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
Contents lists available at HCTM, Technical Campus
Pure and Applied Mathematics Letters Journal homepage: www.pamletters.org
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 π‘
π
π·0+ π’ π‘ = π π‘, π’, 1
πΌπ’ 0 + π½π’β² 0 = 0
π π , π’ π ππ , 0 < π‘ < 1,1 < π < 2, 0
π1 π’ π ππ , πΌπ’ 1 + π½π’β² 1 =
1 0
π2 π’ π ππ .
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
π’β²β² π‘ = π π‘, π’ π‘ , 0 < π‘ < 1, 1
π’β² 0 = 0, π’ 1 =
π π , π’ π ππ , 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 π
π·0+ π’ π‘ = π π‘, π’ π‘ , 0 < π‘ < 1,1 < π < 2 1
π’ 0 = 0, π’ 0 β πΌπ’ 1 =
π π , π’ π ππ .
β²
0 π
π·0+ π’ π‘ = π π‘, π’ π‘ , 0 < π‘ < 1,1 < π < 2 1
π
π’β² 0 = 0, π’ 0 β πΌπ’ 1 =
π π , π’ π ππ .
;
0 π β, π·0+
where π: [0,1] Γ β β β is a given function,πΌ β denotes the Caputo's fractional derivative of order π. 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 πΌ of a function π is defined by πΌ0πΌ+ π π‘ =
1 π€(πΌ)
π‘ π
π(π ) ππ . (π‘ β π )1βπΌ
Definition 2.2 The Caputo fractional derivative of order π of g is defined by π
π·0+ π π‘ =
1 ; π€(π β π)
where n= [q] +1 ([q] is the entire part of q).
Lemma 2.1 For q > 0, the homogenous fractional differential equation β― ππ π‘ πβ1 , where, ππ β β; π = 0, . . . , π and π = [π] + 1. π
π
π
π
π·0+ π π‘ = 0 has a solution π π‘ = π1 + π2 π‘ +
π+π
π
π
Lemma 2.2 Let π, π β₯ 0, π β πΏ1 π, π . Then πΌ0π+ πΌ0+ π π‘ = πΌ0+ πΌ0+ π π‘ = πΌ0+ π π‘ and π·0+ πΌ0+ π π‘ = π π‘ , πππ πππ π‘ β π, π . Define the functional space πΈ as the Banach space of all function π’ β πΆ 0,1 into β with the norm πππ₯π‘β 0,1 π’ π‘ .
, where
π’ =
Now assume that Ξ± β 1 and consider the nonlocal boundary value problem for the linear non-homogeneous equation π
π·0+ π’ π‘ = π¦ π‘ ,
1
π’β² 0 = 0, π’ 0 β πΌπ’ 1 =
π π ππ . 0
Using Lemma 2.1, we get π
π’ π‘ = πΌ0+ π¦ π‘ + π + ππ‘. From the nonlocal boundary conditions, it yields π
π’ π‘ = πΌ0+ π¦ π‘ +
πΌ π 1 πΌ0+ π¦ 1 + 1βπΌ 1βπΌ
1
π π ππ . 0
Therefore π’ π‘ =
1 π€(π)
π‘
π‘βπ 0
πβ1
π¦ π ππ +
πΌ 1βπΌ
1
1βπ 0
πβ1
π¦ π ππ +
1 1βπΌ
1
π π ππ , 0
2
that can be written as π’ π‘ =
1 π€(π)
1
πΊ(π‘, π )π¦ π ππ + 0
1 1βπΌ
1
π π ππ , 0
where π‘βπ
πΊ π‘, π =
πβ1
+
πΌ 1βπ 1βπΌ
πΌ 1βπ 1βπΌ
πβ1
πβ1
,0 β€ π β€ π‘ β€ 1
, 0 β€ π‘ β€ π β€ 1.
Define the integral operator π: πΈ β πΈ by ππ’ π‘ =
1 π€(π)
1
πΊ(π‘, π )π π , π’(π ) ππ + 0
1 1βπΌ
1
π π , π’(π ) ππ 0
. Lemma 2.3 The function π’ β πΈ is a solution of the boundary value problem (P) if and only if ππ’ π‘ = π’ π‘ for all π‘ β 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 π’ is called positive solution of problem (P) if π’ π‘ β₯ 0, β π‘ β 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 πΌπ₯ β π for all π₯ β π and πΌ β β+ π₯, βπ₯ β π imply π₯ = 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 < πΌ < 1: (i)
πΊ π‘, π β β 0,1 Γ 0,1 , πΊ π‘, π β₯ 0 πππ π‘, π β 0,1
(ii)
πΌπΎ π β€ πΊ π‘, π β€ πΎ π for all π‘, π β 0,1 , where πΎ π =
(1βπ )π β1 . 1βπΌ
Proof From the expression of πΊ we easily show that πΊ is continuous and positive for π‘, π β 0,1 . The statement (ii) is proved by using the fact that π‘ β€ 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 < πΌ < 1, then πππtβ 0,1 u t β₯ Ξ± u . From here we can choose the cone K asπΎ = π’ β πΈ, π’ β₯ 0, πππtβ 0,1 u(t) β₯ Ξ± u . Now we state the assumptions that will be used to prove the existence of positive solutions: π»1 π»2
π π‘, π’ π‘ 1 0
= π π‘ π1 π’ π‘
and π π‘, π’ π‘
= π π‘ π1 π’ π‘ , where π, π β πΏ1 0,1 , β+ and π1 , π1 β β β+ , β+ .
1 β s q a s ds > 0, there exists a constante πΏ > 0, such that π1 π₯ β€ πΏπ₯, βπ₯ β β+ and π
πΏ1