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

1 Department of Mathematics, Faculty of Science and Letters, Namik Kemal University, 59030 Tekirdağ, Turkey 2Department of Mathematics Engineering, Istanbul Technical University, Maslak, 34469 Istanbul, Turkey 3 Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, 27310 Gaziantep, Turkey 4Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea 5 Department of Mathematics, College of Natural Sciences, Kwangwoon University, Seoul 139-701, Republic of Korea 6Atatürk Street, 31290 Hatay, Turkey


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  √  : , an obvious paradox.In recent years, fractional calculus has been studied by many mathematicians from Leibniz's time to the present.
Although the fractional differential equation boundaryvalue 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 -Laplacian operator when a parameter  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 boundaryvalue problems with -Laplacian operator on the parameter .Han et al. [29] studied the existence and uniqueness of positive solutions for the fractional differential equation with -Laplacian operator where 0 <  ⩽ 1, 2 <  ⩽ 3 are real numbers;   0+ ,   0+ are the standard Caputo fractional derivatives;   () = || −2 ,  > 1.Therefore, to enrich the theoretical knowledge of the above, in this paper, we investigate the following -Laplacian fractional differential equation boundary-value problem: where 1 <  ⩽ 2, 3 <  ⩽ 4 are real numbers,   0+ ,   0+ are the standard Caputo fractional derivatives,   () = || −2 ,  > 1,  −1  =   , 1/ + 1/ = 1, 0 ⩽  < 1, 0 ⩽ ℎ ⩽ 1, ,  > 0 are parameters,  : (0, 1) → [0, +∞), and  : [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  and  are obtained.The uniqueness of positive solution on the parameters  and , is also studied.
Definition 2 (see [4]).The Caputo fractional derivative of order  > 0 of a continuous function  : (0, +∞) → R is given by where  is the smallest integer greater than or equal to , provided that the right side is pointwise defined on (0, +∞).
Remark 3 (see [8]).By Definition 2, under natural conditions on the function (), for  → , the Caputo derivative becomes a conventional th derivative of the function ().
From the definition of the Caputo derivative and Remark 4, we can obtain the following statement.
Lemma 5 (see [4]).Let  > 0.Then, the fractional differential equation has a unique solution where  is the smallest integer greater than or equal to .
Lemma 10 (Schauder fixed point theorem [30]).Let (, ) be a complete metric space,  be a closed convex subset of , and  :  →  be a mapping such that the set { :  ∈ } is relatively compact in .Then,  has at least one fixed point.
To prove our main results, we use the following assumptions.

Existence
and an operator   :  1 → [0, 1] by Then,  1 is a closed convex set.From Lemma 8,  is a solution of fractional differential equation boundary-value problem (3) if and only if  is a fixed point of   .Moreover, a standard argument can be used to show that   is compact.For any  ∈  1 , from ( 28) and ( 29), we obtain Proof.Let  > 0 be fixed and  > 0 be given in (H3).Define  = max 0⩽⩽ ().Then  () ⩽ , for 0 ⩽  ⩽ .

Uniqueness
Definition 16 (see [32]).A cone  in a real Banach space  is called solid if its interior   is not empty.
The proof is complete.

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 where usual convention about replacing   by   in is used.Also, we note that the Bernoulli polynomials is analytic on the region  = { ∈ C | || < 2} (see [33]).
Let / be familiar normal derivative, then we can obtain the following identity Differentiating in both sides of (61), we have (see [33]).
When  = 0 in (61), we have   (0) :=   are called Bernoulli numbers, which can be generated by By ( 61) and (64), we have the following functional equation: and this equation yields to (see [33]).
Owing to (69) and (70), we readily see that Therefore, we can state the following theorem.