Existence and Uniqueness of Positive Solutions of Boundary-Value Problems for Fractional Differential Equations with 𝑝-Laplacian Operator and Identities on the Some Special Polynomials

Abstract

We consider the following boundary-value problem of nonlinear fractional differential equation with 𝑝-Laplacian operator 𝐷^𝛽₀₊(𝜙_𝑝(𝐷^𝛼₀₊𝑢(𝑡)))+𝑎(𝑡)𝑓(𝑢)=0, 0<𝑡<1, 𝑢(0)=𝛾𝑢(ℎ)+𝜆, 𝑢^′(0)=𝜇, 𝜙_𝑝(𝐷^𝛼₀₊𝑢(0))=(𝜙_𝑝(𝐷^𝛼₀₊𝑢(1)))^′=(𝜙_𝑝(𝐷^𝛼₀₊𝑢(0)))^′′=(𝜙_𝑝(𝐷^𝛼₀₊𝑢(0)))^′′′=0, where 1<𝛼⩽2, 3<𝛽⩽4 are real numbers, 𝐷^𝛼₀₊,𝐷^𝛽₀₊ are the standard Caputo fractional derivatives, 𝜙_𝑝(𝑠)=|𝑠|^𝑝−2𝑠, 𝑝>1, 𝜙_𝑝⁻¹=𝜙_𝑞, 1/𝑝+1/𝑞=1,0⩽𝛾<1, 0⩽ℎ⩽1, 𝜆,𝜇>0 are parameters, 𝑎∶(0,1)→[0,+∞), and 𝑓∶[0,+∞)→[0,+∞) are continuous. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results for positive solutions, in terms of the parameters 𝜆 and 𝜇 are obtained. The uniqueness of positive solution on the parameters 𝜆 and 𝜇 is also studied. In the final section of this paper, we derive not only new but also interesting identities related special polynomials by which Caputo fractional derivative.

1. Introduction

In 1695, L’Hôpital asked Leibniz: what if the order of the derivative is 1/2? To which Leibniz considered in a useful means, thus it follows that will be equal to 𝑥√𝑑𝑥∶𝑥, an obvious paradox. In recent years, fractional calculus has been studied by many mathematicians from Leibniz’s time to the present.

Also, fractional differential equations arise in many engineering and scientific disciplines as the mathematical modelling of systems and processes in the fields of physics, fluid flows, electrical networks, viscoelasticity, aerodynamics, and many other branches of science. For details, see [1–9].

In the last few decades, fractional-order models are found to be more adequate than integer-order models for some real world problems. Recently, there have been some papers dealing with the existence and multiplicity of solutions (or positive solutions) of nonlinear initial fractional differential equations by the use of techniques of nonlinear analysis [10–21], upper and lower solutions method [22–24], fixed point index [25, 26], coincidence theory [27], Banach contraction mapping principle [28], and so forth.

Chai [11] investigated the existence and multiplicity of positive solutions for a class of boundary-value problem of fractional differential equation with 𝑝-Laplacian operator 𝐷^𝛽₀₊𝜙_𝑝𝐷^𝛼₀₊𝑢(𝑡)+𝑓𝑡,𝑢(𝑡),𝐷^𝜌₀₊𝑢(𝑡)𝑢(0)(1)=0,0<𝑡<1,=0,𝑢+𝜎𝐷^𝛾₀₊𝑢(1)=0,𝐷^𝛼₀₊𝑢(0)=0, (1) where 1<𝛼≤2, 0<𝛾≤1, 0≤𝛼−𝛾−1, 𝜎 is a positive constant number, and 𝐷^𝛼₀₊,𝐷^𝛽₀₊,𝐷^𝛾₀₊ are the standard Riemann-Liouville derivatives. By means of the fixed point theorem on cones, some existence and multiplicity results of positive solutions are obtained.

Although the fractional differential equation boundary-value problems have been studied by several authors, very little is known in the literature on the existence and nonexistence of positive solutions of fractional differential equation boundary-value problems with 𝑝-Laplacian operator when a parameter 𝜆 is involved in the boundary conditions. We also mention that, there is very little known about the uniqueness of the solution of fractional differential equation boundary-value problems with 𝑝-Laplacian operator on the parameter 𝜆. Han et al. [29] studied the existence and uniqueness of positive solutions for the fractional differential equation with 𝑝-Laplacian operator 𝐷^𝛽₀₊𝜙_𝑝𝐷^𝛼₀₊𝑢(𝑡)(𝑡)𝑓(𝑢)𝑢(0)(𝜉)𝜙+𝑎=0,0<𝑡<1,=𝛾𝑢+𝜆,_𝑝𝐷^𝛼₀₊𝑢(0)=𝜙_𝑝𝐷^𝛼₀₊𝑢(1)^′=𝜙_𝑝𝐷^𝛼₀₊𝑢(0)^′′=0, (2) where 0<𝛼⩽1, 2<𝛽⩽3 are real numbers; 𝐷^𝛼₀₊,𝐷^𝛽₀₊ are the standard Caputo fractional derivatives; 𝜙_𝑝(𝑠)=|𝑠|^𝑝−2𝑠, 𝑝>1. Therefore, to enrich the theoretical knowledge of the above, in this paper, we investigate the following 𝑝-Laplacian fractional differential equation boundary-value problem: 𝐷^𝛽₀₊𝜙_𝑝𝐷^𝛼₀₊𝑢(𝑡)(𝑡)𝑓(𝑢)𝑢(0)(ℎ)+𝑎=0,0<𝑡<1,=𝛾𝑢+𝜆,𝑢^′(0)𝜙=𝜇,_𝑝𝐷^𝛼₀₊𝑢(0)=𝜙_𝑝𝐷^𝛼₀₊𝑢(1)^′=𝜙_𝑝𝐷^𝛼₀₊𝑢(0)^′′=𝜙_𝑝𝐷^𝛼₀₊𝑢(0)^′′′=0, (3) where 1<𝛼⩽2, 3<𝛽⩽4 are real numbers, 𝐷^𝛼₀₊,𝐷^𝛽₀₊ are the standard Caputo fractional derivatives, 𝜙_𝑝(𝑠)=|𝑠|^𝑝−2𝑠, 𝑝>1, 𝜙_𝑝⁻¹=𝜙_𝑞, 1/𝑝+1/𝑞=1, 0⩽𝛾<1, 0⩽ℎ⩽1, 𝜆,𝜇>0 are parameters, 𝑎∶(0,1)→[0,+∞), and 𝑓∶[0,+∞)→[0,+∞) are continuous. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results for positive solutions, in terms of the parameters 𝜆 and 𝜇 are obtained. The uniqueness of positive solution on the parameters 𝜆 and 𝜇, is also studied.

2. Preliminaries and Related Lemmas

Definition 1 (see [4]). The Riemann-Liouville fractional integral of order 𝛼>0 of a function 𝑦∶(0,+∞)→ℝ is given by 𝐼^𝛼₀₊𝑦(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑦(𝑠)𝑑𝑠 (4) provided the right side is pointwise defined on (0,+∞).

Definition 2 (see [4]). The Caputo fractional derivative of order 𝛼>0 of a continuous function 𝑦∶(0,+∞)→ℝ is given by 𝐷^𝛼₀₊𝑦(𝑡)=1Γ(𝑛−𝛼)^𝑡₀𝑦^(𝑛)(𝑠)(𝑡−𝑠)^{𝛼−𝑛+1}𝑑𝑠, (5) where 𝑛 is the smallest integer greater than or equal to 𝛼, provided that the right side is pointwise defined on (0,+∞).

Remark 3 (see [8]). By Definition 2, under natural conditions on the function 𝑓(𝑡), for 𝛼→𝑛, the Caputo derivative becomes a conventional 𝑛th derivative of the function 𝑓(𝑡).

Remark 4 (see [4]). As a basic example, 𝐷^𝛼₀⁺𝑡^𝜇⋯×Γ=𝜇𝜇−1𝜇−𝑛+11+𝜇−𝑛Γ𝑡1+𝜇−𝛼^𝜇−𝛼,for𝑡∈(0,∞). (6) In particular, 𝐷^𝛼₀⁺𝑡^𝜇=0, 𝜇=0,1,…,𝑛−1, where 𝐷^𝛼₀⁺ is the Caputo fractional derivative, 𝑛 is the smallest integer greater than or equal to 𝛼.

From the definition of the Caputo derivative and Remark 4, we can obtain the following statement.

Lemma 5 (see [4]). Let 𝛼>0. Then, the fractional differential equation 𝐷^𝛼₀₊𝑢(𝑡)=0 (7) has a unique solution 𝑢(𝑡)=𝑐₀+𝑐₁𝑡+𝑐₂𝑡²+⋯+𝑐_𝑛−1𝑡^𝑛−1,𝑐_𝑖∈ℝ,𝑖=0,1,2,…,𝑛−1, (8) where 𝑛 is the smallest integer greater than or equal to 𝛼.

Lemma 6 (see [4]). Let 𝛼>0. Assume that 𝑢∈𝐶^𝑛[0,1]. Then 𝐼^𝛼₀₊𝐷^𝛼₀₊𝑢(𝑡)=𝑢(𝑡)+𝑐₀+𝑐₁𝑡+𝑐₂𝑡²+⋯+𝑐_𝑛−1𝑡^𝑛−1, (9) for some 𝑐_𝑖∈ℝ, 𝑖=0,1,2,…,𝑛−1, where 𝑛 is the smallest integer greater than or equal to 𝛼.

Lemma 7. Let 𝑦∈𝐶[0,1] and 1<𝛼≤2. Then, fractional differential equation boundary-value problem 𝐷^𝛼₀₊𝑢(𝑡)=𝑦(𝑡),0<𝑡<1,𝑢(0)=𝛾𝑢(ℎ)+𝜆,𝑢^′(0)=𝜇 (10)(11) has a unique solution 𝑢(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)+𝛾𝑑𝑠1−𝛾^ℎ₀(ℎ−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝜆+𝛾𝜇ℎ.1−𝛾 (12)

Proof. We apply Lemma 6 to reduce (10) to an equivalent integral equation, 𝑢(𝑡)=𝐼^𝛼₀₊𝑦(𝑡)+𝑐₀+𝑐₁𝑡,𝑐₀,𝑐₁∈ℝ. (13) Consequently, the general solution of (10) is 𝑢(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑐₀+𝑐₁𝑡,𝑐₀,𝑐₁∈ℝ. (14) By (11), we have 𝑐₀=𝛾1−𝛾^ℎ₀(ℎ−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝛾𝑐₁ℎ+𝜆1−𝛾,1−𝛾 (15) and since 𝑢^′(0)=𝑐₁, we have by (11) 𝑐₁=𝜇. (16) Therefore, the unique solution of problem (10) and (11) is 𝑢(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)+𝛾𝑑𝑠1−𝛾^ℎ₀(ℎ−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝜆𝑑𝑠++1−𝛾𝛾𝜇ℎ.1−𝛾 (17)

Lemma 8. Let 𝑦∈𝐶[0,1] and 1<𝛼≤2, 3<𝛽≤4. Then, fractional differential equation boundary-value problem 𝐷^𝛽₀₊𝜙_𝑝𝐷^𝛼₀₊𝑢(𝑡)+𝑦(𝑡)=0,0<𝑡<1,𝑢(0)=𝛾𝑢(ℎ)+𝜆,𝑢^′(0)𝜙=𝜇,_𝑝𝐷^𝛼₀₊𝑢(0)=𝜙_𝑝𝐷^𝛼₀₊𝑢(1)^′=𝜙_𝑝𝐷^𝛼₀₊𝑢(0)^′′=𝜙_𝑝𝐷^𝛼₀₊𝑢(0)^′′′=0, (18)(19) has a unique solution 𝑢(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞¹₀𝐻()𝑦(𝜏)+𝛾𝑠,𝜏𝑑𝜏𝑑𝑠1−𝛾^ℎ₀(ℎ−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞¹₀𝐻()𝑦(𝜏)+𝑠,𝜏𝑑𝜏𝑑𝑠𝜆+𝛾𝜇ℎ,1−𝛾 (20) where 𝐻()=𝑡()𝑡,𝑠𝛽−11−𝑠^𝛽−2−(𝑡−𝑠)^𝛽−1Γ𝛽𝑡(),0≤𝑠≤𝑡≤1,𝛽−11−𝑠^𝛽−2Γ𝛽,0≤𝑡≤𝑠≤1. (21)

Proof. From Lemma 6, the boundary-value problems (18) and (19) are equivalent to the integral equation 𝜙_𝑝𝐷^𝛼₀₊𝑢(𝑡)=−𝐼^𝛽₀₊𝑦(𝑡)+𝑐₀+𝑐₁𝑡+𝑐₂𝑡²+𝑐₃𝑡³, (22) for some 𝑐₀,𝑐₁,𝑐₂,𝑐₃∈ℝ; that is, 𝜙_𝑝𝐷^𝛼₀₊𝑢(𝑡)=−^𝑡₀(𝑡−𝜏)^𝛽−1Γ𝛽𝑦(𝜏)𝑑𝜏+𝑐₀+𝑐₁𝑡+𝑐₂𝑡²+𝑐₃𝑡³. (23) By the boundary conditions 𝜙_𝑝(𝐷^𝛼₀₊𝑢(0))=(𝜙_𝑝(𝐷^𝛼₀₊𝑢(1)))^′=(𝜙_𝑝(𝐷^𝛼₀₊𝑢(0)))^′′=(𝜙_𝑝(𝐷^𝛼₀₊𝑢(0)))^′′′=0, we have 𝑐₀=𝑐₂=𝑐₃=0,𝑐₁=¹₀𝛽−1(1−𝜏)^𝛽−2Γ𝛽𝑦(𝜏)𝑑𝜏. (24) Therefore, the solution 𝑢(𝑡) of fractional differential equation boundary-value problems (18) and (19) satisfies 𝜙_𝑝𝐷^𝛼₀₊𝑢(𝑡)=−^𝑡₀(𝑡−𝜏)^𝛽−1Γ𝛽𝑦(𝜏)𝑑𝜏+𝑡¹₀𝛽−1(1−𝜏)^𝛽−2Γ𝛽𝑦(𝜏)=𝑑𝜏₁₀𝐻(𝑡,𝜏)𝑦(𝜏)𝑑𝜏. (25) Consequently, 𝐷^𝛼₀₊𝑢(𝑡)=𝜙_𝑞(∫¹₀𝐻(𝑡,𝜏)𝑦(𝜏)𝑑𝜏). Thus, fractional differential equation boundary-value problem (18) and (19) is equivalent to the problem 𝐷^𝛼₀₊𝑢(𝑡)=𝜙_𝑞¹₀𝐻()𝑦(𝜏)𝑢(0)(ℎ)𝑡,𝜏𝑑𝜏,0<𝑡<1,=𝛾𝑢+𝜆,𝑢^′(0)=𝜇. (26) Lemma 7 implies that the fractional differential equation boundary-value problems (18) and (19) have a unique solution, 𝑢(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞¹₀𝐻()𝑦(𝜏)+𝛾𝑠,𝜏𝑑𝜏𝑑𝑠1−𝛾^ℎ₀(ℎ−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞¹₀𝐻()𝑦(𝜏)+𝑠,𝜏𝑑𝜏𝑑𝑠𝜆+𝛾𝜇ℎ.1−𝛾 (27) The proof is complete.

Lemma 9 (see [15]). Let 1<𝛼⩽2,3<𝛽⩽4. The function 𝐻(𝑡,𝑠) is continuous on [0,1]×[0,1] and satisfies (1)𝐻(𝑡,𝑠)⩾0,𝐻(𝑡,𝑠)⩽𝐻(1,𝑠), for 𝑡,𝑠∈[0,1];(2)𝐻(𝑡,𝑠)⩾𝑡^𝛽−1𝐻(1,𝑠), for 𝑡,𝑠∈(0,1).

Lemma 10 (Schauder fixed point theorem [30]). Let (𝐸,𝑑) be a complete metric space, 𝑈 be a closed convex subset of 𝐸, and 𝐴∶𝑈→𝑈 be a mapping such that the set {𝐴𝑢∶𝑢∈𝑈} is relatively compact in 𝐸. Then, 𝐴 has at least one fixed point.

To prove our main results, we use the following assumptions.(H1)0<∫¹₀𝐻(1,𝜏)𝑎(𝜏)𝑑𝜏<+∞;(H2) there exist 0<𝜎<1 and 𝑐>0 such that 𝑓(𝑥)⩽𝜎𝐿𝜙_𝑝(𝑥),for0⩽𝑥⩽𝑐, (28) where 𝐿 satisfies 0<𝐿⩽𝜙_𝑝1+𝛾ℎ^𝛼−1Γ(𝛼+1)1−𝛾¹₀𝐻(1,𝜏)𝑎(𝜏)𝑑𝜏⁻¹; (29)(H3) there exist 𝑑>0 such that 𝑓(𝑥)⩽𝑀𝜙_𝑝(𝑥),for𝑑<𝑥<+∞, (30) where 𝑀 satisfies <𝜙0<𝑀_𝑝1+𝛾ℎ^𝛼−1Γ()2𝛼+11−𝛾^𝑞−11₀𝐻(1,𝜏)𝑎(𝜏)𝑑𝜏⁻¹; (31)(H4) there exist 0<𝛿<1 and 𝑒>0 such that 𝑓(𝑥)⩾𝑁𝜙_𝑝(𝑥),for𝑒<𝑥<+∞, (32) where 𝑁 satisfies 𝑁>𝜙_𝑝𝑐_𝛿¹₀(1−𝑠)^𝛼−2Γ(𝛼)𝜙_𝑞𝑠^𝛽−1×𝑑𝑠_1𝛿𝐻(1,𝜏)𝑎(𝜏)𝑑𝜏⁻¹; (33) with 𝑐_𝛿=^𝛿₀𝛼(1−𝑠)^𝛼−2𝜙_𝑞𝑠^𝛽−1𝑑𝑠∈(0,1); (34)(H5)𝑓(𝑥) is nondecreasing in 𝑥;(H6) there exist 0⩽𝜃<1 such that 𝑓(𝑘𝑥)⩾𝜙_𝑝(𝑘)^𝜃𝑓(𝑥),forany0<𝑘<1,0<𝑥<+∞. (35)

Remark 11. Let 𝑓₀=lim_𝑥→0+𝑓(𝑥)𝜙_𝑝(𝑥),𝑓_∞=lim_𝑥→+∞𝑓(𝑥)𝜙_𝑝(𝑥). (36) Then, (H2) holds if 𝑓₀=0, (H3) holds if 𝑓_∞=0, and (H4) holds if 𝑓_∞=+∞.

3. Existence

Theorem 12. Assume that (H1), (H2) hold. Then, the fractional differential equation boundary-value problem (3) has at least one positive solution for 0<𝜆+𝛾𝜇≤(1−𝛾)(1−𝜙_𝑞(𝜎))𝑐.

Proof. Let 𝑐>0 be given in (H2). Define 𝐾₁={𝑢∈𝐶[0,1]∶0⩽𝑢(𝑡)⩽𝑐on[0,1]} (37) and an operator 𝑇_𝜆∶𝐾₁→𝐶[0,1] by 𝑇_𝜆𝑢(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞¹₀𝐻()𝑎(𝜏)𝑓(𝑢(𝜏+𝛾𝑠,𝜏))𝑑𝜏1−𝛾^ℎ₀(ℎ−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞×¹₀𝐻()𝑎(𝜏)𝑓(𝑢(𝜏+𝑠,𝜏))𝑑𝜏𝑑𝑠𝜆+𝛾𝜇ℎ.1−𝛾 (38) Then, 𝐾₁ is a closed convex set. From Lemma 8, 𝑢 is a solution of fractional differential equation boundary-value problem (3) if and only if 𝑢 is a fixed point of 𝑇_𝜆. Moreover, a standard argument can be used to show that 𝑇_𝜆 is compact.
For any 𝑢∈𝐾₁, from (28) and (29), we obtain 𝑓(𝑢(𝑡))⩽𝜎𝐿𝜙_𝑝(𝑢(𝑡))⩽𝜎𝐿𝜙_𝑝(𝑐)[],ℎ,on0,11+𝛾^𝛼−1Γ()𝜙𝛼+11−𝛾^𝑞(𝐿)𝜙_𝑞¹₀𝐻(1,𝜏)𝑎(𝜏)𝑑𝜏⩽1. (39) Let 0<𝜆+𝛾𝜇⩽(1−𝛾)(1−𝜙_𝑞(𝜎))𝑐. Then, from Lemma 6 and (38), it follows that 0⩽𝑇_𝜆𝑢(𝑡)⩽^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞¹₀𝐻()𝑎(𝜏)𝑓(𝑢(𝜏+𝛾1,𝜏))𝑑𝜏𝑑𝑠×1−𝛾_ℎ0(ℎ−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞¹₀𝐻()𝑎(𝜏)𝑓(𝑢(𝜏+1,𝜏))𝑑𝜏𝑑𝑠𝜆+𝛾𝜇ℎ⩽11−𝛾Γ()𝜙𝛼+1^𝑞1₀𝐻()𝑎(𝜏)𝑓(𝑢(𝜏+1,𝜏))𝑑𝜏𝛾ℎ^𝛼Γ()𝜙𝛼+11−𝛾^𝑞×¹₀𝐻(1,𝜏)𝑎(𝜏)𝑓(𝑢(𝜏))𝑑𝜏+1−𝜙_𝑞(𝜎)𝑐=1+𝛾ℎ^𝛼−𝛾Γ()𝜙𝛼+11−𝛾^𝑞1₀𝐻()𝑎(𝜏)𝑓(𝑢(𝜏+1,𝜏))𝑑𝜏1−𝜙_𝑞(𝜎)𝑐⩽1+𝛾ℎ^𝛼−1Γ()𝜙𝛼+11−𝛾^𝑞(𝐿)𝜙_𝑞¹₀𝐻(1,𝜏)𝑎(𝜏)𝑑𝜏𝜙_𝑞(𝜎)𝑐+1−𝜙_𝑞(𝜎)𝑐⩽𝜙_𝑞(𝜎)𝑐+1−𝜙_𝑞(𝜎)𝑐=𝑐,𝑡∈[0,1]. (40) Thus, 𝑇_𝜆(𝐾₁)⊆𝐾₁, by Schauder fixed point theorem, 𝑇_𝜆 has a fixed point 𝑢∈𝐾₁; that is, the fractional differential equation boundary-value problem (3) has at least one positive solution. The proof is complete.

Corollary 13. Assume that (H1) holds and 𝑓₀=0. Then, the fractional differential equation boundary-value problem (3) has at least one positive solution for sufficiently small 𝜆>0.

Theorem 14. Assume that (H1), (H3) hold. Then, the fractional differential equation boundary-value problem (3) has at least one positive solution for all 𝜆>0.

Proof. Let 𝜆>0 be fixed and 𝑑>0 be given in (H3). Define 𝐷=max_{0⩽𝑥⩽𝑑}𝑓(𝑥). Then 𝑓(𝑥)⩽𝐷,for0⩽𝑥⩽𝑑. (41) From (31), we have 1+𝛾ℎ^𝛼−1Γ()2𝛼+11−𝛾^𝑞−1𝜙_𝑞(𝑀)𝜙_𝑞¹₀𝐻(1,𝜏)𝑎(𝜏)𝑑𝜏<1. (42) Thus, there exists 𝑑^∗>𝑑 large enough so that 1+𝛾ℎ^𝛼−1Γ()2𝛼+11−𝛾^𝑞−1𝜙_𝑞(𝐷)+𝜙_𝑞(𝑀)𝑑^∗𝜙_𝑞×¹₀𝐻(1,𝜏)𝑎(𝜏)𝑑𝜏+𝜆+𝛾𝜇ℎ1−𝛾⩽𝑑^∗. (43) Let 𝐾₂=𝑢∈𝐶[0,1]∶0⩽𝑢(𝑡)⩽𝑑^∗on[0,1]. (44) For 𝑢∈𝐾₂, define 𝐼^𝑢₁={[](𝑡)},𝐼𝑡∈0,1∶0⩽𝑢⩽𝑑_𝑢2=𝑡∈[0,1]∶𝑑<𝑢(𝑡)⩽𝑑^∗. (45) Then, 𝐼^𝑢₁∪𝐼^𝑢₂=[0,1], 𝐼^𝑢₁∩𝐼^𝑢₂=∅, and in view of (30), we have 𝑓(𝑢(𝑡))⩽𝑀𝜙_𝑝(𝑢(𝑡))⩽𝑀𝜙_𝑝𝑑^∗,for𝑡∈𝐼^𝑢₂. (46) Let the compact operator 𝑇_𝜆 be defined by (38). Then, from Lemma 6, (30), and (41), we have 0⩽𝑇_𝜆𝑢(𝑡)⩽^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞¹₀𝐻()𝑎(𝜏)𝑓(𝑢(𝜏+𝛾1,𝜏))𝑑𝜏𝑑𝑠×1−𝛾_ℎ0(ℎ−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞¹₀𝐻()𝑎(𝜏)𝑓(𝑢(𝜏+1,𝜏))𝑑𝜏𝑑𝑠𝜆+𝛾𝜇ℎ⩽11−𝛾Γ()𝜙𝛼+1^𝑞1₀𝐻()𝑎(𝜏)𝑓(𝑢(𝜏+1,𝜏))𝑑𝜏𝛾ℎ^𝛼Γ()𝜙𝛼+11−𝛾^𝑞1₀𝐻()𝑎(𝜏)𝑓(𝑢(𝜏+1,𝜏))𝑑𝜏𝜆+𝛾𝜇ℎ=ℎ1−𝛾1+𝛾^𝛼−1Γ()𝜙𝛼+11−𝛾^𝑞×_𝐼𝑢1𝐻()𝑎(𝜏)𝑓(𝑢(𝜏+1,𝜏))𝑑𝜏_𝐼𝑢2𝐻()𝑎(𝜏)𝑓(𝑢(𝜏+1,𝜏))𝑑𝜏𝜆+𝛾𝜇ℎ⩽ℎ1−𝛾1+𝛾^𝛼−1Γ()𝜙𝛼+11−𝛾^𝑞×𝐷_𝐼𝑢1𝐻(1,𝜏)𝑎(𝜏)𝑑𝜏+𝑀𝜙_𝑝𝑑^∗_𝐼𝑢2𝐻()𝑎(𝜏)+1,𝜏𝑑𝜏𝜆+𝛾𝜇ℎ⩽ℎ1−𝛾1+𝛾^𝛼−1Γ()𝜙𝛼+11−𝛾^𝑞𝐷+𝑀𝜙_𝑝𝑑^∗𝜙_𝑞×¹₀𝐻(1,𝜏)𝑎(𝜏)𝑑𝜏+𝜆+𝛾𝜇ℎ.1−𝛾 (47) From (43) and the inequality (𝑎+𝑏)^𝑟⩽2^𝑟(𝑎^𝑟+𝑏^𝑟) for any 𝑎,𝑏,𝑟>0 (see, e.g., [31]), we obtain 0⩽𝑇_𝜆𝑢(𝑡)⩽1+𝛾ℎ^𝛼−1Γ()2𝛼+11−𝛾^𝑞−1𝜙_𝑞(𝐷)+𝜙_𝑞(𝑀)𝑑^∗𝜙_𝑞×¹₀𝐻(1,𝜏)𝑎(𝜏)𝑑𝜏+𝜆+𝛾𝜇ℎ1−𝛾⩽𝑑^∗. (48) Thus, 𝑇_𝜆∶𝐾₂→𝐾₂. Consequently, by Schauder fixed point theorem, 𝑇_𝜆 has a fixed point 𝑢∈𝐾₂, that is, the fractional differential equation boundary-value problem (3) has at least one positive solution. The proof is complete.

Corollary 15. Assume that (H1) holds and 𝑓_∞=0. Then, the fractional differential equation boundary-value problem (3) has at least one positive solution for all 𝜆>0.

4. Uniqueness

Definition 16 (see [32]). A cone 𝑃 in a real Banach space 𝑋 is called solid if its interior 𝑃^𝑜 is not empty.

Definition 17 (see [32]). Let 𝑃 be a solid cone in a real Banach space 𝑋,𝑇∶𝑃^𝑜→𝑃^𝑜 be an operator, and 0⩽𝜃<1. Then, 𝑇 is called a 𝜃-concave operator if 𝑇(𝑘𝑢)⩾𝑘^𝜃𝑇𝑢forany0<𝑘<1,𝑢∈𝑃^𝑜. (49)

Lemma 18 (see [32, Theorem 2.2.6]). Assume that 𝑃 is a normal solid cone in a real Banach space 𝑋, 0⩽𝜃<1, and 𝑇∶𝑃^𝑜→𝑃^𝑜 is a 𝜃-concave increasing operator. Then, 𝑇 has only one fixed point in 𝑃^𝑜.

Theorem 19. Assume that (H1), (H5), (H6) hold. Then, the fractional differential equation boundary-value problem (3) has a unique positive solution for any 𝜆>0.

Proof. Define 𝑃={𝑢∈𝐶[0,1]∶𝑢(𝑡)⩾0on[0,1]}. Then, 𝑃 is a normal solid cone in 𝐶[0,1] with 𝑃^𝑜={𝑢∈𝐶[0,1]∶𝑢(𝑡)>0on[0,1]}. (50) For any fixed 𝜆>0, let 𝑇_𝜆∶𝑃→𝐶[0,1] be defined by (38). Define 𝑇∶𝑃→𝐶[0,1] by 𝑇𝑢(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞¹₀𝐻()𝑎(𝜏)𝑓(𝑢(𝜏+𝛾𝑠,𝜏))𝑑𝜏𝑑𝑠1−𝛾^ℎ₀(ℎ−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞×¹₀𝐻(𝑠,𝜏)𝑎(𝜏)𝑓(𝑢(𝜏))𝑑𝜏𝑑𝑠. (51) Then, from (H5), we have 𝑇 increasing in 𝑢∈𝑃^𝑜 and 𝑇_𝜆𝑢(𝑡)=𝑇𝑢(𝑡)+𝜆+𝛾𝜇ℎ.1−𝛾 (52) Clearly, 𝑇_𝜆∶𝑃^𝑜→𝑃^𝑜. Next, we prove that 𝑇_𝜆 is a 𝜃-concave increasing operator. In fact, for 𝑢₁,𝑢₂∈𝑃 with 𝑢₁(𝑡)⩾𝑢₂(𝑡) on [0,1], we obtain 𝑇_𝜆𝑢₁(𝑡)⩾𝑇𝑢₂(𝑡)+𝜆+𝛾𝜇ℎ1−𝛾=𝑇_𝜆𝑢₂(𝑡); (53) that is, 𝑇_𝜆 is increasing. Moreover, (H6) implies 𝑇_𝜆(𝑘𝑢)(𝑡)⩾𝑘^𝜃𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)×𝜙_𝑞¹₀𝐻(𝑠,𝜏)𝑎(𝜏)𝑓(𝑢(𝜏))𝑑𝜏𝑑𝑠+𝑘^𝜃𝛾×1−𝛾_ℎ0(ℎ−𝑠)^𝛼−1Γ(𝛼)×𝜙_𝑞¹₀𝐻()𝑎(𝜏)𝑓(𝑢(𝜏+𝑠,𝜏))𝑑𝜏𝑑𝑠𝜆+𝛾𝜇ℎ1−𝛾=𝑘^𝜃𝑇𝑢(𝑡)+𝜆+𝛾𝜇ℎ1−𝛾⩾𝑘^𝜃𝑇𝑢(𝑡)+𝜆+𝛾𝜇ℎ1−𝛾=𝑘^𝜃𝑇_𝜆𝑢(𝑡), (54) that is, 𝑇_𝜆 is 𝜃-concave. By Lemma 8, 𝑇_𝜆 has a unique fixed point 𝑢_𝜆 in 𝑃^𝑜, that is, the fractional differential equation boundary-value problem (3) has a unique positive solution. The proof is complete.

5. Nonexistence

In this section, we let the Banach space 𝐶[0,1] be endowed with the norm ‖𝑢‖=max_0⩽𝑡⩽1|𝑢(𝑡)|.

Lemma 20. Assume (H1) holds and let 0<𝛿<1 be given in (H4). Then, the unique solution 𝑢(𝑡) of fractional differential equation boundary-value problem (18) and (19) satisfies 𝑢(𝑡)⩾𝑐_𝛿‖𝑢‖𝑓𝑜𝑟𝛿⩽𝑡⩽1, (55) where 𝑐_𝛿 is defined by (34).

Proof. In view of Lemma 9 and (19), we have 𝑢(𝑡)⩽^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞¹₀𝐻()𝑦(𝜏)+𝛾1,𝜏𝑑𝜏𝑑𝑠×1−𝛾_ℎ0(ℎ−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞¹₀𝐻()𝑦(𝜏)+𝑠,𝜏𝑑𝜏𝑑𝑠𝜆+𝛾𝜇ℎ⩽11−𝛾Γ()𝜙𝛼+1^𝑞1₀𝐻()𝑦(𝜏)+𝛾1,𝜏𝑑𝜏×1−𝛾_ℎ0(ℎ−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞¹₀𝐻()𝑦(𝜏)+𝑠,𝜏𝑑𝜏𝑑𝑠𝜆+𝛾𝜇ℎ1−𝛾 (56) for 𝑡∈[0,1], and 𝑢(𝑡)⩾^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞¹₀𝑠^𝛽−1𝐻()𝑦(𝜏)+𝛾1,𝜏𝑑𝜏𝑑𝑠×1−𝛾_ℎ0(ℎ−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞¹₀𝐻()𝑦(𝜏)+𝑠,𝜏𝑑𝜏𝑑𝑠𝜆+𝛾𝜇ℎ1−𝛾=^𝑡₀𝛼(𝑡−𝑠)^𝛼−2𝜙_𝑞𝑠^𝛽−11𝑑𝑠Γ()𝜙𝛼+1^𝑞×¹₀𝐻()𝑦(𝜏)+𝛾1,𝜏𝑑𝜏1−𝛾^ℎ₀(ℎ−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞×¹₀𝐻(𝑠,𝜏)𝑦(𝜏)𝑑𝜏𝑑𝑠+𝜆+𝛾𝜇ℎ1−𝛾⩾𝑐_𝛿1Γ()𝜙𝛼+1^𝑞1₀𝐻()𝑦(𝜏)+𝛾1,𝜏𝑑𝜏×1−𝛾_ℎ0(ℎ−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞¹₀𝐻()𝑦(𝜏)+𝑠,𝜏𝑑𝜏𝑑𝑠𝜆+𝛾𝜇ℎ1−𝛾⩾𝑐_𝛿1Γ()𝜙𝛼+1^𝑞1₀𝐻()𝑦(𝜏)+𝛾1,𝜏𝑑𝜏1−𝛾^ℎ₀(ℎ−𝑠)^𝛼−1Γ(𝛼)×𝜙_𝑞¹₀𝐻()𝑦(𝜏)+𝑠,𝜏𝑑𝜏𝑑𝑠𝜆+𝛾𝜇ℎ1−𝛾 (57) for 𝑡∈[𝛿,1]. Therefore, 𝑢(𝑡)⩾𝑐_𝛿‖𝑢‖ for 𝛿⩽𝑡⩽1. The proof is complete.

Theorem 21. Assume that (H1), (H4) hold. Then, the fractional differential equation boundary-value problem (3) has no positive solution for 𝜆+𝛾𝜇ℎ>(1−𝛾)𝑒.

Proof. Assume, to the contrary, the fractional differential equation boundary-value problem (3) has a positive solution 𝑢(𝑡) for 𝜆+𝛾𝜇ℎ>(1−𝛾)𝑒. Then, by Lemma 8, we have 𝑢(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞×¹₀𝐻()𝑎(𝜏)𝑓(𝑢(𝜏+𝛾𝑠,𝜏))𝑑𝜏𝑑𝑠×1−𝛾_ℎ0(ℎ−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞¹₀𝐻()𝑎(𝜏)𝑓(𝑢(𝜏+𝑠,𝜏))𝑑𝜏𝑑𝑠𝜆+𝛾𝜇ℎ.1−𝛾 (58) Therefore, 𝑢(𝑡)>𝑒 on [0,1]. In view of (32) and (33), we obtain 𝑓(𝑢(𝑡))⩾𝑁𝜙_𝑝(𝑢(𝑡[],𝑐))on0,1_𝛿𝜙_𝑞(𝑁)𝜙_𝑞¹_𝛿𝐻()𝑎(𝜏)×1,𝜏𝑑𝜏₁₀(1−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞𝑠^𝛽−1𝑑𝑠>1. (59) Then, by Lemmas 6 and 9, we obtain ‖𝑢‖(1)>≥𝑢₁₀(1−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞×¹₀𝐻()𝑎(𝜏)𝑓(𝑢(𝜏⩾𝑠,𝜏))𝑑𝜏𝑑𝑠₁₀(1−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞𝑠^𝛽−1𝑑𝑠𝜙_𝑞×¹₀𝐻()𝑎(𝜏)𝑓(𝑢(𝜏⩾1,𝜏))𝑑𝜏₁₀(1−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞𝑠^𝛽−1𝑑𝑠𝜙_𝑞(𝑁)𝜙_𝑞×¹_𝛿𝐻(1,𝜏)𝑎(𝜏)𝜙_𝑝(𝑢(𝜏⩾‖𝑢‖𝑐))𝑑𝜏_𝛿10(1−𝑠)^𝛼−1Γ(𝛼)𝜙_𝑞𝑠^𝛽−1𝑑𝑠𝜙_𝑞(𝑁)𝜙_𝑞×¹_𝛿𝐻()𝑎(𝜏)>‖𝑢‖.1,𝜏𝑑𝜏 (60) This contradiction completes the proof.

Corollary 22. Assume that (H1) holds and 𝑓_∞=+∞. Then, the fractional differential equation boundary-value problem (3) has no positive solution for sufficiently large 𝜆>0.

6. Conclusion: Identities on the Special Polynomials whereby Caputo Fractional Derivative

In this final part, we will focus on the new interesting identities related to special polynomials by means of Caputo fractional derivative.

As well known, the Bernoulli polynomials may be defined to be 𝐹()=𝑧𝑡,𝑧𝑒^𝑧𝑒−1^𝑡𝑧=𝑒^𝐵𝑧=^∞_𝑛=0𝐵_𝑛(𝑡)𝑧^𝑛,𝑛! (61) where usual convention about replacing 𝐵^𝑛 by 𝐵_𝑛 in is used. Also, we note that the Bernoulli polynomials is analytic on the region 𝐷={𝑧∈ℂ∣|𝑧|<2𝜋} (see [33]).

Let 𝑑/𝑑𝑡 be familiar normal derivative, then we can obtain the following identity 𝑑𝑡𝑑𝑡^𝑛=𝑛𝑡^𝑛−1. (62)

Differentiating in both sides of (61), we have 𝑑𝐵𝑑𝑡^𝑛(𝑡)=𝑛𝐵_𝑛−1(𝑡) (63) (see [33]).

When 𝑡=0 in (61), we have 𝐵_𝑛(0)∶=𝐵_𝑛 are called Bernoulli numbers, which can be generated by 𝐹(𝑧)=𝑧𝑒^𝑧=−1^∞_𝑛=0𝐵_𝑛𝑧^𝑛.𝑛! (64)

By (61) and (64), we have the following functional equation: 𝐹(𝑡,𝑧)=𝑒^𝑡𝑧𝐹(𝑧) (65) and this equation yields to 𝐵_𝑚(𝑡)=^𝑚_𝑘=0𝑚𝑘𝑡^𝑚−𝑘𝐵_𝑘=^𝑚_𝑘=0𝑚𝑘𝑡^𝑘𝐵_𝑚−𝑘 (66) (see [33]).

Let us now take 𝑦(𝑡)=𝐵_𝑚(𝑡) in Definition 2 leads to 𝐷^𝛼₀₊𝐵_𝑚(𝑡)=1Γ(𝑛−𝛼)^𝑡₀𝑑^𝑛/𝑑𝑡^𝑛𝐵_𝑚(𝑡)_𝑡=𝑠(𝑡−𝑠)^{𝛼−𝑛+1}()⋯()×𝑑𝑠=𝑚𝑚−1𝑚−𝑛+1_{𝑚−𝑛𝑘=0}𝑘𝐵𝑚−𝑛_{𝑚−𝑛−𝑘}×1Γ(𝑛−𝛼)^𝑡₀𝑠^𝑘(𝑡−𝑠)^{𝛼−𝑛+1}=Γ()𝑑𝑠𝑚+1Γ(𝑚−𝑛+1)^𝑚−𝑛_𝑘=0𝑘!(_𝑘^𝑚−𝑛)𝐵_{𝑚−𝑛−𝑘}Γ()𝑡𝑛+𝑘−𝛼+1^{𝑘−𝛼+𝑛}. (67)

Therefore, we procure the following theorem.

Theorem 23. The following identity holds true: 𝐷^𝛼₀₊𝐵_𝑚(𝑡)=Γ(𝑚+1)Γ()×𝑚−𝑛+1_{𝑚−𝑛𝑘=0}𝑘!(_𝑘^𝑚−𝑛)𝐵_{𝑚−𝑛−𝑘}Γ()𝑡𝑛+𝑘−𝛼+1^{𝑘−𝛼+𝑛}. (68)

In [34], the Bernoulli polynomials of higher order are defined by 𝑧𝑒^𝑧𝑧−1𝑒^𝑧⋯𝑧−1𝑒^𝑧−1_𝑙-times𝑒^𝑡𝑧=^∞_𝑚=0𝐵_𝑚^(𝑙)(𝑡)𝑧^𝑛,𝑛! (69) we note that 𝐵_𝑚^(𝑙)(𝑡) is analytic on 𝐷. It follows from (69), we have 𝑑𝐵𝑑𝑡^𝑚(𝑙)(𝑡)=𝑚𝐵^(𝑙)_𝑚−1(𝑡),𝑑^𝑛𝑑𝑡^𝑛𝐵_𝑚^(𝑙)(𝑡)=Γ(𝑚+1)Γ()𝐵𝑚−𝑛+1^{(𝑙)𝑚−𝑛}(𝑡) (70) (see [34]).

Substituting 𝑥=0 into (69), 𝐵_𝑚^(𝑙)(0)∶=𝐵_𝑚^(𝑙) are called Bernoulli polynomials of higher order.

Owing to (69) and (70), we readily see that 𝐷^𝛼₀₊𝐵_𝑚^(𝑙)(𝑡)=1Γ(𝑛−𝛼)^𝑡₀𝑑^𝑛/𝑑𝑡^𝑛𝐵_𝑚^(𝑙)(𝑡)_𝑡=𝑠(𝑡−𝑠)^{𝛼−𝑛+1}()⋯()×𝑑𝑠=𝑚𝑚−1𝑚−𝑛+1_{𝑚−𝑛𝑘=0}𝑘𝐵𝑚−𝑛_{(𝑙)𝑚−𝑛−𝑘}×1Γ(𝑛−𝛼)^𝑡₀𝑠^𝑘(𝑡−𝑠)^{𝛼−𝑛+1}=Γ()𝑑𝑠𝑚+1Γ(𝑚−𝑛+1)^𝑚−𝑛_𝑘=0𝑘!(_𝑘^𝑚−𝑛)𝐵^(𝑙)_{𝑚−𝑛−𝑘}Γ()𝑡𝑛+𝑘−𝛼+1^{𝑘−𝛼+𝑛}=Γ(𝑚+1)Γ(𝑚−𝑛+1)^𝑚−𝑛_𝑘=0𝑘!(_𝑘^𝑚−𝑛)Γ()𝑡𝑛+𝑘−𝛼+1^{𝑘−𝛼+𝑛}×_{𝑠1+𝑠2+⋯+𝑠𝑙=𝑚−𝑛−𝑘𝑠𝑙≥0}𝑠𝑚−𝑛−𝑘₁,𝑠₂,…,𝑠_𝑙^𝑙𝑗=1𝐵_𝑠𝑗. (71)

Therefore, we can state the following theorem.

Theorem 24. The following identity holds true: 𝐷^𝛼₀₊𝐵_𝑚^(𝑙)(𝑡)=Γ(𝑚+1)Γ(𝑚−𝑛+1)^𝑚−𝑛_𝑘=0𝑘!(_𝑘^𝑚−𝑛)Γ()𝑡𝑛+𝑘−𝛼+1^{𝑘−𝛼+𝑛}×_{𝑠1+𝑠2+⋯+𝑠𝑙=𝑚−𝑛−𝑘𝑠𝑙≥0}𝑠𝑚−𝑛−𝑘₁,𝑠₂,…,𝑠_𝑙^𝑙𝑗=1𝐵_𝑠𝑗, (72) in which 𝐵_𝑠𝑗 and 𝑠𝑚−𝑛−𝑘₁,𝑠₂,…,𝑠_𝑙 are Bernoulli numbers and multibinomial coefficients.

In the region 𝑇={𝑧∈ℂ∣|𝑧|<𝜋}, the Euler polynomials and the Euler polynomials of higher order are given, respectively, with the help of the following generating functions: 2𝑒^𝑧𝑒+1^𝑡𝑧=^∞_𝑚=0𝐸_𝑚(𝑡)𝑧^𝑚,𝑚!2𝑒^𝑧2+1𝑒^𝑧⋯2+1𝑒^𝑧+1_𝑙-times𝑒^𝑡𝑧=^∞_{𝑚=0𝑠1+𝑠2+⋯+𝑠𝑙=𝑚}𝑚𝑠₁,𝑠₂,…,𝑠_𝑙^{𝑙−1𝑗=1}𝐸_𝑠𝑗𝑡^𝑠𝑙𝑧^𝑚=𝑚!_∞𝑚=0𝐸_𝑚^(𝑙)(𝑡)𝑧^𝑛,𝑛! (73)(74) where 𝐸_𝑠𝑗 are Euler numbers in (see [33–36]). From the last equation, we discover the following: 𝑑𝐸𝑑𝑡^𝑚(𝑡)=𝑚𝐸_𝑚−1(𝑡),𝑑𝐸𝑑𝑡^𝑚(𝑙)(𝑡)=𝑚𝐸^(𝑙)_𝑚−1(𝑡) (75) (see [35]).

Obviously, we have that 𝐸_𝑚⁽¹⁾(𝑡)∶=𝐸_𝑚(𝑡). (76)

Taking 𝑦(𝑡)=𝐸_𝑚^(𝑙)(𝑡) in Definition 2, by (73) and (75), we compute 𝐷^𝛼₀₊𝐸_𝑚^(𝑙)(𝑡)=1Γ(𝑛−𝛼)^𝑡₀𝑑^𝑛/𝑑𝑡^𝑛𝐸_𝑚^(𝑙)(𝑡)_𝑡=𝑠(𝑡−𝑠)^{𝛼−𝑛+1}()⋯()×𝑑𝑠=𝑚𝑚−1𝑚−𝑛+1_{𝑚−𝑛𝑘=0}𝑘𝐸𝑚−𝑛_{(𝑙)𝑚−𝑛−𝑘}×1Γ(𝑛−𝛼)^𝑡₀𝑠^𝑘(𝑡−𝑠)^{𝛼−𝑛+1}=Γ()𝑑𝑠𝑚+1Γ(𝑚−𝑛+1)^𝑚−𝑛_𝑘=0𝑘!(_𝑘^𝑚−𝑛)𝐸^(𝑙)_{𝑚−𝑛−𝑘}Γ()𝑡𝑛+𝑘−𝛼+1^{𝑘−𝛼+𝑛}=Γ(𝑚+1)Γ(𝑚−𝑛+1)^𝑚−𝑛_𝑘=0𝑘!(_𝑘^𝑚−𝑛)Γ()𝑡𝑛+𝑘−𝛼+1^{𝑘−𝛼+𝑛}×_{𝑠1+𝑠2+⋯+𝑠𝑙=𝑚−𝑛−𝑘𝑠𝑙≥0}𝑠𝑚−𝑛−𝑘₁,𝑠₂,…,𝑠_𝑙^𝑙𝑗=1𝐸_𝑠𝑗. (77)

Therefore, we obtain the following theorem.

Theorem 25. The following identity 𝐷^𝛼₀₊𝐸_𝑚^(𝑙)(𝑡)=Γ(𝑚+1)Γ(𝑚−𝑛+1)^𝑚−𝑛_𝑘=0𝑘!(_𝑘^𝑚−𝑛)Γ()×𝑛+𝑘−𝛼+1_{𝑠1+𝑠2+⋯+𝑠𝑙=𝑚−𝑛−𝑘𝑠𝑙≥0}𝑠𝑚−𝑛−𝑘₁,𝑠₂,…,𝑠_𝑙^𝑙𝑗=1𝐸_𝑠𝑗𝑡^{𝑘−𝛼+𝑛}. (78) is true. Obviously, we have that 𝐷^𝛼₀₊𝐸_𝑚(𝑡)=Γ(𝑚+1)Γ(𝑚−𝑛+1)^𝑚−𝑛_𝑘=0𝑘!(_𝑘^𝑚−𝑛)𝐸_{𝑚−𝑛−𝑘}Γ()𝑡𝑛+𝑘−𝛼+1^{𝑘−𝛼+𝑛}. (79)

In the region 𝑇={𝑧∈ℂ∣|𝑧|<𝜋}, Genocchi polynomials, 𝐺_𝑚(𝑥), and Genocchi polynomials of higher order, 𝐺_𝑚^(𝑙)(𝑥), are defined as an extension of Genocchi numbers 𝐺_𝑚 defined in [33, 37, 38], respectively, 2𝑧𝑒^𝑧𝑒+1^𝑡𝑧=^∞_𝑚=0𝐺_𝑚(𝑡)𝑧^𝑚,𝑚!2𝑧𝑒^𝑧+12𝑧𝑒^𝑧⋯+12𝑧𝑒^𝑧+1_𝑙-times𝑒^𝑡𝑧=^∞_𝑚=0𝐺_𝑚^(𝑙)(𝑡)𝑧^𝑛.𝑛! (80)

In this final section, by the similar method, we arrive at the following theorem.

Theorem 26. The following identity 𝐷^𝛼₀₊𝐺_𝑚^(𝑙)(𝑡)=Γ(𝑚+1)Γ(𝑚−𝑛+1)^𝑚−𝑛_𝑘=0𝑘!(_𝑘^𝑚−𝑛)Γ()×𝑛+𝑘−𝛼+1_{𝑠1+𝑠2+⋯+𝑠𝑙=𝑚−𝑛−𝑘𝑠𝑙≥0}𝑠𝑚−𝑛−𝑘₁,𝑠₂,…,𝑠_𝑙^𝑙𝑗=1𝐺_𝑠𝑗𝑡^{𝑘−𝛼+𝑛} (81) is true. Obviously, we have that, 𝐷^𝛼₀₊𝐺_𝑚(𝑡)=Γ(𝑚+1)Γ(𝑚−𝑛+1)^𝑚−𝑛_𝑘=0𝑘!(_𝑘^𝑚−𝑛)𝐺_{𝑚−𝑛−𝑘}Γ()𝑡𝑛+𝑘−𝛼+1^{𝑘−𝛼+𝑛}. (82)