Open Access

Existence and Uniqueness of Positive and Nondecreasing Solutions for a Class of Singular Fractional Boundary Value Problems

Boundary Value Problems20092009:421310

DOI: 10.1155/2009/421310

Received: 24 April 2009

Accepted: 14 June 2009

Published: 19 July 2009

Abstract

We establish the existence and uniqueness of a positive and nondecreasing solution to a singular boundary value problem of a class of nonlinear fractional differential equation. Our analysis relies on a fixed point theorem in partially ordered sets.

1. Introduction

Many papers and books on fractional differential equations have appeared recently. Most of them are devoted to the solvability of the linear fractional equation in terms of a special function (see, e.g., [1, 2]) and to problems of analyticity in the complex domain [3]. Moreover, Delbosco and Rodino [4] considered the existence of a solution for the nonlinear fractional differential equation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq1_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq2_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq4_HTML.gif is a given continuous function in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq5_HTML.gif . They obtained results for solutions by using the Schauder fixed point theorem and the Banach contraction principle. Recently, Zhang [5] considered the existence of positive solution for equation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq6_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq7_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq8_HTML.gif is a given continuous function by using the sub- and super-solution methods.

In this paper, we discuss the existence and uniqueness of a positive and nondecreasing solution to boundary-value problem of the nonlinear fractional differential equation

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq9_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq10_HTML.gif is the Caputo's differentiation and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq11_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq12_HTML.gif (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq13_HTML.gif is singular at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq14_HTML.gif ).

Note that this problem was considered in [6] where the authors proved the existence of one positive solution for (1.1) by using Krasnoselskii's fixed point theorem and nonlinear alternative of Leray-Schauder type in a cone and assuming certain hypotheses on the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq15_HTML.gif . In [6] the uniqueness of the solution is not treated.

In this paper we will prove the existence and uniqueness of a positive and nondecreasing solution for the problem (1.1) by using a fixed point theorem in partially ordered sets.

Existence of fixed point in partially ordered sets has been considered recently in [712]. This work is inspired in the papers [6, 8].

For existence theorems for fractional differential equation and applications, we refer to the survey [13]. Concerning the definitions and basic properties we refer the reader to [14].

Recently, some existence results for fractional boundary value problem have appeared in the literature (see, e.g., [1517]).

2. Preliminaries and Previous Results

For the convenience of the reader, we present here some notations and lemmas that will be used in the proofs of our main results.

Definition 2.1.

The Riemman-Liouville fractional integral of order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq16_HTML.gif of a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq17_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ2_HTML.gif
(2.1)

provided that the right-hand side is pointwise defined on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq18_HTML.gif .

Definition 2.2.

The Caputo fractional derivative of order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq19_HTML.gif of a continuous function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq20_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ3_HTML.gif
(2.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq21_HTML.gif , provided that the right-hand side is pointwise defined on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq22_HTML.gif .

The following lemmas appear in [14].

Lemma 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq23_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq24_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ4_HTML.gif
(2.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq25_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq26_HTML.gif

Lemma 2.4.

The relation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ5_HTML.gif
(2.4)

is valid when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq27_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq28_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq29_HTML.gif .

The following lemmas appear in [6].

Lemma 2.5.

Given https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq30_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq31_HTML.gif , the unique solution of
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ6_HTML.gif
(2.5)
is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ7_HTML.gif
(2.6)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ8_HTML.gif
(2.7)

Remark 2.6.

Note that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq32_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq33_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq34_HTML.gif (see [6]).

Lemma 2.7.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq35_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq36_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq37_HTML.gif is a continuous function with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq38_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq39_HTML.gif is a continuous function on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq40_HTML.gif . Then the function defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ9_HTML.gif
(2.8)

is continuous on [0,1], where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq41_HTML.gif is the Green function defined in Lemma 2.5.

Now, we present some results about the fixed point theorems which we will use later. These results appear in [8].

Theorem 2.8.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq42_HTML.gif be a partially ordered set and suppose that there exists a metric https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq43_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq44_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq45_HTML.gif is a complete metric space. Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq46_HTML.gif satisfies the following condition: if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq47_HTML.gif is a non decreasing sequence in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq48_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq49_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq50_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq51_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq52_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq53_HTML.gif be a nondecreasing mapping such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ10_HTML.gif
(2.9)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq54_HTML.gif is continuous and nondecreasing function such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq55_HTML.gif is positive in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq56_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq57_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq58_HTML.gif . If there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq59_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq60_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq61_HTML.gif has a fixed point.

If we consider that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq62_HTML.gif satisfies the following condition:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ11_HTML.gif
(2.10)

then we have the following theorem [8].

Theorem 2.9.

Adding condition (2.10) to the hypotheses of Theorem 2.8 one obtains uniqueness of the fixed point of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq63_HTML.gif .

In our considerations, we will work in the Banach space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq64_HTML.gif with the standard norm https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq65_HTML.gif .

Note that this space can be equipped with a partial order given by

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ12_HTML.gif
(2.11)

In [10] it is proved that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq66_HTML.gif with the classic metric given by

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ13_HTML.gif
(2.12)

satisfies condition (2) of Theorem 2.8. Moreover, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq67_HTML.gif , as the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq68_HTML.gif is continuous in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq69_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq70_HTML.gif satisfies condition (2.10).

3. Main Result

Theorem 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq71_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq72_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq73_HTML.gif is continuous and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq74_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq75_HTML.gif is a continuous function on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq76_HTML.gif . Assume that there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq77_HTML.gif such that for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq78_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq79_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq80_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ14_HTML.gif
(3.1)

Then one's problem (1.1) has an unique nonnegative solution.

Proof.

Consider the cone
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ15_HTML.gif
(3.2)

Note that, as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq81_HTML.gif is a closed set of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq82_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq83_HTML.gif is a complete metric space.

Now, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq84_HTML.gif we define the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq85_HTML.gif by

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ16_HTML.gif
(3.3)
By Lemma 2.7, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq86_HTML.gif . Moreover, taking into account Remark 2.6 and as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq87_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq88_HTML.gif by hypothesis, we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ17_HTML.gif
(3.4)

Hence, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq89_HTML.gif .

In what follows we check that hypotheses in Theorems 2.8 and 2.9 are satisfied.

Firstly, the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq90_HTML.gif is nondecreasing since, by hypothesis, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq91_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ18_HTML.gif
(3.5)
Besides, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq92_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ19_HTML.gif
(3.6)
As the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq93_HTML.gif is nondecreasing then, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq94_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ20_HTML.gif
(3.7)
and from last inequality we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ21_HTML.gif
(3.8)

Put https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq95_HTML.gif . Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq96_HTML.gif is continuous, nondecreasing, positive in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq97_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq98_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq99_HTML.gif .

Thus, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq100_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ22_HTML.gif
(3.9)

Finally, take into account that for the zero function, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq101_HTML.gif , by Theorem 2.8 our problem (1.1) has at least one nonnegative solution. Moreover, this solution is unique since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq102_HTML.gif satisfies condition (2.10) (see comments at the beginning of this section) and Theorem 2.9.

Remark 3.2.

In [6, lemma  3.2] it is proved that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq103_HTML.gif is completely continuous and Schauder fixed point theorem gives us the existence of a solution to our problem (1.1).

In the sequel we present an example which illustrates Theorem 3.1.

Example 3.3.

Consider the fractional differential equation (this example is inspired in [6])
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ23_HTML.gif
(3.10)
In this case, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq104_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq105_HTML.gif . Note that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq106_HTML.gif is continuous in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq107_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq108_HTML.gif . Moreover, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq109_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq110_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ24_HTML.gif
(3.11)
because https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq111_HTML.gif is nondecreasing on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq112_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ25_HTML.gif
(3.12)

Note that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq113_HTML.gif .

Theorem 3.1 give us that our fractional differential (3.10) has an unique nonnegative solution.

This example give us uniqueness of the solution for the fractional differential equation appearing in [6] in the particular case https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq114_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq115_HTML.gif

Remark 3.4.

Note that our Theorem 3.1 works if the condition (3.1) is changed by, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq116_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq117_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq118_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ26_HTML.gif
(3.13)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq119_HTML.gif is continuous and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq120_HTML.gif satisfies

(a) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq121_HTML.gif and nondecreasing;

(b) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq122_HTML.gif ;

(c) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq123_HTML.gif is positive in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq124_HTML.gif ;

(d) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq125_HTML.gif .

Examples of such functions are https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq126_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq127_HTML.gif .

Remark 3.5.

Note that the Green function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq128_HTML.gif is strictly increasing in the first variable in the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq129_HTML.gif . In fact, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq130_HTML.gif fixed we have the following cases

Case 1.

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq131_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq132_HTML.gif as, in this case,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ27_HTML.gif
(3.14)
It is trivial that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ28_HTML.gif
(3.15)

Case 2.

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq133_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq134_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ29_HTML.gif
(3.16)
Now, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq135_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq136_HTML.gif then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ30_HTML.gif
(3.17)

Hence, taking into account the last inequality and (3.16), we obtain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq137_HTML.gif .

Case 3.

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq138_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq139_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ31_HTML.gif
(3.18)

and, as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq140_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq141_HTML.gif , it can be deduced that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq142_HTML.gif and consequently, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq143_HTML.gif .

This completes the proof.

Remark 3.5 gives us the following theorem which is a better result than that [6, Theorem  3.3] because the solution of our problem (1.1) is positive in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq144_HTML.gif and strictly increasing.

Theorem 3.6.

Under assumptions of Theorem 3.1, our problem (1.1) has a unique nonnegative and strictly increasing solution.

Proof.

By Theorem 3.1 we obtain that the problem (1.1) has an unique solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq145_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq146_HTML.gif . Now, we will prove that this solution is a strictly increasing function. Let us take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq147_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq148_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ32_HTML.gif
(3.19)

Taking into account Remark 3.4 and the fact that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq149_HTML.gif , we get https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq150_HTML.gif .

Now, if we suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq151_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq152_HTML.gif and as, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq153_HTML.gif we deduce that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq154_HTML.gif a.e.

On the other hand, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq155_HTML.gif a.e. then

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ33_HTML.gif
(3.20)
Now, as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq156_HTML.gif , then for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq157_HTML.gif there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq158_HTML.gif such that for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq159_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq160_HTML.gif we get https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq161_HTML.gif . Observe that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq162_HTML.gif , consequently,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ34_HTML.gif
(3.21)

and this contradicts that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq163_HTML.gif a.e.

Thus, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq164_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq165_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq166_HTML.gif . Finally, as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq167_HTML.gif we have that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq168_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq169_HTML.gif .

Declarations

Acknowledgment

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

Authors’ Affiliations

(1)
Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria

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Copyright

© J. Caballero Mena et al. 2009

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