Research Article | Open Access
The Existence of Solutions for a Fractional 2-Point Boundary Value Problems
By using the coincidence degree theory, we consider the following 2-point boundary value problem for fractional differential equation , , where and are the standard Riemann-Liouville fractional derivative and fractional integral, respectively. A new result on the existence of solutions for above fractional boundary value problem is obtained.
Fractional differential equations have been of great interest recently. This is because of the intensive development of the theory of fractional calculus itself as well as its applications. Apart from diverse areas of mathematics, fractional differential equations arise in a variety of different areas such as rheology, fluid flows, electrical networks, viscoelasticity, chemical physics, and many other branches of science (see [1–4] and references cited therein). The research of fractional differential equations on boundary value problems, as one of the focal topics has attained a great deal of attention from many researchers (see [5–13]).
However, there are few papers which consider the boundary value problem at resonance for nonlinear ordinary differential equations of fractional order. In , Hu and Liu studied the following BVP of fractional equation at resonance: where , is the standard Caputo fractional derivative.
In , Zhang and Bai investigated the nonlinear nonlocal problem where , they consider the case , that is, the resonance case.
In , Bai investigated the boundary value problem at resonance is considered, where is a real number, and are the standard Riemann-Liouville fractional derivative and fractional integral, respectively, and is continuous and are given constants such that
In this paper, we study the 2-point boundary value problem where , , satisfies Carathéodory conditions, and are the standard Riemann-Liouville fractional derivative and fractional integral, respectively.
In this paper, we will always suppose that the following conditions hold:(): ():
The rest of this paper is organized as follows. Section 2 contains some necessary notations, definitions, and lemmas. In Section 3, we establish a theorem on existence of solutions for BVP (1.4)-(1.5) under nonlinear growth restriction of , basing on the coincidence degree theory due to Mawhin (see ).
Now, we will briefly recall some notation and an abstract existence result.
Let be real Banach spaces, a Fredholm map of index zero,s and continuous projectors such that It follows that is invertible. We denote the inverse of the map by . If is an open-bounded subset of such that , the map will be called -compact on if is bounded and is compact.
The lemma that we used is [17, Theorem 2.4].
Lemma 1.1. Let be a Fredholm operator of index zero and let be L-compact on . Assume that the following conditions are satisfied: (i), for all ; (ii), for all ; (iii),where is a projection as above with and is any isomorphism. Then the equation has at least one solution in .
Definition 2.1. The fractional integral of order of a function is given by provided the right side is pointwise defined on , where is the Gamma function.
Definition 2.2. The fractional derivative of order of a function is given by where , provided the right side is pointwise defined on .
Definition 2.3. We say that the map satisfies Carathéodory conditions with respect to if the following conditions are satisfied: (i)for each , the mapping is Lebesgue measurable; (ii)for almost every , the mapping is continuous on ; (iii) for each , there exists such that for a.e. and every , we have .
Lemma 2.4 (see ). Assume that with a fractional derivative of order that belongs to . Then for some , where is the smallest integer greater than or equal to .
We use the classical Banach space with the norm with the norm
Definition 2.5. For , we denote by the space of functions which have continuous derivatives up to order on such that is absolutely continuous: = and is absolutely continuous in .
Lemma 2.6 (see ). Given and we can define a linear space where . By means of the linear functional analysis theory, we can prove that with the is a Banach space.
Remark 2.7. If is a natural number, then is in accordance with the classical Banach space .
Lemma 2.8 (see ). is a sequentially compact set if and only if is uniformly bounded and equicontinuous. Here uniformly bounded means there exists , such that for every and equicontinuous means that , , such that
Lemma 2.9 (see ). Let . Assume that with a fractional integration of order that belongs to . Then the equality holds almost everywhere on .
Definition 2.10 (see ). Let denote the space of functions , represented by fractional integral of order of a summable function: .
Let , with the norm , defined by Lemma 2.6, with the norm , where is a Banach space.
3. Main Results
Lemma 3.1. Let L be defined by (2.12), then
Proof. In the following lemma, we use the unified notation of both for fractional integrals and fractional derivatives assuming that for .
Let , by Lemma 2.9, has solution Combine with (1.5), So, Let and let
Then a.e. and, if hold, then satisfies the boundary conditions (1.5). That is, and we have
Let . Then for , we have where which, due to the boundary value condition (1.5), implies that satisfies (3.5). In fact, from we have , from , , we have Hence, Therefore, The proof is complete.
Lemma 3.2. The mapping is a Fredholm operator of index zero, and where define by by and .
Proof. Consider the continuous linear mapping and defined by
Using the above definitions, we construct the following auxiliary maps and :
Since the condition (C2) holds, the mapping defined by
is well defined.
Recall (C2) and note that and similarly we can derive that So, for , it follows from the four relations above that that is, the map is idempotent. In fact is a continuous linear projector.
Note that implies . Conversely, if , so but then we must have ; since the condition (C2) holds, this can only be the case if , that is, . In fact , take in the form so that , thus, , Let and assume that is not identically zero on . Then, since , from (3.5) and the condition (C2), we have So but we derive , which is a contradiction. Hence, ; thus .
Now, and so is a Fredholm operator of index zero.
Let be defined by Note that is a continuous linear projector and It is clear that .
Note that the projectors and are exact. Define by by Hence we have then and thus
In fact, if , then Also, if , then where and from the boundary value condition (1.5) and the fact that , , , we have , thus This shows that . The proof is complete. Using (3.16), we write
By Lemma 2.8 and a standard method, we obtain the following lemma.
Lemma 3.3 (see ). For every given is completely continuous.
Assume that the following conditions on the function are satisfied.(H1)There exist functions , and a constant such that for all , , one of the following inequalities is satisfied:(H2)There exists a constant , such that for satisfying for all , we have (H3)There exists a constant such that for every satisfying then either
Remark 3.4. and from (H3) stand for the images of under the maps and , respectively.
Lemma 3.5. Suppose (H1)-(H2) hold, then the set is bounded.
Then for , thus , and hence for all . By the definition of , we have . It follows from (H2) that there exists , such that .
Now and so Therefore, we have Note that for all . Then, by Lemma 3.2, we have so, we have where is a constant. This is for all . If the first condition of (H1) is satisfied, then, we have where , and consequently, for so But and , so there exists such that Therefore, for all , we can prove that is also bounded.
If (3.38) or (3.39) holds, similar to the above argument, we can prove that is bounded too.
Lemma 3.6. Suppose (H3) holds, then the set is bounded.
Proof. Let for and ; thus . By (H3), , that is, is bounded.
Lemma 3.7. Suppose (H3) holds, then the set is bounded.
Proof. We define the isomorphism by
If the first part of (H3) is satisfied, let For every , If , then , and if , then by (H3) which, in either case, is a contradiction. If the other part of (H3) is satisfied, then we take and, again, obtain a contradiction. Thus, in either case for all , that is, is bounded.
Remark 3.8. Suppose the second part of (H3) holds, then the set is bounded.
Proof. Set to be a bounded open set of such that . It follows from Lemmas 3.2 and 3.3 that is a Fredholm operator of index zero, and the operator is compact , thus, is L-compact on . By Lemmas 3.5 and 3.6, we get that the following two conditions are satisfied: (i) for every ; (ii), for every .
Finally, we will prove that (iii) of Lemma 1.1 is satisfied. Let , where is the identity operator in the Banach space . According to Lemma 3.7 (or Remark 3.8), we know that , for all , and thus, by the homotopy property of degree, Then by Lemma 1.1, has at least one solution in , so the boundary value problem (1.4) and (1.5) has at least one solution in the space . The proof is finished.
4. An Example
Let us consider the following boundary value problem: where Corresponding to the problem (1.4)-(1.5), we have that , , , , , , , , , and then there is Again, taking , , then therefore Take , . By simple calculation, we can get that (C1)-(C2) and (H1)–(H3) hold. By Lemma 1.1, we obtain that (4.1) has at least one solution.
This work is sponsored by NNSF of China (10771212) and the Fundamental Research Funds for the Central Universities (2010LKSX09).
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B. V., Amsterdam, The Netherland, 2006.
- J. Sabatier, O.P. Agrawal, and J. A. T. Machado, Eds., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherland, 2007.
- V. Lakshmikantham, S. Leela, and J. V. Devi, Theory of Fractional Dynamic Systems, Academic Publishers, Oxford, UK, 2009.
- I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press Inc., San Diego, Calif, USA, 1999.
- V. Lakshmikantham and A. S. Vatsala, “Theory of fractional differential inequalities and applications,” Communications in Applied Analysis, vol. 11, no. 3-4, pp. 395–402, 2007.
- Z. Bai, “On positive solutions of a nonlocal fractional boundary value problem,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 2, pp. 916–924, 2010.
- Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005.
- Z. Bai and T. Qiu, “Existence of positive solution for singular fractional differential equation,” Applied Mathematics and Computation, vol. 215, no. 7, pp. 2761–2767, 2009.
- X. Su, “Boundary value problem for a coupled system of nonlinear fractional differential equations,” Applied Mathematics Letters, vol. 22, no. 1, pp. 64–69, 2009.
- N. Kosmatov, “A boundary value problem of fractional order at resonance,” Electronic Journal of Differential Equations, vol. 135, pp. 1–10, 2010.
- V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 8, pp. 2677–2682, 2008.
- C. Bai, “Positive solutions for nonlinear fractional differential equations with coefficient that changes sign,” Nonlinear Analysis. Theory, Methods & Applications, vol. 64, no. 4, pp. 677–685, 2006.
- M. Benchohra and F. Berhoun, “Impulsive fractional differential equations with variable times,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1245–1252, 2010.
- Z. Hu and W. Liu, “Solvability for fractional order boundary value problem at resonance,” Boundary Value Problem, vol. 20, pp. 1–10, 2011.
- Y. Zhang and Z. Bai, “Existence of positive solutions for s nonlinear fractional three-point boundary value problen at resonance,” Journal of Applied Mathematics and Computing, vol. 36, no. 1-2, pp. 417–440, 2010.
- Z. Bai, “On solutions of some fractional m-point boundary value problems at resonance,” Electronic Journal of Qualitative Theory of Differential Equations, p. No. 37, 15, 2010.
- J. Mawhin, “Topological degree and boundary value problems for nonlinear differential equations,” in Topological Methods for Ordinary Differential Equations, P. M. Fitzpatrick, M. Martelli, J. Mawhin, and R. Nuss-baum, Eds., vol. 1537 of Lecture Notes in Mathematics, pp. 74–142, Springer, Berlin, Germany, 1991.
- A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems. I,” Applicable Analysis, vol. 78, no. 1-2, pp. 153–192, 2001.
Copyright © 2012 Gang Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.