Solutions of Sign-Changing Fractional Differential Equation with the Fractional Derivatives

Abstract

We study the singular fractional-order boundary-value problem with a sign-changing nonlinear term −𝒟^𝛼𝑥(𝑡)=𝑝(𝑡)𝑓(𝑡,𝑥(𝑡),𝒟^𝜇1𝑥(𝑡),𝒟^𝜇2𝑥(𝑡),…,𝒟^𝜇𝑛−1𝑥(𝑡)),0<𝑡<1,𝒟^𝜇𝑖𝑥(0)=0,1≤𝑖≤𝑛−1,𝒟^{𝜇𝑛−1+1}𝑥(0)=0, 𝒟^𝜇𝑛−1𝑥(1)=∑^{𝑝−2𝑗=1}𝑎_𝑗𝒟^𝜇𝑛−1𝑥(𝜉_𝑗), where 𝑛−1<𝛼≤𝑛, 𝑛∈ℕ and 𝑛≥3 with 0<𝜇₁<𝜇₂<⋯<𝜇_𝑛−2<𝜇_𝑛−1 and 𝑛−3<𝜇_𝑛−1<𝛼−2, 𝑎_𝑗∈ℝ,0<𝜉₁<𝜉₂<⋯<𝜉_𝑝−2<1 satisfying 0<∑^{𝑝−2𝑗=1}𝑎_𝑗𝜉^{𝛼−𝜇𝑛−1𝑗−1}<1, 𝒟^𝛼 is the standard Riemann-Liouville derivative, 𝑓∶[0,1]×ℝ^𝑛→ℝ is a sign-changing continuous function and may be unbounded from below with respect to 𝑥_𝑖, and 𝑝∶(0,1)→[0,∞) is continuous. Some new results on the existence of nontrivial solutions for the above problem are obtained by computing the topological degree of a completely continuous field.

1. Introduction

Fractional differential equations arise in many engineering and scientific disciplines, and particularly in the mathematical modeling of systems and processes in physics, chemistry, aerodynamics, electrodynamics of complex medium, and polymer rheology [1–6]. Fractional-order models have proved to be more accurate than integer-order models, that is, there are more degrees of freedom in the fractional-order models. Hence fractional differential equations have attracted great research interest in recent years, and for more details we refer the reader to [7–16] and the references cited therein.

In this paper, we consider the existence of nontrivial solutions for the following singular fractional-order boundary-value problem with a sign-changing nonlinear term and fractional derivatives: −𝒟𝒟^𝛼𝑥(𝑡)=𝑝(𝑡)𝑓(𝑡,𝑥(𝑡),𝒟^𝜇1𝑥(𝑡),𝒟^𝜇2𝑥(𝑡),…,𝒟^𝜇𝑛−1𝑥(𝑡)),0<𝑡<1,_𝜇𝑖𝑥(0)=0,1≤𝑖≤𝑛−1,𝒟^{𝜇𝑛−1+1}𝑥(0)=0,𝒟^𝜇𝑛−1𝑥(1)=^𝑝−2_𝑗=1𝑎_𝑗𝒟^𝜇𝑛−1𝑥𝜉_𝑗,(1.1) where 𝑛−1<𝛼≤𝑛,𝑛∈ℕ and 𝑛≥3 with 0<𝜇₁<𝜇₂<⋯<𝜇_𝑛−2<𝜇_𝑛−1 and 𝑛−3<𝜇_𝑛−1<𝛼−2, 𝑎_𝑗∈ℝ,0<𝜉₁<𝜉₂<⋯<𝜉_𝑝−2<1 satisfying 0<∑^{𝑝−2𝑗=1}𝑎_𝑗𝜉^{𝛼−𝜇𝑛−1𝑗−1}<1, 𝒟^𝛼 is the standard Riemann-Liouville derivative, 𝑓∶[0,1]×ℝ^𝑛→ℝ is a sign-changing continuous function and may be unbounded from below with respect to 𝑥_𝑖, and 𝑝∶(0,1)→[0,∞) is continuous.

In this paper, we assume that 𝑓∶[0,1]×ℝ^𝑛→ℝ, which implies that the problem (1.1) is changing sign (or semipositone particularly). Differential equations with changing-sign arguments are found to be important mathematical tools for the better understanding of several real-world problems in physics, chemistry, mechanics, engineering, and economics [17–19]. In general, the cone theory is difficult to handle this type of problems since the operator generated by 𝑓 is not a cone mapping. So to find a new method to solve changing-sign problems is an interesting, important, and difficult work. An effective approach to this problem was recently suggested by Sun [20] based on the topological degree of a completely continuous field. Then, Han and Wu [21, 22] obtained a new Leray-Schauder degree theorem by improving the results of Sun [20]. In [22], Han et al. also investigated a kind of singular two-point boundary-value problems with sign-changing nonlinear terms by applying the new Leray-Schauder degree theorem obtained in [22].

To our knowledge, very few results have been established when 𝑓 is changing sign [20–24]. In [20, 21, 23], 𝑓 permits sign changing but required to be bounded from below. In [22, Theorem 1.1], 𝑓 may be a sign-changing and unbounded function, but the Green function must be symmetric and 𝑓 is controlled by a special function ℎ(𝑢)=−𝑏−𝑐|𝑢|^𝜇, where 𝑏>0,𝑐>0 and 𝜇∈(0,1). Recently, by improving and generalizing the main results of Sun [20] and Han et al. [21, 22], Liu et al. [24] established a generalized Leray-Schauder degree theorem of a completely continuous field for solving 𝑚-point boundary-value problems for singular second-order differential equations.

Motivated by [20–24], we established some new results on the existence of nontrivial solutions for the problem (1.1) by computing the topological degree of a completely continuous field. The conditions used in the present paper are weaker than the conditions given in previous works [20–24], and particularly we drop the assumption of even function in [24]. The new features of this paper mainly include the following aspects. Firstly, the nonlinear term 𝑓(𝑡,𝑥₁,𝑥₂,…,𝑥_𝑛) in the BVP (1.1) is allowed to be sign changing and unbounded from below with respect to 𝑥_𝑖. Secondly, the nonlinear term 𝑓 involves fractional derivatives of unknown functions. Thirdly, the boundary conditions involve fractional derivatives of unknown functions which is a more general case, and include the two-point, three-point, multipoint, and some nonlocal problems as special cases of (1.1).

2. Preliminaries and Lemmas

In this section, we give some preliminaries and lemmas.

Definition 2.1. Let 𝐸 be a real Banach space. A nonempty closed convex set 𝑃⊂𝐸 is called a cone of 𝐸 if it satisfies the following two conditions:
(1) 𝑥∈𝑃,𝜎>0 implies 𝜎𝑥∈𝑃;
(2) 𝑥∈𝑃,−𝑥∈𝑃 implies 𝑥=𝜃.

Definition 2.2. An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.
Let 𝐸 be a real Banach space, 𝐸^∗ the dual space of 𝐸, 𝑃 a total cone in 𝐸, that is, 𝐸=𝑃−𝑃, and 𝑃^∗ the dual cone of 𝑃.

Lemma 2.3 (Deimling [25]). Let 𝐿∶𝐸→𝐸 be a continuous linear operator, 𝑃 a total cone, and 𝐿(𝑃)⊂𝑃. If there exist 𝜓∈𝐸⧵(−𝑃) and a positive constant 𝑐 such that 𝑐𝐿(𝜓)≥𝜓, then the spectral radius 𝑟(𝐿)≠0 and has a positive eigenfunction corresponding to its first eigenvalue 𝜆=𝑟(𝐿)⁻¹.

Lemma 2.4 (see [25]). Let 𝑃 be a cone of the real Banach space 𝐸, and Ω a bounded open subsets of 𝐸. Suppose that 𝑇∶Ω∩𝑃→𝑃 is a completely continuous operator. If there exists 𝑥₀∈𝑃⧵{𝜃} such that 𝑥−𝑇𝑥≠𝜇𝑥₀,∀𝑥∈𝜕Ω∩𝑃,𝜇≥0, then the fixed-point index 𝑖(𝑇,Ω∩𝑃,𝑃)=0.
Let 𝐿∶𝐸→𝐸 be a completely continuous linear positive operator with the spectral radius 𝑟₁≠0. On account of Lemma 2.3, there exist 𝜑₁∈𝑃⧵{𝜃} and 𝑔₁∈𝑃^∗⧵{𝜃} such that 𝑟𝐿₁𝜑₁=𝑟₁𝜑₁,𝐿^∗𝑔₁=𝑔₁,(2.1) where 𝐿^∗ is the dual operator of 𝐿. Choose a number 𝛿>0 and let 𝑃𝑔₁,𝛿=𝑢∈𝑃∶𝑔₁(𝑢)≥𝛿|||𝑢|||,(2.2) then 𝑃(𝑔₁,𝛿) is a cone in 𝐸.

Lemma 2.5 (see [24]). Suppose that the following conditions are satisfied. (A1) 𝑇∶𝐸→𝑃 is a continuous operator satisfying lim_{‖𝑢‖→+∞}‖𝑇𝑢‖‖𝑢‖=0;(2.3) (A2) 𝐹∶𝐸→𝐸 is a bounded continuous operator and there exists 𝑢₀∈𝐸 such that 𝐹𝑢+𝑢₀+𝑇𝑢∈𝑃, for all 𝑢∈𝐸; (A3) 𝑟₁>0 and there exist 𝑣₀∈𝐸 and 𝜂>0 such that 𝐿𝐹𝑢≥𝑟₁(1+𝜂)𝐿𝑢−𝐿𝑇𝑢−𝑣₀,∀𝑢∈𝐸.(2.4)Let 𝐴=𝐿𝐹, then there exists 𝑅>0 such that deg𝐼−𝐴,𝐵_𝑅,𝜃=0,(2.5) where 𝐵_𝑅={𝑢∈𝐸∶‖𝑢‖<𝑅} is the open ball of radius 𝑅 in 𝐸.

Remark 2.6. If the operator 𝑇 which satisfies the conditions of Lemma 2.5 is a null operator, then Lemma 2.5 turns into Theorem  1 in [20]. On the other hand, if the operator 𝑇 in Lemma 2.5 is such that there exist constants 𝛼∈(0,1) and 𝑁>0 satisfying ||𝑇𝑢||≤𝑁||𝑢||^𝛼 for all 𝑢∈𝐸, then Lemma 2.5 turns into Theorem  2.1 in [22] or Theorem  1 in [21]. So Lemma 2.5 is an improvement of the results of paper [20–22].
Now we present the necessary definitions from fractional calculus theory. These definitions can be found in some recent literatures, for example, [26, 27].

Definition 2.7 (see [26, 27]). The Riemann-Liouville fractional integral of order 𝛼>0 of a function 𝑥∶(0,+∞)→ℝ is given by 𝐼^𝛼𝑥(𝑡)=1Γ(𝛼)^𝑡0(𝑡−𝑠)^𝛼−1𝑥(𝑠)𝑑𝑠(2.6) provided that the right-hand side is pointwisely defined on (0,+∞).

Definition 2.8 (see [26, 27]). The Riemann-Liouville fractional derivative of order 𝛼>0 of a function 𝑥∶(0,+∞)→ℝ is given by 𝒟^𝛼𝑥(𝑡)=1Γ(𝑛−𝛼)𝑑𝑑𝑡^𝑛^𝑡0(𝑡−𝑠)^{𝑛−𝛼−1}𝑥(𝑠)𝑑𝑠,(2.7) where 𝑛=[𝛼]+1, and [𝛼] denotes the integer part of the number 𝛼, provided that the right-hand side is pointwisely defined on (0,+∞).

Remark 2.9. If 𝑥,𝑦∶(0,+∞)→ℝ with order 𝛼>0, then 𝒟^𝛼(𝑥(𝑡)+𝑦(𝑡))=𝒟^𝛼𝑥(𝑡)+𝒟^𝛼𝑦(𝑡).(2.8)

Lemma 2.10 (see [27]). (1) If 𝑥∈𝐿¹(0,1),𝜌>𝜎>0, then 𝐼^𝜌𝐼^𝜎𝑥(𝑡)=𝐼^𝜌+𝜎𝑥(𝑡),𝒟^𝜎𝐼^𝜌𝑥(𝑡)=𝐼^𝜌−𝜎𝑥(𝑡),𝒟^𝜎𝐼^𝜎𝑥(𝑡)=𝑥(𝑡).(2.9)
(2) If 𝜌>0,𝜎>0, then 𝒟^𝜌𝑡^𝜎−1=Γ(𝜎)Γ(𝜎−𝜌)𝑡^{𝜎−𝜌−1}.(2.10)

Lemma 2.11 (see [27]). Assume that 𝑥∈𝐶(0,1)∩𝐿¹(0,1) with a fractional derivative of order 𝛼>0. Then 𝐼^𝛼𝒟^𝛼𝑥(𝑡)=𝑥(𝑡)+𝑐₁𝑡^𝛼−1+𝑐₂𝑡^𝛼−2+⋯+𝑐_𝑛𝑡^𝛼−𝑛,(2.11) where 𝑐_𝑖∈ℝ(𝑖=1,2,…,𝑛), 𝑛 is the smallest integer greater than or equal to 𝛼.

Noticing that 2<𝛼−𝜇_𝑛−1≤𝑛−𝜇_𝑛−1<3, let 𝑘(𝑡,𝑠)=⎧⎪⎪⎨⎪⎪⎩(𝑡(1−𝑠))^{𝛼−𝜇𝑛−1−1}−(𝑡−𝑠)^{𝛼−𝜇𝑛−1−1}(𝑡(1−𝑠))Γ𝛼−_{𝛼−𝜇𝑛−1−1}𝜇_𝑛−1,0≤𝑠≤𝑡≤1,Γ𝛼−𝜇_𝑛−1,0≤𝑡≤𝑠≤1,(2.12) by [28], for 𝑡,𝑠∈[0,1], one has 𝑡^{𝛼−𝜇𝑛−1−1}(1−𝑡)𝑠(1−𝑠)^{𝛼−𝜇𝑛−1−1}Γ𝛼−𝜇_𝑛−1≤𝑘(𝑡,𝑠)≤𝑠(Γ1−𝛼𝑠−)𝜇_𝑛−1^{𝛼−𝜇𝑛−1−1}.(2.13)

Lemma 2.12. If 2<𝛼−𝜇_𝑛−1<3 and 𝜌∈𝐿¹[0,1], then the boundary-value problem 𝑤(0)=𝑤_𝒟^{𝛼−𝜇𝑛−1}𝑤(𝑡)+𝜌(𝑡)=0,(0)=0,𝑤(1)=^𝑝−2_𝑗=1𝑎_𝑗𝑤𝜉_𝑗,(2.14) has the unique solution 𝑤(𝑡)=¹⁰𝐾(𝑡,𝑠)𝜌(𝑠)𝑑𝑠,(2.15) where 𝐾(𝑡,𝑠)=𝑘(𝑡,𝑠)+𝑡^{𝛼−𝜇𝑛−1−1}1−∑^{𝑝−2𝑗=1}𝑎_𝑗𝜉^𝛼−𝜇_𝑛−1^{𝑗−1𝑝−2}_𝑗=1𝑎_𝑗𝑘𝜉_𝑗,𝑠,(2.16) is the Green function of the boundary-value problem (2.14).

Proof. By applying Lemma 2.11, we may reduce (2.14) to an equivalent integral equation: 𝑤(𝑡)=−𝐼^{𝛼−𝜇𝑛−1}𝜌(𝑡)+𝑐₁𝑡^{𝛼−𝜇𝑛−1−1}+𝑐₂𝑡^{𝛼−𝜇𝑛−1−2}+𝑐₃𝑡^{𝛼−𝜇𝑛−1−3},𝑐₁,𝑐₂,𝑐₃∈ℝ.(2.17) Note that 𝑤(0)=𝑤′(0)=0 and (2.17), we have 𝑐₂=𝑐₃=0. Consequently the general solution of (2.14) is 𝑤(𝑡)=−𝐼^{𝛼−𝜇𝑛−1}𝜌(𝑡)+𝑐₁𝑡^{𝛼−𝜇𝑛−1−1}.(2.18) By (2.18) and Lemma 2.10, we have 𝑤(𝑡)=−𝐼^{𝛼−𝜇𝑛−1}𝜌(𝑡)+𝑐₁𝑡^{𝛼−𝜇𝑛−1−1}=−^𝑡0(𝑡−𝑠)^{𝛼−𝜇𝑛−1−1}Γ𝛼−𝜇_𝑛−1𝜌(𝑠)𝑑𝑠+𝑐¹𝑡^{𝛼−𝜇𝑛−1−1}.(2.19) So, 𝑤(1)=−¹⁰(1−𝑠)^{𝛼−𝜇𝑛−1−1}Γ𝛼−𝜇_𝑛−1𝜌(𝑠)𝑑𝑠+𝑐¹,(2.20) and for 𝑗=1,2,…,𝑝−2, 𝑤𝜉_𝑗=−^𝜉𝑗₀𝜉_𝑗−𝑠^{𝛼−𝜇𝑛−1−1}Γ𝛼−𝜇_𝑛−1𝜌(𝑠)𝑑𝑠+𝑐¹𝜉^{𝛼−𝜇𝑛−1𝑗−1}.(2.21) By 𝑤(1)=∑^{𝑝−2𝑗=1}𝑎_𝑗𝑤(𝜉_𝑗), combining with (2.20) and (2.21), we obtain 𝑐₁=∫¹⁰(1−𝑠)^{𝛼−𝜇𝑛−1−1}𝜌(𝑠)𝑑𝑠−∑^{𝑝−2𝑗=1}𝑎_𝑗∫^𝜉𝑗₀𝜉_𝑗−𝑠^{𝛼−𝜇𝑛−1−1}𝜌(𝑠)𝑑𝑠Γ𝛼−𝜇_𝑛−11−∑^{𝑝−2𝑗=1}𝑎_𝑗𝜉^{𝛼−𝜇𝑛−1𝑗−1}.(2.22) So, the unique solution of problem (2.14) is 𝑤(𝑡)=−^𝑡0(𝑡−𝑠)^{𝛼−𝜇𝑛−1−1}+Γ𝑡𝛼−𝜇_𝑛−1𝜌(𝑠)𝑑𝑠_{𝛼−𝜇𝑛−1−1}1−∑^{𝑝−2𝑗=1}𝑎_𝑗𝜉^{𝛼−𝜇𝑛−1𝑗−1}¹⁰(1−𝑠)^{𝛼−𝜇𝑛−1−1}Γ𝛼−𝜇_𝑛−1𝜌(𝑠)𝑑𝑠−^𝑝−2_𝑗=1𝑎_𝑗^𝜉𝑗₀𝜉_𝑗−𝑠^{𝛼−𝜇𝑛−1−1}=−_𝑡0Γ𝛼−𝜇_𝑛−1𝜌(𝑠)𝑑𝑠(𝑡−𝑠)^{𝛼−𝜇𝑛−1−1}Γ𝛼−𝜇_𝑛−1𝜌(𝑠)𝑑𝑠+¹⁰(1−𝑠)^{𝛼−𝜇𝑛−1−1}𝑡^{𝛼−𝜇𝑛−1−1}+𝑡Γ𝛼−𝜇_𝑛−1𝜌(𝑠)𝑑𝑠_{𝛼−𝜇𝑛−1−1}1−∑^{𝑝−2𝑗=1}𝑎_𝑗𝜉^𝛼−𝜇_𝑛−1^{𝑗−1𝑝−2}_𝑗=1𝑎_𝑗¹⁰(1−𝑠)^{𝛼−𝜇𝑛−1−1}𝜉^{𝛼−𝜇𝑛−1𝑗−1}−𝑡Γ𝛼−𝜇_𝑛−1𝜌(𝑠)𝑑𝑠_{𝛼−𝜇𝑛−1−1}1−∑^{𝑝−2𝑗=1}𝑎_𝑗𝜉^𝛼−𝜇_𝑛−1^{𝑗−1𝑝−2}_𝑗=1𝑎_𝑗^𝜉𝑗₀𝜉_𝑗−𝑠^{𝛼−𝜇𝑛−1−1}=Γ𝛼−𝜇_𝑛−1𝜌(𝑠)𝑑𝑠₁₀⎛⎜⎜⎝𝑘(𝑡,𝑠)+𝑡_{𝛼−𝜇𝑛−1−1}1−∑^{𝑝−2𝑗=1}𝑎_𝑗𝜉^𝛼−𝜇_𝑛−1^{𝑗−1𝑝−2}_𝑗=1=𝑎_𝑗𝑘𝜉_𝑗,𝑠⎞⎟⎟⎠𝜌(𝑠)𝑑𝑠₁₀𝐾(𝑡,𝑠)𝜌(𝑠)𝑑𝑠.(2.23) The proof is completed.

Lemma 2.13. The function 𝐾(𝑡,𝑠) has the following properties:
(1) 𝐾(𝑡,𝑠)>0,𝑓𝑜𝑟𝑡,𝑠∈(0,1);
(2)  𝐾(𝑡,𝑠)≤𝑀𝑠(1−𝑠)^{𝛼−𝜇𝑛−1−1},𝑓𝑜𝑟𝑡,𝑠∈[0,1], where 𝑀=1Γ𝛼−𝜇_𝑛−1⎛⎜⎜⎝1+∑^{𝑝−2𝑗=1}𝑎_𝑗1−∑^{𝑝−2𝑗=1}𝑎_𝑗𝜉^{𝛼−𝜇𝑛−1𝑗−1}⎞⎟⎟⎠.(2.24)

Proof. It is obvious that (1) holds. In the following, we will prove (2). In fact, by (2.13), we have 𝐾(𝑡,𝑠)=𝑘(𝑡,𝑠)+𝑡^{𝛼−𝜇𝑛−1−1}1−∑^{𝑝−2𝑗=1}𝑎_𝑗𝜉^𝛼−𝜇_𝑛−1^{𝑗−1𝑝−2}_𝑗=1≤𝑠(1−𝑠)_{𝛼−𝜇𝑛−1−1}𝑎_𝑗𝑘𝜉_𝑗,𝑠Γ𝛼−𝜇_𝑛−1+∑^{𝑝−2𝑗=1}𝑎_𝑗1−∑^{𝑝−2𝑗=1}𝑎_𝑗𝜉^{𝛼−𝜇𝑛−1𝑗−1}𝑠(1−𝑠)^{𝛼−𝜇𝑛−1−1}≤⎛⎜⎜⎝1+∑_{𝑝−2𝑗=1}Γ𝛼−𝜇_𝑛−1𝑎_𝑗1−∑^{𝑝−2𝑗=1}𝑎_𝑗𝜉^{𝛼−𝜇𝑛−1𝑗−1}⎞⎟⎟⎠𝑠(1−𝑠)^𝛼−𝜇_𝑛−1⁻¹Γ𝛼−𝜇_𝑛−1.(2.25) This completes the proof.

Now let us consider the following modified problems of the BVP (1.1): −𝒟^{𝛼−𝜇𝑛−1}𝑢𝑢((0𝑡))==0𝑝,(𝑡)𝑓𝑢(𝑡_,𝐼^𝜇𝑛−1𝑢(𝑡),𝐼^{𝜇𝑛−1−𝜇1}𝑢(𝑡),…,𝐼^{𝜇𝑛−1−𝜇𝑛−2}𝑢(𝑡),𝑢(𝑡)),(0)=0,𝒟^{𝜇−𝜇𝑛−1}𝑢(1)=^𝑝−2_𝑗=1𝑎_𝑗𝒟^{𝜇−𝜇𝑛−1}𝑢𝜉_𝑗.(2.26)

Lemma 2.14. Let 𝑥(𝑡)=𝐼^𝜇𝑛−1𝑢(𝑡),𝑢(𝑡)∈𝐶[0,1], Then (1.1) can be transformed into (2.26). Moreover, if 𝑢∈𝐶([0,1],[0,+∞) is a solution of problem (2.26), then, the function 𝑥(𝑡)=𝐼^𝜇𝑛−1𝑢(𝑡) is a positive solution of problem (1.1).

Proof. Substituting 𝑥(𝑡)=𝐼^𝜇𝑛−1𝑢(𝑡) into (1.1), by the definition of the Riemann-Liouville fractional derivative and Lemmas 2.10 and 2.11, we obtain that 𝒟^𝛼𝑥(𝑡)=𝑑^𝑛𝑑𝑡^𝑛𝐼^𝑛−𝛼𝑥(𝑡)=𝑑^𝑛𝑑𝑡^𝑛=𝑑𝐼^𝑛−𝛼𝐼^𝜇𝑛−1𝑢(𝑡)_𝑛𝑑𝑡^𝑛𝒟𝐼^{𝑛−𝛼+𝜇𝑛−1}𝑢(𝑡)=𝒟^{𝛼−𝜇𝑛−1}𝑢(𝑡),_𝜇1𝒟𝑥(𝑡)=𝒟^𝜇1𝐼^𝜇𝑛−1𝑢(𝑡)=𝐼^{𝜇𝑛−1−𝜇1}𝑢(𝑡),_𝜇2𝒟𝑥(𝑡)⋮=𝒟^𝜇2𝐼^𝜇𝑛−1𝑢(𝑡)=𝐼^{𝜇𝑛−1−𝜇2}𝑢⋮_𝜇𝑛−2(𝑡),𝒟𝑥(𝑡)=𝒟^𝜇𝑛−2𝐼^𝜇𝑛−1𝑢(𝑡)=𝐼^{𝜇𝑛−1−𝜇𝑛−2}𝑢(𝑡),_𝜇𝑛−1𝒟𝑥(𝑡)=𝒟^𝜇𝑛−1𝐼^𝜇𝑛−1𝑢(𝑡)=𝑢(𝑡),_{𝜇𝑛−1+1}𝑥(𝑡)=𝒟^𝜇𝑛−1𝐼^𝜇𝑛−1𝑢(𝑡)^=𝑢^(𝑡).(2.27) Also, we have 𝒟^𝜇𝑛−1𝑥(0)=𝑢(0)=0,𝒟^{𝜇𝑛−1+1}𝑥(0)=𝑢′(0)=0, and it follows from 𝒟^𝜇𝑛−1𝑥(𝑡)=𝑢(𝑡) that 𝑢(1)=∑^{𝑝−2𝑗=1}𝑎_𝑗𝑢(𝜉_𝑗). Hence, by 𝑥(𝑡)=𝐼^𝜇𝑛−1𝑢(𝑡),𝑢∈𝐶[0,1], (1.1) is transformed into (2.14).
Now, let 𝑢∈𝐶([0,1,0,+∞)) be a solution of problem (2.26). Then, by Lemma 2.10, (2.26), and (2.27), one has −𝒟^𝛼𝑥(𝑡)=−𝑑^𝑛𝑑𝑡^𝑛𝐼^𝑛−𝛼𝑥(𝑡)=−𝑑^𝑛𝑑𝑡^𝑛=−𝑑_𝑛𝐼^𝑛−𝛼𝐼^𝜇𝑛−1𝑢(𝑡)𝑑𝑡^𝑛=𝑝(𝑡)𝑓𝐼(𝑡,𝐼^{𝑛−𝛼+𝜇𝑛−1}_𝜇𝑛−1𝑢(𝑡)=−𝒟^{𝛼−𝜇𝑛−1}𝑢(𝑡)=𝑝(𝑡)𝑓(𝑡,𝑥(𝑡),𝑢𝒟(𝑡)_𝜇1,𝐼^{𝜇𝑛−1−𝜇1}𝑢(𝑡),…,𝐼^{𝜇𝑛−1−𝜇𝑛−2}𝑢(𝑡),𝑢(𝑡))𝑥(𝑡),𝒟^𝜇2𝑥(𝑡),…,𝒟^𝜇𝑛−1𝑥(𝑡)),0<𝑡<1.(2.28) Noticing that 𝐼^𝛼𝑢(𝑡)=1Γ(𝛼)^𝑡0(𝑡−𝑠)^𝛼−1𝑢(𝑠)𝑑𝑠,(2.29) which implies that 𝐼^𝛼𝑢(0)=0, from (2.27), for 𝑖=1,2,…,𝑛−1, we have 𝒟^𝜇𝑖𝑥(0)=0,𝒟^{𝜇𝑛−1+1}𝑥(0)=0,𝒟^𝜇𝑛−1𝑥(1)=^𝑝−2_𝑗=1𝑎_𝑗𝒟^𝜇𝑛−1𝑥𝜉_𝑗.(2.30)
Moreover, it follows from the monotonicity and property of 𝐼^𝜇𝑛−1 that 𝐼^𝜇𝑛−1𝑢∈𝐶([0,1],[0,+∞)).(2.31)
Consequently, 𝑥(𝑡)=𝐼^𝜇𝑛−1𝑢(𝑡) is a positive solution of problem (1.1).

In the following let us list some assumptions to be used in the rest of this paper.

(H1)  𝑝∶(0,1)→[0,+∞) is continuous, 𝑝(𝑡)≢0 on any subinterval of (0,1), and ¹⁰𝑝(𝑠)𝑑𝑠<+∞.(2.32)

(H2)  𝑓∶[0,1]×ℝ^𝑛→(−∞,+∞) is continuous.

In order to use Lemma 2.5, let 𝐸=𝐶[0,1] be our real Banach space with the norm ‖𝑢‖=max_𝑡∈[0,1]|𝑢(𝑡)| and 𝑃={𝑢∈𝐶[0,1]∶𝑢(𝑡)≥0,forall𝑡∈[0,1]}, then 𝑃 is a total cone in 𝐸.

Define two linear operators 𝐿,𝐽∶𝐶[0,1]→𝐶[0,1] by ((𝐿𝐽𝑢𝑢))((𝑡𝑡))==¹⁰₁₀𝐾(𝑡,𝑠)𝑝(𝑠)𝑢(𝑠)𝑑𝑠,𝑡∈[0,1],𝑢∈𝐶[0,1],(2.33)𝐾(𝑠,𝑡)𝑝(𝑠)𝑢(𝑠)𝑑𝑠,𝑡∈[0,1],𝑢∈𝐶[0,1],(2.34) and define a nonlinear operator 𝐴∶𝐶[0,1]→𝐶[0,1] by (𝐴𝑢)(𝑡)=¹⁰𝐾(𝑡,𝑠)𝑝(𝑠)𝑓(𝑡,𝐼^𝜇𝑛−1𝑢(𝑠),𝐼^{𝜇𝑛−1−𝜇1}𝑢(𝑠),…,𝐼^{𝜇𝑛−1−𝜇𝑛−2}𝑢(𝑠),𝑢(𝑠))𝑑𝑠,𝑡∈[0,1].(2.35)

Lemma 2.15. Assume (H1) holds. Then(i)𝐿,𝐽∶𝐶[0,1]→𝐶[0,1] are completely continuous positive linear operators with the first eigenvalue 𝑟>0 and 𝜆>0, respectively.(ii)𝐿 satisfies 𝐿(𝑃)⊂𝑃(𝑔₁,𝛿).

Proof. (i) By using the similar method of paper [22], it is easy to know that 𝐿,𝐽∶𝐶[0,1]→𝐶[0,1] are completely continuous positive linear operators. In the following, by using the Krein-Rutmann’s theorem, we prove that 𝐿,𝐽 have the first eigenvalue 𝑟>0 and 𝜆>0, respectively.
In fact, it is obvious that there is 𝑡₁∈(0,1) such that 𝐾(𝑡₁,𝑡₁)𝑝(𝑡₁)>0. Thus there exists [𝑎,𝑏]⊂(0,1) such that 𝑡₁∈(𝑎,𝑏) and 𝐾(𝑡,𝑠)𝑝(𝑠)>0 for all 𝑡,𝑠∈[𝑎,𝑏]. Choose 𝜓∈𝑃 such that 𝜓(𝑡₁)>0 and 𝜓(𝑡)=0 for all 𝑡∉[𝑎,𝑏]. Then for 𝑡∈[𝑎,𝑏], (𝐿𝜓)(𝑡)=¹⁰𝐾(𝑡,𝑠)𝑝(𝑠)𝜓(𝑠)𝑑𝑠≥^𝑏𝑎𝐾(𝑡,𝑠)𝑝(𝑠)𝜓(𝑠)𝑑𝑠>0.(2.36)
So there exists 𝜈>0 such that 𝜈(𝐿𝜓)(𝑡)≥𝜓(𝑡) for 𝑡∈[0,1]. It follows from Lemma 2.3 that the spectral radius 𝑟₁≠0. Thus corresponding to 𝑟=𝑟¹⁻¹, the first eigenvalue of 𝐿, and 𝐿 has a positive eigenvector 𝜑₁ such that 𝑟𝐿𝜑₁=𝜑₁.(2.37)
In the same way, 𝐽 has a positive first eigenvalue 𝜆 and a positive eigenvector 𝜑₂ corresponding to the first eigenvalue 𝜆, which satisfy 𝜆𝐽𝜑₂=𝜑₂.(2.38)
(ii) Notice that 𝐾(𝑡,0)=𝐾(𝑡,1)≡0 for 𝑡∈[0,1], by 𝜆𝐽𝜑₂=𝜑₂ and (2.12)–(2.16), we have 𝜑₂(0)=𝜑₂(1)=0. This implies that 𝜑^2(0)>0 and 𝜑^2(1)<0 (see [29]). Define a function 𝜒 on [0,1] by 𝜒(𝑠)=⎧⎪⎪⎨⎪⎪⎩𝜑^2𝜑(0),𝑠=0,₂(𝑠)(−1𝜑−_2𝑠)𝑠,0<𝑠<1,(1),𝑠=1.(2.39)
Then 𝜒 is continuous on [0,1] and 𝜒(𝑠)>0 for all 𝑠∈[0,1]. So, there exist 𝛿₁,𝛿₂>0 such that 𝛿₁≤𝜒(𝑠)≤𝛿₂ for all 𝑠∈[0,1]. Thus 𝛿₁(1−𝑠)𝑠≤𝜑₂(𝑠)≤𝛿₂(1−𝑠)𝑠,(2.40) for all 𝑠∈[0,1].
Let 𝐿^∗ be the dual operator of 𝐿, we will show that there exists 𝑔₁∈𝑃^∗⧵{𝜃} such that 𝜆𝐿^∗𝑔₁=𝑔₁.(2.41) In fact, let 𝑔₁(𝑢)=¹⁰𝑝(𝑡)𝜑₂(𝑡)𝑢(𝑡)𝑑𝑡∀𝑢∈𝐸.(2.42) Then by (H1) and (2.40), we have ¹⁰𝑝(𝑡)𝜑₂(𝑡)𝑢(𝑡)𝑑𝑡≤𝛿₂‖𝑢‖¹⁰𝑡(1−𝑡)𝑝(𝑡)𝑑𝑡≤𝛿₂‖𝑢‖¹⁰𝑝(𝑡)𝑑𝑡<+∞,(2.43) which implies that 𝑔₁ is well defined. We state that 𝑔₁ of (2.42) satisfies (2.41). In fact, by (2.40), (2.41), and interchanging the order of integration, for any 𝑠,𝑡∈[0,1], we have 𝜆⁻¹𝑔₁(𝑢)==¹⁰𝑝(𝑡)𝜆⁻¹𝜑₂(𝑡)𝑢(𝑡)𝑑𝑡=¹⁰𝑝(𝑡)𝐽𝜑₂(𝑡)𝑢(𝑡)𝑑𝑡₁₀=𝑝(𝑡)𝑢(𝑡)¹⁰𝐾(𝑠,𝑡)𝑝(𝑠)𝜑₂(𝑠)𝑑𝑠𝑑𝑡₁₀=𝑝(𝑠)𝜑₂(𝑠)¹⁰𝐾(𝑠,𝑡)𝑝(𝑡)𝑢(𝑡)𝑑𝑡𝑑𝑠₁₀=𝑔₁𝑝(𝑠)𝜑₂(𝑠)(𝐿𝑢)(𝑠)𝑑𝑠(𝐿𝑢)=𝐿^∗𝑔₁(𝑢)∀𝑢∈𝐸.(2.44) So (2.41) holds.
In the following we prove that 𝐿(𝑃)⊂𝑃(𝑔₁,𝛿). In fact, by (2.41) and 𝛼−𝜇_𝑛−1−1>1, we have 𝜑₂(𝑠)≥𝛿₁(1−𝑠)𝑠≥𝛿₁(1−𝑠)^{𝛼−𝜇𝑛−1−1}𝑠≥𝛿₁𝑀⁻¹𝐾(𝑡,𝑠),𝑡,𝑠∈[0,1].(2.45) Take 𝛿=𝛿₁𝑀⁻¹𝜆⁻¹>0 in (2.2). For any 𝑢∈𝑃, by (2.44), (2.45), we have 𝑔₁(𝐿𝑢)=𝜆⁻¹𝑔₁(𝑢)=𝜆⁻¹¹⁰≥𝛿₁𝑝(𝑠)𝜑₂(𝑠)𝑢(𝑠)𝑑𝑠𝑀⁻¹𝜆⁻¹¹⁰𝐾(𝑡,𝑠)𝑝(𝑠)𝑢(𝑠)𝑑𝑠=𝛿(𝐿𝑢)(𝑡)∀𝑡∈[0,1].(2.46) Hence, 𝑔₁(𝐿𝑢)≥𝛿||𝐿𝑢||, that is, 𝐿(𝑃)⊂𝑃(𝑔₁,𝛿). The proof is completed.

3. Main Result

Theorem 3.1. Assume that (H1)(H2) hold, and the following conditions are satisfied.
(H3) There exist nonnegative continuous functions 𝑏,𝑐∶[0,1]→(0,+∞) and a nondecreasing continuous function ℎ∶ℝ^𝑛→[0,+∞) satisfying lim_{∑𝑛𝑖=1||𝑥𝑖||→+∞}ℎ𝑥₁,𝑥₂,…,𝑥_𝑛𝑓𝑡,𝑥₁∑^𝑛𝑖=1||𝑥_𝑖||=0,,𝑥₂,…,𝑥_𝑛≥−𝑏(𝑡)−𝑐(𝑡)ℎ𝑥₁,𝑥₂,…,𝑥_𝑛,∀𝑥_𝑖∈ℝ.(3.1)
(H4) 𝑓 also satisfies liminf_{∑𝑛𝑥𝑖𝑖=1→+∞∑𝑛−1𝑖=1𝑥𝑖≥0}𝑓𝑡,𝑥₁,𝑥₂,…,𝑥_𝑛∑^𝑛𝑖=1𝑥_𝑖>𝜆limsup_{∑𝑛𝑖=1|𝑥𝑖|→0}||𝑓𝑡,𝑥₁,𝑥₂,…,𝑥_𝑛||∑^𝑛𝑖=1||𝑥_𝑖||<𝜆,(3.2) uniformly on 𝑡∈[0,1], where 𝜆 is the first eigenvalue of the operator 𝐽 defined by (2.34).
Then the singular fractional-order boundary-value problem (1.1) has at least one nontrivial solution.

Proof. According to Lemma 2.15, 𝐿 satisfies 𝐿(𝑃)⊂𝑃(𝑔₁,𝛿). Let (𝑇𝑢)(𝑡)=𝑐ℎ(𝐼^𝜇𝑛−1𝑢(𝑡),𝐼^{𝜇𝑛−1−𝜇1}𝑢(𝑡),…,𝐼^{𝜇𝑛−1−𝜇𝑛−2}𝑢(𝑡),𝑢(𝑡))for𝑢∈𝐸,(3.3) where 𝑐=max_𝑡∈[0,1]𝑐(𝑡). It follows from (H3) that 𝑇∶𝐸→𝑃 is a continuous operator. Note that 𝐼^𝜇𝑛−1𝑢(𝑡)=^𝑡0(𝑡−𝑠)^{𝜇𝑛−1−1}𝑢(𝑠)𝐼_{𝜇𝑛−1−𝜇𝑖}Γ𝜇_𝑛−1𝑑𝑠≤Γ‖𝜇𝑢‖_𝑛−1,𝑢(𝑡)=^𝑡0(𝑡−𝑠)^{𝜇𝑛−1−𝜇𝑖−1}𝑢(𝑠)Γ𝜇_𝑛−1−𝜇_𝑖𝑑𝑠≤Γ𝜇‖𝑢_𝑛−1‖−𝜇_𝑖,𝑖=1,2,…,𝑛−2.(3.4) Thus from the monotone assumption of ℎ on 𝑥_𝑖, we have ℎ(≤𝐼ℎ^𝜇𝑛−1𝑢(𝑡)‖,𝑢𝐼‖^{𝜇𝑛−1−𝜇1}𝑢(𝑡),…,𝐼^{𝜇𝑛−1−𝜇𝑛−2}𝑢(𝑡),𝑢(𝑡))Γ𝜇_𝑛−1,Γ𝜇‖𝑢_𝑛−1‖−𝜇₁,…,Γ𝜇‖_𝑛−1𝑢−‖𝜇_𝑛−2,‖𝑢‖,forany𝑢∈𝐸,𝑡∈[0,1],(3.5) which implies that ‖𝑇𝑢‖=≤𝑐mℎax_[]𝑡∈0,1{𝑐‖ℎ𝑢(‖𝐼^𝜇𝑛−1𝑢(𝑡),𝐼^{𝜇𝑛−1−𝜇1}𝑢(𝑡),…,𝐼^{𝜇𝑛−1−𝜇𝑛−2}𝑢(𝑡),𝑢(𝑡))}Γ𝜇_𝑛−1,Γ𝜇‖𝑢_𝑛−1‖−𝜇₁,…,Γ𝜇‖_𝑛−1𝑢−‖𝜇_𝑛−2,‖𝑢‖forany𝑢∈𝐸.(3.6) Let 𝜏=1Γ𝜇_𝑛−1+Γ𝜇1_𝑛−1−𝜇₁+⋯+Γ𝜇_𝑛−11−𝜇_𝑛−2+1,(3.7) then lim_{‖𝑢‖→+∞}‖𝑇𝑢‖‖𝑢‖≤lim^{‖𝑢‖→+∞}𝑐ℎ‖𝑢‖/Γ𝜇_𝑛−1,‖𝑢‖/Γ𝜇_𝑛−1−𝜇₁,…,‖𝑢‖/Γ𝜇_𝑛−1−𝜇_𝑛−2,‖𝑢‖=lim_{‖𝑢‖→+∞}‖𝑢‖𝑐𝜏ℎ‖𝑢‖/Γ𝜇_𝑛−1,‖𝑢‖/Γ𝜇_𝑛−1−𝜇₁,…,‖𝑢‖/Γ𝜇_𝑛−1−𝜇_𝑛−2,‖𝑢‖≤lim_{‖𝑢‖→+∞}𝜏‖𝑢‖𝑐𝜏ℎ‖𝑢‖/Γ𝜇_𝑛−1,‖𝑢‖/Γ𝜇_𝑛−1−𝜇₁,…,‖𝑢‖/Γ𝜇_𝑛−1−𝜇_𝑛−2,‖𝑢‖=𝑐𝜏lim‖𝑢‖_{∑𝑛𝑖=1||𝑥𝑖||→+∞}/Γ𝜇_𝑛−1+‖𝑢‖/Γ𝜇_𝑛−1−𝜇₁+⋯+‖𝑢‖/Γ𝜇_𝑛−1−𝜇_𝑛−2+‖𝑢‖ℎ𝑥₁,𝑥₂,…,𝑥_𝑛∑^𝑛𝑖=1||𝑥_𝑖||=0,(3.8) that is, lim_{‖𝑢‖→+∞}‖𝑇𝑢‖‖𝑢‖=0.(3.9) Hence 𝑇 satisfies condition (A1) in Lemma 2.5.
Next take 𝑢₀(𝑡)≡𝑏(𝑡), and (𝐹𝑢)(𝑡)=𝑓(𝑡,𝐼^𝜇𝑛−1𝑢(𝑡),𝐼^{𝜇𝑛−1−𝜇1}𝑢(𝑡),…,𝐼^{𝜇𝑛−1−𝜇𝑛−2}𝑢(𝑡),𝑢(𝑡)) for 𝑢∈𝐸. Then it follows from (H3) that 𝐹𝑢+𝑢₀+𝑇𝑢=+𝑓𝑏(𝑡(,𝑡𝐼)+𝑐^𝜇𝑛−1𝑢ℎ((𝑡𝐼),𝐼^{𝜇𝑛−1−𝜇1}_𝜇𝑛−1𝑢(𝑡),…,𝐼^{𝜇𝑛−1−𝜇𝑛−2}𝑢(𝑡),𝑢(𝑡))𝑢(𝑡),𝐼^{𝜇𝑛−1−𝜇1}𝑢(𝑡),…,𝐼^{𝜇𝑛−1−𝜇𝑛−2}𝑢(𝑡),𝑢(𝑡))≥0,(3.10) that yields 𝐹𝑢+𝑢₀+𝑇𝑢∈𝑃,∀𝑢∈𝐸,(3.11) namely, condition (A2) in Lemma 2.5 holds.
From (H4), there exists 𝜀>0 and a sufficiently large 𝑙₁>0 such that, for any 𝑡∈[0,1], 𝑓𝑡,𝑥₁,𝑥₂,…,𝑥_𝑛≥≥𝜆𝜆((11++𝜀𝜀))𝑥𝑥_1𝑛+𝑥₂+⋯+𝑥_𝑛,𝑥₁+𝑥₂+⋯+𝑥_𝑛>𝑙₁.(3.12) Combining (H3) with (3.12), there exists 𝑏₁≥0 such that 𝐹𝑢=≥𝑓𝜆((1𝑡,+𝐼𝜀)^𝜇𝑛−1𝑢𝑢((𝑡𝑡))−,𝐼𝑏^{𝜇𝑛−1−𝜇1}₁𝑢(𝑡),…,𝐼^{𝜇𝑛−1−𝜇𝑛−2}𝑢(𝑡),𝑢(𝑡))−𝑐ℎ(𝐼^𝜇𝑛−1𝑢(𝑡),…,𝐼^{𝜇𝑛−1−𝜇𝑛−2}𝑢(𝑡),𝑢(𝑡)),∀𝑢∈𝐸,(3.13) that is, F𝑢≥𝜆(1+𝜀)𝑢−𝑏₁−𝑇𝑢∀𝑢∈𝐸.(3.14) As 𝐿 is a positive linear operator, it follows from (3.14) that (𝐿𝐹𝑢)(𝑡)≥𝜆(1+𝜀)(𝐿𝑢)(𝑡)−𝐿𝑏₁−(𝐿𝑇𝑢)(𝑡),∀𝑡∈[0,1].(3.15) So condition (A3) in Lemma 2.5 holds. According to Lemma 2.5, there exists a sufficiently large number 𝑅>0 such that deg𝐼−𝐴,𝐵_𝑅,𝜃=0.(3.16)
On the other hand, it follows from (H4) that there exist 0<𝜀<1 and 0<𝑟<𝑅, for any 𝑡∈[0,1], such that ||𝑓𝑡,𝑥₁,𝑥₂,…,𝑥_𝑛||≤1−𝜀𝑀𝜏∫¹⁰𝑝(𝑠)𝑑𝑠||𝑥¹||𝑥₁||+||𝑥₂||+⋯+||𝑥_𝑛||,||+||𝑥₂||+⋯+||𝑥_𝑛||<𝑟.(3.17) Thus for any 𝑢∈𝐸 with ||𝑢||≤𝑟/𝜏≤𝑟≤𝑅, we have ||𝐼^𝜇𝑛−1𝑢(𝑡)||+≤||𝐼‖𝑢‖^{𝜇𝑛−1−𝜇1}𝑢(𝑡)||+⋯+||𝐼^{𝜇𝑛−1−𝜇𝑛−2}𝑢(𝑡)||+||𝑢(𝑡)||≤𝜏Γ‖𝑢𝜇‖≤_𝑛−1𝑟+.Γ𝜇‖𝑢_𝑛−1‖−𝜇₁+⋯+Γ𝜇‖_𝑛−1𝑢−‖𝜇_𝑛−2+‖𝑢‖(3.18) By (3.17), for any 𝑡∈[0,1], we have ||𝑓≤(𝑡,𝐼^𝜇𝑛−11𝑢−(𝑡)𝜀,𝐼^{𝜇𝑛−1−𝜇1}𝑢(𝑡),…,𝐼^{𝜇𝑛−1−𝜇𝑛−2}𝑢(𝑡),𝑢(𝑡))||𝑀𝜏∫¹⁰𝑝(𝑠)𝑑𝑠||𝐼^𝜇𝑛−1𝑢(𝑡)||+||𝐼^{𝜇𝑛−1−𝜇1}𝑢(𝑡)||+⋯+||𝐼^{𝜇𝑛−1−𝜇𝑛−2}𝑢(𝑡)||+||𝑢(𝑡)||.(3.19) Thus if there exist 𝑢₁∈𝜕𝐵_𝑟/𝜏 and 𝜇₁∈[0,1] such that 𝑢₁=𝜇₁𝐴𝑢₁, then by (3.19), we have 𝑔₁‖‖𝑢₁‖‖==𝑔𝜇₁₁𝜇₁‖‖𝐴𝑢₁‖‖=𝜇₁𝑔₁‖‖𝐴𝑢₁‖‖𝑔₁max^0≤𝑡≤1||||¹⁰≤1−𝜀𝐾(𝑡,𝑠)𝑝(𝑠)𝑓𝑠,𝐼^𝜇𝑛−1𝑢₁(𝑠),…,𝐼^{𝜇𝑛−1−𝜇𝑛−2}𝑢₁(𝑠),𝑢₁(𝑠)𝑑𝑠||||𝑀𝜏∫¹⁰𝑝(𝑠)𝑑𝑠𝑔¹¹⁰≤1−𝜀𝑀𝑝(𝑠)||𝐼^𝜇𝑛−1𝑢₁(𝑠)||+⋯+||𝐼^{𝜇𝑛−1−𝜇𝑛−2}𝑢₁(𝑠)||+||𝑢₁(𝑠)||𝑑𝑠𝑀𝜏∫¹⁰𝑝(𝑠)𝑑𝑠𝑀𝜏¹⁰𝑝(𝑠)𝑑𝑠𝑔₁‖‖𝑢₁‖‖=(1−𝜀)𝑔₁‖‖𝑢₁‖‖.(3.20) Therefore, 𝑔₁(‖𝑢₁‖)≤0.
But 𝜑₂(𝑡)>0 for all 𝑡∈(0,1) by the maximum principle, and 𝑢₁(𝑡) attains zero on isolated points by the Sturm theorem. Hence, from (2.42), 𝑔₁‖‖𝑢₁‖‖=¹⁰𝑝(𝑡)𝜑₂(𝑡)‖‖𝑢₁‖‖𝑑𝑡>0.(3.21) This is a contradiction. Thus 𝑢≠𝜇𝐴𝑢,∀𝑢∈𝜕𝐵_𝑟,𝜇∈[0,1].(3.22) It follows from the homotopy invariance of the Leray-Shauder degree that deg𝐼−𝐴,𝐵_𝑟/𝜏,𝜃=1.(3.23) By (3.16), (3.23), and the additivity of Leray-Shauder degree, we obtain deg𝐼−𝐴,𝐵_𝑅⧵𝐵_𝑟/𝜏,𝜃=deg𝐼−𝐴,𝐵_𝑅,𝜃−deg𝐼−𝐴,𝐵_𝑟/𝜏,𝜃=−1.(3.24) As a result, 𝐴 has at least one fixed point 𝑢 on 𝐵_𝑅⧵𝐵_𝑟/𝜏, namely, the BVP (1.1) has at least one nontrivial solution 𝑥(𝑡)=𝐼^𝜇𝑛−1𝑢(𝑡).

Corollary 3.2. Assume that (H1), (H2), and (H4) hold. If the following condition is satisfied.
(H*3) There exists a nonnegative continuous functions 𝑏∶[0,1]→(0,+∞) such that 𝑓𝑡,𝑥₁,𝑥₂,…,𝑥_𝑛≥−𝑏(𝑡),𝑓𝑜𝑟𝑎𝑛𝑦𝑡∈[0,1],∀𝑥_𝑖∈ℝ,(3.25) then the singular higher multipoint boundary-value problems (1.1) have at least one nontrivial solution.

Corollary 3.3. Assume that (H1), (H2), and (H4) hold. If the following condition is satisfied.
(H3**) There exist three constants 𝑏>0,𝑐>0, and 𝛼_𝑖∈(0,1) such that 𝑓𝑡,𝑥₁,𝑥₂,…,𝑥_𝑛≥−𝑏−𝑐^𝑛_𝑖=1||𝑥_𝑖||^𝛼𝑖,𝑓𝑜𝑟𝑎𝑛𝑦𝑡∈[0,1],∀𝑥_𝑖∈ℝ,(3.26) then the singular higher multipoint boundary-value problems (1.1) have at least one nontrivial solution.

Remark 3.4. Noticing that the Green function of the BVP (1.1) is not symmetrical, which implies that the existence of nontrivial solutions of the BVP (1.1) cannot be obtained by Theorem  2.1 in [22] and Theorem  1 in [20]. It is interesting that we construct a new linear operator 𝐽 instead of 𝐾 in paper [22] and use its first eigenvalue and its corresponding eigenfunction to seek a linear continuous functional 𝑔 of 𝑃. As a result, we overcome the difficulty caused by the nonsymmetry of the Green function. In [24], the nonlinearity does not contain derivatives and a stronger condition is required, that is, ℎ must be an even function; here we omit this stronger assumption.

Remark 3.5. The results of [20–22] is a special case of the Corollary 3.2 and Corollary 3.3 when 𝛼₁(𝑖=1,2,…,𝑛) are integer and the nonlinear term does not involve derivatives of unknown functions.

Acknowledgments

This project is supported financially by Scientific Research Project of Zhejiang Education Department (no. Y201016244), also by Scientific Research Project of Wenzhou (no. G20110004), Natural Science Foundation of Zhejiang Province (No. 2012C31025) and the National Natural Science Foundation of China (11071141, 11126231, 21207103), and the Natural Science Foundation of Shandong Province of China (ZR2010AM017).