Abstract
By using two fixed-point theorems on cone, we discuss the existence results of positive solutions for the following boundary value problem of fractional differential equation with integral boundary conditions: , , , and .
1. Introduction
Boundary value problem for fractional differential equation has aroused much attention in the past few years; many professors devoted themselves to the solvability of fractional differential equations, especially to the study of the existence of solutions for boundary value problems of fractional differential equation (see [1–28]). For example, Wang et al. [19] studied the existence of positive solutions for the following problem: with the boundary conditions where is the Riemann-Liouville differential operator of , is a positive parameter, and may be singular at and may change sign. And Ma [14] discussed the positive solutions of where is integer and , .
There have already been lots of books and papers involving the positive solutions for boundary value problems of fractional differential equation; however, only a few papers cover that for fractional differential equation boundary value problems with integral boundary conditions. Motivated by [14], we shall investigate the positive solutions of the following boundary value problem:where is the Riemann-Liouville differential operator of , is right continuous on , left continuous at , and nondecreasing on with , and denotes the Riemann-Stieltjes integrals of with respect to . And , satisfies the following conditions:(H1) is nonnegative and not identically zero on any compact subset of , .(H2) is continuous.
This paper consists of four sections. After the introduction, we recall some definitions, lemmas, and theorems in Section 2. And the main results of this paper are stated in Section 3. In the last section, we give two examples of the main results.
2. Preliminaries
Firstly, for convenience we recall some definitions, lemmas, and theorems.
Definition 1 (see [29, 30]). Let define the Riemann-Liouville fractional integral of order for as where is Euler gamma function.
Definition 2 (see [29, 30]). Define the Riemann-Liouville fractional derivative of order for as where has absolutely continuous derivatives up to order on .
Lemma 3. Let , ; then the boundary value problem has the unique solution , where
Proof. The boundary value problem can be converted to an equivalent integral equation: Then the solution is It follows from the boundary conditions that and Thus we get Then we can obtain which means So
Lemma 4 (see [20]). defined in (9) has the following properties:(i), .(ii), .(iii), where
Lemma 5. If (H2) is satisfied, then defined in (8) has the following properties: where .
Proof. The proof can be easily accomplished by Lemma 4, so we omitted it.
Theorem 6 (see [31]). Let be a Banach space and be a cone in . Suppose that are two bounded open sets of with . Assume that is a completely continuous operator such that either(i) for any and for any or (ii) for any and for any . Then has a fixed point in .
Theorem 7 (see [32]). Let be a cone of a Banach space . . is a nonnegative continuous concave function on , such that, for any , , and . Assume that is completely continuous, and there exist constants such that(c1) and for ;(c2) for ;(c3) for any with . Then has at least three fixed points , , with , , , and .
Let be a Banach space with the maximum norm ; define the cone as
Define a continuous operator as
Lemma 8. Assume that (H1) and (H2) hold; then is a completely continuous operator.
Proof. The lemma can be easily proven, so we omitted it.
3. Main Results
We define the following notation: given , takeNow we can obtain the following theorems.
Theorem 9. Suppose that (H1) and (H2) are satisfied; there exist two positive constants such that(H3), ;(H4), . Then boundary value problem (4) has at least one positive solution such that .
Proof. The solution of boundary value problem (4) is equivalent to the fixed point of operator . Let ; when , for any we have . By Lemma 5 and (H3) we get which means when , .
Let ; when , for any we have . By Lemma 5 and (H4) we get which means when , .
It follows from Theorem 6 that we know that has at least one fixed point in , which means that the boundary value problem (4) has at least one solution.
Theorem 10. Suppose that (H1) and (H2) are satisfied; there exist four positive constants , , , with , such that(H5), ;(H6), ;(H7), . Then boundary value problem (4) has at least three positive solutions , , , such that
Proof. Define a nonnegative continuous concave function on as If , then ; it follows from (H7) that ; hence Thus, . It follows from Lemma 8 that is completely continuous. In the same way, let ; it follows from (H5) that for any , which shows that condition (c2) of Theorem 7 is fulfilled.
Let ; it is easy to know that and . If , we have for any . We know for by (H6). So we get So condition (c1) of Theorem 7 holds.
When with , noting that thus so we obtainThat is to say, (c3) is satisfied.
All conditions of Theorem 7 are satisfied, so has at least three fixed points , , , which means that the boundary value problem (4) has at least three positive solutions , , , such that The proof of this theorem is finished.
4. Some Examples
Now we present two examples to illustrate our main results.
Example 1. Let us see the following problem:
Choose ; we obtain that , , , and Then for any , we have , and for any , we get
Then the boundary value problem has at least one positive solution such that .
Example 2. We now study the following problem: where
Choose ; we have , , , and . Let , , , and ; then for any , we have , for any , we have , and for any , we have . Then by Theorem 10, we conclude that this boundary value problem has at least three positive solutions , , , such that
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This project is supported by NNSF of China (11371221, 11571207), SDNSF (ZR2018MA011), and the Tai’shan Scholar Engineering Construction Fund of Shandong Province of China.