The field of fractional differential equations has been subjected to an intensive development of the theory and the applications (see, for example, [1–6] and the references therein). It should be noted that most of papers and books on fractional calculus are devoted to the solvability of linear initial fractional differential equations on terms of special functions. There are some papers dealing with the existence of solutions of nonlinear initial value problems of fractional differential equations by using the techniques of nonlinear analysis such as fixed point results, the Leray-Schauder theorem, stability, etc. (see, for example, [7–19] and the references therein). In fact, fractional differential equations arise in many engineering and scientific disciplines such as physics, chemistry, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena and aerodynamics (see, for example, [20–23] and the references therein). The main advantage of using the fractional nonlinear differential equations is related to the fact that we can describe the dynamics of complex non-local systems with memory. In this line of taught, the equations involving various fractional orders are important from both theoretical and applied view points. We need the following notions.
Definition 1.1 ([1, 4])
For a continuous function
, the Caputo derivative of fractional order α
is defined by
where , and denotes the integer part of α.
Definition 1.2 ([1, 4])
The Riemann-Liouville fractional derivative of order α
for a continuous function f
is defined by
where the right-hand side is pointwise defined on .
Definition 1.3 ([1, 4])
be an interval in ℝ and
. The Riemann-Liouville fractional order integral of a function
is defined by
whenever the integral exists.
Suppose that E is a Banach space which is partially ordered by a cone , that is, if and only if . We denote the zero element of E by θ. A cone P is called normal if there exists a constant such that implies (see ). Also, we define the order interval for all . We say that an operator is increasing whenever implies . Also, means that there exist and such that (see ). Finally, put for all . It is easy to see that is convex and for all . We recall the following in our results. Let E be a real Banach space and let P be a cone in E. Let be an interval and let τ and φ be two positive-valued functions such that for all and is a surjection. We say that an operator is τ-φ-concave whenever for all and . We say that A is φ-concave whenever for all t . We recall the following result.
Theorem 1.1 ()
Let E be a Banach space, let P be a normal cone in E, and let be an increasing and τ-φ-concave operator. Suppose that there exists such that . Then there are and such that and , the operator A has a unique fixed point , and for and the sequence with , we have .