟,0≤𝐼^𝜇𝑛−1[𝑢(𝑠)−𝑤(𝑠)]^∗≤𝑟Γ𝜇_𝑛−1,0≤𝐼^{𝜇𝑛−1−𝜇𝑖}[𝑢(𝑠)−𝑤(𝑠)]^∗≤𝑟Γ𝜇_𝑛−1−𝜇_𝑖,𝑖=1,…,𝑛−2. (3.3) Choose 𝜆^∗𝑟=min,14𝜂,𝑁+1 (3.4) where 𝑁=max_{𝑢1,𝑢2,…,𝑢𝑛∈[]×𝜇0,10,𝑟/Γ𝑛−1𝜇×⋯×0,𝑟/Γ𝑛−1−𝜇𝑛−2×[]0,𝑟}ℎ𝑢₁,𝑢₂,…,𝑢_𝑛, (3.5) and 𝜂 is defined by Lemma 2.7. Thus, for any 𝑢∈𝜕Ω₁,  𝑠∈[0,1], and 𝜆∈(0,𝜆^∗), by (3.3), we have ‖𝑇𝑢‖=max_[]𝑡∈0,1(𝑇𝑢)(𝑡)=𝜆max_[]𝑡∈0,1¹₀𝐾(𝑡,𝑠)𝑓𝑠,𝐼^𝜇𝑛−1[𝑢(𝑠)−𝑤(𝑠)]^∗,𝐼^{𝜇𝑛−1−𝜇1}[𝑢(𝑠)−𝑤(𝑠)]^∗𝐼,…,_{𝜇𝑛−1−𝜇𝑛−2}[𝑢(𝑠)−𝑤(𝑠)]^∗,[𝑢(𝑠)−𝑤(𝑠)]^∗+𝛼(𝑠)𝑑𝑠≤𝜆𝜏¹₀𝛽(𝑠)ℎ𝐼^𝜇𝑛−1[𝑢(𝑠)−𝑤(𝑠)]^∗,𝐼^{𝜇𝑛−1−𝜇1}[𝑢(𝑠)−𝑤(𝑠)]^∗𝐼,…,_{𝜇𝑛−1−𝜇𝑛−2}[𝑢(𝑠)−𝑤(𝑠)]^∗,[𝑢(𝑠)−𝑤(𝑠)]^∗(𝑠()+𝛼)𝑑𝑠≤𝜆𝜏𝑁+1¹₀𝛽(𝑠)+𝛼(𝑠)𝑑𝑠≤𝑟=‖𝑢‖. (3.6) Therefore, ‖𝑇𝑢‖≤‖𝑢‖,𝑢∈𝑃∩𝜕Ω₁. (3.7)
On the other hand, choose a real number 𝐿>0 such that 𝜆𝐿^3/4_1/4𝐾12,𝑠𝑑𝑠1−^𝑚−2_𝑗=1𝑎_𝑗𝜉^{𝜇−𝜇𝑛−1}_𝑗14^{𝜇−𝜇𝑛−1+1}≥1. (3.8) By (3.1), for any 𝑡∈[1/4,3/4], there exists a constant 𝐵>0 such that 𝑓𝑡,𝑥₁,𝑥₂,…,𝑥_𝑛∑^𝑛_𝑖=1𝑥_𝑖>𝐿,for^𝑛_𝑖=1𝑥_𝑖≥𝐵. (3.9) Take 𝑅=max2𝑟,4𝜆𝜂,1−^𝑚−2_𝑗=1𝑎_𝑗𝜉^{𝜇−𝜇𝑛−1}_𝑗⁻¹4^{𝜇−𝜇𝑛−1+1}𝐵, (3.10) let Ω₂={𝑢∈𝑃∶‖𝑢‖<𝑅} and 𝜕Ω₂={𝑢∈𝑃∶‖𝑢‖=𝑅}. Then for any 𝑢∈𝜕Ω₂,  𝑠∈[0,1], by Lemma 2.7, we have 𝑢(𝑠)(𝑠)(𝑠)(𝑠)≥1−𝑤≥𝑢−𝜆𝜂𝛾2(𝑠)(𝑠)≥1𝑅𝛾−𝜆𝜂𝛾4𝑅𝛾(𝑠)≥0. (3.11) And then, for any 𝑢∈𝜕Ω₂,  𝑠∈[1/4,3/4], one gets ^𝑛−2_𝑖=1𝐼^{𝜇𝑖𝑛−1−𝜇}[𝑢(𝑠)−𝑤(𝑠)]^∗+𝐼^𝜇𝑛−1[𝑢(𝑠)−𝑤(𝑠)]^∗+[𝑢(𝑠)−𝑤(𝑠)]^∗(𝑠)(𝑠)≥1≥𝑢−𝑤4𝑅𝛾(𝑡)≥1−^𝑚−2_𝑗=1𝑎_𝑗𝜉^{𝜇−𝜇𝑛−1}_𝑗⁻¹14^{𝜇−𝜇𝑛−1+1}𝑅+^𝑚−2_𝑗=1𝑎_𝑗𝜉^{𝜇−𝜇𝑛−1}_𝑗⁻¹1−𝜉_𝑗14^{𝜇−𝜇𝑛−1}𝑅≥1−^𝑚−2_𝑗=1𝑎_𝑗𝜉^{𝜇−𝜇𝑛−1}_𝑗14^{𝜇−𝜇𝑛−1+1}𝑅≥𝐵. (3.12) It follows from (3.12) that, for any 𝑢∈𝜕Ω₂, ‖𝑇𝑢‖≥𝜆¹₀𝐾12,𝑠𝑓𝑠,𝐼^𝜇𝑛−1[𝑢(𝑠)−𝑤(𝑠)]^∗,𝐼^{𝜇𝑛−1−𝜇1}[𝑢(𝑠)−𝑤(𝑠)]^∗𝐼,…,_{𝜇𝑛−1−𝜇𝑛−2}[𝑢(𝑠)−𝑤(𝑠)]^∗,[𝑢(𝑠)−𝑤(𝑠)]^∗+𝛼(𝑠)𝑑𝑠≥𝜆^3/4_1/4𝐾12,𝑠𝑓𝑠,𝐼^𝜇𝑛−1[𝑢(𝑠)−𝑤(𝑠)],𝐼^{𝜇𝑛−1−𝜇1}[𝑢(𝑠)(𝑠𝐼−𝑤)],…,_{𝜇𝑛−1−𝜇𝑛−2}[𝑢(𝑠)−𝑤(𝑠)],[𝑢(𝑠)−𝑤(𝑠)]𝑑𝑠≥𝜆^3/4_1/4𝐾12,𝑠𝐿𝐼^𝜇𝑛−1[𝑢(𝑠)−𝑤(𝑠)]+𝐼^{𝜇𝑛−1−𝜇1}[𝑢(𝑠)(𝑠𝐼−𝑤)]+⋯+^{𝜇𝑛−1−𝜇𝑛−2}[𝑢(𝑠)−𝑤(𝑠)]+|[𝑢(𝑠)−𝑤(𝑠)]|𝑑𝑠≥𝜆𝐿^3/4_1/4𝐾12,𝑠|[𝑢(𝑠)−𝑤(𝑠)]|𝑑𝑠≥𝜆𝐿^3/4_1/4𝐾12,𝑠𝑑𝑠1−^𝑚−2_𝑗=1𝑎_𝑗𝜉^{𝜇−𝜇𝑛−1}_𝑗14^{𝜇−𝜇𝑛−1+1}𝑅=𝑅=‖𝑢‖. (3.13) So, we have ‖𝑇𝑢‖≥‖𝑢‖,𝑢∈𝑃∩𝜕Ω₂. (3.14) By Lemma 2.9, 𝑇 has at least a fixed point 𝑢∈(𝑃∩Ω₂)⧵Ω₁ such that 𝑟≤||𝑢||≤𝑅.
It follows from 𝜆^∗≤𝑟/4𝜂 and 𝜆∈(0,𝜆^∗) that 𝑢(𝑡)(𝑡)(𝑡)(𝑡)≥1−𝑤≥𝑢−𝜆𝜂𝛾2(𝑡)(𝑡)≥𝑟𝑟𝛾−𝜆𝜂𝛾4𝛾(𝑡)>0,𝑡∈(0,1). (3.15)
Let 𝑥(𝑡)=𝐼^𝜇𝑛−1[𝑢(𝑡)−𝑤(𝑡)], then 𝑥(𝑡)>0,𝑡∈(0,1). (3.16) By Lemma 2.6, we know that the differential equation (1.1) has at least a positive solutions 𝑥.

Theorem 3.2. Suppose (𝐁𝟏) holds and liminf_{∑𝑛𝑖=1𝑢𝑖→∞}min_{[]𝑡∈1/4,3/4}𝑓𝑡,𝑢₁,𝑢₂,…,𝑢_𝑛=+∞,lim_{∑𝑛𝑖=1|𝑢𝑖|→∞}ℎ𝑢₁,𝑢₂,…,𝑢_𝑛∑^𝑛_𝑖=1𝑢_𝑖=0. (3.17) Then there exists 𝜆^∗>0 such that the higher order multipoint boundary value problem (1.1) has at least one positive solution for any 𝜆∈(𝜆^∗,+∞).

Proof. By (3.17), there exists 𝑀>0 such that for any 𝑡∈[1/4,3/4] we have 𝑓𝑡,𝑢₁,𝑢₂,…,𝑢_𝑛≥4𝜂∫^3/4_1/4𝐾(1/2,𝑠)𝑑𝑠,if^𝑛_𝑖=1𝑢_𝑖≥𝑀. (3.18) Let 𝜆^∗=𝑀𝜂1−∑^𝑚−2_𝑗=1𝑎_𝑗𝜉^{𝜇−𝜇𝑛−1}_𝑗(1/4)^{𝜇−𝜇𝑛−1+1}. (3.19) In the following of the proof, we suppose 𝜆>𝜆^∗. Take 𝑟=4𝜆𝜂, (3.20) and let Ω₁={𝑢∈𝑃∶‖𝑢‖<𝑟} and 𝜕Ω₁={𝑢∈𝑃∶‖𝑢‖=𝑟}. Then for any 𝑢∈𝜕Ω₁,  𝑠∈[0,1], by Lemma 2.7, we have 𝑢(𝑠)(𝑠)(𝑠)(𝑠)≥1−𝑤≥𝑢−𝜆𝜂𝛾2𝑟𝛾(𝑠)−𝜆𝜂𝛾(𝑠)=𝜆𝜂𝛾(𝑠)≥0. (3.21)
So for any 𝑢∈𝜕Ω₁,  𝑠∈[1/4,3/4], one gets ^𝑛−2_𝑖=1𝐼^{𝜇𝑖𝑛−1−𝜇}[𝑢(𝑠)−𝑤(𝑠)]^∗+𝐼^𝜇𝑛−1[𝑢(𝑠)−𝑤(𝑠)]^∗+[𝑢(𝑠)−𝑤(𝑠)]^∗≥𝑢(𝑠)−𝑤(𝑠)≥𝜆𝜂𝛾(𝑠)≥𝜆𝜂1−^𝑚−2_𝑗=1𝑎_𝑗𝜉^{𝜇−𝜇𝑛−1}_𝑗14^{𝜇−𝜇𝑛−1+1}≥𝜆^∗𝜂1−^𝑚−2_𝑗=1𝑎_𝑗𝜉^{𝜇−𝜇𝑛−1}_𝑗14^{𝜇−𝜇𝑛−1+1}≥𝑀. (3.22) Thus, by (3.22), for any 𝑢∈𝜕Ω₁, we have ‖𝑇𝑢‖≥𝜆¹₀𝐾12,𝑠𝑓𝑠,𝐼^𝜇𝑛−1[𝑢(𝑠)−𝑤(𝑠)]^∗,𝐼^{𝜇𝑛−1−𝜇1}[𝑢(𝑠)−𝑤(𝑠)]^∗I,…,_{𝜇𝑛−1−𝜇𝑛−2}[𝑢(𝑠)−𝑤(𝑠)]^∗,[𝑢(𝑠)−𝑤(𝑠)]^∗+𝛼(𝑠)𝑑𝑠≥𝜆^3/4_1/4𝐾12,𝑠𝑓𝑠,𝐼^𝜇𝑛−1[𝑢(𝑠)−𝑤(𝑠)]^∗,𝐼^{𝜇𝑛−1−𝜇1}[𝑢(𝑠)−𝑤(𝑠)]^∗𝐼,…,_{𝜇𝑛−1−𝜇𝑛−2}[𝑢(𝑠)−𝑤(𝑠)]^∗,[𝑢(𝑠)−𝑤(𝑠)]^∗𝑑𝑠≥𝜆^3/4_1/4𝐾12,𝑠𝑑𝑠×4𝜂∫^3/4_1/4𝐾()‖𝑢‖.1/2,𝑠𝑑𝑠=𝑟= (3.23) So, we have ‖𝑇𝑢‖≥‖𝑢‖,𝑢∈𝑃∩𝜕Ω₁. (3.24)
Next, take 𝜎=^𝑛−2_𝑖=11Γ𝜇_𝑛−1−𝜇_𝑖+1Γ𝜇_𝑛−1+1. (3.25) Let us choose 𝜀>0 such that 𝜆𝜏𝜎𝜀¹₀𝛽(𝑠)𝑑𝑠<1. (3.26) Then for the above 𝜀, by (3.17), there exists 𝑁>𝑟>0 such that, for any 𝑡∈[0,1], ℎ𝑢₁,𝑢₂,…,𝑢_𝑛≤𝜀^𝑛_𝑖=1𝑢_𝑖,if^𝑛_𝑖=1𝑢_𝑖>𝑁. (3.27) Thus, by (3.3) and (3.27), if ^𝑛−2_𝑖=1𝐼^{𝜇𝑖𝑛−1−𝜇}[𝑢(𝑠)−𝑤(𝑠)]^∗+𝐼^𝜇𝑛−1[𝑢(𝑠)−𝑤(𝑠)]^∗+[𝑢(𝑠)−𝑤(𝑠)]^∗>𝑁, (3.28) we have ℎ𝐼^𝜇𝑛−1[𝑢(𝑠)−𝑤(𝑠)]^∗,𝐼^{𝜇𝑛−1−𝜇1}[𝑢(𝑠)−𝑤(𝑠)]^∗𝐼,…,_{𝜇𝑛−1−𝜇𝑛−2}[𝑢(𝑠)−𝑤(𝑠)]^∗,[𝑢(𝑠)−𝑤(𝑠)]^∗≤𝐼𝑑𝑠_𝜇𝑛−1[𝑢(𝑠)−𝑤(𝑠)]^∗+𝐼^{𝜇𝑛−1−𝜇1}[𝑢(𝑠)−𝑤(𝑠)]^∗+𝐼+⋯_{𝜇𝑛−1−𝜇𝑛−2}[𝑢(𝑠)−𝑤(𝑠)]^∗+[𝑢(𝑠)−𝑤(𝑠)]^∗𝜀≤^𝑛−2_𝑖=11Γ𝜇_𝑛−1−𝜇_𝑖+1Γ𝜇_𝑛−1+1‖𝑢‖𝜀=𝜎‖𝑢‖𝜀. (3.29)
Take 𝑅=𝜆𝜏𝜃∫¹₀𝛽(𝑠)+𝛼(𝑠)𝑑𝑠+𝜆𝜏∫¹₀𝛼(𝑠)𝑑𝑠1−𝜆𝜏𝜎𝜀∫¹₀𝛽(𝑠)𝑑𝑠+𝑁, (3.30) where 𝜃=max_{∑𝑛𝑖=1𝑢𝑖≤𝑁}ℎ𝑢₁,𝑢₂,…,𝑢_𝑛+1. (3.31) Then 𝑅>𝑁>𝑟.
Now let Ω₂={𝑢∈𝑃∶‖𝑢‖<𝑅} and 𝜕Ω₂={𝑢∈𝑃∶‖𝑢‖=𝑅}. Then, for any 𝑢∈𝑃∩𝜕Ω₂, we have ‖𝑇𝑢‖=max_[]𝑡∈0,1(𝑇𝑢)(𝑡)=𝜆max_[]𝑡∈0,1¹₀𝐾(𝑡,𝑠)𝑓𝑠,𝐼^𝜇𝑛−1[𝑢(𝑠)−𝑤(𝑠)]^∗,𝐼^{𝜇𝑛−1−𝜇1}[𝑢(𝑠)−𝑤(𝑠)]^∗𝐼,…,_{𝜇𝑛−1−𝜇𝑛−2}[𝑢(𝑠)−𝑤(𝑠)]^∗,[𝑢(𝑠)−𝑤(𝑠)]^∗+𝛼(𝑠)𝑑𝑠≤𝜆𝜏¹₀𝛽(𝑠)ℎ𝐼^𝜇𝑛−1[𝑢(𝑠)−𝑤(𝑠)]^∗,𝐼^{𝜇𝑛−1−𝜇1}[𝑢(𝑠)−𝑤(𝑠)]^∗𝐼,…,_{𝜇𝑛−1−𝜇𝑛−2}[𝑢(𝑠)−𝑤(𝑠)]^∗,[𝑢(𝑠)−𝑤(𝑠)]^∗+𝛼(𝑠)𝑑𝑠≤𝜆𝜏max_{∑𝑛𝑖=1𝑢𝑖≤𝑁}ℎ𝑢₁,𝑢₂,…,𝑢_𝑛+1¹₀𝛽(𝑠)+𝛼(𝑠)𝑑𝑠+𝜆𝜏¹₀𝛽(𝑠)𝜎𝜀‖𝑢‖+𝛼(𝑠)𝑑𝑠≤𝜆𝜏𝜃¹₀𝛽(𝑠)+𝛼(𝑠)𝑑𝑠+𝜆𝜏¹₀𝛼(𝑠)𝑑𝑠+𝜆𝜏𝜎𝜀¹₀𝛽(𝑠)𝑑𝑠𝑅≤𝑅=‖𝑢‖, (3.32) which implies that ‖𝑇𝑢‖≤‖𝑢‖,𝑢∈𝑃∩𝜕Ω₂. (3.33) By Lemma 2.9, 𝑇 has at least a fixed point 𝑢∈(𝑃∩Ω₂)⧵Ω₁ such that 𝑟≤||𝑢||≤𝑅.
It follows from 𝑟=4𝜆𝜂 that 𝑢(𝑡)(𝑡)≥1−𝑤2‖𝑢‖𝛾(𝑡)−𝜆𝜂𝛾(𝑡)=𝜆𝜂𝛾(𝑡)>0,𝑡∈(0,1). (3.34)
Let 𝑥(𝑡)=𝐼^𝜇𝑛−1[𝑢(𝑡)−𝑤(𝑡)], then 𝑥(𝑡)>0,𝑡∈(0,1). (3.35) By Lemma 2.6, we know that the differential equation (1.1) has at least a positive solutions 𝑥.

Example 3.3. Consider the existence of positive solutions for the nonlinear higher order fractional differential equation with four-point boundary condition −𝓓_𝐭^5/2𝑥(𝑡)=|𝑥(𝑡)|+𝓓_𝐭^1/4𝑥(𝑡)+𝓓_𝐭^3/8𝑥(𝑡)^1/2𝓓+ln𝑡,_𝐭1/4𝑥(0)=𝓓_𝐭^3/8𝑥(0)=𝓓_𝐭^11/8𝑥(0)=0,𝓓_𝐭^3/8𝑥(1)=2𝓓_𝐭^3/8𝑥12−𝓓_𝐭^3/8𝑥34. (3.36) Then there exists 𝜆^∗>0 such that the higher order four-point boundary value problem (1.1) has at least one positive solution for any 𝜆∈(𝜆^∗,+∞).

Proof. Let 𝑓𝑡,𝑥,𝑦,𝑧=|𝑥|+𝑦+|𝑧|^1/2+ln𝑡,𝑡∈(0,1)×ℝ³, (3.37) then −|ln𝑡|=ln𝑡≤𝑓𝑡,𝑥,𝑦,𝑧=|𝑥|+𝑦+|𝑧|^1/2+ln𝑡≤|𝑥|+𝑦+|𝑧|^1/2,𝛼(t)=|ln𝑡|,𝛽(𝑡)=1,ℎ𝑥,𝑦,𝑧=|𝑥|+𝑦+|𝑧|^1/2,liminf_{|𝑥|+𝑦+|𝑧|→+∞}min_{[]𝑡∈1/4,3/4}𝑓𝑡,𝑥,𝑦,𝑧=+∞,lim_{|𝑥|+|𝑦|+|𝑧|→+∞}ℎ𝑥,𝑦,𝑧|𝑥|+𝑦+|𝑧|=0. (3.38) Clearly, 𝛼,𝛽∈𝐿¹[(0,1),(0,+∞)], and 0<^𝑚−2_𝑗=1𝑎_𝑗𝜉^{𝜇−𝜇𝑛−1}_𝑗⁻¹1=1−22^9/8+34^9/8=0.8065<1. (3.39) By Theorem 3.2, there exists 𝜆^∗>0 such that the higher order multipoint boundary value problem (3.36) has at least one positive solution for any 𝜆∈(𝜆^∗,+∞).