INVOLVING RIEMANN LIOUVILLE FRACTIONAL DERIVATIVE

. This paper is devoted to studying the multiple positive solutions for a system of nonlinear fractional boundary value problems. Our analysis is based upon the Avery Peterson (cid:133)xed point theorem. In addition, we include an example for the demonstration of our main result.

At the same time, boundary value problems for integer order di¤erential systems are widely studied, despite fractional di¤erential systems have emerged as a signi…cant …eld of investigation quite recently. Thus intensive study of the existence theory of fractional systems has been carried out by means of methods of nonlinear analysis such as …xed point theory, lower and upper solutions, monotone iterative methods, see [11,13,14,6,7,8,5,10,22,23,24] and the references therein.
In this paper, we discuss the multiple positive solutions for the following systems of nonlinear fractional di¤erential equations : 0 v(s)dA 2 (s)); (4) in which D is the Riemann-Liouville fractional derivative, 2 < q i 3 and 0 < p i 1; 0 < q i p i 1 for i = 1; 2; 0 < < 1; 2 (0; 1); qi pi 1 < 1, Motivated by the above papers, our goal is to obtain the existence of multiple positive solutions for the fractional di¤erential system (1)-(4). Here, we employ Riemann-Stieltjes integral boundary conditions. As they include multi-point and integral conditions as special cases, the system (1)-(4) is more general than the problems mentioned in some literature. Applying the Avery Peterson …xed point theorem, multiple positive solutions are established. An example is also presented to illustrate our main result.
In order to present our main result, we will make use of the following concepts and the Avery Peterson …xed point theorem.

ANALYSIS OF FRACTIONAL DIFFERENTIAL SYSTEM S 1347
Let ' and be nonnegative continuous convex functionals on the cone P, be a nonnegative continuous concave functional on P, and be a nonnegative continuous functional on P. Then, for positive numbers a,b,c,d we de…ne the following sets: Let P be a cone in a real Banach space E. and ', , , be de…ned as above, furthermore holds (kx) k (x) for 0 k 1 such that, for some positive numbers M and d, Then, T has at least three …xed points

Existence Results
During the last decade, many de…nitions on the fractional calculus have been carried out. In our paper, our work is based upon the Riemann Liouville fractional operator de…ned by where g : (0; 1) ! R is a function, n is the smallest integer greater than or equal to whenever the right hand side is de…ned. In particular, for = n, D g(t) = D n g(t): In order to derive the main result of the system (1)-(4), we present the following lemma: with the boundary conditions (3) and (4) has the solution where and i = 1 qi pi 1 , (i 2 f1; 2g).