# On existence and uniqueness of positive solutions to a class of fractional boundary value problems

- J Caballero
^{1}Email author, - J Harjani
^{1}and - K Sadarangani
^{1}

**2011**:25

**DOI: **10.1186/1687-2770-2011-25

© Caballero et al; licensee Springer. 2011

**Received: **28 February 2011

**Accepted: **18 September 2011

**Published: **18 September 2011

## Abstract

The purpose of this paper is to investigate the existence and uniqueness of positive solutions for the following fractional boundary value problem

where 2 < *α* ≤ 3 and ${D}_{{0}^{+}}^{\alpha}$ is the Riemann-Liouville fractional derivative.

Our analysis relies on a fixed-point theorem in partially ordered metric spaces. The autonomous case of this problem was studied in the paper [Zhao et al., Abs. Appl. Anal., to appear], but in Zhao et al. (to appear), the question of uniqueness of the solution is not treated.

We also present some examples where we compare our results with the ones obtained in Zhao et al. (to appear).

**2010 Mathematics Subject Classification**: 34B15

### Keywords

fractional boundary value problem fixed-point theorem positive solution## 1 Introduction

Differential equations of fractional order occur more frequently on different research areas and engineering such as physics, chemistry, economics, etc. Indeed, we can find numerous applications in viscoelasticity, electrochemistry control, porous media, electromagnetic, etc. [1–6].

For an extensive collection of results about this type of equations, we refer the reader to the monograph by Kilbas and Trujillo [7], Samko, Kilbas, and Marichev [8], Miller and Ross [9], and Podlubny [10].

On the other hand, some basic theory for the initial value problems of fractional differential equations involving the Riemann-Lioville differential operator has been discussed by Lakshmikantham et al. [11, 12], Bai et al. [13–16], Zhang [17], etc.

and they proved the existence of positive solutions by means of the Krasnosel'skii fixed-point theorem and Legget-Williams fixed-point theorem.

where 2 < *α* ≤ 3, *λ* is a positive parameter, and *f* : (0, ∞) → (0, ∞) is continuous.

where 2 < *α* ≤ 3 and *f* : [0, 1] × [0, ∞) → [0, ∞) is a continuous function.

Notice that in [18] the question of uniqueness of solutions is not treated.

In our study, the main tool is a fixed-point theorem in partially ordered sets, which gives us uniqueness of the solution. This result appears in [19].

## 2 Basic facts

For the convenience of the reader, we present some definitions, lemmas, and basic results that will be used later.

**Definition 1**. [7] The Riemann-Liouville fractional derivative of order

*α*> 0 of a function

*f*: (0, ∞) → ℝ is given by

where *n* = [*α*] + 1 and [*α*] denotes the integer part of *α* and Γ(*α*) denotes the gamma function, provided that the right side is pointwise defined on (0, ∞).

**Definition 2**. [7] The Riemman-Liouville fractional integral of order

*α*> 0 of a function

*f*: (0, ∞) → ℝ is defined by

provided that the right side is pointwise defined on (0, ∞).

The following two lemmas can be found in [20].

**Lemma 1**.

*Let α*> 0

*and u*∈

*C*(0, 1) ∩

*L*

^{1}(0, 1)

*. Then, the fractional differential equation*

*has*

*where c*_{
i
} ∈ ℝ (*i* = 1, ... *n*) *and n* = [*α*] + 1 *as unique solution*.

**Lemma 2**.

*Assume that u*∈

*C*(0, 1) ∩

*L*

^{1}(0, 1)

*with a fractional derivative of order α*> 0

*that belongs to C*(0, 1) ∩

*L*

^{1}(0, 1)

*. Then*,

*for some c*_{
i
} ∈ ℝ (*i* = 1, ..., *n*) *and n* = [*α*] + 1.

In [21], it is proved the following result by using Lemmas 1 and 2.

**Lemma 3**.

*Given f*∈

*C*[0, 1]

*and*2 <

*α*≤ 3

*. The unique solution of*

*is*

*where*

is the Green's function associated to the boundary value problem (3).

*Remark* 1. In [21], it is proved that *G*(*t*, *s*) ≥ 0, for *t*, *s* ∈ [0, 1].

Now, we present the fixed-point theorems that we will use later. These results appear in [19].

**Theorem 1**.

*Let*(

*X*, ≤)

*be a partially ordered set and suppose that there exists a metric d in X such that*(

*X*,

*d*)

*is a complete metric space. Assume that X satisfies the following condition*

*Let T*:

*X*→

*X be a nondecreasing mapping such that*

*where ψ* : [0, ∞) → [0, ∞) *is a continuous and nondecreasing function such that ψ is positive in* (0, ∞)*, ψ*(0) = 0*, and* lim_{t→∞}*ψ*(*t*) = ∞*. If there exists x*_{0} ∈ *X with x*_{0} ≤ *Tx*_{0} *then T has a fixed point*.

*X*, ≤) satisfies the following condition:

we have the following result.

**Theorem 2**. *Adding condition (5) to the hypotheses of Theorem 1, we obtain uniqueness of the fixed point*.

*Remark* 2. In Theorems 1 and 2, the condition lim_{t→∞}*ψ*(*t*) = ∞ is superfluous.

In our considerations, we will work in the Banach space *C*[0, 1] = {*x* : [0, 1] → ℝ, continuous} with the standard distance given by *d*(*x*, *y*) = sup_{0≤t≤1}{|*x*(*t*) - *y*(*t*)|}.

In [22], it is proved that (*C*[0, 1], ≤) with the above-mentioned metric satisfies condition (4) of Theorem 1. Moreover, for *x*, *y* ∈ *C*[0, 1], as the function max(*x*, *y*) ∈ *C*[0, 1], (*C*[0, 1], ≤) satisfies condition (5).

By $\mathcal{F}$, we denote the class of functions *ψ* : [0, ∞) → [0, ∞) continuous, nondecreasing, positive in (0, ∞) and *ψ*(0) = 0, and by $\mathcal{J}$ the class of functions *φ* : [0, ∞) → [0, ∞) continuous, nondecreasing, and satisfying that $I-\phi \in \mathcal{F}$, where *I* denotes the identity mapping on [0, ∞).

## 3 Main result

Our starting point of this section is the following result about Green's function appearing in Section 2.

**Lemma 4.**
${max}_{t\in \left[0,1\right]}\underset{0}{\overset{1}{\int}}G\left(t,s\right)ds=\frac{1}{\Gamma \left(\alpha +1\right)}\left[{\left(\frac{\alpha -1}{\alpha}\right)}^{\alpha -1}-\left(\frac{\alpha -1}{\alpha}\right)\right]$

*Proof*. In fact,

□

In the sequel, we present the main result of this paper.

For convenience, we put $A=\frac{1}{\Gamma \left(\alpha +1\right)}\left[{\left(\frac{\alpha -1}{\alpha}\right)}^{\alpha -1}-{\left(\frac{\alpha -1}{\alpha}\right)}^{\alpha}\right]$.

**Theorem 3**. *Our Problem (2) has a unique nonnegative solution u*(*t*) *if the following conditions are satisfied:*

(H1) *f* : [0, 1] × [0, ∞) → [0, ∞) *is continuous and nondecreasing respect to the second argument*.

*There exists*$0<\lambda \le \frac{1}{A}$

*such that, for x*,

*y*∈ [0, ∞)

*with y*≥

*x and t*∈ [0, 1],

*where*$\phi \in \mathcal{J}$.

*Proof*. Consider the cone

Obviously, (*P*, *d*) with *d*(*x*, *y*) = sup{|*x*(*t*) - *y*(*t*)|: *t* ∈ [0, 1]} is a complete metric space satisfying conditions (4) and (5).

where *G*(*t*, *s*) is the Green's function appearing in Section 2. Obviously, *T* applies *P* into itself since *f*(*t*, *x*) and *G*(*t*, *s*) are nonnegative continuous functions.

In what follows we check that assumptions in Theorem 2 are satisfied.

Firstly, the operator *T* is nondecreasing.

*u*,

*v*∈

*P*,

*u*≥

*v*, and

*t*∈ [0, 1], we have

Now, we prove that *T* satisfies the contractive condition appearing in Theorem 1.

*u*,

*v*∈

*P*and

*u*≥

*v*and, taking into account assumption (H2), we get

*φ*is nondecreasing, and, taking into account (H2) and Lemma 4, we obtain

*ψ*(

*x*) =

*x*-

*φ*(

*x*). As $\phi \in \mathcal{J}$, this means that $\psi \in \mathcal{F}$ and from the last inequality

This proves that *T* satisfies the contractive condition of Theorem 1.

*G*(

*t*,

*s*) and

*f*(

*t*,

*x*) [assumption (H1)] gives us

where 0 denotes the zero function.

Therefore, Theorem 2 says us that Problem (2) has a unique nonnegative solution.

□

In the sequel, we present a sufficient condition for the existence and uniqueness of positive solutions for Problem (2) (positive solution means *x*(*t*) > 0 for *t* ∈ (0, 1)). The proof of this condition is similar to the proof of Theorem 2.3 of [23]. We present this proof for completeness.

**Theorem 4**. *Under assumptions of Theorem 3 and suppose that f*(*t*_{0}, 0) ≠ 0 *for certain t*_{0} ∈ [0, 1]. *Then, Problem (2) has a unique positive solution*.

*Proof*. Consider the nonnegative solution *x*(*t*) for Problem (2) whose existence is guaranteed by Theorem 3.

In the sequel, we will prove that *x*(*t*) is a positive solution.

*x*(

*t*) is a fixed point of the operator $\left(Tu\right)\left(t\right)={\int}_{0}^{1}G\left(t,s\right)f\left(s,u\left(s\right)\right)\mathsf{\text{d}}s$ and, consequently,

*t** < 1 such that

*x*(

*t**) = 0. This means that

*x*(

*t*) is a nonnegative function,

*f*(

*t*,

*y*) is nondecreasing with respect to the second argument and the nonnegative character of

*G*(

*t*,

*s*), we get

This gives us $x\left({t}^{*}\right)={\int}_{0}^{1}G\left({t}^{*},s\right)f\left(s,0\right)\mathsf{\text{d}}s=0$.

*G*(

*t*,

*s*) ≥ 0 and

*f*(

*s*, 0) ≥ 0, the last expression implies

*G*(

*t**,

*s*) ≠ 0 a.e (

*s*) (because

*G*(

*t**,

*s*) is given by a polynomial), we can obtain

On the other hand, as *f*(*t*_{0}, 0) ≠ 0 for certain *t*_{0} ∈ [0, 1], the nonnegative character of *f*(*t*, *y*) gives us *f*(*t*_{0}, 0) > 0. As *f*(*t*, *y*) is a continuous function, we can find a set *A* ⊂ [0, 1] with *t*_{0} ∈ *A*, *μ*(*A*) > 0, where *μ* is the Lebesgue measure and *f*(*t*, 0) > 0 for any *t* ∈ *A*. This contradicts (6).

Therefore, *x*(*t*) > 0 for *t* ∈ (0, 1). This finishes the proof. □

*Remark*3. In Theorem 4, the condition

*f*(

*t*

_{0}, 0) ≠ 0 for certain

*t*

_{0}∈ [0, 1] seems to be a strong condition in order to obtain a positive solution for Problem (2), but when the solution is unique, we will see that this condition is very adjusted one. In fact, suppose that Problem (2) has a unique nonnegative solution

*x*(

*t*) then

In fact, if *f*(*t*, 0) = 0 for each *t* ∈ [0, 1], it is easily seen that the zero function satisfies Problem (2) and the uniqueness of the solution gives us *x*(*t*) = 0. The reverse implication is obvious.

*Remark*4. Notice that the hypotheses in Theorem 3 are invariant by continuous perturbation. More precisely, if

*f*(

*t*, 0) = 0 for any

*t*∈ [0, 1] and

*f*satisfies (H1) and (H2) of Theorem 3 then

*g*(

*t*,

*x*) =

*a*(

*t*) +

*f*(

*t*,

*x*) with

*a*: [0, 1] → [0, ∞) continuous and

*a*≠ 0, satisfies assumptions of Theorem 4, and this means that the following boundary value problem

has a unique positive solution.

Now, we present an example that illustrates our results.

*Example*1. Consider the boundary value problem

In this case, $\alpha =\frac{5}{2}$ and *f*(*t*, *u*) = *c* + *λ* · *arctg u*. It is easily seen that *f*(*t*, *u*) satisfies (H1) of Theorem 3.

In the sequel, we prove that *f*(*t*, *u*) satisfies (H2) of Theorem 3.

*ϕ*: [0, ∞) → [0, ∞) given by

*ϕ*(

*u*) =

*arctg u*and we will see that

*ϕ*satisfies

In fact, put *ϕ*(*u*) = *arctag u* = *α* and *ϕ*(*v*) = *arctg v* = *β* (notice that, as *u* ≥ *v* and *ϕ* is nondecreasing, *α* ≥ *β*).

*tgα*,

*tgβ*∈ [0, ∞), we can obtain

*ϕ*to the last inequality and taking into account the nondecreasing character of

*ϕ*, we obtain

This proof our previous claim.

*u*≥

*v*and

*t*∈ [0, 1], we have,

Now, we prove that *ϕ*(*u*) = *arctg u* belongs to $\mathcal{J}$. Obviously, *ϕ* : [0, ∞) → [0, ∞) is a continuous and nondecreasing function. Moreover, *ψ*(*u*) = *u* - *ϕ*(*u*) = *u* - *arctg u* is also continuous and nondecreasing and satisfies *ψ*(*u*) > 0 for *u* > 0 and *ψ*(0) = 0. Consequently, $\varphi \in \mathcal{J}$.

*f*(

*t*, 0) =

*c*+

*arctg*0 =

*c*> 0, by Theorem 4, Problem (7) has a unique positive solution for

## 4 Some remarks

where 2 < *α* ≤ 3, *λ* is a positive parameter and *f* : (0, ∞) → (0, ∞) is continuous. The main tool used by the authors in this paper is Guo-Kranosel'skii fixed-point theorem on cones. In [18], the question about the uniqueness of solutions is not treated.

One of the results of [18] is the following theorem.

**Theorem 5**. [[18], Theorem 3.2] *If there exists l* ∈ (0, 1) *such that q*(*l*)*c*_{2}*f*_{0} > *F*_{∞}*c*_{1} *holds then, for each λ* ∈ ((*q*(*l*)*c*_{2}*f*_{0})^{-1}, (*F*_{∞}*c*_{1})^{-1})*, the boundary value problem (8) has at least one positive solution*.

Here, we consider (*q*(*l*)*c*_{2}*f*_{0})^{-1} = 0 if *f*_{0} = ∞ and (*F*_{∞}*c*_{1})^{-1} = ∞ if *F*_{∞} = 0, where ${f}_{0}={lim}_{u\to {0}^{+}}inf\frac{f\left(u\right)}{u}$, ${F}_{\infty}={lim}_{u\to +\infty}sup\frac{f\left(u\right)}{u}$, *q*(*t*) = *t*^{α-1}(1 - *t*), *k*(*s*) = *s*(1 - *s*)^{α-1}, ${c}_{1}=\frac{1}{\Gamma \left(\alpha \right)}{\int}_{0}^{1}\left(\alpha -1\right)k\left(s\right)\mathsf{\text{d}}s$, and ${c}_{2}=\frac{1}{\Gamma \left(\alpha \right)}{\int}_{0}^{1}\frac{1}{\alpha -1}q\left(s\right)k\left(s\right)\mathsf{\text{d}}s$.

Now, we present the following example.

*Example*2. Consider the boundary value problem that is a variant of Example 1.

In this case, $\alpha =\frac{5}{2}$ and *f*(*u*) = *c* + *arctg u*. Then, we have *F*_{∞} = 0 and *f*_{0} = ∞. Moreover, *c*_{1} = 0.129, *c*_{2} = 0.0077, and $q\left(1/2\right)=\frac{\sqrt{2}}{8}=0.1768$ [[18], Example 5.1]. Thus, *q*(1/2)*c*_{2}*f*_{0} > *F*_{∞}*c*_{1} holds. Theorem 5 gives us the existence of a positive solution for Problem (9) for each *λ* ∈ (0, ∞). The question of uniqueness cannot be treated by the results of [18].

On the other hand, following a similar reasoning that in Example 1, Theorem 4 gives us the existence of a unique positive solution for Problem (9) when $0<\lambda \le {\left(\frac{1}{\Gamma \left(5/2+1\right)}\left[{\left(\frac{3}{5}\right)}^{3/2}-{\left(\frac{3}{5}\right)}^{5/2}\right]\right)}^{-1}\approx 17.8682$.

Our main contribution is the uniqueness of positive solution for Problem (9) when 0 < *λ* ≤ 17.8682.

Now, we present an example that cannot be studied by the results of [18], and it can be treated by the ones obtained in this paper.

*Example*3. Consider the following boundary value problem

In this case, the boundary value problem is nonautonomous, and thus, this problem cannot be studied by the results of [18].

On the other hand, using a similar argument that in example 1, and using Theorem 4, we obtain the existence of a unique positive solution for Problem (10) when 0 < *λ* ≤ 17.868.

## Declarations

### Acknowledgements

This research was partially supported by "Ministerio de Educación y Ciencia" Project MTM 2007/65706.

## Authors’ Affiliations

## References

- Diethelm K, Freed AD: On the solutions of nonlinear fractional order differential equations used in the modelling of viscoplasticity. In
*Scientific Computing in Chemical Engineering II--Computational Fluid Dynamics, Reaction Engineering and Molecular Properties*. Edited by: Keil F, Mackens W, Voss H, Werthers J. Springer, Heidelberg; 1999:217-224. - Gaul L, Klein P, Kempffe S: Damping description involving fractional operators.
*Mech Syst Signal Process*1991, 5: 81-88. 10.1016/0888-3270(91)90016-XView Article - Glockle WG, Nonnenmacher TF: A fractional calculus approach of self-similar protein dynamics.
*Biophys J*1995, 68: 46-53. 10.1016/S0006-3495(95)80157-8View Article - Mainardi F: Fractional calculus: Some basic problems in continuum and statistical mechanics. In
*Fractal and Fractional Calculus in Continuum Mechanics*. Edited by: Carpinteri CA, Mainardi F. Springer, Wien; 1997:291-348.View Article - Metzler F, Schick W, Kilian HG, Nonnenmache TF: Relaxation in filled polymers: A fractional calculus approach.
*J Chem Phys*1995, 103: 7180-7186. 10.1063/1.470346View Article - Oldham KB, Spanier J:
*The Fractional Calculus.*Academic Press, New York; 1974. - Kilbas AA, Trujillo JJ: Differential equations of fractional order: Methods, results and problems-I.
*Appl Anal*2001, 78: 153-192. 10.1080/00036810108840931View ArticleMathSciNetMATH - Samko SG, Marichev OI, Kilbas AA:
*Fractional Integral and Derivative, Theory and Applications.*Gordon and Breach, Yverdon, Switzerland; 1993. - Miller KS, Ross B:
*An Introduction to the Fractional Calculus and Fractional Differential Equations.*Wiley, New York; 1993. - Podlubny I:
*Fractional Differential Equations.*Academic Press, San Diego; 1999. - Lakshmikantham V, Vatsala AS: Basic theory of fractional differential equations.
*Nonlinear Anal*2008, 69: 2677-2682. 10.1016/j.na.2007.08.042View ArticleMathSciNetMATH - Lakshmikantham V: Theory of fractional functional differential equations.
*Nonlinear Anal*2008, 69: 3337-3343. 10.1016/j.na.2007.09.025View ArticleMathSciNetMATH - Bai C: Positive solutions for nonlinear fractional differential equations with coefficient that changes sign.
*Nonlinear Anal*2006, 64: 677-685. 10.1016/j.na.2005.04.047View ArticleMathSciNetMATH - Bai Z, Ge W: Existence of three positive solutions for some second-order boundary value problems.
*Comput Math Appl*2004, 48: 699-707. 10.1016/j.camwa.2004.03.002View ArticleMathSciNetMATH - Bai Z, Lü H: Positive solutions for boundary value problem of nonlinear fractional differential equation.
*J Math Anal Appl*2005, 311: 495-505. 10.1016/j.jmaa.2005.02.052View ArticleMathSciNetMATH - Bai Z: On positive solutions of a nonlocal fractional boundary value problem.
*Nonlinear Anal*2010, 72: 916-924. 10.1016/j.na.2009.07.033View ArticleMathSciNetMATH - Zhang S: Existence of solution for a boundary value problem of fractional order.
*Acta Math Sci*2006, 26: 220-228.View ArticleMATH - Zhao Y, Sun S, Han Z, Li Q: Positive solutions to boundary value problems of nonlinear fractional differential equations.
*Abs Appl Anal*2011, 2011: Article ID390543.MathSciNet - Harjani J, Sadarangani K: Fixed point theorems for weakly contractive mappings in partially ordered sets.
*Nonlinear Anal*2009, 71: 3403-3410. 10.1016/j.na.2009.01.240View ArticleMathSciNetMATH - Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. In
*North-Holland Mathematics Studies*.*Volume 204*. Elsevier, Amsterdam; 2006. - Yu Y, Jiang D:
*Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation.*Northeast Normal University, Changchun, Jilin Province, Republic of China PRC; 2009. - Nieto JJ, Rodríguez-López R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations.
*Order*2005, 22: 223-239. 10.1007/s11083-005-9018-5View ArticleMathSciNetMATH - Harjani J, López B, Sadarangani K: On positive solutions of a nonlinear fourth order boundary value problem via a fixed point theorem in ordered sets.
*Dyn Syst Appl*2010, 19: 625-634.MATH

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.