- Research Article
- Open Access
© 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.
- Fractional Derivative
- Fractional Calculus
- Fractional Differential Equation
- Real Banach Space
- Topological Degree
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.
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.
In the following, we present the useful lemmas which are fundamental in the proof of our main results.
Lemma 2.4 (see ).
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
where is the standard sup norm of the space . Throughout, we denote and . Then is a Banach space (see ).
We divide the proof into three steps.
Then the BVP (1.1)-(1.2) has at least one positive solution.
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).
- Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York, NY, USA; 1993:xxxvi+976.MATHGoogle Scholar
- Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.MATHGoogle Scholar
- Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication. John Wiley & Sons, New York, NY, USA; 1993:xvi+366.Google Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Volume 204. Elsevier Science, Amsterdam, The Netherlands; 2006:xvi+523.Google Scholar
- Lakshmikantham V, Vatsala AS: Basic theory of fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008,69(8):2677-2682. 10.1016/j.na.2007.08.042MathSciNetView ArticleMATHGoogle Scholar
- Lakshmikantham V: Theory of fractional functional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008,69(10):3337-3343. 10.1016/j.na.2007.09.025MathSciNetView ArticleMATHGoogle Scholar
- Lakshmikantham V, Vatsala AS: General uniqueness and monotone iterative technique for fractional differential equations. Applied Mathematics Letters 2008,21(8):828-834. 10.1016/j.aml.2007.09.006MathSciNetView ArticleMATHGoogle Scholar
- El-Sayed AMA, El-Mesiry AEM, El-Saka HAA: On the fractional-order logistic equation. Applied Mathematics Letters 2007,20(7):817-823. 10.1016/j.aml.2006.08.013MathSciNetView ArticleMATHGoogle Scholar
- El-Sayed AMA, El-Maghrabi EM: Stability of a monotonic solution of a non-autonomous multidimensional delay differential equation of arbitrary (fractional) order. Electronic Journal of Qualitative Theory of Differential Equations 2008, (16):1-9.Google Scholar
- Diethelm K, Ford NJ: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 2002,265(2):229-248. 10.1006/jmaa.2000.7194MathSciNetView ArticleMATHGoogle Scholar
- Bai C: Positive solutions for nonlinear fractional differential equations with coefficient that changes sign. Nonlinear Analysis: Theory, Methods & Applications 2006,64(4):677-685. 10.1016/j.na.2005.04.047MathSciNetView ArticleMATHGoogle Scholar
- Su X: Boundary value problem for a coupled system of nonlinear fractional differential equations. Applied Mathematics Letters 2009,22(1):64-69. 10.1016/j.aml.2008.03.001MathSciNetView ArticleMATHGoogle Scholar
- Bai Z, Lü H: Positive solutions for boundary value problem of nonlinear fractional differential equation. Journal of Mathematical Analysis and Applications 2005,311(2):495-505. 10.1016/j.jmaa.2005.02.052MathSciNetView ArticleMATHGoogle Scholar
- Zhang S: Existence of solution for a boundary value problem of fractional order. Acta Mathematica Scientia 2006,26(2):220-228. 10.1016/S0252-9602(06)60044-1MathSciNetView ArticleMATHGoogle Scholar
- Zhang S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electronic Journal of Differential Equations 2006,2006(36):1-12.MathSciNetGoogle Scholar
- Mawhin J: Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Regional Conference Series in Mathematics. Volume 40. American Mathematical Society, Providence, RI, USA; 1979:v+122.View ArticleGoogle Scholar
- Bai Z, Ge W: Existence of three positive solutions for some second-order boundary value problems. Computers & Mathematics with Applications 2004,48(5-6):699-707. 10.1016/j.camwa.2004.03.002MathSciNetView ArticleMATHGoogle Scholar
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