Positive Solution of Fractional Differential Equationswith Integral Boundary Condition

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Lilia ZENKOUFI 
 Department of Mathematics. Faculty of Sciences
University 8 may 1945 Guelma, Algeria
Laboratory of Applied Mathematics and Modeling “LAMM”
e-mail: zenkoufi@yahoo.fr

Abstract: This present work concerns the study of a class of nonlinear fractional differential
equations with an integral condition, by the help of some fixed point theorems. We establish the
uniqueness result by the Banach contraction principle and to prove the existence of positive
solution we use a cone fixed point theorem due to Guo-Krasnoselskii by introducing height
functions of the nonlinear term on some bounded sets and considering integrations of these height
functions. Two examples are also included to illustrate our results.
Keywords: Cone, fractional differential equations, fixed-point theorem, Integral condition,
Riemann-Liouville fractional derivative.
Mathematics Subject Classifications: 34B10, 34B15.

Introduction
Fractional derivatives provide an excellent instrument for the description of
memory and hereditary properties of various materials and processes. Fractional
boundary value problems have been widely studied in the last decades and many
monographs and books are devoted to this subject we refer to [1,4,5,7-13,15-19,21,23
24] and their references.
Fractional and ordinary boundary value problems with integral conditions have
been investigated by many authors see   ,… 20 , 14 ,6,3,2 . The history of fractional
calculus can be traced back to the 17th century, when the German mathematician
Gottfried Leibniz first mentioned the concept of fractional differentiation in a letter
to his colleague Johann Bernoulli. However, the development of fractional calculus
as a field of study actually began in the 19th century, with the work of several
mathematicians, including Augustin-Louis Cauchy, Liouville, and Riemann. In the
early 20th century, the French mathematician Paul Lévy used fractional calculus to
model random processes, and it was subsequently used in the study of fractals and
other areas of mathematics. We can cite the paper where Wenxia Wang
studied the following fractional integral boundary value problem (BVP) with a

so, the contraction principle ensures the uniqueness of a solution for the fractional
boundary value problem  .1.1 This finishes the proof.

Let  ,1,0C E so that E is a Banach space endowed with the norm . max
1 0
tu u
t

The main result of this section is the following well-known Guo-Krasnosel’skii
fixed point theorem on cone.
Theorem 10:  9 Let E be a Banach space, and let ,E K be a cone. Assume
2 1
, are open subsets of E with , 0 1
 ,2 1
 and let
  , \ : 1 2
K K   A
be a completely continuous operator. In addition suppose either
i ,u u A ,1
 K u and ,u u A ;2
 K u or
  ,u u ii  A ,1
 K u and ,u u A ,2
 K u
holds. Then A has a fixed point in  . \ 1 2
 K
Definition 11: A function tu is called positive solution for the boundary value
problem  1.1 if  ,0 tu  1,0t .
Lemma 12: Let E u , then the solution u of the fractional boundary value
problem  1.1 is nonnegative and satisfies
 
 . min
1, E t
u tu 




Hence
                . , 1 , 1
1
1
0 1
1
0
ds susg s G ds sus f s G uE
        
On the other hand, for all  ,1,t we obtain
                 . , 1 , 1
1
1
0 1
1
0
1  


 


       ds susg sG ds sus f sG tu    
Therefore, we have
 
 . min
1, E t
u tu 




The proof is complete.

Let K be the cone of nonnegative function in  1,0C with the following form
    .1, , ,       t u tu E u K E

The height functions  r t, and  r t, satisfy the following inequality:
   , 2
1 : 3
1 max , 3
1
6
5
2
7 5 5    
  
       r t r r u r t t u r t

   , 2
1 : 3
1 min , 3
1
2
25
2
7 5 5   
  
       r r t r u r t t u r t
and ) , ( u t g is nondecreasing on u for any ].1,0[t
It follows that
              .1 , 1 1, 1
1
1
0 1
1
0
      ds susg s G ds s s G    

And
            . 100
1 , 1
125
1 , 1
1
1
0 1
1
0
    
  

   ds susg s G ds s s G    

By Theorem14, we get that  2
P has at least one strictly increasing positive solution
K u such that .1 125
1   u

Conclusion
In this paper, we considered a fractional differential equation involving Riemann
Liouville fractional derivative of order , 3 2   . We studied the uniqueness of
solution by using the Banach contraction principle and, we established the
existence of positive solution by Guo-Krasnosel’skii fixed point theorem by
introducing height functions of the nonlinear term on some bounded sets and
considering integrations of these height functions. As application, examples are
presented to illustrate the main results.

Author:
Lilia ZENKOUFI
Department of Mathematics. Faculty of Sciences
University 8 may 1945 Guelma, Algeria
Laboratory of Applied Mathematics and Modeling “LAMM”
E-mail: zenkoufi@yahoo.fr

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