The Eigenvalue Problem for Caputo Type Fractional Differential Equation with Riemann-Stieltjes Integral Boundary Conditions

. In this paper, we investigate the eigenvalue problem for Caputo fractional differential equation with Riemann-Stieltjes integral boundary conditions 𝑐 𝐷 𝜃0+ 𝑝(𝑦) + 𝜇𝑓(𝑡, 𝑝(𝑦)) = 0 , 𝑦 ∈ [0,1] , 𝑝(0) = 𝑝 󸀠󸀠 (0) = 0 , 𝑝(1) = ∫ 1 0 𝑝(𝑦)𝑑𝐴(𝑦) , where 𝑐 𝐷 𝜃0+ is Caputo fractional derivative, 𝜃 ∈ (2,3] , and 𝑓 : [0, 1] × [0,+∞) 󳨀→ [0,+∞) is continuous. By using the Guo-Krasnoselskii’s fixed point theoremonconeandthepropertiesoftheGreen’sfunction,somenewresultsontheexistenceandnonexistenceofpositivesolutions forthefractionaldifferentialequationareobtained.


Introduction
The experience of the last few years has fully borne out the fact that the integer order calculus is not as widely used as fractional order calculus in some fields such as chemistry, control theory, and signal processing.On the remarkable survey of Agarwal, Benchohra, and Hamani [1] it is pointed out that fractional differential equations constitute a fundamental tool in the modeling of some phenomena (see also [2][3][4]).The use of fractional order is more accurate for the description of phenomena, so the study of fractional differential equations becomes the mainstream with the help of techniques of nonlinear analysis.We refer the reader to  for recent results.For example, in [9], the author studied the following fractional differential equation:
They solved the above problem by means of classical fixed point theorems.

Preliminaries
In order to solve problem (7), we provide the properties related to problem (7).
Definition 1 (see [3]).The Caputo's fractional derivative of order  > 0 for a function  ∈   [0, +∞) is defined as where  is the smallest integer greater than or equal to .
Lemma 5.The Green's function (, ) has the following properties: Proof.(i) Obviously, the inequality For  ≤ , we have For  ≤ , we have Thus, the above two inequalities yield the inequality in (ii).
The proof is completed.
Let   :  →  be the operator defined as Thus, the fixed point of the above integral equation is equivalent to the solution of the BVP (7).Lemma 6.   () ⊂  and   :  →  is a completely continuous operator, where   is defined in (26).
The proof is completed.
The following Guo-Krasnoselskii's fixed point theorem is used to prove the existence of positive solution of (7).

Existence of Positive Solutions
In this section, we investigate the existence of positive solutions for integral boundary value problems of fractional differential equation (7).

Nonexistence of Positive Solutions
In this section, we present some sufficient conditions for nonexistence of positive solution to integral boundary value problems of fractional differential equation (7).
One can easily see that, for all  > 0,