On a fractional differential equation with infinitely many solutions

Bäƒleanu Dumitru, Mustafa, O’Regan, O’Regan Donal

Description

We present a set of restrictions on the fractional differential equation x ( α ) ( t ) = g ( x ( t ) ) , t ≥ 0 , where α ∈ ( 0 , 1 ) and g ( 0 ) = 0 , that leads to the existence of an infinity of solutions (a continuum of solutions) starting from x ( 0 ) = 0 . The operator x ( α ) is the Caputo differential operator. We present a set of restrictions on the fractional differential equation x ( α ) ( t ) = g ( x ( t ) ) , t ≥ 0 , where α ∈ ( 0 , 1 ) and g ( 0 ) = 0 , that leads to the existence of an infinity of solutions (a continuum of solutions) starting from x ( 0 ) = 0 . The operator x ( α ) is the Caputo differential operator.

Sujets

• Fractional calculus

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 4
• Langue: English

Extrait

B ˘ aleanuetal.Advances in Difference Equations2012,2012:145 http://www.advancesindifferenceequations.com/content/2012/1/145

R E S E A R C H

On a fractional differential equation with infinitely many solutions 1,2* 34 Dumitru Ba˘leanu , Octavian G Mustafaand Donal O’Regan

* Correspondence: [e-mail] 1 Department of Mathematics and Computer Science, Çankaya University, Ögretmenler Cad. 14, Balgat, Ankara 06530, Turkey 2 Institute of Space Sciences, M ˘ agurele-Bucure¸sti,Romania Full list of author information is available at the end of the article

Open Access

Abstract ( α ) We present a set of restrictions on the fractional differential equationx(t) =g(x(t)), t≥0, where α ∈(0, 1) andg(0) = 0, that leads to the existence of an infinity of ( α ) solutions (a continuum of solutions) starting fromx(0) = 0. The operatorxis the Caputo differential operator. Keywords:fractional differential equation; multiplicity of solutions; Caputo differential operator

1 Introduction The issue of multiplicity for solutions of an initial value problem that is associated to some nonlinear differential equation is essential in the modeling of complex phenomena. Typically, when the nonlinearity of an equation is not of Lipschitz type [], there are only a few techniques to help us decide whether an initial value problem has more than one √  solution. As an example, the equationx=f(x) =x∙χ (,+∞) (x) has an infinity of solutions  (t–T) (a continuum of solutions [, p.])x T (t) =∙χ (T,+∞) (t) defined on the nonnegative  half-line which start fromx() = . Here, byχwe denote the characteristic function of a Lebesgue-measurable set. An interesting classical result [, ], which generalizes the example, asserts that the initial value problem    x(t) =g(x(t)),t≥, ()  x() =x  ,x  ∈R,

where the continuous functiong:R→Rhas a zero atx  and is positive everywhere else,  du possesses an infinity of solutions if and only if< +∞. x  + g(u) Recently, variants of this result have been employed in establishing various facts regard- ing some mathematical models [, ]. In particular, if the functiongis allowed to have two zerosx  <x  while remaining positive everywhere else and   x–  du du < +∞, =+∞, x+ g(u)g(u) 

then the problem () has an infinity of solutions (x T ) T> such thatlim t→+∞ x T (t) =x  . Our intention in the following is to discuss a particular case of the above non-uniqueness theorem in the framework of fractional differential equations. To the best of our knowl-

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