- Research Article
- Open Access
Existence results for fractional differential inclusions arising from real estate asset securitization and HIV models
Advances in Difference Equations volume 2013, Article number: 216 (2013)
This paper studies a new class of boundary value problems of nonlinear differential inclusions with Riemann-Liouville integral boundary conditions arising from real estate asset securitization and HIV models. Some new existence results are obtained by using standard fixed point theorems when the right-hand side of the inclusion has convex as well as non-convex values. Some illustrative examples are also discussed.
In this paper, we study the following boundary value problem:
where is the standard Riemann-Liouville fractional derivative of order , , , is a multivalued map, is the family of all subsets of ℝ, denotes the Riemann-Stieltjes integral, and A is a function of bounded variation.
The subject of fractional differential equations has evolved as an interesting and important field of research in view of its numerous applications in physics, mechanics, chemistry, engineering (like traffic, transportation, logistics etc.), and so forth [1–3]. The tools of fractional calculus have played a key role in improving the mathematical modeling of many real world processes based on classical calculus. The nonlocal characteristic of a fractional order differential operator distinguishes it from a classical integer-order differential operator. In fact, differential equations of arbitrary order are capable of describing memory and hereditary properties of some important and useful materials and processes. For some recent development on the topic, see [4–19] and the references cited therein.
On the other hand, the nonlocal condition given by a Riemann-Stieltjes integral is due to Webb and Infante in  and gives a unified approach to many BVPs. Since covers a multipoint BVP and an integral BVP as special cases, the fractional differential equations with the Riemann-Stieltjes integral condition were extensively studied by many authors, see [21, 22].
The problems of type (1.1) are referred to as semipositone problems that arise naturally in chemical reactor theory in the literature . Recently, by SWOT analysis method, one has found that many mathematical models arising from real estate asset securitization can be interpreted by fractional-order differential or difference equations under suitable initial conditions or boundary conditions. The existence and uniqueness of solution of the fractional-order mechanical model are important and useful. Especially, by examining the numerical simulation and analysis of solution, one can undertake macroscopical analysis and comparative research into advantages and disadvantages of real estate securitization process and find that there may exist problems and risks with real estate asset securitization, and then one can put forward optimizing the views on traditional risk control process. In recent years, fractional-order models have been proved to be more accurate than integer order models, i.e., there are more degrees of freedom in the fractional-order models. Tao and Qian  discussed the existence and uniqueness of positive solutions for the following differential equation with nonlocal Riemann-Stieltjes integral condition arising from the real estate asset securitization:
Mathematical models have been proven valuable in understanding the dynamics of HIV infection [25–27]. Recently Perelson  introduced fractional order into a model of HIV infection. Motivated by these HIV models, Yang  considered the existence of nontrivial solution for the fractional differential system
where λ is a parameter, , , and A, B are functions of bounded variation.
In the present paper, we are motivated by some recent papers [18, 24, 29], which considered problem (1.1) with F being single-valued and provided results on the existence and nonexistence of positive solutions. Here we extend the results to cover the multivalued case.
We establish existence results for problem (1.1), when the right-hand side is convex as well as non-convex valued. The first result relies on the nonlinear alternative of Leray-Schauder type. In the second result, we shall combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values; while in the third result, we shall use the fixed point theorem for contraction multivalued maps due to Covitz and Nadler.
The paper is organized as follows. In Section 2 we recall some preliminary facts that we need in the sequel, and in Section 3 we prove our main results. Examples illustrating the obtained results are presented in Section 4.
In this section, we introduce notations, definitions and preliminary facts that will be used in the remainder of this paper.
Let denote a Banach space of continuous functions from into ℝ with the norm . Let be the Banach space of measurable functions which are Lebesgue integrable and normed by .
For a normed space , let
Let . The Pompeiu-Hausdorff distance of A, B is defined by
where and .
A multivalued map :
is convex (closed) valued if is convex (closed) for all ;
is bounded on bounded sets if is bounded in X for all (i.e., );
is called upper semicontinuous (u.s.c.) on X if, for each , the set is a nonempty closed subset of X, and if, for each open set N of X containing , there exists an open neighborhood of such that ;
G is lower semicontinuous (l.s.c.) if the set is open for any open set B in E;
is said to be completely continuous if is relatively compact for every ;
is said to be measurable if, for every , the function
has a fixed point if there is such that . The fixed point set of the multivalued operator G will be denoted by FixG.
Definition 2.1 A multivalued operator is called:
γ-Lipschitz if and only if there exists such that
a contraction if and only if it is γ-Lipschitz with .
Definition 2.2 A multivalued map is said to be Carathéodory if
is measurable for each ;
is upper semicontinuous for almost all .
Further, a Carathéodory function F is called -Carathéodory if
for each , there exists such that
for all and for a.e. .
For each , define the set of selections of F by
We define the graph of G to be the set and recall two useful results regarding closed graphs and upper-semicontinuity.
Lemma 2.3 [, Proposition 1.2]
If is u.s.c., then is a closed subset of ; i.e., for every sequence and , if when , , and , then . Conversely, if G is completely continuous and has a closed graph, then it is upper semicontinuous.
Lemma 2.4 
Let X be a Banach space. Let be an -Carathéodory multivalued map and let Θ be a linear continuous mapping from to . Then the operator
is a closed graph operator in .
Definition 2.5 The Riemann-Liouville derivative of fractional order q is defined as
provided the integral exists, where denotes the integer part of the real number q.
Definition 2.6 The Riemann-Liouville fractional integral of order q is defined as
provided the integral exists.
Lemma 2.7 
For , the general solution of the fractional differential equation is given by
where , ().
Lemma 2.8 
If , , then(2.1)
If , , then(2.2)
In view of Lemma 2.7, it follows that
for some , ().
Let , . By standard discussion, one can easily reduce BVP (1.1) to the following modified problem:
and BVP (2.4) is equivalent to BVP (1.1).
Lemma 2.9 
Given , then the unique solution of the problem
is given by
where is the Green function of BVP (2.5) and is given by
Lemma 2.10 For any , satisfies
By Lemma 2.9, the unique solution of the problem
is . Let
as in , we can get that the Green function for BVP (2.4) is given by
Throughout the paper we always assume that the following holds.
() A is an increasing function of bounded variation such that for and , where is defined by (2.9).
Lemma 2.11 
Let . Assume that () holds. Then satisfies
We end this section by recalling two well-known fixed point theorems which will be used in the sequel, the nonlinear alternative of Leray-Schauder for multivalued maps and Covitz and Nadler fixed point theorem.
Lemma 2.12 (Nonlinear alternative for Kakutani maps) 
Let E be a Banach space, let C be a closed convex subset of E, U be an open subset of C and . Suppose that is an upper semicontinuous compact map. Then either
F has a fixed point in , or
there is a and with .
Lemma 2.13 
Let be a complete metric space. If is a contraction, then .
3 Existence results
Now we are in a position to present our main results. The methods used to prove the existence results are standard; however, their exposition in the framework of problem (1.1) is new.
3.1 Convex case
Theorem 3.1 Assume that () holds. In addition we assume that:
() is Carathéodory and has nonempty compact and convex values;
() there exist two continuous nondecreasing functions , , and a function such that
for each ;
() there exists a constant such that
Then boundary value problem (1.1) has at least one solution on .
Proof Define the operator by
We will show that satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As the first step, we show that is convex for each . This step is obvious since is convex (F has convex values), and therefore we omit the proof.
In the second step, we show that maps bounded sets (balls) into bounded sets in . For a positive number ρ, let be a bounded ball in . Then, for each , , there exists such that
Then for , and notice that
Now we show that maps bounded sets into equicontinuous sets of . Let with and . For each , we obtain
Obviously the right-hand side of the above inequality tends to zero independently of as . As satisfies the above three assumptions, it follows by the Arzelá-Ascoli theorem that is completely continuous.
In our next step, we show that has a closed graph. Let , and . Then we need to show that . Associated with , there exists such that for each ,
Thus it suffices to show that there exists such that for each ,
Let us consider the linear operator given by
Thus, it follows by Lemma 2.4 that is a closed graph operator. Further, we have . Since , therefore, we have
for some .
Finally, we show that there exists an open set with for any and all . Let and . Then there exists with such that, for , we have
and, using the computations of the second step above, we have
Consequently, we have
In view of (), there exists M such that . Let us set
Note that the operator is upper semicontinuous and completely continuous. From the choice of U, there is no such that for some . Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 2.12), we deduce that has a fixed point , which is a solution of problem (1.1). This completes the proof. □
3.2 Non-convex case
In this subsection, we study the case when F is not necessarily convex-valued.
Definition 3.2 Let A be a subset of . A is measurable if A belongs to the σ-algebra generated by all sets of the form , where is Lebesgue measurable in and is Borel measurable in ℝ.
Definition 3.3 A subset of is decomposable if, for all and measurable , the function , where stands for the characteristic function of .
Theorem 3.4 Assume that (), (), () and the following condition holds:
() is a nonempty compact-valued multivalued map such that
is lower semicontinuous for each .
Then boundary value problem (1.1) has at least one solution on .
Proof It follows from (), () and Lemma 4.1 of  that F is of l.s.c. type. Then, from the selection theorem due to Bressan and Colombo  for lower semicontinuous maps with decomposable values, there exists a continuous function such that for all .
Consider the problem
Observe that if is a solution of (3.1), then x is a solution to problem (1.1). In order to transform problem (3.1) into a fixed point problem, we define the operator as
It can easily be shown that is continuous and completely continuous and satisfies all the conditions of the nonlinear alternative of Leray-Schauder type for single-valued maps . The remaining part of the proof is similar to that of Theorem 3.1. So we omit it. This completes the proof. □
Now we prove our second existence result for problem (1.1) with a non-convex-valued right-hand side by applying a fixed point theorem for a multivalued map due to Covitz and Nadler .
Theorem 3.5 Assume that the following conditions hold:
() is such that is measurable for each ;
() for almost all and with and for almost all .
Then boundary value problem (1.1) has at least one solution on if
Proof We transform boundary value problem (1.1) into a fixed point problem. Define the operator by
We show that the operator satisfies the assumptions of Lemma 2.13. The proof will be given in two steps.
Step 1. is nonempty and closed for every . Note that since the set-valued map is measurable with the measurable selection theorem (e.g., [, Theorem III.6]), it admits a measurable selection . Moreover, by the assumption (), we have
i.e., and hence F is integrably bounded. Therefore, .
To show that for each , let be such that () in . Then and there exists such that, for each ,
As F has compact values, we pass onto a subsequence (if necessary) to obtain that converges to v in . Thus, and for each , we have
Step 2. The multivalued map is a contraction. We show that there exists such that
Let and . Then there exists such that, for each ,
By (), we have
So, there exists such that
Since the multivalued operator is measurable (Proposition III.4 in ), there exists a function which is a measurable selection for U. So, and for each , we have .
For each , let us define
Analogously, interchanging the roles of x and , we obtain
Since is a contraction, it follows by Lemma 2.13 that has a fixed point x which is a solution of (1.1). This completes the proof. □
In this section, we give two examples to show the applicability of our results.
Example 4.1 Consider the following fractional boundary value problem:
and is a multivalued map given by
Thus BVP (4.1) becomes the four-point BVP
Thus , and is increasing, so () holds.
It is clear that F is convex, compact-valued and is of Carathéodory type. Let and , , we get for ,
and hence () holds.
Using the above values in the condition ()
we find that
Clearly, all the conditions of Theorem 3.1 are satisfied. Hence the conclusion of Theorem 3.1 applies to problem (4.1).
Example 4.2 Consider the following fractional boundary value problem:
where as in Example 4.1 and
Then we have
Let . Then
Hence all the assumptions of Theorem 3.5 are satisfied and by the conclusion of it, BVP (4.3) has at least one solution on .
Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) - Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.
Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.
Baleanu D, Diethelm K, Scalas E, Trujillo JJ Series on Complexity, Nonlinearity and Chaos. In Fractional Calculus Models and Numerical Methods. World Scientific, Boston; 2012.
Agarwal RP, Andrade B, Cuevas C: Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations. Nonlinear Anal., Real World Appl. 2010, 11: 3532-3554. 10.1016/j.nonrwa.2010.01.002
Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Bound. Value Probl. 2009., 2009: Article ID 708576
Ahmad B, Ntouyas SK: Fractional differential inclusions with fractional separated boundary conditions. Fract. Calc. Appl. Anal. 2012, 15: 362-382.
Ahmad B, Ntouyas SK, Alsaedi A: A nonlocal three-point inclusion problem of Langevin equation with two different fractional orders. Adv. Differ. Equ. 2012., 2012: Article ID 54
Ahmad B, Ntouyas SK: Existence results for higher order fractional differential inclusions with multi-strip fractional integral boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2013., 2013: Article ID 20
Ahmad B, Ntouyas SK, Alsaedi A: On fractional differential inclusions with anti-periodic type integral boundary conditions. Bound. Value Probl. 2013., 2013: Article ID 82
Ahmad B, Ntouyas SK, Alsaedi A: A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multi-strip boundary conditions. Math. Probl. Eng. 2013., 2013: Article ID 320415
Bai ZB: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 2010, 72: 916-924. 10.1016/j.na.2009.07.033
Balachandran K, Trujillo JJ: The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces. Nonlinear Anal. 2010, 72: 4587-4593. 10.1016/j.na.2010.02.035
Baleanu D, Mustafa OG: On the global existence of solutions to a class of fractional differential equations. Comput. Math. Appl. 2010, 59: 1835-1841. 10.1016/j.camwa.2009.08.028
Baleanu D, Mustafa OG, O’Regan D: A Nagumo-like uniqueness theorem for fractional differential equations. J. Phys. A, Math. Theor. 2011., 44(39): Article ID 392003
El-Shahed M, Shammakh WM: Multiple positive solutions for nonlinear fractional eigenvalue problem with non local conditions. Fract. Calc. Appl. Anal. 2012, 3: 1-13.
Nyamoradi N, Baleanu D, Agarwal RP: On a multipoint boundary value problem for a fractional order differential inclusion on an interval. Adv. Math. Phys. 2013., 2013: Article ID 823961
Razminia A, Baleanu D, Majd VJ: Conditional optimization problems: fractional order case. J. Optim. Theory Appl. 2013, 156: 45-55. 10.1007/s10957-012-0211-6
Wang W, Huang L: Existence of positive solution for semipositone fractional differential equations involving Riemann-Stieltjes integral conditions. Abstr. Appl. Anal. 2012., 2012: Article ID 723507
Zhang X, Liu L: Positive solutions of superlinear semipositone singular Dirichlet boundary value problems. J. Math. Anal. Appl. 2006, 316: 525-537. 10.1016/j.jmaa.2005.04.081
Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems involving integral conditions. NoDEA Nonlinear Differ. Equ. Appl. 2008, 15: 45-67. 10.1007/s00030-007-4067-7
Wang Y, Liu L, Wu Y: Positive solutions for a nonlocal fractional differential equation. Nonlinear Anal. 2011, 74: 3599-3605. 10.1016/j.na.2011.02.043
Zhang X, Han Y: Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations. Appl. Math. Lett. 2012, 25: 555-560. 10.1016/j.aml.2011.09.058
Aris R: Introduction to the Analysis of Chemical Reactors. Prentice Hall, Englewood Cliffs; 1965.
Tao H, Fu M, Qian R: Positive solutions for fractional differential equations from real estate asset securitization via new fixed point theorem. Abstr. Appl. Anal. 2012., 2012: Article ID 842358
Merdan M, Khan T: Homotopy perturbation method for solving viral dynamical model. Fen Bilimleri Dergisi 2010, 31: 65-77.
Petrovic L, Spasic D, Atanackovic TM: On a mathematical model of a human root dentin. Dent. Mater. 2005, 21: 125-128. 10.1016/j.dental.2004.01.004
Tuckwell C, Frederic Y: On the behavior of solutions in viral dynamical models. Biosystems 2004, 73: 157-161. 10.1016/j.biosystems.2003.11.004
Perelson A: Modeling the interaction of the immune system with HIV. Lecture Notes in Biomathematics 83. In Mathematical and Statistical Approaches to AIDS Epidemiology. Edited by: Castillo-Chavez C. Springer, New York; 1989:350.
Yang G: Nontrivial solution of fractional differential system involving Riemann-Stieltjes integral condition. Abstr. Appl. Anal. 2012., 2012: Article ID 719192
Deimling K: Multivalued Differential Equations. de Gruyter, Berlin; 1992.
Lasota A, Opial Z: An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 1965, 13: 781-786.
Granas A, Dugundji J: Fixed Point Theory. Springer, New York; 2005.
Covitz H, Nadler SB Jr.: Multivalued contraction mappings in generalized metric spaces. Isr. J. Math. 1970, 8: 5-11. 10.1007/BF02771543
Frigon M: Théorèmes d’existence de solutions d’inclusions différentielles. NATO ASI Series C 472. In Topological Methods in Differential Equations and Inclusions. Edited by: Granas A, Frigon M. Kluwer Academic, Dordrecht; 1995:51-87.
Bressan A, Colombo G: Extensions and selections of maps with decomposable values. Stud. Math. 1988, 90: 69-86.
Castaing C, Valadier M Lecture Notes in Mathematics 580. In Convex Analysis and Measurable Multifunctions. Springer, Berlin; 1977.
The authors thank the reviewers for their useful comments. This paper was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
The authors declare that they have no competing interests.
Each of the authors, BA and SKN, contributed to each part of this work equally and read and approved the final version of the manuscript.