pure and applied mathematics letters

P A M L December 2014

PURE AND APPLIED MATHEMATICS LETTERS An International Journal

Fixed point theorem applied to a fractional boundary value problem by Assia Guezane-Lakoud, Allaberen Ashyralyeval

HCTM Technical Campus P A M L December 2 0 1 4

Pure and Applied Mathematics Letters 2(2014)1-6

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ISSN 2349-4956

Fixed point theorem applied to a fractional boundary value problem Assia Guezane-Lakoud1, Allaberen Ashyralyev2 1

Departement of Mathematics, Faculty of Sciences, Badji Mokhtar-Annaba University, P.O. Box 12, 23000 Annaba, Algeria

2

Departe Department of Mathematics, Fatih University 34500Buyucekmece, Turkey, ITTU, Ashgabat, Turkmenistan

Abstract This paper concerns the existence of positive of solutions for a class of fractional boundary value problems with an integral condition. To prove our results, we use a cone fixed point theorem due to Guo-Krasnoselskii. Keywords: Integral condition, Fractional Caputo derivative, Positive solution, Fixed point theorem. AMS subject classification: 34B10, 26A33, 34B15.

Article Information: Received 08.06.2014

Revised 11.07.2014

Accepted 11.07.2014

Manuscript No. PAML-0801062014

Communicated by Prof. V. K. Kaushik

1. Introduction and Background Fractional boundary value problems have been widely studied in the last decades and many monographs and books are devoted to this subject see [1-11, 14, 15, 17-20, 24, 27, 28, 31, 33], this is due to their numerous advantages such description of memory and hereditary properties of various materials and processes. Many phenomena in viscoelasticity, electrochemistry, electromagnetism, control theory, etc., can be modelled as fractional differential equations. We refer the reader to [12, 24, 29, 30, 32] and the references therein for some their applications. Fractional and ordinary boundary value problems with integral conditions have been investigated by many authors [3,9,1517,19,35,36], we can cite the paper [3], where Ahmad et al. studied a Caputo fractional boundary value problem involving integro-differential equation with integral conditions 𝑡

𝑞

𝐷0+ 𝑢 𝑡 = 𝑓 𝑡, 𝑢, 1

𝛼𝑢 0 + 𝛽𝑢′ 0 = 0

𝑔 𝑠, 𝑢 𝑠 𝑑𝑠 , 0 < 𝑡 < 1,1 < 𝑞 < 2, 0

𝑞1 𝑢 𝑠 𝑑𝑠, 𝛼𝑢 1 + 𝛽𝑢′ 1 =

1 0

𝑞2 𝑢 𝑠 𝑑𝑠.

Under some conditions on the given terms, the authors investigated the existence and nonexistence of positive solutions with the application of some fixed point theorems. Note that many problems in natural sciences require a notion of positivity, only non-negative densities, population sizes or probability make sense in real life. Integral conditions come up when values of the function on the boundary are connected to its values inside the domain or when direct measurements on the boundary are not possible. The presence of an integral term in the boundary condition leads to great difficulties. In the interesting paper [16], Infante considered the following nonlinear ordinary boundary value problem ___________________ Corresponding author: Assia Guezane-Lakoud, Mob. No. , E-mail addresses: [email protected]

1

𝑢′′ 𝑡 = 𝑓 𝑡, 𝑢 𝑡 , 0 < 𝑡 < 1, 1

𝑢′ 0 = 0, 𝑢 1 =

𝑔 𝑠, 𝑢 𝑠 𝑑𝑠, 0

and proved the existence of positive solutions by means of the well known fixed point index for compact maps. In present paper, we will investigate the existence of positivity of solution of the following fractional boundary value problem 𝑞

𝐷0+ 𝑢 𝑡 = 𝑓 𝑡, 𝑢 𝑡 , 0 < 𝑡 < 1,1 < 𝑞 < 2 1

𝑢 0 = 0, 𝑢 0 − 𝛼𝑢 1 =

𝑔 𝑠, 𝑢 𝑠 𝑑𝑠.

′

0 𝑞

𝐷0+ 𝑢 𝑡 = 𝑓 𝑡, 𝑢 𝑡 , 0 < 𝑡 < 1,1 < 𝑞 < 2 1

𝑃

𝑢′ 0 = 0, 𝑢 0 − 𝛼𝑢 1 =

𝑔 𝑠, 𝑢 𝑠 𝑑𝑠.

;

0 𝑞 ℝ, 𝐷0+

where 𝑓: [0,1] × ℝ → ℝ is a given function,𝛼 ∈ denotes the Caputo's fractional derivative of order 𝑞. To solve the problem (P), we apply a fixed point theorem due to Guo- Krasnoselskii. The organization of this paper is as follows. In section 2, we provide necessary background. Section 3, is devoted to the existence of positive solutions on a cone under some sufficient conditions on the nonlinear term. We achieve the paper with an example.

2. Background and Preliminaries We present some definitions from fractional calculus theory [24], which will be needed later. Definition 2.1 The Riemann-Liouville fractional integral of order 𝛼 of a function 𝑔 is defined by 𝐼0𝛼+ 𝑔 𝑡 =

1 𝛤(𝛼)

𝑡 𝑎

𝑔(𝑠) 𝑑𝑠. (𝑡 − 𝑠)1−𝛼

Definition 2.2 The Caputo fractional derivative of order 𝑞 of g is defined by 𝑞

𝐷0+ 𝑔 𝑡 =

1 ; 𝛤(𝑛 − 𝑞)

where n= [q] +1 ([q] is the entire part of q).

Lemma 2.1 For q > 0, the homogenous fractional differential equation ⋯ 𝑐𝑛 𝑡 𝑛−1 , where, 𝑐𝑖 ∈ ℝ; 𝑖 = 0, . . . , 𝑛 and 𝑛 = [𝑞] + 1. 𝑞

𝑞

𝑝

𝑞

𝐷0+ 𝑔 𝑡 = 0 has a solution 𝑔 𝑡 = 𝑐1 + 𝑐2 𝑡 +

𝑝+𝑞

𝑞

𝑞

Lemma 2.2 Let 𝑝, 𝑞 ≥ 0, 𝑓 ∈ 𝐿1 𝑎, 𝑏 . Then 𝐼0𝑃+ 𝐼0+ 𝑓 𝑡 = 𝐼0+ 𝐼0+ 𝑓 𝑡 = 𝐼0+ 𝑓 𝑡 and 𝐷0+ 𝐼0+ 𝑓 𝑡 = 𝑓 𝑡 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 ∈ 𝑎, 𝑏 . Define the functional space 𝐸 as the Banach space of all function 𝑢 ∈ 𝐶 0,1 into ℝ with the norm 𝑚𝑎𝑥𝑡∈ 0,1 𝑢 𝑡 .

, where

𝑢 =

Now assume that α ≠ 1 and consider the nonlocal boundary value problem for the linear non-homogeneous equation 𝑞

𝐷0+ 𝑢 𝑡 = 𝑦 𝑡 ,

1

𝑢′ 0 = 0, 𝑢 0 − 𝛼𝑢 1 =

𝑔 𝑠 𝑑𝑠. 0

Using Lemma 2.1, we get 𝑞

𝑢 𝑡 = 𝐼0+ 𝑦 𝑡 + 𝑎 + 𝑏𝑡. From the nonlocal boundary conditions, it yields 𝑞

𝑢 𝑡 = 𝐼0+ 𝑦 𝑡 +

𝛼 𝑞 1 𝐼0+ 𝑦 1 + 1−𝛼 1−𝛼

1

𝑔 𝑠 𝑑𝑠. 0

Therefore 𝑢 𝑡 =

1 𝛤(𝑞)

𝑡

𝑡−𝑠 0

𝑞−1

𝑦 𝑠 𝑑𝑠 +

𝛼 1−𝛼

1

1−𝑠 0

𝑞−1

𝑦 𝑠 𝑑𝑠 +

1 1−𝛼

1

𝑔 𝑠 𝑑𝑠 , 0

2

that can be written as 𝑢 𝑡 =

1 𝛤(𝑞)

1

𝐺(𝑡, 𝑠)𝑦 𝑠 𝑑𝑠 + 0

1 1−𝛼

1

𝑔 𝑠 𝑑𝑠 , 0

where 𝑡−𝑠

𝐺 𝑡, 𝑠 =

𝑞−1

+

𝛼 1−𝑠 1−𝛼

𝛼 1−𝑠 1−𝛼

𝑞−1

𝑞−1

,0 ≤ 𝑠 ≤ 𝑡 ≤ 1

, 0 ≤ 𝑡 ≤ 𝑠 ≤ 1.

Define the integral operator 𝑇: 𝐸 → 𝐸 by 𝑇𝑢 𝑡 =

1 𝛤(𝑞)

1

𝐺(𝑡, 𝑠)𝑓 𝑠, 𝑢(𝑠) 𝑑𝑠 + 0

1 1−𝛼

1

𝑔 𝑠, 𝑢(𝑠) 𝑑𝑠 0

. Lemma 2.3 The function 𝑢 ∈ 𝐸 is a solution of the boundary value problem (P) if and only if 𝑇𝑢 𝑡 = 𝑢 𝑡 for all 𝑡 ∈ 0,1 . From here we see that to solve the problem (P) it remains to prove that the map T has a fixed point in E.

3. Existence of positive solutions Definition 3.1 A function 𝑢 is called positive solution of problem (P) if 𝑢 𝑡 ≥ 0, ∀ 𝑡 ∈ 0,1 and it satisfies the equation and the boundary conditions in (P). To study the existence of positive solution of problem (P), first, we will introduce a positive cone constituting of continuous positive functions or some suitable subset of it. Second, we will impose suitable assumptions on the nonlinear terms such that the hypotheses of the cone theorem are satisfied. Third, we will apply a fixed point theorem to conclude the existence of a positive solution in the annular region. We give the definition of a cone

Definition 3.2 A nonempty subset P of a Banach space E is called a cone if P is convex, closed and satisfies the conditions 𝛼𝑥 ∈ 𝑃 for all 𝑥 ∈ 𝑃 and 𝛼 ∈ ℝ+ 𝑥, −𝑥 ∈ 𝑃 imply 𝑥 = 0.

(i) (ii)

To define a cone for our study we need the properties of function G. Lemma 3.1 We have the following two properties in the case0 < 𝛼 < 1: (i)

𝐺 𝑡, 𝑠 ∈ ∁ 0,1 × 0,1 , 𝐺 𝑡, 𝑠 ≥ 0 𝑓𝑜𝑟 𝑡, 𝑠 ∈ 0,1

(ii)

𝛼𝛾 𝑠 ≤ 𝐺 𝑡, 𝑠 ≤ 𝛾 𝑠 for all 𝑡, 𝑠 ∈ 0,1 , where 𝛾 𝑠 =

(1−𝑠)𝑞 −1 . 1−𝛼

Proof From the expression of 𝐺 we easily show that 𝐺 is continuous and positive for 𝑡, 𝑠 ∈ 0,1 . The statement (ii) is proved by using the fact that 𝑡 ≤ 1. The following result can be easily proved by using Lemma 3.1. Lemma 3.2 If u is solution of the problem (P) and0 < 𝛼 < 1, then 𝑚𝑖𝑛t∈ 0,1 u t ≥ α u . From here we can choose the cone K as𝐾 = 𝑢 ∈ 𝐸, 𝑢 ≥ 0, 𝑚𝑖𝑛t∈ 0,1 u(t) ≥ α u . Now we state the assumptions that will be used to prove the existence of positive solutions: 𝐻1 𝐻2

𝑓 𝑡, 𝑢 𝑡 1 0

= 𝑎 𝑡 𝑓1 𝑢 𝑡

and 𝑔 𝑡, 𝑢 𝑡

= 𝑏 𝑡 𝑔1 𝑢 𝑡 , where 𝑎, 𝑏 ∈ 𝐿1 0,1 , ℝ+ and 𝑓1 , 𝑔1 ∈ ∁ ℝ+ , ℝ+ .

1 − s q a s ds > 0, there exists a constante 𝛿 > 0, such that 𝑔1 𝑥 ≤ 𝛿𝑥, ∀𝑥 ∈ ℝ+ and 𝑏

𝐿1