Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces
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Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces Dumitru Baleanu1,2,3* , Ravi P Agarwal4 , Hakimeh Mohammadi5 and Shahram Rezapour5 *
Correspondence: [email protected] 1 Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia 2 Department of Mathematics, Cankaya University, Ogretmenler Cad. 14 06530, Balgat, Ankara, Turkey Full list of author information is available at the end of the article
Abstract By using fixed point results on cones, we study the existence and uniqueness of positive solutions for some nonlinear fractional differential equations via given boundary value problems. Examples are presented in order to illustrate the obtained results.
1 Introduction The field of fractional differential equations has been subjected to an intensive development of the theory and the applications (see, for example, [–] 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, [–] 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, [–] 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 . ([, ]) For a continuous function f : [, ∞) → R, the Caputo derivative of fractional order α is defined by c α
D f (t) =
(n – α)
t
(t – s)n–α– f (n) (s) ds,
where n – < α < n, n = [α] + and [α] denotes the integer part of 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.
Baleanu et al. Boundary Value Problems 2013, 2013:112 http://www.boundaryvalueproblems.com/content/2013/1/112
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Definition . ([, ]) The Riemann-Liouville fractional derivative of order α for a continuous function f is defined by n t f (s) d ds n = [α] + , D f (t) = α–n– (n – α) dt (t – s) α
where the right-hand side is pointwise
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