Positive Solutions for Fractional Differential Equations from Real Estate Asset Securitization via New Fixed Point Theorem

and Applied Analysis 3 Remark 2.3. If x, y : 0, ∞ → R with order α > 0, then Dt ( x t y t ) Dtx t Dty t . 2.3 Proposition 2.4 see 18, 19 . 1 If x ∈ L1 0, 1 , ν > σ > 0, then IIx t I x t , DtIx t Iν−σx t , DtIx t x t . 2.4 2 If ν > 0, σ > 0, then Dttσ−1 Γ σ Γ σ − ν t σ−ν−1. 2.5 Proposition 2.5 see 18, 19 . Let α > 0, and f x is integrable, then IDtf x f x c1xα−1 c2xα−2 · · · cnxα−n, 2.6 where ci ∈ R (i 1, 2, . . . , n), n is the smallest integer greater than or equal to α. Let x t Iy t , y t ∈ C 0, 1 , by standard discuss, we easily reduce the BVP 1.1 to the following modified problems: −Dty t f ( t, Iy t ,−y t ) , y 0 y′ 0 0, y 1 ∫1


Introduction
Real estate asset securitization is a kind of important financial derivatives in the world. In the international capital market, by combining with different development pattern of the financial industry and real estate industry, real estate asset securitization has become a class of financial products with rapid development and great vitality. 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 for real estate securitization process advantages and disadvantages, and find real estate asset securitization may exist problems and risks, and then one can put forward to optimize the views on traditional risk control process. In recent years, fractional-order models have been proved to be more accurate than integer order models; 2 Abstract and Applied Analysis that is, there are more degrees of freedom in the fractional-order models, see 1, 2 . For recent works about fractional-order models, we refer the reader to 3-9 and the references therein.
In this paper, we discuss the existence and uniqueness of positive solutions for the following fractional differential equation with nonlocal Riemann-Stieltjes integral condition arising from the real estate asset securitization i 1 μ i u η i , μ i > 0 is often required. Thus the BVP 1.1 include more generalized boundary value conditions. For a detailed description of multipoint and integral boundary conditions on fractional differential equation, we refer the reader to some recent papers see 10-15 . Note that the problem 1.1 has been considered by Wang and Li 16 , and the authors obtained the existence of one positive solution for 1.1 by using Krasnoselskii's fixed point theorem on a cone under semipositone case, but the uniqueness of the solution is not treated.
The paper is organized as follows. In Section 2, we extend and improve some fixed point theorems for weakly contractive mappings in partially ordered sets by Harjani and Sadarangani 17 , we omit many redundant conditions of Theorems 2 and 3 in 17 and obtain better results than those of 17 . In Section 3, the existence and uniqueness of a positive solution for the problem 1.1 are obtained by using a fixed point theorem in partially ordered sets. An example is also given to illuminate the application of the main result.

Preliminaries and Lemmas
Definition 2.1 see 18, 19 . The Riemann-Liouville fractional integral of order α > 0 of a function x : 0, ∞ → R is given by provided that the right-hand side is pointwise defined on 0, ∞ .
where n α 1, α denotes the integer part of number α, provided that the right-hand side is pointwise defined on 0, ∞ .

Abstract and Applied Analysis
where c i ∈ R (i 1, 2, . . . , n), n is the smallest integer greater than or equal to α.
Let x t I β y t , y t ∈ C 0, 1 , by standard discuss, we easily reduce the BVP 1.1 to the following modified problems:

Lemma 2.6 see 3 .
Given y ∈ L 1 0, 1 , then the problem has the unique solution where G t, s is given by which is the Green function of the BVP 2.8 .

Lemma 2.7 see 3 .
For any t, s ∈ 0, 1 , G t, s satisfies By Lemma 2.6, the unique solution of the problem and define 2.14 as in 4 , we can get that the Green function for the nonlocal BVP 2.7 is given by Throughout the paper, we always assume the following holds.
H0 A is a increasing function of bounded variation such that G A s ≥ 0 for s ∈ 0, 1 and 0 ≤ C < 1, where C is defined by 2.13 .
Proof. By 2.11 and that A t is a increasing function of bounded variation, we have Consequently, where ψ : 0, ∞ → 0, ∞ is a nondecreasing function. If there exists x ∈ X with x 0 ≤ Tx 0 , then T has a fixed point.
Proof. If T x 0 x 0 , then the proof is finished. Suppose that x 0 < T x 0 . Since T is a nondecreasing mapping, we obtain by induction that 2.20 Put x n 1 T n x 0 Tx n . Then for each integer n ≥ 1, from 2.20 , the elements x n and x n 1 are comparable, we get d x n 1 , x n d Tx n , Tx n−1 ≤ λd x n , x n−1 − ψ λd x n , x n−1 ≤ λd x n , x n−1 .

2.21
If there exists n 0 ∈ N such that d x n 0 , x n 0 −1 0 then x n 0 T n 0 −1 x 0 x n 0 −1 and x n 0 −1 is a fixed point and the proof is finished. In other case, suppose that d x n 0 , x n 0 −1 / 0 for all n ∈ N. Then by 2.21 , we have d x n 1 , x n ≤ λd x n , x n−1 ≤ · · · ≤ λ n−n 0 1 d x n 0 , x n 0 −1 −→ 0, as n −→ ∞, 2.22 that is, ρ n d x n 1 , x n −→ 0, as n −→ ∞.

2.23
Similar to 17 , x n is a Cauchy sequence and then there exists z ∈ X such that lim n → ∞ x n z since X, d is a complete metric space. We claim that T z z. In fact,  10. In Theorem 2.9, We do not require that ψ is continuous and ψ is positive in 0, ∞ , ψ 0 0 and lim t → ∞ ψ t ∞. This in essence improve and generalize the corresponding results of Theorem 2 and Theorem 3 in paper 17 .

Abstract and Applied Analysis
If we consider that X, ≤ satisfies the following condition: for x, y ∈ X, there exists z ∈ X which is comparable to x and y, 2.25 then we have the following theorem, see 17 .
Theorem 2.11. Adding condition 2.25 to the hypotheses of Theorem 2.9, one obtains uniqueness of the fixed point of T .
In our considerations, we will work in the Banach space C 0, 1 {x : 0, 1 → R is continuous} with the standard norm ||x|| max 0≤t≤1 |x t |.
Note that this space can be equipped with a partial order given by x, y ∈ C 0, 1 , x ≤ y ⇐⇒ x t ≤ y t , for t ∈ 0, 1 .

2.26
In 20, 21 , it is proved that C 0, 1 , ≤ with the classic metric given by satisfies the following condition. If x n is a nondecreasing sequence in X such that x n → x then x n ≤ x for all n ∈ N. Moreover, for x, y ∈ C 0, 1 , as the function max{x, y} is continuous in 0, 1 , and C 0, 1 , ≤ satisfies condition 2.25 .
and there exist a function φ ∈ A and constants 0 for x i , y i ∈ 0, ∞ , i 1, 2 with x 1 ≥ x 2 , y 1 ≤ y 2 and t ∈ 0, 1 . Then problem 1.1 has a unique nonnegative solution.

3.3
Note that, as P is a closed set of C 0, 1 , P is a complete metric space.
Abstract and Applied Analysis 7 Now, for y ∈ P , we define the operator T by Then from the assumption on f and Lemma 2.8, we have In what follows, we check that hypotheses in Theorems 2.9 and 2.11 are satisfied. Firstly, the operator T is nondecreasing since, by hypothesis, for any u ≥ v

3.11
Finally, taking into account that the zero function, 0 ≤ T 0, by Theorem 2.11, problem 1.1 has a unique nonnegative solution. Proof. By Theorem 3.1, the problem 1.1 has a unique nonnegative solution, which also is positive.

3.19
Then the BVP 3.17 has a unique positive solution.

3.25
Thus φ ∈ A and all of condition of Theorem 3.1 are satisfied. On the other hand, f t, 0, 0 e t / 0 for any t ∈ 0, 1 , by Theorem 3.2, the BVP 3.17 has a unique positive solution.
In the end, we claim this process and conclusion of paper can be applied to the forefront of research of the real estate asset securitization.