## Recent Advance in Function Spaces and their Applications in Fractional Differential Equations

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# Existence of Uniqueness and Nonexistence Results of Positive Solution for Fractional Differential Equations Integral Boundary Value Problems

**Academic Editor:**Yong H. Wu

#### Abstract

In this paper, we consider a class of fractional differential equations with conjugate type integral conditions. Both the existence of uniqueness and nonexistence of positive solution are obtained by means of the iterative technique. The interesting point lies in that the assumption on nonlinearity is closely associated with the spectral radius corresponding to the relevant linear operator.

#### 1. Introduction

In this paper, we consider the existence of uniqueness and nonexistence of positive solution for the following fractional differential equations:where , , , , , is the standard Riemann-Liouville derivative, is continuous, and is nonnegative.

In the recent years, many results were obtained to deal with the existence of solutions for nonlinear differential equations by using nonlinear analysis methods; see [1ā16] and references therein. The fractional nonlocal boundary value problems have particularly attracted a great deal of attention (see [17ā27]). While there are a lot of works dealing with the existence and multiplicity of solutions for nonlinear fractional differential equations, the results dealing with the uniqueness of solution are relatively scarce (see [28ā35]). The main tool used in most of the papers dealing with the uniqueness of solution is the Banach contraction map principle provided that the nonlinearity is a Lipschitz continuous function. When , and is continuous on , Zhang and Zhong [34] established the uniqueness results of solution to problem (1) by using the Banach contraction map principle. It is worth mentioning that only positive solutions are meaningful in most practical problems. As far as we know, the nonexistence of positive solution has seldom been considered up to now.

Motivated by the above work, the aim of this paper is to establish the existence of uniqueness and nonexistence of positive solution to problem (1). Our analysis relies on the iterative technique on the cone derived from the properties of the Green function. This article provides some new insights. Firstly, the uniqueness results are obtained under some conditions concerning the spectral radius with respect to the relevant linear operator. In addition, the error estimation of the iterative sequences is given. Secondly, we impose weaker positivity conditions on ; that is, the Lipschitz constant is generalized to a function and may be singular at . Finally, the nonexistence results of positive solution are obtained under conditions concerning the spectral radius of the relevant linear operator.

#### 2. Preliminaries

For the convenience of the reader, we present here the necessary definitions from fractional calculus theory and lemmas.

*Definition 1. *The fractional integral of order of a function is given byprovided that the right-hand side is point-wise defined on .

*Definition 2. *The Riemann-Liouville fractional derivative of order of a function is given by where ; denotes the integer part of number , provided that the right-hand side is point-wise defined on .

For convenience, we here list the assumptions to be used throughout the paper.

āā is nonnegative, and

āā is continuous.

Lemma 3 ([34]). *For any , the unique solution of the boundary value problemiswhere*

Lemma 4. *The function has the following properties:*(1)*;*(2)*;*(3)*.*

*Proof. *It is obvious that (1) and (2) hold. In the following, we will prove (3).*Case (i)* (). Noticing , we have which impliesNoticing , we have On the other hand, it follows fromthatTherefore,*Case (ii)* (). It is easy to see that On the other hand, we have It follows from (10), (13), (14), and (15) that (3) holds.

Lemma 5. *The function has the following properties:*(1)*;*(2)*;*(3)*,** where*

*Proof. *It can be directly deduced from Lemma 4 and the definition of , so we omit the proof.

Let be endowed with the maximum norm , . Define cones , byIt is clear that is nonempty set since .

For convenience, we list here one more assumption to be used later:

There exists such thatMoreover,

Define operators and as follows:

Lemma 6. *Assume that holds; then .*

*Proof. *It is clear that ,āā. For any , we have Then is well-defined on . It follows from Lemma 5 that .

By virtue of the Krein-Rutmann theorem and Lemma 5, we have the following lemma.

Lemma 7. *Assume that holds. Then is a completely continuous linear operator. Moreover, the spectral radius and has a positive eigenfunction corresponding to its first eigenvalue ; that is, .*

#### 3. Main Results

Theorem 8. *Assume that holds. Then (1) has a unique positive solution if the spectral radius .*

*Proof. *It follows from that is not a fixed point of . Then we only need to prove that has a unique fixed point in .

Firstly, we will prove has a fixed point in .

For any , let Then By Lemma 7, we have that . It is easy to see that For any , set We may suppose that (otherwise, the proof is finished). It follows from (23) and (24) that Then, we have By induction, we can get Then, for any , one has It follows from thatwhich implies is a Cauchy sequence. Therefore, there exists such that converges to . Clearly is a fixed point of .

In the following, we will prove the fixed point of is unique.

Suppose is a positive fixed point of . Then there exists such that Therefore,By induction, we can get It follows from that which implies the positive fixed point of is unique.

*Remark 9. *The unique positive solution of (1) can be approximated by the iterative schemes: for any , let and then . Furthermore, we have error estimationand with the rate of convergence

*Remark 10. *The spectral radius satisfies and . Particularly,

Theorem 11. *Assume that the following condition holds:** There exists satisfying such thatThen (1) has no positive solution if the spectral radius , where *

*Proof. *We only need to prove that has no fixed point in . Otherwise, there exists , such that .

By Lemma 7, we have that the spectral radius and has a positive eigenfunction satisfying It is clear that . Therefore, there exists such that It follows from that . It is obvious that is increasing on . By induction, we can get , . Thus, Noticing , we have , which contradicts with .

Theorem 12. *Assume that the following condition holds:** There exists satisfying such thatThen (1) has no positive solution if the spectral radius , where *

*Proof. *Suppose that there exists , such that .

By Lemma 7, we have that the spectral radius and has a positive eigenfunction satisfying It is clear that . Therefore, there exists such that It follows from that . Noticing that is increasing on , by induction, we have , . Thus, It follows from that , which contradicts with .

#### 4. Example

*Example 1. *Consider the following integral boundary value problem: withwhere

By direct calculations, we have It is clear that and hold. Clearly, we have Let ; then we have It is easy to get that holds.

Denote By direct calculations, we have , we have

Therefore,

It is obvious that andwhich implies that Lemma 7 and Remark 10 can guarantee that So all of the assumptions of Theorem 8 are satisfied. As a result, BVP (51) has a unique positive solution.

*Example 2. *Consider BVP (51) with

Clearly, we haveIt is not difficult to check that holds.

It follows from Example 1 that So all of the assumptions of Theorem 11 are satisfied. As a result, BVP (51) has no positive solution.

#### 5. Conclusions

In this paper, we consider the existence of positive solution for fractional differential equations with conjugate type integral conditions. Both the existence of uniqueness and nonexistence of positive solution are established under conditions closely associated with the spectral radius with respect to the relevant linear operator. In addition, the unique positive solution can be approximated by an iterative scheme, and the error estimation of the iterative sequences is also given.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by the Natural Science Foundation of Shandong Province of China (ZR2017MA036) and the National Natural Science Foundation of China (11571296).

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#### Copyright

Copyright © 2018 Yongqing Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.