Abstract:In this paper we consider nonlinear boundary value problems for differential equations of fractional order α, 0 < α < 1. We study the existence and uniqueness of the solution and extend existing published results. In the last part of the paper we study a class of prototype methods to determine their numerical solution.MSC 2010 : Primary 65L05; Secondary 34A08, 26A33
“…[4,7,8]. It would be of interest to find out whether the conditions of our Nagumo-type result, Theorem 3.1, can be modified to prove a corresponding uniqueness theorem for this class of problems too.…”
Dedicated to the memory of my teacher, Professor Dr. Helmut BraßWe generalize the classical mean value theorem of differential calculus by allowing the use of a Caputo-type fractional derivative instead of the commonly used first-order derivative. Similarly, we generalize the classical mean value theorem for integrals by allowing the corresponding fractional integral, viz. the Riemann-Liouville operator, instead of a classical (firstorder) integral. As an application of the former result we then prove a uniqueness theorem for initial value problems involving Caputo-type fractional differential operators. This theorem generalizes the classical Nagumo theorem for first-order differential equations.MSC 2010 : Primary 26A33; Secondary 34A08, 34A12 Key Words and Phrases: Caputo derivative, mean value theorem of differential calculus, mean value theorem of integral calculus, fractional differential equation, Nagumo's theorem, uniqueness of solutions
“…[4,7,8]. It would be of interest to find out whether the conditions of our Nagumo-type result, Theorem 3.1, can be modified to prove a corresponding uniqueness theorem for this class of problems too.…”
Dedicated to the memory of my teacher, Professor Dr. Helmut BraßWe generalize the classical mean value theorem of differential calculus by allowing the use of a Caputo-type fractional derivative instead of the commonly used first-order derivative. Similarly, we generalize the classical mean value theorem for integrals by allowing the corresponding fractional integral, viz. the Riemann-Liouville operator, instead of a classical (firstorder) integral. As an application of the former result we then prove a uniqueness theorem for initial value problems involving Caputo-type fractional differential operators. This theorem generalizes the classical Nagumo theorem for first-order differential equations.MSC 2010 : Primary 26A33; Secondary 34A08, 34A12 Key Words and Phrases: Caputo derivative, mean value theorem of differential calculus, mean value theorem of integral calculus, fractional differential equation, Nagumo's theorem, uniqueness of solutions
“…[2,5,7,10,11]. A rigorous theory of FDE has been started quite recently, see for example -the books [4,9,11] and to mention only few papers related to ordinary FDE as [3,6,8]. Separately, there are many recent works related c 2012 Diogenes Co., Sofia pp.…”
In this paper we initiate the oscillation theory for fractional differential equations. Oscillation criteria are obtained for a class of nonlinear fractional differential equations of the formwhere D q a denotes the Riemann-Liouville differential operator of order q, 0 < q ≤ 1. The results are also stated when the Riemann-Liouville differential operator is replaced by Caputo's differential operator.MSC 2010 : Primary 34A08: Secondary 34C10, 26A33
“…For some recent works on the topic, see ( [1, 2, 3, 4, 5, 7, c 2012 Diogenes Co., Sofia pp. 362-382 , DOI: 10.2478/s13540-012-0027-y 8,9,10,11,12,13,14,15,16,17,18,21,22,23,24,27,31,35,36,37,38]) and references therein.…”
This paper studies a new class of boundary value problems of nonlinear fractional differential inclusions of order q ∈ (1, 2] with fractional separated boundary conditions. New existence results are obtained for this class of problems by using some standard fixed point theorems. A possible generalization for the inclusion problem with fractional separated integral boundary conditions is also discussed. Some illustrative examples are presented.MSC 2010 : Primary 34A60: Secondary 26A33, 34A08
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