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Positive solutions and monotone iterative sequences for a fractional differential equation with integral boundary conditions
Advances in Difference Equations volume 2014, Article number: 29 (2014)
Using a monotone iterative method combined with some inequalities associated with the Green’s function, we investigate the existence of positive solutions for a fractional differential equation with integral boundary conditions. In addition, two examples are given to illustrate the results.
MSC:34A08, 34B10, 34B15.
This paper discusses the existence and iterative method of positive solutions for the following nonlinear fractional differential equations with integral boundary condition:
where is a real number and is an integer, μ is a parameter and , is the standard Riemann-Liouville fractional derivative of order α. A function u is called a positive solution of the problem (1.1) if satisfies (1.1) and on .
Fractional differential equations arise in many engineering and scientific disciplines such as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electro-dynamics of a complex medium, polymer rheology, and so on. Recently, the subject of fractional differential equations has gained much more importance and attention. Some excellent work in the study of fractional differential equations can be found in [1–22] and the references cited therein. Integral boundary conditions have various applications in chemical engineering, thermo-elasticity, population dynamics, and so on. Boundary value problems for fractional differential equations with integral boundary conditions are very interesting and largely unknown. Recently, by using Guo-Krasnoselskii’s fixed point theorem, Cabada and Wang in  investigated the existence of positive solutions for the fractional boundary value problem
where , , is the Caputo fractional derivative and is a continuous function. Karakostas  provided sufficient conditions for the non-existence of solutions of the boundary-value problems with fractional derivative of order in the Caputo sense,
Motivated by the works mentioned above, our purpose in this paper is to show the existence and iteration of positive solutions to the problem (1.1) by using a monotone iterative method. The method used in this paper is different from that used in . We not only obtain the existence of positive solutions, but also give two iterative schemes approximating the solutions, and the iterative scheme starts off with a known simple function or the zero function, which is interesting because it gives a numerical method to compute approximate solutions. The monotone iterative method has been successfully applied to boundary-value problems of integer-order ordinary differential equations (see [23–27] and the references therein). To our knowledge, there is still little utilization of the monotone iterative method to study the existence of positive solutions for nonlinear fractional boundary-value problems. So, it is worth investigating the problem (1.1) by using monotone iterative method.
Let us recall some basic definitions on fractional calculus.
The Riemann-Liouville fractional derivative of order of a continuous function is defined to be
where Γ denotes the Euler gamma function and denotes the integer part of number α, provided that the right side is pointwise defined on .
The Riemann-Liouville fractional integral of order α is defined as
provided the integral exists.
In , the author obtained the Green’s function associated with the problem (1.1). More precisely, the author proved the following lemma.
Lemma 2.1 ()
Let be a given function, then the boundary-value problem
has a unique solution,
and is continuous on the unit square .
Lemma 2.2 ()
The function defined by (2.2) has the following properties:
Lemma 2.3 The Green’s function defined by (2.1) has the following properties:
Proof It is obvious from (2.3) that the right inequality of (2.5) holds. Relation (2.4) implies that . Thus by (2.1) we know that the left inequality of (2.5) is correct. Now we show that (2.6) holds. In fact, by (2.1) and (2.4), we have
On the other hand, by (2.1) and (2.4), we get
Then the proof is completed. □
3 Main results
Now, we consider the problem (1.1). Obviously, u is a solution of the problem (1.1) if and only if u is a solution of the following nonlinear integral equation:
where is the Green’s function defined by (2.3). For the forthcoming analysis, we need the following assumptions:
(H1) is continuous and on ;
(H2) is continuous and .
The basic space used in this paper is a real Banach space with the norm , where . Then, define a set by
It is obvious that is a cone. We define the operator by
It is clear that the existence of a positive solution for the problem (1.1) is equivalent to the existence of a nontrivial fixed point of in .
Lemma 3.1 is a completely continuous operator and .
Proof Applying the Arzela-Ascoli theorem and a standard argument, we can prove that is a completely continuous operator. We conclude that . In fact, for any , it follows from (H1), (H2), and (2.6) that
which implies that
On the other hand, by (H1), (H2), and (2.6) we have
which together with (3.2) implies
Therefore, . The proof is completed. □
For convenience, we denote
By (H2) we know that is well defined.
Theorem 3.1 Suppose (H1) and (H2) hold. In addition, we assume that there exists , such that
where Λ is given by (3.3). Then the problem (1.1) has two positive solutions and satisfying . In addition, the iterative sequences , , , converge, in C-norm, to positive solutions and , respectively, where , , . Moreover,
Remark 3.1 The iterative schemes in Theorem 3.1 start off with the zero function and a known simple function, respectively.
Proof The proof will be given in several steps.
Step 1. Let , then .
In fact, if , then
Thus by (2.5) and (3.4), we get
which implies that , thus .
Step 2. The iterative sequence is increasing, and there exists such that , and is a positive solution of the problem (1.1).
Obviously, . Since , we have , . Since is completely continuous, we assert that is a sequentially compact set. Since , we have
It follows from (3.4) that is nondecreasing, and then
Thus, by the induction, we have
Hence, there exists such that . By the continuity of and equation , we get . Moreover, since the zero function is not a solution of the problem (1.1), . It follows from the definition of the cone , that we have , , i.e. is a positive solution of the problem (1.1).
Step 3. The iterative sequence is decreasing, and there exists such that , and is a positive solution of the problem (1.1).
Obviously, . Since , we have , . Since is completely continuous, we assert that is a sequentially compact set. Since , by (2.5) and (3.4), we have
Thus we obtain
which together with (3.4) implies that
By the induction, we have
Hence, there exists such that . Applying the continuity of and the definition of , we can concluded that is a positive solution of the problem (1.1).
Step 4. From , we get
By the induction, we have
The proof is complete. □
Remark 3.2 Certainly, may happen and then the problem (1.1) has only one solution in .
Corollary 3.1 Suppose that (H1) and (H2) hold. Suppose further that is nondecreasing in x for each and
Then the problem (1.1) has at least two positive solutions.
To illustrate the usefulness of the results, we give two examples.
Example 4.1 Consider the fractional boundary-value problem
In this problem,
It is easy to see that (H1) and (H2) hold. If we let , by simple computation, we have
Then (3.4) is satisfied. Consequently, Theorem 3.1 guarantees that the problem (4.1) has at least two positive solutions and , satisfying .
Moreover, the two iterative schemes are
After direct calculations by Matlab 7.5, the second and third terms of the two schemes are as follows:
Example 4.2 Consider the fractional boundary value
In this problem,
Obviously, and satisfy the conditions (H1) and (H2). In addition, is increasing in x, and
Let , then for any , direct computations give
Therefore, all assumptions of Theorem 3.1 are satisfied. Thus Theorem 3.1 ensures that the problem (4.2) has two positive solutions and , satisfying and , , where
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The authors sincerely thanks the anonymous referees for their valuable suggestions and comments which have greatly helped improve this article. The first author was supported financially by the Natural Science Foundation of Zhejiang Province of China (Y12A01012). The second author was supported financially second by the Foundation of Shanghai Natural Science (13ZR1430100) and the Foundation of Shanghai Municipal Education Commission (DYL201105).
The authors declare that they have no competing interests.
All authors contributed equally in this article. They read and approved the final manuscript.
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Sun, Y., Sun, Y. Positive solutions and monotone iterative sequences for a fractional differential equation with integral boundary conditions. Adv Differ Equ 2014, 29 (2014). https://doi.org/10.1186/1687-1847-2014-29
- positive solution
- fractional differential equations
- boundary value problems
- integral boundary conditions
- monotone iterative method