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TWO POINT FRACTIONAL BOUNDARY VALUE PROBLEM AT RESONANCE
TWO POINT FRACTIONAL BOUNDARY VALUE PROBLEM AT RESONANCE
Journal of Applied Mathematics & Informatics. 2015. May, 33(3_4): 425-434
Copyright © 2015, Korean Society of Computational and Applied Mathematics
  • Received : August 20, 2014
  • Accepted : November 27, 2014
  • Published : May 30, 2015
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About the Authors
A. GUEZANE-LAKOUD
S. KOUACHI
F. ELLAGGOUNE

Abstract
In this paper, a two-point fractional boundary value problem at resonance is considered. By using the coincidence degree theory some existence results of solutions are established. AMS Mathematics Subject Classification : 34B15.
Keywords
1. Introduction
In this paper, we investigate the existence of solutions for the following twopoint fractional boundary value problem (BVP) at resonance
PPT Slide
Lager Image
where f : [0, 1] × IR 3 → IR is continuous, 2 < α < 3 and
PPT Slide
Lager Image
denotes the Caputo’s fractional derivative. The two-point boundary value problem (1.1) happens to be at resonance in the sense that its associated linear homogeneous boundary value problem
PPT Slide
Lager Image
has a nontrivial solution u ( t ) = ct 2 , c ∈ IR. To solve this problem, we use the coincidence degree of Mawhin [12 , 13] . This method is based on an equivalent formulation in an abstract space and a theory of topological degree. This formulation generally leads to an abstract operator of the form N + L , where L is a Fredholm operator of index zero and N is generally a nonlinear operator having some compactness properties with respect to L . Fractional differential equations arise in different areas of sciences such as in rheology, fluid flows, viscoelasticity, chemical physics, and so on [1 , 6 , 8 , 11 , 14 - 19] . Recently, boundary value problems for fractional differential equations at nonresonance have been studied by many authors [1 , 3 , 4 , 10] by using fixed point theorems, lower and upper solution. Moreover, boundary value problems for differential equations at resonance have also been studied in some papers, see [2 , 7] . In [20] , the authors studied, by using the coincidence degree theory, the following BVP of fractional equation at resonance
PPT Slide
Lager Image
where
PPT Slide
Lager Image
denotes the Caputo’s fractional differential operator of order α, 1 < α ≤ 2 and f : [0, 1] × IR 2 → IR is a continuous function.
A similar boundary value problem at resonance involving Riemann-Liouville fractional derivative is considered in [7] . The author solved the problem
PPT Slide
Lager Image
by applying degree theory theorem for coincidences.
By the same method, in [5] the authors established the existence of solutions for the following third-order differential equation
PPT Slide
Lager Image
where f : [0, 1] × IR 2 → IR is a Caratheodory function, and η ∈ (0, 1).
The organization of this paper is as follows. We present in Section 2 some notation and some basic results involved in the reformulation of the problem. In section 3, we give the main theorem and some lemmas, then we will see that the proof of the main theorem is an immediate consequence of these lemmas and the coincidence degree of Mawhin. At the end of this section, we give an example to illustrate the main result.
2. Preliminaries
We begin by introducing the fundamental tools of fractional calculus and the coincidence degree theory which will be used throughout this paper.
Definition 2.1. The Riemann-Liouville fractional integral is defined by
PPT Slide
Lager Image
for g C ([ a , b ]) and α > 0.
Definition 2.2. Let f Cn ([ a , b ]), the Caputo fractional derivative of order α ≥ 0 of f is defined by
PPT Slide
Lager Image
, where n = [ α ] + 1 ([ α ] is the integer part of α ).
Lemma 2.3. For α > 0, g C ([0, 1], IR) , the homogenous fractional differential equation
PPT Slide
Lager Image
has a solution g ( t ) = c 0 + c 1 t + c 2 t 2 +…+ c n−1 t n−1 , where ci ∈ IR , i = 0,…, n −1, here n is the smallest integer greater than or equal to α .
Definition 2.4. Let X and Y be real normed spaces. A linear mapping L : domL X Y is called a Fredholm mapping if the following two conditions hold:
(i) kerL has a finite dimension, and
(ii) ImL is closed and has a finite codimension.
If L is a Fredholm mapping, its index is the integer IndL = dimkerL codimImL .
From here if L Fredholm mapping of index zero then there exist continuous projectors P : X X and Q : Y Y such that ImP = KerL , KerQ = ImL , X = KerL KerP , Y = ImL ImQ and that the mapping L | domLKerP : domL KerP ImL is one-to-one and onto. We denote its inverse by KP . Moreover, since dimImQ = codimImL , there exists an isomorphism J : ImQ KerL .
If Ω is an open bounded subset of X , and
PPT Slide
Lager Image
, the map N : X Y will be called L−compact on
PPT Slide
Lager Image
if QN (
PPT Slide
Lager Image
) is bounded and KP ( I Q ) N :
PPT Slide
Lager Image
X is compact.
Lemma 2.5 ( [13] ). Let L : domL X Y be a Fredholm operator of index zero and N : X Y , L compact on
PPT Slide
Lager Image
. Assume that the following conditions are satisfied
(1) Lu Nu for every ( u , λ ) ∈ [( domL KerL )] ∩ ∂Ω × (0, 1);
(2) Nx ImL for every x KerL ∩ ∂Ω;
(3) deg ( JQN | KerL , KerL ∩ Ω, 0) ≠ 0, where Q : Y Y is a projection such that ImL = KerQ .
Then the equation Lx = Nx has at least one solution in domL ∩
PPT Slide
Lager Image
.
In this paper, we denote by X and Y the Banach spaces: X = C 2 ([0, 1], IR) equipped with the norm ║ u X = max {║ u ; | u , ║ u′′ ║∞} and Y = C ([0, 1], IR) equipped with the norm ║ y Y = ║ y , where ║ u = maxt ∈[0,1] | u ( t )|.
Define the operator L : domL X Y by
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Let N : X Y be the operator
PPT Slide
Lager Image
Then BVP (1.1) is equivalent to the operator equation
PPT Slide
Lager Image
It is known [12] that the coincidence equation Lu = Nu is equivalent to
PPT Slide
Lager Image
3. Main results
We can now state our result on the existence of a solution for the BVP (1.1).
Theorem 3.1. Let f : [0, 1] × IR 3 → IR be continuous. Assume that
( H 1 ) there exists nonnegative functions p , q , r , z C [0, 1] with Γ( α −1)−2 q1 −2 r1 − 2 z1 > 0 such that
| f ( t , u , v , w )| ≤ p ( t ) + q ( t )| u | + r ( t )| v | + z ( t )| w |, ∀ t ∈ [0, 1], ( u , v , w ) ∈ IR 3
where
PPT Slide
Lager Image
( H 2 ) there exists a constant B > 0 such that for all c ∈ IR with |2 c | > B either
PPT Slide
Lager Image
or
PPT Slide
Lager Image
Then BVP(1.1) has at least one solution in X .
In order to prove Theorem 3.1, we need to prove some lemmas below.
Lemma 3.2. Let L be defined by (2.1), then
PPT Slide
Lager Image
PPT Slide
Lager Image
Proof . By Lemma 2.3,
PPT Slide
Lager Image
has solution
PPT Slide
Lager Image
Combining with boundary condition of BVP (1.1), (3.1) holds. For y ImL , there exists u domL such that y = Lu Y . We have
PPT Slide
Lager Image
Differentiating two times and using the boundary conditions for BVP (1.1), we get
PPT Slide
Lager Image
On the other hand, let y Y satisfying
PPT Slide
Lager Image
, so y ImL . □
Lemma 3.3. If L be defined by (2.1), then L is a Fredholm operator of index zero and the linear continuous projector operators P : X X and Q : Y Y can be defined as
PPT Slide
Lager Image
Furthermore, the operator Kp : ImL domL KerP can be written as
PPT Slide
Lager Image
Proof . Obviously, ImP = KerL and P 2 u = Pu . It follows from u = ( u Pu ) + Pu that X = KerP + KerL . By simple calculation, we get that KerL KerP = 0. Hence
PPT Slide
Lager Image
For y Y , it yields
PPT Slide
Lager Image
Let y = ( y Qy )+ Qy , where y Qy KerQ = ImL , Qy ImQ . It follows from KerQ = ImL and Q 2 y = Qy that ImQ ImL = 0. Then, we have
PPT Slide
Lager Image
Thus
PPT Slide
Lager Image
This means that L is a Fredholm operator of index zero. From the definitions of P and KP , it is easy to see that the generalized inverse of L is KP . In fact, for y ImL , we have
PPT Slide
Lager Image
Moreover, for u domL KerP , we get u (0) = u (0) = u′′ (0) = 0. By Lemma 2.3, we obtain that
PPT Slide
Lager Image
This together with u (0) = u (0) = u′′ (0) = 0 yields
PPT Slide
Lager Image
Combining (3.3) with (3.4), we get KP = ( L | domLKerP ) −1 . □
Lemma 3.4. If Ω ⊂ X is an open bounded subset such that
PPT Slide
Lager Image
, then N is L − compact on
PPT Slide
Lager Image
.
Proof . By the continuity of f , we conclude that QN (
PPT Slide
Lager Image
) and KP ( I Q ) N (
PPT Slide
Lager Image
) are bounded. In view of Arzelà-Ascoli theorem, we need only to prove that KP ( I Q ) N (
PPT Slide
Lager Image
) ⊂ X is equicontinuous. The continuity of f implies that exists a constant A > 0 such that |( I Q ) Nu | ≤ A , ∀ u
PPT Slide
Lager Image
, ∀ t ∈ [0, 1]. Denote KP,Q = KP ( I Q ) N and let 0 ≤ t 1 t 2 ≤ 1, u ∈ (
PPT Slide
Lager Image
), then
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Since tα , t α−1 and t α−2 are uniformly continuous on [0, 1], we get that
KP,Q (
PPT Slide
Lager Image
), ( KP,Q ) (
PPT Slide
Lager Image
)( KP,Q ) ′′ (
PPT Slide
Lager Image
) ∈ C [0, 1] are equicontinuous. Hence KP,Q :
PPT Slide
Lager Image
X is compact. □
Lemma 3.5. Suppose ( H 1 ) holds, then the set
PPT Slide
Lager Image
is bounded .
Proof . Take u ∈ Ω 1 , then Nu ImL . By (3.2), it yields
PPT Slide
Lager Image
By the integral mean value theorem, we conclude that there exists a constant h ∈ (0, 1) such that f ( h , u ( h ), u ( h ), u′′ ( h )) = 0. Then from ( H 2 ), we have | u′′ ( h )| ≤ B .
Since u domL , then u (0) = u (0) = 0. Therefore
PPT Slide
Lager Image
and
PPT Slide
Lager Image
That is
PPT Slide
Lager Image
By Lu = λNu and u domL , we have
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Take t = h , we get
PPT Slide
Lager Image
Together with | u′′ ( h )| ≤ B , ( H 1 ) and (3.5), we have
PPT Slide
Lager Image
Then we have
PPT Slide
Lager Image
Since Γ(α − 1) − 2 q1 − 2 r1 − 2 z1 > 0, we obtain
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Therefore, ║ u X M , consequently Ω 1 is bounded. □
Lemma 3.6. Suppose ( H 2 ) holds, then the set
PPT Slide
Lager Image
is bounded .
Proof . For u ∈ Ω 2 , we have u ( t ) = ct 2 , c ∈ IR, and Nu ImL . Then we get
PPT Slide
Lager Image
this together with ( H 2 ) implies |2 c | ≤ B . Thus, we have ║ u X B . Hence, Ω 2 is bounded. □
Lemma 3.7. Suppose the first part of ( H 2 ) holds, then the set
PPT Slide
Lager Image
is bounded. Where J : KerL ImQis an isomorphism defined by J ( ct 2 ) = ct 2 ; ∀ c ∈ IR; t ∈ [0, 1]:
Proof . For u ∈ Ω 3 , we have u ( t ) = ct 2 , c ∈ IR, and
PPT Slide
Lager Image
If λ = 0, then
PPT Slide
Lager Image
f ( s , cs 2 , 2 cs , 2 c ) ds = 0, thus |2 c | ≤ B in view of the first part of ( H 2 ). If λ ∈ (0, 1], we can also obtain |2 c | ≤ B . Otherwise, if |2 c | > B , in view of the first part of ( H 2 ), one has
PPT Slide
Lager Image
which contradicts (3.6). Therefore, Ω 3 is bounded. □
Lemma 3.8. Suppose the second part of ( H 2 ) hold, the set
PPT Slide
Lager Image
is bounded .
Proof . Using similar argument as in the proof of Lemma 3.7, we prove that Ω 3 is bounded. □
Now we are able to give the proof of Theorem 3.1, which is an immediate consequence of Lemmas 3.2-3.8 and Lemma 2.5.
Proof . Set Ω = { u X | ║ u X < max { M , B } + 1}. It follows from Lemma 3.2 and Lemma 3.3 that L is a Fredholm operator of index zero and N is L−compact on
PPT Slide
Lager Image
. By Lemma 3.4 and Lemma 3.5, we get that the following two conditions are satisfied
(1) Lu Nu for every ( u , λ ) ∈ [( domL KerL )] ∩ ∂Ω × (0, 1);
(2) Nu ImL for every u KerL ∩ ∂Ω;
Define H ( u , λ ) = λIdu + (1 − λ ) JQNu : According to Lemma 3.6 (or Lemma 3.7), we see that H ( u , λ ) ≠ 0 for u KerL ∩ ∂Ω. By the degree property of invariance under a homotopy, it yields
PPT Slide
Lager Image
So that, the condition (3) of Lemma 2.1 is satisfied. By Lemma 2.1, we can get that Lu = Nu has at least one solution in domL
PPT Slide
Lager Image
. Therefore by Lemma 2.5, the BVP (1.1) has at least one solution. The proof is complete. □
Example 3.9. Let us consider the following fractional boundary value problem
PPT Slide
Lager Image
We have f ( t , u , v , w ) = (
PPT Slide
Lager Image
, then
PPT Slide
Lager Image
where p ( t ) = 3 t , q ( t ) = 0,
PPT Slide
Lager Image
. We get that q 1 = 0,
PPT Slide
Lager Image
and
PPT Slide
Lager Image
If we choose B = 15 then for |2 c | > B it yields
PPT Slide
Lager Image
Then, the conditions of Theorem 3.1 are satisfied, so BVP(3.7) has at least one solution.
BIO
A. Guezane-Lakoud Her research interests are on differential equations and their applications.
Department of Mathematics, Faculty of Sciences, Badji Mokhtar-Annaba University, P.O. Box 12, 23000 Annaba, Algeria.
e-mail: a_guezane@yahoo.fr
S. Kouachi Her research interests are on differential equations and their applications.
Department of Mathematics, Faculty of Sciences, Badji Mokhtar-Annaba University, P.O. Box 12, 23000 Annaba, Algeria.
e-mail: sa.kouachi@gmail.com
F. Ellaggoune His research interests in difference equations and its applications.
Department of Mathematics, University 8 mai 1945 - Guelma P.O. Box 401, Guelma 24000, Algeria.
e-mail: fellaggoune@gmail.com
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