AA Kilbas

D Băleanu

Donal O’Regan

Dumitru Băleanu

E Kamke

H Delavari

I Podlubny

KS Miller

MAE Herzallah

Octavian G Mustafa

OG Mustafa

OP Agrawal

P Hartman

RP Agarwal

S Bhalekar

SG Samko

W Rudin

Advances in Difference Equations

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Ba˘leanu et al. Advances in Diﬀerence Equations 2012, 2012:145http://www.advancesindifferenceequations.com/content/2012/1/145RESEARCH Open AccessOn a fractional differential equation withinﬁnitely many solutionsDumitru Ba˘leanu1,2*, Octavian G Mustafa3 and Donal O’Regan4*Correspondence:dumitru@cankaya.edu.tr1Department of Mathematics andComputer Science, ÇankayaUniversity, Ögretmenler Cad. 14,Balgat, Ankara 06530, Turkey2Institute of Space Sciences,Ma˘gurele-Bucures¸ti, RomaniaFull list of author information isavailable at the end of the articleAbstractWe 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 ofsolutions (a continuum of solutions) starting from x(0) = 0. The operator x(α) is theCaputo diﬀerential operator.Keywords: fractional diﬀerential equation; multiplicity of solutions; Caputodiﬀerential operator1 IntroductionThe issue of multiplicity for solutions of an initial value problem that is associated to somenonlinear diﬀerential equation is essential in the modeling of complex phenomena.Typically, when the nonlinearity of an equation is not of Lipschitz type [], there are onlya few techniques to help us decide whether an initial value problem has more than onesolution. As an example, the equation x′ = f (x) =√x ·χ(,+∞)(x) has an inﬁnity of solutions(a continuum of solutions [, p.]) xT (t) = (t–T) · χ(T ,+∞)(t) deﬁned on the nonnegativehalf-line which start from x() = . Here, by χ we denote the characteristic function of aLebesgue-measurable set.An interesting classical result [, ], which generalizes the example, asserts that theinitial value problem⎧⎨⎩x′(t) = g(x(t)), t ≥ ,x() = x, x ∈R,()where the continuous function g :R→R has a zero at x and is positive everywhere else,possesses an inﬁnity of solutions if and only if∫x+dug(u) < +∞.Recently, variants of this result have been employed in establishing various facts regard-ing some mathematical models [, ]. In particular, if the function g is allowed to havetwo zeros x < x while remaining positive everywhere else and∫x+dug(u) < +∞,∫ x– dug(u) = +∞,then the problem () has an inﬁnity of solutions (xT )T> such that limt→+∞ xT (t) = x.Our intention in the following is to discuss a particular case of the above non-uniquenesstheorem in the framework of fractional diﬀerential equations. To the best of our knowl-© 2012 Ba˘leanu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly cited.Ba˘leanu et al. Advances in Diﬀerence Equations 2012, 2012:145 Page 2 of 6http://www.advancesindifferenceequations.com/content/2012/1/145edge, the result has not been established in its full generality yet for any generalized dif-ferential equation. We mention at this point the closely connected investigation [].In the last number of years, it became evident that diﬀerential equations of non-integerorder, also called fractionals (FDE’s), can capture better in models many of the relevantfeatures of complex phenomena fromengineering, physics or chemistry, see the referencesin [–, , , , , , ].Let us consider a function h ∈ C(I,R) ∩ C(I,R) with limt↘[t–αh′(t)] ∈ R for someα ∈ (, ), where I = (, ]. The Caputo derivative of order α of h is deﬁned ash(α)(t) = ( – α) ·∫ th′(s)(t – s)α ds, t ∈ I,where  is Euler’s function Gamma, cf. [, p.]. To have an idea about the eventualsmoothness of this quantity, we mention that by letting the function h′ be at least abso-lutely continuous [, Chapter ] the (usual) derivative of h(α) will exist almost everywherewith respect to the Lebesgue measurem onR, see [, p., Lemma .]. Further, we havethath(t) = h() + (α)∫ th(α)(s)(t – s)–α ds, t ∈ I, ()provided that h(α) is in L∞(m).The initial value problem we investigate in this paper is⎧⎨⎩x(α)(t) = g(x(t)), t ∈ I,x() = ,()where the function g :R→R is continuous, g() =  and g(u) >  when u ∈ (, ]. Furtherrestrictions will be imposed on g to ensure that∫+dug(u) < +∞.By means of (), we deduce thatx(t) =∫ tx′(s)ds= (α)∫ t(t – s)–α[ ( – α)∫ sx′(τ )(s – τ )α dτ]dsand so the problem () can be recast as⎧⎨⎩y(t) = g(∫ ty(s)(t–s)β ds), t ∈ [, ],y() = ,()where y = x(α), β =  – α and the (general) function g has absorbed the constant (α) .In the next section, we look for a family (yT )T>, with yT ∈ C([, ],R), of (non-trivial)solutions to ().Ba˘leanu et al. Advances in Diﬀerence Equations 2012, 2012:145 Page 3 of 6http://www.advancesindifferenceequations.com/content/2012/1/1452 Inﬁnitely many solutions to (4)We start by noticing that the function f : [, ] → (, ) with the formula f (x) = +x+x–β = – –β+x–β is increasing. Introduce now the numbers δ and δ such thatf () =  – β > δ ≥ δ > f () = – β . ()Obviously, δ, δ ∈ (, ).Set c ≤ c in (,+∞) and assume thatc · yδ ≤ g(y)≤ c · yδ , y ∈ [, ]. ()As a by-product,∫ +dyg(y) ≤c––δ < +∞. Further, suppose that there exists c >  such that∣∣g(y) – g(y)∣∣≤ c(min{y, y})–δ · |y – y|, y, y ∈ (, ]. ()The latter condition has been inspired by the analysis in [].Introduce now the numbers Y,Y ≥  and T ∈ (, ) with(Y + Y)( – T)–β <  – β ()andY ≤ c ≤ c ≤ ( – β)Y –δ ()andk = c( – β)δ ·( Y)–δ< . ()These will be used in the following for describing the solution yT .Several simple estimates, of much help in the proof of our result, are established next.Notice ﬁrst that, via the change of variables s = T + u(t – T), we get∫ tT(s – T)+ε(t – s)β ds = (t – T)+ε–β ·∫ u+ε( – u)β du= B( + ε,  – β)(t – T)+ε–β , ()where ε ∈ (, ) and B represents Euler’s function Beta []. Also,B( + ε,  – β) ≥∫ u+ε( – u)β du≥+ε ·∫ du( – u)β= –(+ε–β) – β ≥( – β) ()andB( + ε,  – β)≤∫ du( – u)β = – β . ()Ba˘leanu et al. Advances in Diﬀerence Equations 2012, 2012:145 Page 4 of 6http://www.advancesindifferenceequations.com/content/2012/1/145Now, returning to (), there exist ε, ε ∈ (, ), with ε ≥ ε, such thatf (ε) = δ ≥ δ = f (ε). ()In particular,  – δ = –β+ε–β and, by means of (),YB( + ε,  – β)( – T)+ε–β+ YB( + ε,  – β)( – T)+ε–β < . ()Taking into account () and (), we deduce thatY –δ ≤ c[ ( – β)]δ≤ cB( + ε,  – β)δ ,which leads toY ≤ c[YB( + ε,  – β)]δ , ()and, via (),cB( + ε,  – β)δ ≤ c( – β)δ ≤c – β ,which implies thatc[YB( + ε,  – β)]δ ≤ Y. ()Let the set Y ⊂ C([T , ],R) be given by the double inequalityY(t – T)+ε ≤ y(t)≤ Y(t – T)+ε , t ∈ [T , ], y ∈ Y . ()Observe that, by means of (), (), (),YB( + ε,  – β)(t – T)+ε–β ≤∫ tTy(s)(t – s)β ds ()andY(t – T)+ε ≤ c[YB( + ε,  – β)]δ · (t – T)+ε= c[YB( + ε,  – β)(t – T)+ε–β]δ≤ g(∫ tTy(s)(t – s)β ds). ()Similarly, via (),∫ tTy(s)(t – s)β ds≤ YB( + ε,  – β)(t – T)+ε–β ()Ba˘leanu et al. Advances in Diﬀerence Equations 2012, 2012:145 Page 5 of 6http://www.advancesindifferenceequations.com/content/2012/1/145andg(∫ tTy(s)(t – s)β ds)≤ c[YB( + ε,  – β)]δ · (t – T)+ε≤ Y(t – T)+ε .In conclusion, the mapping t → g(∫ tT y(s)(t–s)β ds) is a member of Y whenever y ∈ Y . Also,taking into account (), we deduce that the quantities y = YB( + ε,  – β)(t – T)+ε–βfrom () and y = YB(+ε, –β)(t–T)+ε–β from () belong to [, ] as imposed in ().We are now ready to state and prove our main result.Theorem Assume that the nonlinearity g of () satisﬁes the restrictions (), (), (). Giventhe numbers Y, Y, T subject to (), (), () and the set Y = Y(Y,Y,T) from (), theproblem () has a unique solution yT in Y .Proof The operator O : Y → Y with the formulaO(y)(t) = g(∫ tTy(s)(t – s)β ds), y ∈ Y , t ∈ [T , ],is well deﬁned.The typical sup-metric d(y, y) = supt∈[T ,] |y(t) – y(t)| provides the set Y with thestructure of a complete metric space.Taking into account (), () and (), we get∣∣O(y)(t) –O(y)(t)∣∣≤ c[YB( + ε,  – β)(t – T)+ε–β ]–δ ·∫ tT|y(s) – y(s)|(t – s)β≤ c[( – β)Y]–δ· (t – T)–β · d(y, y)∫ tTds(t – s)β= k · d(y, y), y, y ∈ Y .The operator O being thus a contraction, its ﬁxed point yT in Y is the solution we arelooking for. Notice that yT is identically null in [,T]. As an easily computed example, consider the initial value problem⎧⎨⎩y(t) = c(∫ tTy(s)√t–s ds)δ , t ∈ [T , ],y(T) = ,()where c >  and δ ∈ (  ,  ). Then, introducing Y >  such that c ·B(  , –δ–δ )δ ·Y δ– = , theproblem () has the solution y(t) = Y (t–T)δ(–δ) throughout [T , ] which can be extendedas a C-function downward to .Competing interestsThe authors declare that they have no competing interests.Ba˘leanu et al. Advances in Diﬀerence Equations 2012, 2012:145 Page 6 of 6http://www.advancesindifferenceequations.com/content/2012/1/145Authors’ contributionsAll authors have equal contributions in preparing this manuscript. All authors read and approved the ﬁnal manuscript.Author details1Department of Mathematics and Computer Science, Çankaya University, Ögretmenler Cad. 14, Balgat, Ankara 06530,Turkey. 2Institute of Space Sciences, Ma˘gurele-Bucures¸ti, Romania. 3Faculty of Exact Sciences, University of Craiova, A. I.Cuza 13, Craiova, 200534, Romania. 4School of Mathematics, Statistics and Applied Mathematics, National University ofIreland, Galway, Ireland.AcknowledgementsThe work of the second author has been supported by a grant of the Romanian National Authority for Scientiﬁc Research,CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0075. The authors would like to thank the referees for theircomments and remarks in order to improve the manuscript.Received: 14 May 2012 Accepted: 6 August 2012 Published: 23 August 2012References1. Agarwal, RP, Lakshmikantham, V: Uniqueness and Nonuniqueness Criteria for Ordinary Diﬀerential Equations. WorldScientiﬁc, Singapore (1993)2. Agrawal, OP, Defterli, O, Ba˘leanu, D: Fractional optimal control problems with several state and control variables. J.Vib. Control 16, 1967-1976 (2010)3. Ba˘leanu, D, Diethelm, K, Scalas, E, Trujillo, JJ: Fractional Calculus Models and Numerical Methods. Series onComplexity, Nonlinearity and Chaos. 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Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Diﬀerential Equations. North-Holland,New York (2006)12. Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Diﬀerential Equations. Wiley, New York(1993)13. Mustafa, OG: Initial value problem with inﬁnitely many linear-like solutions for a second-order diﬀerential equation.Appl. Math. Lett. 18, 931-934 (2005)14. Mustafa, OG: On smooth traveling waves of an integrable two-component Camassa-Holm shallow water system.Wave Motion 46, 397-402 (2009)15. Mustafa, OG: On the uniqueness of ﬂow in a recent tsunami model. Appl. Anal. (2012).doi:10.1080/00036811.2011.56949916. Podlubny, I: Fractional Diﬀerential Equations. Academic Press, San Diego (1999)17. Rudin, W: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)18. Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integrals and Derivatives, Theory and Applications. Gordon & Breach,Switzerland (1993)doi:10.1186/1687-1847-2012-145Cite this article as: Ba˘leanu et al.: On a fractional differential equation with inﬁnitely many solutions. Advances inDiﬀerence Equations 2012 2012:145.

On a fractional differential equation with infinitely many solutions

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On a fractional differential equation with infinitely many solutions

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