On a nonlinear fractional differential equation on partially ordered metric spaces
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RESEARCH
Open Access
On a nonlinear fractional differential equation on partially ordered metric spaces Dumitru Baleanu1,2,3* , Hakimeh Mohammadi4 and Shahram Rezapour4 *
Correspondence: [email protected] 1 Department of Mathematics, Cankaya University, Ogretmenler Cad. 14 06530, Balgat, Ankara, Turkey 2 Institute of Space Sciences, Magurele-Bucharest, Romania Full list of author information is available at the end of the article
Abstract In this paper, by using a fixed point result on ordered metric spaces, we prove the existence and uniqueness of a solution of the nonlinear fractional differential equation Dα u(t) = f (t, u(t)) (t ∈ I = [0, T], 0 < α < 1) via the periodic boundary condition u(0) = 0, where T > 0 and f : I × R → R is a continuous increasing function and c Dα denotes the Caputo fractional derivative of order α . Also, we solve it by using the anti-periodic boundary conditions u(0) + u(T) = 0 with u(0) ≤ 0 and u(0) + μu(T) = 0 with u(0) ≤ 0 and μ > 0 separately.
1 Introduction Fractional calculus started to be used intensively as an important tool in several branches of science and engineering (see, for example, [–] and the references therein). This type of calculus has an important impact in describing the dynamics of complex phenomena [–, ]. During the last few years, some new experimental confirmations have appeared in the literature in addition to the ones already established in chemistry, engineering, biology, physics, etc. As a result, the fractional differential equations were investigated intensively during the last few years. A special attention was devoted to the solvability of linear initial fractional differential equations on terms of special functions. On the other hand, the fixed point theory has wide applications in several fields (see Ref. [] and the references therein) and it is continuously developing. Having these above mentioned things in mind, in this manuscript we have the main aim to prove the existence and uniqueness of a nonlinear fractional differential equation of Caputo type with the help of results obtained within ordered metric spaces. The paper is organized as follows. After the introductory part, in the second section, we present some of the basic tools needed in the rest of the manuscript. The third section is devoted to the main result as well as to the illustrative examples. Finally, the manuscript ends with our conclusions. 2 Basic tools Recall that for a continuous function f : [, ∞) → R, the Caputo derivative of fractional order α is defined as follows. Definition . The Caputo fractional derivative of order α for a continuous function f is defined by c
Dα f (t) =
(n – α)
t
f (n) (s) ds, (t – s)α–n+
n = [α] + .
© 2013 Baleanu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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