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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 691651, 13 pages
Positive Mild Solutions of Periodic Boundary Value Problems for Fractional Evolution Equations
1School of Mathematics and Computer Science Institute, Northwest University for Nationalities, Gansu, Lanzhou 730000, China
2Department of Mathematics, Northwest Normal University, Gansu, Lanzhou 730000, China
Received 29 October 2011; Accepted 14 February 2012
Academic Editor: Shiping Lu
Copyright © 2012 Jia Mu and Hongxia Fan. 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.
The periodic boundary value problem is discussed for a class of fractional evolution equations. The existence and uniqueness results of mild solutions for the associated linear fractional evolution equations are established, and the spectral radius of resolvent operator is accurately estimated. With the aid of the estimation, the existence and uniqueness results of positive mild solutions are obtained by using the monotone iterative technique. As an application that illustrates the abstract results, an example is given.
In this paper, we investigate the existence and uniqueness of positive mild solutions of the periodic boundary value problem (PBVP) for the fractional evolution equation in an ordered Banach space where is the Caputo fractional derivative of order is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators on , and is a continuous function.
The origin of fractional calculus goes back to Newton and Leibnitz in the seventieth century. We observe that fractional order can be complex in viewpoint of pure mathematics and there is much interest in developing the theoretical analysis and numerical methods to fractional equations, because they have recently proved to be valuable in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, electromagnetism, biology, and hydrogeology. For example space-fractional diffusion equations have been used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium [1, 2] or to model activator-inhibitor dynamics with anomalous diffusion .
Fractional evolution equations, which is field have abundant contents. Many differential equations can turn to semilinear fractional evolution equations in Banach spaces. For example, fractional diffusion equations are abstract partial differential equations that involve fractional derivatives in space and time. The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order , namely, we can take , for , or for , where , , are the fractional derivatives of order , , , respectively. Recently, fractional evolution equations are attracting increasing interest, see El-Borai [4, 5], Zhou and Jiao [6, 7], Wang et al. [8, 9], Shu et al.  and Mu et al. [11, 12]. They established various criteria on the existence of solutions for some fractional evolution equations by using the Krasnoselskii fixed point theorem, the Leray-Schauder fixed point theorem, the contraction mapping principle, or the monotone iterative technique. However, no papers have studied the periodic boundary value problems for abstract fractional evolution equations (1.1), though the periodic boundary value problems for ordinary differential equations have been widely studied by many authors (see [13–18]).
In this paper, without the assumptions of lower and upper solutions, by using the monotone iterative technique, we obtain the existence and uniqueness of positive mild solutions for PBVP (1.1). Because in many practical problems such as the reaction diffusion equations, only the positive solution has the significance, we consider the positive mild solutions in this paper. The characteristics of positive operator semigroup play an important role in obtaining the existence of the positive mild solutions. Positive operator semigroup are widely appearing in heat conduction equations, the reaction diffusion equations, and so on (see ). It is worth noting that our assumptions are very natural and we have tested them in the practical context. In particular to build intuition and throw some light on the power of our results, we examine sufficient conditions for the existence and uniqueness of positive mild solutions for periodic boundary value problem for fractional parabolic partial differential equations (see Example 4.1).
We now turn to a summary of this work. Section 2 provides the definitions and preliminary results to be used in theorems stated and proved in the paper. In particular to facilitate access to the individual topics, the existence and uniqueness results of mild solutions for the associated linear fractional evolution equations are established and the spectral radius of resolvent operator is accurately estimated. In Section 3, we obtain very general results on the existence and uniqueness of positive mild solutions for PBVP (1.1), when the nonlinear term satisfies some conditions related to the growth index of the operator semigroup . The main method is the monotone iterative technique. In Section 4, we give also an example to illustrate the applications of the abstract results.
Definition 2.1. The fractional integral of order with the lower limit zero for a function is defined as: provided the right side is point-wise defined on , where is the gamma function.
Definition 2.2. The Riemann-Liouville derivative of order with the lower limit zero for a function can be written as:
Definition 2.3. The Caputo fractional derivative of order for a function can be written as:
Remark 2.4. (i) If , then
(ii) The Caputo derivative of a constant is equal to zero.
(iii) If is an abstract function with values in , then the integrals and derivatives which appear in Definitions 2.1–2.3 are taken in Bochner's sense.
Throughout this paper, let be an ordered Banach space with norm and partial order ≤, whose positive cone ( is the zero element of ) is normal with normal constant . Let be the Banach space of all continuous -value functions on interval with norm . Evidently, is also an ordered Banach space with the partial ≤ reduced by the positive function cone . is also normal with the same constant . For if for all . For , denote the ordered interval in , and in . Set exists and . denotes the Banach space with the graph norm . Suppose that is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators . This means there exists such that
Lemma 2.5 (see ). If satisfies a uniform Hölder condition, with exponent , then the unique solution of the linear initial value problem (LIVP) for the fractional evolution equation, is given by where is a probability density function defined on .
Remark 2.6. (i) See [6, 7]
(ii) see [6, 24], for ,
(iii) see [4, 5], the Laplace transform of is given by where is the Mittag-Leffler function (see ),
(iv) see  by (i) and (ii), we can obtain that for where , are the Mittag-Leffler functions.
(v) see  for ,
(vi) see  if and , then .
Remark 2.7. See [6, 8], the operators and , given by (2.8), have the following properties:
(i) For any fixed , and are linear and bounded operators, that is, for any , (ii) and are strongly continuous.
We also introduce some basic theories of the operator semigroups. For an analytic semigroup , there exist and such that (see ) Then is called the growth index of the semigroup . Furthermore, can also be obtained by the following formula:
Definition 2.9 (see ). A -semigroup is called a compact semigroup if is compact for .
Definition 2.10. An analytic semigroup is called positive if for all and .
Definition 2.12. A bounded linear operator on is called to be positive if for all .
Remark 2.13. By Remark 2.6(ii), we obtain that and are positive for if is a positive semigroup.
Lemma 2.14. Let be an ordered Banach space, whose positive cone is normal. If is an exponentially stable analytic semigroup, that is, . Then the linear periodic boundary value problem (LPBVP), has a unique mild solution where and are given by (2.8), is a bounded linear operator, and the spectral radius .
Proof. For any , by there exists such that In , give the equivalent norm by then . By we denote the norm of in , then for , Thus, . Then by Remark 2.6, Therefore, has bounded inverse operator and Set then is the unique mild solution of LIVP (2.6) satisfing . So set then is the unique mild solution of LPBVP (2.16). By Remark 2.7, is a bounded linear operator. Furthermore, by Remark 2.6, we obtain that By (2.24), (2.28) and Remark 2.6, for we have that where . Thus, . Then and the spectral radius . By the randomicity of , we obtain that .
Remark 2.15. For sufficient conditions of exponentially stable operator semigroups, one can see .
Remark 2.16. If is a positive and exponentially stable analytic semigroup generated by , by Remark 2.13, then the resolvent operator is also a positive bounded linear operator.
Remark 2.17. For the applications of Lemma 2.14, it is important to estimate the growth index of . If is continuous in the uniform operator topology for, it is well known that can be obtained by: the spectrum of (see ) We know that is continuous in the uniform operator topology for if is a compact semigroup, see . Assume that is a regeneration cone, is a compact and positive analytic semigroup. Then by the characteristic of positive semigroups (see ), for sufficiently large , we have that has positive bounded inverse operator . Since , the spectral radius . By the Krein-Rutmann theorem (see [34, 35]), has the first eigenvalue , which has a positive eigenfunction , and that is, .
Corollary 2.18. Let be an ordered Banach space, whose positive cone is a regeneration cone. If is a compact and positive analytic semigroup, and its first eigenvalue of is then LPBVP (2.16) has a unique mild solution , is a bounded linear operator, and the spectral radius
Proof. By (2.32), we know that the growth index of is , that is, is exponentially stable. By Lemma 2.14, is a bounded linear operator, and the spectral radius . On the other hand, since has a positive eigenfunction , in LPBVP (3.17) we set , then is the corresponding mild solution. By the definition of the operator , , that is, is an eigenvalue of . Then . Thus, .
3. Main Results
Theorem 3.1. Let be an ordered Banach space, whose positive cone is normal with normal constant . If is a positive analytic semigroup, , and the following conditions are satisfied. For any , there exists such that for any , , , . There exists ( is the growth index of ), such that for any , . Then PBVP (1.1) has a unique positive mild solution.
Proof. Let , then , . Consider LPBVP
generates a positive analytic semigroup , whose growth index is . By Lemma 2.14 and Remark 2.16, LPBVP (3.3) has a unique mild solution and .
Set is the corresponding constant in . We may suppose , otherwise substitute for , is also satisfied. Then we consider LPBVP generates a positive analytic semigroup , whose growth index is . By Lemma 2.14 and Remark 2.16, for LPBVP (3.4) has a unique mild solution is a positive bounded linear operator and the spectral radius .
Set , then is continuous, . By , is an increasing operator on . Set , we can define the sequences By (3.4), we have that In , we set , then By (3.6) and (3.8), the definition and the positivity of , we have that Since is an increasing operator on , in view of (3.5), we have that Therefore, we obtain that By induction, In view of the normality of the cone , we have that On the other hand, since , for some , we have that . By the Gelfand formula, . Then there exist , for , we have that . By (3.13), we have that By (3.10) and (3.14), similarly to the nested interval method, we can prove that there exists a unique , such that By the continuity of the operator and (3.5), we have that By the definition of and (3.10), we know that is a positive mild solution of (3.4) when . Then is the positive mild solution of PBVP (1.1).
In the following, we prove that the uniqueness. If , are the positive mild solutions of PBVP (1.1). Substitute and for , respectively, then (). By (3.14), we have that Thus, , PBVP (1.1) has a unique positive mild solution.
Corollary 3.2. Let be an ordered Banach space, whose positive cone is a regeneration cone. If is a compact and positive analytic semigroup, for , satisfies and the following condition: There exist , where is the first eigenvalue of , such that for any . Then PBVP (1.1) has a unique positive mild solution.
Example 4.1. Consider the following periodic boundary value problem for fractional parabolic partial differential equations in : where is the Caputo fractional partial derivative of order is a bounded domain with a sufficiently smooth boundary , is the Laplace operator, is continuous.
Let , . Then is an Banach space with the partial order “≤” reduced by the normal cone . Define the operator as follows: Then generates an operator semigroup which is compact, analytic, and uniformly bounded. By the maximum principle, we can find that is a positive semigroup. Denote , then the system (4.1) can be reformulated as the problem (1.1) in .
Theorem 4.2. Assume that for , the partial derivative is continuous on any bounded domain and , where is the first eigenvalue of under the condition . Then the problem (4.1) has a unique positive mild solution.
This research was supported by NNSFs of China (10871160, 11061031) and project of NWNU-KJCXGC-3-47.
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