Positive Solutions of Nonlocal Boundary Value Problem for High-Order Nonlinear Fractional 𝑞-Difference Equations

Abstract

We study the nonlinear 𝑞-difference equations of fractional order (𝐷^𝛼_𝑞𝑢)(𝑡)+𝑓(𝑡,𝑢(𝑡))=0, 0<𝑡<1, (𝐷^𝑖_𝑞𝑢)(0)=0, (𝐷^𝛽_𝑞𝑢)(1)=𝑎(𝐷^𝛽_𝑞𝑢)(𝜂), 0≤𝑖≤𝑛−2, where 𝐷^𝛼_𝑞 is the fractional 𝑞-derivative of the Riemann-Liouville type of order 𝛼, 𝑛−1<𝛼≤𝑛, 𝛼>2, 1≤𝛽≤𝑛−2, and 0≤𝑎≤1. We obtain the existence and multiplicity results of positive solutions by using some fixed point theorems. Finally, we give examples to illustrate the results.

1. Introduction

The 𝑞-difference calculus or quantum calculus is an old subject that was initially developed by Jackson [1, 2]. It is rich in history and in applications as the reader can find in the work by Ernst [3]. For some recent existence results on 𝑞-difference equations, see [4–7] and the references therein.

The fractional 𝑞-difference calculus had its origin in the works by Al-Salam and Agarwal. Henceforth, fractional 𝑞-difference equations have gained considerable importance due to their application in various sciences, such as physics, chemistry, aerodynamics, biology, economics, control theory, mechanics, electricity, signal and image processing, biophysics, blood flow phenomena, and fitting of experimental data. It has been a significant development in difference equations involving fractional 𝑞-derivatives; see [8–11] and references therein. As well known, fractional differential equations boundary value problems is currently under strong research, see [12–21] and references therein. In particular, in recent years, fractional 𝑞-difference boundary value problem (BVP) was in its infancy, and many people begin to study the existence of positive solutions for this kind of BVP; see [22–27] and references therein. However, there are few related results available. Lots of work and development should be done in the future.

Recently, in [16], Li et al. considered the BVP of nonlinear fractional difference equation 𝐷^𝛼₀₊𝑢(𝑡)((𝑡𝑢(0)+𝑓𝑡,𝑢))=0,0<𝑡<1,=0,𝐷^𝛽₀₊𝑢(1)=𝑎𝐷^𝛽₀₊𝑢(𝜉), (1) where 1<𝛼≤2, 0≤𝛽≤1, 0≤𝑎≤1, 𝜉∈(0,1), and 𝑓∶[0,1]×[0,∞)→[0,∞) satisfies Caratheodory type conditions.

More recently, in [23], Ferreira considered the BVP of fractional 𝑞-difference equation 𝐷^𝛼_𝑞𝑦(𝑥)(𝑥𝑦(0)=𝐷=−𝑓𝑥,𝑦),0<𝑥<1,_𝑞𝑦(0)=0,𝐷_𝑞𝑦(1)=𝛽≥0, (2) where 2<𝛼≤3 and 𝑓∶[0,1]×𝑅→𝑅 is a nonnegative continuous function.

Motivated by the work above, in this paper, we will discuss the following BVP: 𝐷^𝛼_𝑞𝑢(𝑡)((𝑡𝐷+𝑓𝑡,𝑢))=0,0<𝑡<1,_𝑖𝑞𝑢(0)=0,𝐷^𝛽_𝑞𝑢(1)=𝑎𝐷^𝛽_𝑞𝑢𝜂,0≤𝑖≤𝑛−2, (3) where 𝑛−1<𝛼≤𝑛(𝑛>2), 1≤𝛽≤𝑛−2, 0<𝜂<1, 0≤𝑎≤1, and 𝑓∶[0,1]×[0,∞)→[0,∞) satisfies Caratheodory type conditions. We discuss the existence of positive solutions for BVP(3) and obtain multiplicity results which extend and improve the known results by using some fixed point theorems.

2. Preliminary Results

In this section, we introduce definitions and preliminary facts which are used throughout this paper.

Let 𝑞∈(0,1) and define [𝑎]_𝑞=1−𝑞^𝑎1−𝑞,𝑎∈𝑅. (4)

The 𝑞-analogue of the power function (𝑎−𝑏)^𝑛 with 𝑛∈𝑁₀ is (𝑎−𝑏)⁽⁰⁾=1,(𝑎−𝑏)^(𝑛)=^𝑛−1_𝑘=0𝑎−𝑏𝑞^𝑘,𝑛∈𝑁,𝑎,𝑏∈𝑅. (5) More generally, if 𝛼∈𝑅, then (𝑎−𝑏)^(𝛼)=𝑎^𝛼𝑛−1_𝑛=0𝑎−𝑏𝑞^𝑛𝑎−𝑏𝑞^𝛼+𝑛. (6) Note that, if 𝑏=0, then 𝑎^(𝛼)=𝑎^𝛼. The 𝑞-gamma function is defined by Γ_𝑞(𝑥)=1−𝑞^(𝑥−1)1−𝑞^𝑥−1,𝑥∈𝑅⧵{0,−1,−2,…}, (7) and satisfies Γ_𝑞(𝑥+1)=[𝑥]_𝑞Γ_𝑞(𝑥).

Then, let us recall some basic concepts of 𝑞-calculus [28].

Definition 1. For 0<𝑞<1, we define the 𝑞-derivative of a real-value function 𝑓 as 𝐷_𝑞𝑓(𝑥)=𝑓(𝑥)−𝑓𝑞𝑥𝑥,𝐷1−𝑞_𝑞𝑓(0)=lim_𝑥→0𝐷_𝑞𝑓(𝑥). (8) Note that lim_𝑞→1⁻𝐷_𝑞𝑓(𝑥)=𝑓^′(𝑥).

Definition 2. The higher-order 𝑞-derivatives are defined inductively as 𝐷⁰_𝑞𝑓(𝑥)(𝑥),𝐷=𝑓_𝑛𝑞𝑓(𝑡)=𝐷_𝑞𝐷_𝑞^𝑛−1𝑓(𝑡),𝑛∈𝑁. (9)

Definition 3. The 𝑞-integral of a function 𝑓 in the interval [0,𝑏] is given by 𝐼_𝑞𝑓(𝑥)=^𝑥₀𝑓(𝑡)𝑑_𝑞𝑡=𝑥1−𝑞^∞_𝑛=0𝑓𝑥𝑞^𝑛𝑞^𝑛,𝑥∈[0,𝑏]. (10) If 𝑎∈[0,𝑏] and 𝑓 is defined in the interval [0,𝑏], its integral from 𝑎 to 𝑏 is defined by ^𝑏_𝑎𝑓(𝑡)𝑑_𝑞𝑡=^𝑏₀𝑓(𝑡)𝑑_𝑞𝑡−^𝑎₀𝑓(𝑡)𝑑_𝑞𝑡. (11) Similarly as done for derivatives, an operator 𝐼^𝑛_𝑞 can be defined; namely, 𝐼⁰_𝑞𝑓(𝑥)(𝑥),𝐼=𝑓_𝑛𝑞𝑓(𝑥)=𝐼_𝑞𝐼_𝑞^𝑛−1𝑓(𝑥),𝑛∈𝑁. (12)
Observe that 𝐷_𝑞𝐼_𝑞𝑓(𝑥)=𝑓(𝑥), (13) and if 𝑓 is continuous at 𝑥=0, then 𝐼_𝑞𝐷_𝑞𝑓(𝑥)=𝑓(𝑥)−𝑓(0).
We now point out three formulas (_𝑖𝐷_𝑞 denotes the derivative with respect to variable 𝑖): [𝑎(𝑡−𝑠)]^(𝛼)=𝑎^𝛼(𝑡−𝑠)^(𝛼),_𝑡𝐷_𝑞(𝑡−𝑠)^(𝛼)=[𝛼]_𝑞(𝑡−𝑠)^(𝛼−1),_𝑥𝐷_𝑞^𝑥₀𝑓(𝑥,𝑡)𝑑_𝑞𝑡=^𝑥_0𝑥𝐷_𝑞𝑓(𝑥,𝑡)𝑑_𝑞𝑡+𝑓𝑞𝑥,𝑥. (14)

Remark 4. We note that if 𝛼≥0 and 𝑎≤𝑏≤𝑡, then (𝑡−𝑎)^(𝛼)≥(𝑡−𝑏)^(𝛼) [19].

Definition 5 (see [9]). Let 𝛼≥0 and let 𝑓 be a function defined on [0,1]. The fractional 𝑞-integral of the Riemann-Liouville type is (𝐼⁰_𝑞𝑓)(𝑥)=𝑓(𝑥) and 𝐼^𝛼_𝑞𝑓(𝑥)=1Γ_𝑞(𝛼)^𝑥₀𝑥−𝑞𝑡^(𝛼−1)𝑓(𝑡)𝑑_𝑞[].𝑡,𝛼>0,𝑥∈0,1 (15)

Definition 6 (see [11]). The fractional 𝑞-derivative of the Riemann-Liouville type of 𝛼≥0 is defined by (𝐷⁰_𝑞𝑓)(𝑥)=𝑓(𝑥) and 𝐷^𝛼_𝑞𝑓(𝑥)=𝐷^𝑚_𝑞𝐼_𝑞^𝑚−𝛼𝑓(𝑥),𝛼>0, (16) where 𝑚 is the smallest integer greater than or equal to 𝛼.

Lemma 7 (see [9, 11]). Let 𝛼,𝛽≥0 and let 𝑓 be a function defined on [0,1]. Then, the next formulas hold:(1)(𝐼^𝛽_𝑞𝐼^𝛼_𝑞𝑓)(𝑥)=(𝐼_𝑞^𝛼+𝛽𝑓)(𝑥), (2)(𝐷^𝛼_𝑞𝐼^𝛼_𝑞𝑓)(𝑥)=𝑓(𝑥).

Remark 8. Assume that 𝑔(𝑡)∈[0,1] and 𝛼,𝛽 are two constants such that 𝛼>2≥𝛽≥1. Then 𝐷^𝛽_𝑞^𝑡₀𝑡−𝑞𝑠^(𝛼−1)𝑔(𝑠)𝑑_𝑞𝑠=Γ_𝑞(𝛼)Γ_𝑞𝛼−𝛽^𝑡₀𝑡−𝑞𝑠^{(𝛼−𝛽−1)}𝑔(𝑠)𝑑_𝑞. (17)

Proof. From Lemma 7, we can get 𝐷^𝛽_𝑞𝐼^𝛼_𝑞𝑔(𝑡)=𝐷^𝛽_𝑞𝐼^𝛽_𝑞𝐼_𝑞^𝛼−𝛽𝑔(𝑡)=𝐼_𝑞^𝛼−𝛽𝑔(𝑡), (18) so 𝐷^𝛽_𝑞1Γ_𝑞(𝛼)^𝑡₀𝑡−𝑞𝑠^(𝛼−1)𝑔(𝑠)𝑑_𝑞𝑠=1Γ_𝑞𝛼−𝛽^𝑡₀𝑡−𝑞𝑠^{(𝛼−𝛽−1)}𝑔(𝑠)𝑑_𝑞; (19) that is, (17) holds. The proof is completed.

Lemma 9 (see [22]). Let 𝛼>0 and let 𝑝 be a positive integer. Then, the following equality holds: 𝐼^𝛼_𝑞𝐷^𝑝_𝑞𝑓(𝑥)=𝐷^𝑝_𝑞𝐼^𝛼_𝑞𝑓(𝑥)−^𝑝−1_𝑘=0𝑥^{𝛼−𝑝+𝑘}Γ_𝑞𝐷𝛼+𝑘−𝑝+1^𝑘𝑞𝑓(0). (20)

Lemma 10. Let 𝑔(𝑡)∈𝐿[0,1]; then the unique solution of 𝐷^𝛼_𝑞𝑢(𝑡)(𝑡)𝐷+𝑔=0,0<𝑡<1,_𝑖𝑞𝑢(0)=0,𝐷^𝛽_𝑞𝑢(1)=𝑎𝐷^𝛽_𝑞𝑢𝜂,0≤𝑖≤𝑛−2, (21) is 𝑢(𝑡)=¹₀𝐺𝑡,𝑞𝑠𝑔(𝑠)𝑑_𝑞𝑠, (22) where 𝐺()=1𝑡,𝑠ΛΓ_𝑞(𝛼)×(1−𝑠)^{(𝛼−𝛽−1)}𝑡^𝛼−1−𝑎𝜂−𝑠^{(𝛼−𝛽−1)}𝑡^𝛼−1−Λ(𝑡−𝑠)^(𝛼−1)(),0≤𝑠≤min𝜂,𝑡≤1,1−𝑠^{(𝛼−𝛽−1)}𝑡^𝛼−1−𝑎𝜂−𝑠^{(𝛼−𝛽−1)}𝑡^𝛼−1(),0≤𝑡≤𝑠≤𝜂≤1,1−𝑠^{(𝛼−𝛽−1)}𝑡^𝛼−1−Λ(𝑡−𝑠)^(𝛼−1)(),0≤𝜂≤𝑠≤𝑡≤1,1−𝑠^{(𝛼−𝛽−1)}𝑡^𝛼−1,0≤max𝑡,𝜂≤𝑠≤1, (23) where Λ=1−𝑎𝜂^{𝛼−𝛽−1}.

Proof. Let 𝑢(𝑡) be a solution of (21); in view of Lemma 7 and Lemma 9, (21) is equivalent to the integral equation 𝑢(𝑡)=𝑐₁𝑡^𝛼−1+𝑐₂𝑡^𝛼−2+⋯+𝑐_𝑛𝑡^𝛼−𝑛−1Γ_𝑞(𝛼)^𝑡₀𝑡−𝑞𝑠^(𝛼−1)𝑔(𝑠)𝑑_𝑞𝑠, (24) where 𝑐₁,𝑐₂,…,𝑐_𝑛 are some constants to be determined. The boundary conditions (𝐷^𝑖_𝑞𝑢)(0)=0, 0≤𝑖≤𝑛−2, imply that 𝑐₂=𝑐₃=⋯=𝑐_𝑛=0. Thus, 𝑢(𝑡)=𝑐₁𝑡^𝛼−1−1Γ_𝑞(𝛼)^𝑡₀𝑡−𝑞𝑠^(𝛼−1)𝑔(𝑠)𝑑_𝑞𝑠. (25) By Remark 8, we have 𝐷^𝛽_𝑞𝑢(𝑡)=𝑐₁Γ_𝑞(𝛼)Γ_𝑞𝑡𝛼−𝛽^{𝛼−𝛽−1}−1Γ_𝑞×𝛼−𝛽_𝑡0𝑡−𝑞𝑠^{(𝛼−𝛽−1)}𝑔(𝑠)𝑑_𝑞𝑠. (26) For (𝐷^𝛽_𝑞𝑢)(1)=𝑎(𝐷^𝛽_𝑞𝑢)(𝜂), 𝑐₁=11−𝑎𝜂^{𝛼−𝛽−1}Γ_𝑞(𝛼)¹₀1−𝑞𝑠^{(𝛼−𝛽−1)}𝑔(𝑠)𝑑_𝑞𝑠−𝑎^𝜂₀𝜂−𝑞𝑠^{(𝛼−𝛽−1)}𝑔(𝑠)𝑑_𝑞𝑠. (27) Hence, 𝑢(𝑡)=𝑡^𝛼−1ΛΓ_𝑞(𝛼)¹₀1−𝑞𝑠^{(𝛼−𝛽−1)}𝑔(𝑠)𝑑_𝑞𝑠−𝑎^𝜂₀𝜂−𝑞𝑠^{(𝛼−𝛽−1)}𝑔(𝑠)𝑑_𝑞𝑠−1Γ_𝑞(𝛼)^𝑡₀𝑡−𝑞𝑠^(𝛼−1)𝑔(𝑠)𝑑_𝑞𝑠=¹₀𝐺𝑡,𝑞𝑠𝑔(𝑠)𝑑_𝑞𝑠. (28) The proof is complete.

Remark 11. For the special case where 𝑎=0, it is easy to see that 𝐺(𝑡,𝑠) can be written as 𝐺()=1𝑡,𝑠Γ_𝑞(𝛼)×(1−𝑠)^{(𝛼−𝛽−1)}𝑡^𝛼−1−(𝑡−𝑠)^(𝛼−1)(),0≤𝑠≤𝑡≤1,1−𝑠^{(𝛼−𝛽−1)}𝑡^𝛼−1,0≤𝑡≤𝑠≤1. (29)

Lemma 12. Green function 𝐺 in Lemma 10 satisfies the following conditions:(i)𝐺(𝑡,𝑞𝑠)≥0 for 𝑡,𝑠∈[0,1];(ii)𝐺(𝑡,𝑞𝑠)≤𝐺(1,𝑞𝑠) for 𝑡,𝑠∈[0,1];(iii)𝐺(𝑡,𝑞𝑠)≥𝑡^𝛼−1𝐺(1,𝑞𝑠) for 𝑡,𝑠∈[0,1].

Proof. Let 𝑔₁(𝑡,𝑠)=(1−𝑠)^{(𝛼−𝛽−1)}𝑡^𝛼−1−𝑎𝜂−𝑠^{(𝛼−𝛽−1)}𝑡^𝛼−1−Λ(𝑡−𝑠)^(𝛼−1)𝑔,0≤𝑠≤min𝜂,𝑡≤1,₂(𝑡,𝑠)=(1−𝑠)^{(𝛼−𝛽−1)}𝑡^𝛼−1−𝑎𝜂−𝑠^{(𝛼−𝛽−1)}𝑡^𝛼−1𝑔,0≤𝑡≤𝑠≤𝜂≤1,₃(𝑡,𝑠)=(1−𝑠)^{(𝛼−𝛽−1)}𝑡^𝛼−1−Λ(𝑡−𝑠)^(𝛼−1)𝑔,0≤𝜂≤𝑠≤𝑡≤1,₄(𝑡,𝑠)=(1−𝑠)^{(𝛼−𝛽−1)}𝑡^𝛼−1,0≤max𝑡,𝜂≤𝑠≤1. (30)
We first prove part (i). For 𝑡,𝑠∈[0,1], from Remark 4, for 𝑡≠0, 𝑔₁𝑡,𝑞𝑠=1−𝑞𝑠^{(𝛼−𝛽−1)}𝑡^𝛼−1−𝑎𝜂^{𝛼−𝛽−1}×1−𝑞𝑠𝜂^{(𝛼−𝛽−1)}𝑡^𝛼−1−Λ1−𝑞𝑠𝑡^(𝛼−1)𝑡^𝛼−1=Λ𝑡^𝛼−11−𝑞𝑠^{(𝛼−𝛽−1)}−1−𝑞𝑠𝑡^(𝛼−1)+𝑎𝜂^{𝛼−𝛽−1}𝑡^𝛼−11−𝑞𝑠^{(𝛼−𝛽−1)}−1−𝑞𝑠𝜂^{(𝛼−𝛽−1)}≥Λ𝑡^𝛼−11−𝑞𝑠^{(𝛼−𝛽−1)}−1−𝑞𝑠^(𝛼−1)+𝑎𝜂^{𝛼−𝛽−1}𝑡^𝛼−11−𝑞𝑠^{(𝛼−𝛽−1)}−1−𝑞𝑠^{(𝛼−𝛽−1)}≥0. (31) Since 𝑔₁(𝑡,𝑞𝑠)≥0, it is easy to know 𝑔₂(𝑡,𝑞𝑠)≥0, 𝑔₃(𝑡,𝑞𝑠)≥0, and 𝑔₄(𝑡,𝑞𝑠)≥0. Therefore, 𝐺(𝑡,𝑞𝑠)≥0.
Next, we prove part (ii). Fix 𝑠∈[0,1], and _𝑡𝐷_𝑞𝑔₁𝑡,𝑞𝑠=[𝛼−1]_𝑞1−𝑞𝑠^{(𝛼−𝛽−1)}𝑡^𝛼−2−𝑎𝜂^{𝛼−𝛽−1}[𝛼−1]_𝑞1−𝑞𝑠𝜂^{(𝛼−𝛽−1)}𝑡^𝛼−2−Λ[𝛼−1]_𝑞1−𝑞𝑠𝑡^(𝛼−2)𝑡^𝛼−2=[𝛼−1]_𝑞𝑡^𝛼−2Λ1−𝑞𝑠^{(𝛼−𝛽−1)}−1−𝑞𝑠𝑡^(𝛼−2)+𝑎𝜂^{𝛼−𝛽−1}×1−𝑞𝑠^{(𝛼−𝛽−1)}−1−𝑞𝑠𝜂^{(𝛼−𝛽−1)}≥[𝛼−1]_𝑞𝑡^𝛼−2Λ1−𝑞𝑠^{(𝛼−𝛽−1)}−1−𝑞𝑠^(𝛼−2)≥0. (32) That is, 𝑔₁(𝑡,𝑞𝑠) is increasing function of 𝑡. By the same way, we can conclude that 𝑔₂(𝑡,𝑞𝑠), 𝑔₃(𝑡,𝑞𝑠), and 𝑔₄(𝑡,𝑞𝑠) are increasing functions of 𝑡 for fixed 𝑠∈[0,1]. Thus, 𝐺(𝑡,𝑞𝑠)≤𝐺(1,𝑞𝑠) for 𝑡,𝑠∈[0,1].
Finally, we prove part (iii). Suppose that 0≤𝑞𝑠≤min{𝑡,𝜂}≤1; then 𝐺𝑡,𝑞𝑠𝐺=1,𝑞𝑠1−𝑞𝑠^{(𝛼−𝛽−1)}𝑡^𝛼−1−𝑎𝜂−𝑞𝑠^{(𝛼−𝛽−1)}𝑡^𝛼−1−Λ𝑡−𝑞𝑠^(𝛼−1)1−𝑞𝑠^{(𝛼−𝛽−1)}−𝑎𝜂−𝑞𝑠^{(𝛼−𝛽−1)}−Λ1−𝑞𝑠^(𝛼−1)=𝑡^𝛼−11−𝑞𝑠^{(𝛼−𝛽−1)}−𝑎𝜂−𝑞𝑠^{(𝛼−𝛽−1)}−Λ1−𝑞𝑠/𝑡^(𝛼−1)1−𝑞𝑠^{(𝛼−𝛽−1)}−𝑎𝜂−𝑞𝑠^{(𝛼−𝛽−1)}−Λ1−𝑞𝑠^(𝛼−1)≥𝑡^𝛼−1. (33)
For other circumstances, we also get 𝐺(𝑡,𝑞𝑠)≥𝑡^𝛼−1𝐺(1,𝑞𝑠) and this completes the proof.

Remark 13. Let 0<𝜏<1; then 0<𝜏^𝛼−1<1 and min_{[]𝑡∈𝜏,1}𝐺𝑡,𝑞𝑠≥𝜏^𝛼−1𝐺1,𝑞𝑠for𝑠∈[0,1]. (34)

Lemma 14 (see [29]). Let 𝑋 be a Banach space with 𝐶⊂𝑋 being closed and convex. Assume that 𝑈 is a relatively open subset of 𝐶 with 0∈𝑈 and 𝑇∶𝑈→𝐶 is complete continuous. Then either(i)𝑇 has a fixed point in 𝑈, or(ii)there exist 𝑢∈𝜕𝑈 and 𝜆∈(0,1) with 𝑢=𝜆𝑇𝑢.

Lemma 15 (see Krasnoselskii’s [30]). 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 0∈Ω₁ and Ω₁⊂Ω₂. Let 𝑇∶𝐾∩(Ω₂⧵Ω₁)→𝐾 be a completely continuous operator. In addition, suppose that either(𝐻₁)‖𝑇𝑢‖≤‖𝑢‖, for all 𝑢∈𝐾∩𝜕Ω₁ and ‖𝑇𝑢‖≥‖𝑢‖, for all 𝑢∈𝐾∩𝜕Ω₂ or(𝐻₂)‖𝑇𝑢‖≤‖𝑢‖, for all 𝑢∈𝐾∩𝜕Ω₂ and ‖𝑇𝑢‖≥‖𝑢‖, for all 𝑢∈𝐾∩𝜕Ω₁holds. Then 𝑇 has a fixed point in 𝐾∩(Ω₂⧵Ω₁).

Lemma 16 (see [31]). Let 𝑃 be a cone in a real Banach space 𝐸, 𝑃_𝑐={𝑥∈𝑃∶‖𝑥‖<𝑐}, 𝜃 is a nonnegative continuous concave functional on 𝑃 such that 𝜃≤‖𝑥‖, for all 𝑥∈𝑃_𝑐, and 𝑃(𝜃,𝑏,𝑑)={𝑥∈𝑃∶𝑏≤𝜃(𝑥),‖𝑥‖≤𝑑}. Suppose that 𝑇∶𝑃_𝑐→𝑃_𝑐 is completely continuous and there exist positive constants 0<𝑎<𝑏<𝑑≤𝑐 such that(𝐻₁){𝑥∈𝑃(𝜃,𝑏,𝑑)∶𝜃(𝑥)>𝑏}≠∅ and 𝜃(𝑇𝑥)>𝑏 for 𝑥∈𝑃(𝜃,𝑏,𝑑),(𝐻₂)‖𝑇𝑥‖<𝑎 for 𝑥∈𝑃_𝑎,(𝐻₃)𝜃(𝑇𝑥)>𝑏 for 𝑥∈𝑃(𝜃,𝑏,𝑑) with ‖𝑇𝑥‖>𝑑.Then 𝑇 has at least three fixed points 𝑥₁,𝑥₂, and 𝑥₃ with ‖𝑥‖<𝑎,𝑏<𝜃𝑥₂,𝑎<𝑥₃with𝜃𝑥₃<𝑏. (35)

Remark 17. If 𝑑=𝑐, then (𝐻₁) implies (𝐻₃).

3. Main Result

In this section, we will consider the question of positive solutions for BVP (3). At first, we prove some lemmas required for the main result.

Let 𝐸=𝐶[0,1] be the Banach space endowed with the norm ‖𝑢‖=sup_𝑡∈[0,1]|𝑢(𝑡)|. Let 𝜏=𝑞^𝑛 for a given 𝑛∈𝑁, and define the cone 𝑃⊂𝐸 by 𝑃=𝑢∈𝐸∶𝑢(𝑡)≥0,min_{[]𝑡∈𝜏,1}𝑢(𝑡)≥𝜏^𝛼−1‖𝑢‖. (36)

Let the nonnegative continuous concave functional 𝜃 on the cone 𝑃 be defined by 𝜃(𝑥)=min_{𝜏≤𝑡≤1}|𝑢(𝑡)|. (37)

In this paper, we assume that 𝑓∶[0,1]×[0,+∞)→[0,+∞] satisfies the following conditions of Caratheodory type:(𝐷₁)𝑓(𝑡,𝑢) is Lebesgue measurable with respect to 𝑡 on [0,1];(𝐷₂)𝑓(𝑡,𝑢) is continuous with respect to 𝑢 on [0,+∞).

Theorem 18. Assume that the conditions (𝐷₁) and (𝐷₂) hold. Suppose further that there exists a real-valued function ℎ(𝑡)∈𝐿[0,1] such that |𝑓(𝑡,𝑢)−𝑓(𝑡,𝑣)|≤ℎ(𝑡)|𝑢−𝑣| for almost every 𝑡∈[0,1] and all 𝑢,𝑣∈[0,+∞). If 0<¹₀𝐺1,𝑞𝑠ℎ(𝑠)𝑑_𝑞𝑠<1, (38) then there exist unique positive solutions of BVP (3) on [0,1].

Proof. Consider the operator 𝑇∶𝑃→𝑃 defined by 𝑇𝑢(𝑡)∶=¹₀𝐺𝑡,𝑞𝑠𝑓(𝑠,𝑢(𝑠))𝑑_𝑞𝑠<1. (39) For any 𝑢,𝑣∈𝑃, we have |𝑇𝑢(𝑡)−𝑇𝑣(𝑡)|=¹₀𝐺𝑡,𝑞𝑠𝑓(𝑠,𝑢(𝑠))−𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠≤¹₀𝐺𝑡,𝑞𝑠𝑓(𝑠,𝑢(𝑠))−𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠≤¹₀𝐺𝑡,𝑞𝑠ℎ(𝑠)|𝑢(𝑠)−𝑣(𝑠)|𝑑_𝑞𝑠≤¹₀𝐺1,𝑞𝑠ℎ(𝑠)|𝑢(𝑠)−𝑣(𝑠)|𝑑_𝑞𝑠≤‖𝑢−𝑣‖¹₀𝐺1,𝑞𝑠ℎ(𝑠)𝑑_𝑞𝑠<‖𝑢−𝑣‖. (40) This implies that 𝑇 is a contraction mapping. By the Banach contraction mapping principle, we deduce that 𝑇 has a unique fixed point which is obviously a solution of BVP (3). The proof is complete.

Corollary 19. Assume that the conditions (𝐷₁) and (𝐷₂) hold. Suppose further that there exists a positive constant 𝐿∈(0,1/𝐴) with |𝑓(𝑡,𝑢)−𝑓(𝑡,𝑣)|≤𝐿|𝑢−𝑣|,𝑡∈[0,1],(𝑢,𝑣)∈[0,+∞), where 𝐴=¹₀𝐺1,𝑞𝑠𝑑_𝑞𝑠; (41) then there exists a unique positive solution of BVP (3) on [0,1].

Corollary 20. Assume that the conditions (𝐷₁) and (𝐷₂) hold. Suppose further that there exists a real-valued function ℎ(𝑡)∈𝐿[0,1] such that |𝑓(𝑡,𝑢)−𝑓(𝑡,𝑣)|≤ℎ(𝑡)|𝑢−𝑣|, 𝑡∈[0,1], (𝑢,𝑣)∈[0,+∞). If 0<¹₀ℎ(𝑠)𝑑_𝑞𝑠<ΛΓ_𝑞(𝛼), (42) then there exists a unique positive solution of BVP (3) on [0,1].

Corollary 21. Assume that the conditions (𝐷₁) and (𝐷₂) hold. Suppose further that there exists a positive constant 𝐿∈(0,ΛΓ_𝑞(𝛼)) with |𝑓(𝑡,𝑢)−𝑓(𝑡,𝑣)|≤𝐿|𝑢−𝑣|, 𝑡∈[0,1], (𝑢,𝑣)∈[0,+∞); then there exists a unique positive solution of BVP (3) on [0,1].

Next, we discuss multiple solutions of BVP (3).

Lemma 22. Assume that the conditions (𝐷₁) and (𝐷₂) hold. Suppose further that there exist two nonnegative real-valued functions 𝑚,𝑛∈𝐿[0,1] such that |𝑓(𝑡,𝑢)|≤𝑛(𝑡)+𝑚(𝑡)𝑢 for almost every 𝑡∈[0,1] and all 𝑢,𝑣∈[0,+∞). Then the operation 𝑇∶𝑃→𝑃 defined by (39) is completely continuous.

Proof. We will divide the proof into three parts.(I) We show that 𝑇∶𝑃→𝑃 is continuous.
For any 𝑢_𝑛,𝑢∈𝑃, 𝑛=1,2,…, with lim_𝑛→∞‖𝑢_𝑛−𝑢‖=0, we have lim_𝑛→∞𝑢_𝑛(𝑡)=𝑢(𝑡),𝑡∈[0,1]. (43) Thus lim_𝑛→∞𝑓𝑡,𝑢_𝑛(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡∈[0,1]. (44) So, we can obtain that sup_𝑠∈[0,1]𝑓𝑠,𝑢_𝑛(𝑠)−𝑓(𝑠,𝑢(𝑠))⟶0,as𝑛⟶∞. (45) On the other hand, we have 𝑇𝑢_𝑛(𝑡)−(𝑡=𝑇𝑢)()₁₀𝐺𝑡,𝑞𝑠𝑓𝑠,𝑢_𝑛(𝑠)−𝑓(𝑠,𝑢(𝑠))𝑑_𝑞𝑠≤sup_[]𝑠∈0,1𝑓𝑠,𝑢_𝑛(𝑠)((𝑠×−𝑓𝑠,𝑢))₁₀𝐺𝑡,𝑞𝑠𝑑_𝑞𝑠. (46) It implies that 𝑇𝑢_𝑛−𝑇𝑢≤sup_[]𝑠∈0,1𝑓𝑠,𝑢_𝑛(𝑠)((𝑠×−𝑓𝑠,𝑢))₁₀𝐺𝑡,𝑞𝑠𝑑_𝑞𝑠. (47) Therefore, ‖𝑇𝑢_𝑛−𝑇𝑢‖→0 as 𝑛→∞. This means that 𝑇 is continuous.(II) We will prove that 𝑇 maps bounded sets into bounded sets in 𝑃.
For any 𝜉>0, there exists a positive constant 𝑙>0 such that for each 𝑢∈𝐵_𝜉={𝑢∈𝑃∶‖𝑢‖≤𝜉}, we have ‖𝑇𝑢‖<𝑙. By the definition of 𝑇, for each 𝑡∈[0,1], we get |𝑇𝑢(𝑡)|=¹₀𝐺𝑡,𝑞𝑠𝑓(𝑠,𝑢(𝑠))𝑑_𝑞𝑠≤¹₀𝐺𝑡,𝑞𝑠(𝑛(𝑠)+𝑚(𝑠)𝑢(𝑠))𝑑_𝑞𝑠≤¹₀𝐺1,𝑞𝑠𝑛(𝑠)𝑑_𝑞𝑠+‖𝑢‖¹₀𝐺1,𝑞𝑠𝑚(𝑠)𝑑_𝑞𝑠≤¹₀𝐺1,𝑞𝑠𝑛(𝑠)𝑑_𝑞𝑠+𝜉¹₀𝐺1,𝑞𝑠𝑚(𝑠)𝑑_𝑞𝑠∶=𝑙. (48) That is, ‖𝑇𝑢‖≤𝑙.(III) We will show that 𝑇 maps bounded sets into equicontinuous sets of 𝑃.
Let 𝐵_𝜉⊂𝑃 be a bounded set, 𝑡₁,𝑡₂∈[0,1] and 𝑡₁<𝑡₂. For any 𝑢∈𝐵_𝜉, we have 𝑇𝑢𝑡₂−𝑇𝑢𝑡₁=¹₀𝐺𝑡₂,𝑞𝑠−𝐺𝑡₁,𝑞𝑠𝑓(𝑠,𝑢(𝑠))𝑑_𝑞𝑠≤¹₀𝐺𝑡₂,𝑞𝑠−𝐺𝑡₁,𝑞𝑠𝑓(𝑠,𝑢(𝑠))𝑑_𝑞𝑠≤¹₀𝐺𝑡₂,𝑞𝑠−𝐺𝑡₁,𝑞𝑠(𝑛(𝑠)+𝑚(𝑠)𝑢(𝑠))𝑑_𝑞𝑠≤¹₀𝐺𝑡₂,𝑞𝑠−𝐺𝑡₁,𝑞𝑠(𝑛(𝑠)+𝜉𝑚(𝑠))𝑑_𝑞𝑠. (49) Since 𝐺(𝑡,𝑞𝑠) is continuous on [0,1]×[0,1], then 𝐺(𝑡,𝑞𝑠) is uniformly continuous in [0,1]×[0,1]. Hence, for any 𝜀>0, there exists 𝛿>0, whenever |𝑡₂−𝑡₁|<𝛿, and we have 𝐺𝑡₂,𝑞𝑠−𝐺𝑡₁<𝜀,𝑞𝑠1+∫¹₀(𝑛(𝑠)+𝜉𝑚(𝑠))𝑑_𝑞𝑠. (50) So |𝑇𝑢(𝑡₂)−𝑇𝑢(𝑡₁)|<𝜀; that is, {𝑇𝑢∶𝑢∈𝐵_𝜉} is equicontinuous.
By Arzela-Ascoli theorem, we can conclude that 𝑇 is completely continuous. This completes the proof.

Remark 23. If 𝑓∶[0,1]×[0,∞)→[0,∞) is continuous, 𝑇 is also completely continuous.

Theorem 24. Assume that all the assumptions of Lemma 22 hold. If 0<¹₀𝐺1,𝑞𝑠𝑚(𝑠)𝑑_𝑞𝑠<1, (51) then BVP (3) has at least one positive solution.

Proof. Let 𝑈={𝑢∈𝑃∶‖𝑢‖<𝑟}, where ∫𝑟=¹⁰𝐺1,𝑞𝑠𝑛(𝑠)𝑑_𝑞𝑠1−∫¹₀𝐺1,𝑞𝑠𝑚(𝑠)𝑑_𝑞𝑠>0 (52) and 𝑇∶𝑈→𝑃,𝑇𝑢(𝑡)∶=∫¹₀𝐺(𝑡,𝑞𝑠)𝑓(𝑠,𝑢(𝑠))𝑑_𝑞𝑠. From Lemma 22, 𝑇 is completely continuous.
Assume that there exist 𝑢∈𝑃 and 𝜆∈(0,1) such that 𝑢=𝜆𝑇𝑢; we claim that ‖𝑢‖≠𝑟: 𝑢(𝑡)=𝜆¹₀𝐺𝑡,𝑞𝑠𝑓(𝑠,𝑢(𝑠))𝑑_𝑞𝑠≤𝜆¹₀𝐺𝑡,𝑞𝑠(𝑛(𝑠)+𝑚(𝑠)𝑢(𝑠))𝑑_𝑞𝑠≤𝜆¹₀𝐺𝑡,𝑞𝑠𝑛(𝑠)𝑑_𝑞𝑠+‖𝑢‖¹₀𝐺𝑡,𝑞𝑠𝑚(𝑠)𝑑_𝑞𝑠, (53) and then ‖𝑢‖≤𝜆¹₀𝐺1,𝑞𝑠𝑛(𝑠)𝑑_𝑞𝑠+𝑟¹₀𝐺1,𝑞𝑠𝑚(𝑠)𝑑_𝑞𝑠<𝑟. (54) That is, 𝑢∉𝜕𝑈. By Lemma 14, 𝑇 has a fixed point 𝑢∈𝑈. Therefore, BVP (3) has at least a positive solution. The proof is complete.

In the following, we set 𝑀=¹₀𝐺1,𝑞𝑠𝑑_𝑞𝑠⁻¹,𝑁=¹_𝜏𝜏^𝛼−1𝐺1,𝑞𝑠𝑑_𝑞𝑠⁻¹. (55)

Theorem 25. Assume that all the assumptions of Lemma 22 hold. If there exist two positive constants 𝑟₂>𝑟₁>0 such that 𝑓(𝑡,𝑢)≥𝑁𝑟₁𝑓𝑜𝑟(𝑡,𝑢)∈[0,1]×0,𝑟₁,𝑓(𝑡,𝑢)≤𝑀𝑟₂𝑓𝑜𝑟(𝑡,𝑢)∈[0,1]×0,𝑟₂, (56)(57) then BVP (3) has at least one positive solution.

Proof. Because 𝑇∶𝑃→𝑃 is completely continuous, we just only show that 𝑢=𝑇𝑢 has a solution 𝑢(𝑡)>0 for 𝑡∈[0,1].
Let Ω₁={𝑢∈𝐸∶‖𝑢‖<𝑟₁}. For any 𝑢∈𝑃∩𝜕Ω₁, we know 𝜏^𝛼−1𝑟₁≤𝑢(𝑡)≤𝑟₁ on [𝜏,1]. Using (56) and (34), we have |𝑇𝑢(𝑡)|=¹₀𝐺𝑡,𝑞𝑠𝑓(𝑠,𝑢(𝑠))𝑑_𝑞𝑠≥¹_𝜏𝐺𝑡,𝑞𝑠𝑓(𝑠,𝑢(𝑠))𝑑_𝑞𝑠≥𝑁𝑟₁¹_𝜏𝐺𝑡,𝑞𝑠𝜏^𝛼−1𝑑_𝑞𝑠=𝑟₁=‖𝑢‖, (58) which implies that ‖𝑇𝑢‖≥‖𝑢‖, 𝑢∈𝑃∩𝜕Ω₁.
Let Ω₂={𝑢∈𝐸∶‖𝑢‖<𝑟₂}. For any 𝑢∈𝑃∩𝜕Ω₂, we get 0≤𝑢(𝑡)≤𝑟₂ on [0,1]. Using (57), we have |𝑇𝑢(𝑡)|=¹₀𝐺𝑡,𝑞𝑠𝑓(𝑠,𝑢(𝑠))𝑑_𝑞𝑠≤𝑀𝑟₂¹₀𝐺1,𝑞𝑠𝑑_𝑞𝑠=𝑟₂=‖𝑢‖; (59) that is, ‖𝑇𝑢‖≤‖𝑢‖, 𝑢∈𝑃∩𝜕Ω₂.
In view of Lemma 15, 𝑇 has a fixed point 𝑢₀∈𝑃∩(Ω₂⧵Ω₁) which is the solution of BVP (3).

Theorem 26. Assume that all assumptions of Lemma 22 hold. If there exist constants 0<𝐴<𝐵<𝐶 such that 𝐺₁𝑓()()∈[]×[],𝐺𝑡,𝑢≤𝑀𝐴,𝑡,𝑢0,10,𝐴₂𝑓()()∈[]×[],𝐺𝑡,𝑢≤𝑀𝐶,𝑡,𝑢0,10,𝐶₃𝑓(𝑡,𝑢)≥𝑁𝐵,(𝑡,𝑢)∈[𝜏,1]×[𝐵,𝐶], (60) hold, then BVP (3) has at least three positive solutions 𝑢₁,𝑢₂, and 𝑢₃ with 𝑢₁<𝐴,𝐵<𝜃𝑢₂<𝑢₂𝑢≤𝐶,𝐴<₃,𝜃𝑢₃≤𝐵. (61)

Proof. First, if 𝑢∈𝑃_𝐶={𝑢∈𝑃∶‖𝑢‖<𝐶}, then ‖𝑢‖≤𝐶. So 0≤𝑢(𝑡)≤𝐶, 𝑡∈[0,1]. By (G₂), we have |𝑇𝑢(𝑡)|=¹₀𝐺𝑡,𝑞𝑠𝑓(𝑠,𝑢(𝑠))𝑑_𝑞𝑠≤𝑀𝐶¹₀𝐺1,𝑞𝑠𝑑_𝑞𝑠=𝐶, (62) which implies that ‖𝑇𝑢‖≤𝐶,𝑢∈𝑃_𝐶. Hence, 𝑇∶𝑃_𝐶→𝑃_𝐶. In view of Lemma 22, 𝑇∶𝑃_𝐶→𝑃_𝐶 is completely continuous.
By using the analogous argument, from (G₁), we can get that if 𝑢∈𝑃_𝐴, then ‖𝑇𝑢‖<𝐴.
Set 𝑢(𝑡)=(𝐵+𝐶)/2, 𝑡∈[0,1], so 𝑢(𝑡)=(𝐵+𝐶)/2∈𝑃(𝜃,𝐵,𝐶), 𝜃(𝑢)=(𝐵+𝐶)/2>𝐵. Therefore, {𝑢∈𝑃(𝜃,𝐵,𝐶)|𝜃(𝑢)>𝐵}≠∅.
On the other hand, if 𝑢∈𝑃(𝜃,𝐵,𝐶), then 𝐵≤𝑢(𝑡)≤𝐶,𝑡∈[𝜏,1]. By (G₃), we have 𝜃(𝑇𝑢)=min_{[]𝑡∈𝜏,1}|𝑇𝑢(𝑡)|=min_{[]𝑡∈𝜏,1}¹₀𝐺𝑡,𝑞𝑠𝑓(𝑠,𝑢(𝑠))𝑑_𝑞𝑠≥min_{𝑡∈[𝜏,1]}¹₀𝐺𝑡,𝑞𝑠𝑁𝐵𝑑_𝑞𝑠,≥𝑁𝐵¹_𝜏𝜏^𝛼−1𝐺1,𝑞𝑠𝑑_𝑞𝑠=𝐵, (63) which implies that 𝜃(𝑇𝑢)>𝐵, for 𝑢∈𝑃(𝜃,𝐵,𝐶).
By Lemma 16, BVP(3) has at least three positive solutions 𝑢₁, 𝑢₂, and 𝑢₃ with 𝑢₁<𝐴,𝐵<𝜃𝑢₂<𝑢₂𝑢<𝐶,𝐴<₃,𝜃𝑢₃<𝐵. (64) The proof is complete.

4. Example

Example 27. Consider the following BVP: 𝐷^4.5_1/2𝑢(𝑡)+𝑒^𝑡𝑢1+𝑒^𝑡+𝑡²𝐷+1=0,0<𝑡<1,_𝑖1/2𝑢(0)=0,𝐷^1.5_1/2𝑢(1)=1𝐷100^1.51/2𝑢9,100≤𝑖≤3. (65)
Let 𝑓(𝑡,𝑢)=𝑒^𝑡𝑢/(1+𝑒^𝑡)+𝑡²+1, (𝑡,𝑢)∈[0,1]×[0,+∞), ℎ(𝑡)=𝑒^𝑡/(1+𝑒^𝑡), so |𝑓(𝑡,𝑢)−𝑓(𝑡,𝑣)|≤ℎ(𝑡)|𝑢−𝑣|, for (𝑡,𝑢),(𝑡,𝑣)∈[0,1]×[0,+∞).
By simple calculation, we get 0<¹₀𝐺1,𝑞𝑠ℎ(𝑠)𝑑_𝑞1𝑠≤ΛΓ_𝑞(𝛼)=0.8471⋯<1. (66) All conditions of Theorem 18 are satisfied. Thus, BVP (65) has a unique positive solution.

Example 28. Consider the following BVP: 𝐷^4.5_1/2𝑢(𝑡)+14()𝑢sin𝑡+11+𝑢²+𝑡²𝐷+1=0,0<𝑡<1,_𝑖1/2𝑢(0)=0,𝐷^1.5_1/2𝑢(1)=1𝐷100^1.51/2𝑢9,100≤𝑖≤3. (67)
Let 𝑛(𝑡)=𝑡²+1, 𝑚(𝑡)=(1/4)(sin𝑡+1). It is easy to check that 𝑓(𝑡,𝑢)≤𝑛(𝑡)+𝑚(𝑡)𝑢 for (𝑡,𝑢)∈[0,1]×[0,+∞). Since 0<¹₀𝐺1,𝑞𝑠𝑚(𝑠)𝑑_𝑞1𝑠≤ΛΓ_𝑞(𝛼)⋅12=0.4235⋯<1, (68) by Theorem 24, BVP (67) has at least one positive solution.

Example 29. Consider the following BVP: 𝐷^2.5_1/2𝑢(𝑡)+𝑢4+sin²𝑡5𝐷+1=0,0<𝑡<1,_𝑖1/2𝑢(0)=0,𝐷_1/2𝑢(1)=12𝐷_1/2𝑢34,0≤𝑖≤1. (69)
Let 𝑓(𝑡,𝑢)=𝑢/4+sin²𝑡/5+1, (𝑡,𝑢)∈[0,1]×[0,+∞). By calculation, we get 𝑀=¹₀𝐺1,𝑞𝑠𝑑_𝑞𝑠⁻¹≥ΛΓ_𝑞(𝛼)=1.4809…,𝑁=¹_𝜏𝜏^𝛼−1𝐺1,𝑞𝑠𝑑_𝑞𝑠⁻¹≤2.414…. (70) Choosing 𝑟₁=1/3, 𝑟₂=1, we have 𝑓()=𝑢𝑡,𝑢4+sin²𝑡5+1≤1.45≤𝑀𝑟₂,(𝑡,𝑢)∈[0,1]×0,𝑟₂,𝑓()=𝑢𝑡,𝑢4+sin²𝑡5+1≥1≥𝑁𝑟₁,(𝑡,𝑢)∈[0,1]×0,𝑟₁. (71) By Theorem 25, BVP (69) has at least one positive solution 𝑢 such that 1/3≤‖𝑢‖≤1.

Example 30. Consider the following BVP: 𝐷^2.5_1/2𝑢(𝑡)()𝐷+𝑓𝑡,𝑢=0,0<𝑡<1,_𝑖1/2𝑢(0)=0,𝐷_1/2𝑢1)=1(2𝐷_1/2𝑢34,0≤𝑖≤1. (72) Here, 𝑓()=𝑡𝑡,𝑢4+4𝑢⁴,(𝑡,𝑢)∈[0,1]×[0,1],𝑡+74+𝑢,(𝑡,𝑢)∈[0,1]×[1,+∞). (73)
By Example 29, we have 𝑀=¹₀𝐺1,𝑞𝑠𝑑_𝑞𝑠⁻¹≥ΛΓ_𝑞(𝛼)=1.4809…,𝑁=¹_𝜏𝜏^𝛼−1𝐺1,𝑞𝑠𝑑_𝑞𝑠⁻¹≤2.414…. (74) Choosing 𝐴=1/2, 𝐵=1, 𝐶=5, we have 𝑓()=𝑡𝑡,𝑢4+4𝑢⁴()∈[]×1≤0.5≤𝑀𝐴,𝑡,𝑢0,10,2,𝑓(𝑡,𝑢)=𝑡+74()∈[]×[],𝑓()=+𝑢≤7≤𝑀𝐶,𝑡,𝑢0,10,5𝑡,𝑢𝑡+74()∈1+𝑢≥2.875≥𝑁𝐵,𝑡,𝑢2,1×[1,5]. (75) By Theorem 26, BVP (72) has at least three solutions 𝑢₁, 𝑢₂, and 𝑢₃ such that 𝑢₁<12,1<𝜃𝑢₂,12<𝑢₃,𝜃𝑢₃<1. (76)

5. Conclusion

In this paper, we obtain the existence and multiplicity results of positive solutions of BVP for high-order fractional 𝑞-difference equations by some fixed point theorems, which enrich the theories for fractional 𝑞-difference equations, and provide the theoretical guarantee for the application of fractional 𝑞-difference equations in every field. In the future, we will use bifurcation theory, critical point theory, variational method, and other methods to continue our works in this area.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

Each of the authors, Changlong Yu and Jufang Wang, contributed to each part of this work equally and read and approved the final version of the paper.

Acknowledgments

This work is supported by the Natural Science Foundation of China ((10901045) and (11201112)) and the Natural Science Foundation of Hebei Province ((A2013208147) and (A2011208012)).