Positive Solutions for Boundary Value Problems of -Dimension Nonlinear Fractional Differential System
© A. Yang andW. Ge. 2008
Received: 27 October 2008
Accepted: 18 December 2008
Published: 5 January 2009
We study the boundary value problem for a kind -dimension nonlinear fractional differential system with the nonlinear terms involved in the fractional derivative explicitly. The fractional differential operator here is the standard Riemann-Liouville differentiation. By means of fixed point theorems, the existence and multiplicity results of positive solutions are received. Furthermore, two examples given here illustrate that the results are almost sharp.
where is the standard Riemann-Liouville fractional derivative of order , and ,
Recently, fractional differential equations (in short FDEs) have been studied extensively. The motivation for those works stems from both the development of the theory of fractional calculus itself and the applications of such constructions in various sciences such as physics, mechanics, chemistry, engineering, and so on. For an extensive collection of such results, we refer the readers to the monographs by Samko et al. , Podlubny , Miller and Ross , and Kilbas et al. .
By using the Schauder fixed point theorem, one existence result was given.
by means of Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem. is the standard Riemann-Liouville fractional derivative.
in [14, 15], respectively. Since conditions (1.6) and (1.7) are nonzero boundary values, the Riemann-Liouville fractional derivative is not suitable. Therefore, the author investigated the BVPs (1.5)-(1.6) and (1.5)–(1.7) by involving in the Caputo fractional derivative .
From above works, we can see a fact, although the BVPs of nonlinear FDE have been studied by some authors, to the best of our knowledge, higher-dimension fractional equation systems are seldom considered. Su in  studied the two-dimension system, however, the Schauder fixed point theorem cannot ensure the solutions to be positive. Since only positive solutions are useful for many applications, we investigate the existence and multiplicity of positive solutions for BVP (1.1)-(1.2) in this paper. In addition, two examples are given to demonstrate our results.
where denotes the integer part of , provided that the right side is pointwise defined on .
In the following, we present the useful lemmas which are fundamental in the proof of our main results.
Lemma 2.4 (see ).
Let be a convex subset of a normed linear space and be an open subset of with . Then every compact continuous map has at least one of the following two properties:
(A1) has a fixed point;
(A2)there is an with for some .
for all and .
The assumptions below about the nonnegative continuous convex functionals , will be used as follows:
(B1)there exists such that for all ;
(B2) for all .
Lemma 2.6 (see ).
Let be a cone in a real Banach space , , and . Assume that and are nonnegative continuous convex functionals satisfying (B1) and (B2), is a nonnegative continuous concave functional on such that for all and is a completely continuous operator. Suppose
(C1) , , for ;
(C2) , , for all ;
(C3) for all with .
3. Related Lemmas
where is the standard sup norm of the space . Throughout, we denote and . Then is a Banach space (see ).
If , then , .
That is, , .
. For the Green functions , , we can obtain
(ii) and .
is completely continuous.
We divide the proof into three steps.
. In fact, for any , since for and for , for . Moreover, implies that .
is continuous on , which is valid due to the continuity of the function .
where . That is, is uniformly bounded. Thus, is relatively compact. By means of the Arzela-Ascoli theorem, is completely continuous.
4. The Existence of One Positive Solution
Then the BVP (1.1)-(1.2) has at least one positive solution.
Lemma 3.2 indicates that is completely continuous.
indicate that , and then . Take in Lemma 2.4, for any , does not hold. Hence, the operator has at least a fixed point, then the BVP (1.1)-(1.2) has at least one positive solution.
It is easy to check that (4.1) holds. Thus, by Theorem 4.1, the BVP (4.5) has at least one positive solution. In fact, is such a solution.
5. The Existence of Triple Positive Solutions
Obviously, and satisfy (B1) and (B2), for all .
Assume that there exist constants such that for . Suppose
(H1) , ;
(H2) , ;
(H3) , ;
Lemma 3.2 has showed that is completely continuous. Now, we will verify that all the conditions of Lemma 2.6 are satisfied. The proof is based on the following steps.
We will show that (H1) implies .
Then and , that is, .
then we can obtain .
It is similar to Step 1 that we can prove by condition (H3), that is, (C2) in Lemma 2.6 holds.
Thus, , (C3) in Lemma 2.6 is satisfied.
The particular case has been studied by  for the existence of one solution, our paper generalizes  for the obtaining of one and three positive solutions. For , we develop [13–15] by the nonlinear terms involved in the -order Riemann-Liouville derivative explicitly.
This work is supported by National Natural Science Foundation of China (NNSF) (10671012) and the Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP) of China (20050007011).
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