Positive solutions and monotone iterative sequences for a fractional differential equation with integral boundary conditions
© Sun and Sun; licensee Springer. 2014
Received: 23 October 2013
Accepted: 6 January 2014
Published: 27 January 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.
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 .
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.
where Γ denotes the Euler gamma function and denotes the integer part of number α, provided that the right side is pointwise defined on .
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 ()
and is continuous on the unit square .
Lemma 2.2 ()
Then the proof is completed. □
3 Main results
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 .
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 .
Therefore, . The proof is completed. □
By (H2) we know that is well defined.
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 .
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).
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).
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).
The proof is complete. □
Remark 3.2 Certainly, may happen and then the problem (1.1) has only one solution in .
Then the problem (1.1) has at least two positive solutions.
To illustrate the usefulness of the results, we give two examples.
Then (3.4) is satisfied. Consequently, Theorem 3.1 guarantees that the problem (4.1) has at least two positive solutions and , satisfying .
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).
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