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# On a fractional differential equation with infinitely many solutions

- Dumitru Băleanu
^{1, 2}Email author, - Octavian G Mustafa
^{3}and - Donal O’Regan
^{4}

**2012**:145

https://doi.org/10.1186/1687-1847-2012-145

© Băleanu et al.; licensee Springer 2012

**Received:**14 May 2012**Accepted:**6 August 2012**Published:**23 August 2012

## Abstract

We present a set of restrictions on the fractional differential equation ${x}^{(\alpha )}(t)=g(x(t))$, $t\ge 0$, where $\alpha \in (0,1)$ 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 ${x}^{(\alpha )}$ is 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 [1], there are only a few techniques to help us decide whether an initial value problem has more than one solution. As an example, the equation ${x}^{\mathrm{\prime}}=f(x)=\sqrt{x}\cdot {\chi}_{(0,+\mathrm{\infty})}(x)$ has an infinity of solutions (a continuum of solutions [8], p.15]) ${x}_{T}(t)=\frac{{(t-T)}^{2}}{4}\cdot {\chi}_{(T,+\mathrm{\infty})}(t)$ defined on the nonnegative half-line which start from $x(0)=0$. Here, by *χ* we denote the characteristic function of a Lebesgue-measurable set.

where the continuous function $g:\mathbb{R}\to \mathbb{R}$ has a zero at ${x}_{0}$ and is positive everywhere else, possesses an infinity of solutions if and only if ${\int}_{{x}_{0}+}\frac{du}{g(u)}<+\mathrm{\infty}$.

*g*is allowed to have two zeros ${x}_{0}<{x}_{1}$ while remaining positive everywhere else and

then the problem (1) has an infinity of solutions ${({x}_{T})}_{T>0}$ such that ${lim}_{t\to +\mathrm{\infty}}{x}_{T}(t)={x}_{1}$.

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 knowledge, the result has not been established in its full generality yet for any generalized differential equation. We mention at this point the closely connected investigation [5].

In the last number of years, it became evident that differential equations of non-integer order, also called *fractionals* (FDE’s), can capture better in models many of the relevant features of complex phenomena from engineering, physics or chemistry, see the references in [2–4, 6, 7, 9, 11, 12, 16].

*Caputo derivative*of order

*α*of

*h*is defined as

*m*on $\mathbb{R}$, see [18], p.35, Lemma 2.2]. Further, we have that

provided that ${h}^{(\alpha )}$ is in ${L}^{\mathrm{\infty}}(m)$.

where the function $g:\mathbb{R}\to \mathbb{R}$ is continuous, $g(0)=0$ and $g(u)>0$ when $u\in (0,1]$. Further restrictions will be imposed on *g* to ensure that ${\int}_{0+}\frac{du}{g(u)}<+\mathrm{\infty}$.

where $y={x}^{(\alpha )}$, $\beta =1-\alpha $ and the (general) function *g* has absorbed the constant $\frac{1}{\mathrm{\Gamma}(\alpha )}$.

In the next section, we look for a family ${({y}_{T})}_{T>0}$, with ${y}_{T}\in C([0,1],\mathbb{R})$, of (non-trivial) solutions to (4).

## 2 Infinitely many solutions to (4)

Obviously, ${\delta}_{1},{\delta}_{2}\in (0,1)$.

The latter condition has been inspired by the analysis in [15].

These will be used in the following for describing the solution ${y}_{T}$.

*via*the change of variables $s=T+u(t-T)$, we get

*B*represents Euler’s function Beta [16]. Also,

*via*(13),

*via*(17),

In conclusion, the mapping $t\mapsto g({\int}_{T}^{t}\frac{y(s)}{{(t-s)}^{\beta}}\phantom{\rule{0.2em}{0ex}}ds)$ is a member of $\mathcal{Y}$ whenever $y\in \mathcal{Y}$. Also, taking into account (15), we deduce that the quantities $y={Y}_{1}B(2+{\epsilon}_{1},1-\beta ){(t-T)}^{2+{\epsilon}_{1}-\beta}$ from (20) and $y={Y}_{2}B(2+{\epsilon}_{2},1-\beta ){(t-T)}^{2+{\epsilon}_{2}-\beta}$ from (21) belong to $[0,1]$ as imposed in (6).

We are now ready to state and prove our main result.

**Theorem 1** *Assume that the nonlinearity* *g* *of* (4) *satisfies the restrictions* (5), (6), (7). *Given the numbers* ${Y}_{1}$, ${Y}_{2}$, *T* *subject to* (8), (9), (10) *and the set* $\mathcal{Y}=\mathcal{Y}({Y}_{1},{Y}_{2},T)$ *from* (18), *the problem* (4) *has a unique solution* ${y}_{T}$ *in* $\mathcal{Y}$.

*Proof*The operator $\mathcal{O}:\mathcal{Y}\to \mathcal{Y}$ with the formula

is well defined.

The typical sup-metric $d({y}_{1},{y}_{2})={sup}_{t\in [T,1]}|{y}_{1}(t)-{y}_{2}(t)|$ provides the set $\mathcal{Y}$ with the structure of a complete metric space.

The operator $\mathcal{O}$ being thus a contraction, its fixed point ${y}_{T}$ in $\mathcal{Y}$ is the solution we are looking for. Notice that ${y}_{T}$ is identically null in $[0,T]$. □

where ${c}_{1}>0$ and $\delta \in (\frac{2}{3},\frac{4}{5})$. Then, introducing $Y>0$ such that ${c}_{1}\cdot B{(\frac{1}{2},\frac{2-3\delta}{2-2\delta})}^{\delta}\cdot {Y}^{\delta -1}=1$, the problem (22) has the solution $y(t)=Y{(t-T)}^{\frac{\delta}{2(1-\delta )}}$ throughout $[T,1]$ which can be extended as a ${C}^{1}$-function downward to 0.

## Declarations

### Acknowledgements

The 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 their comments and remarks in order to improve the manuscript.

## Authors’ Affiliations

## References

- Agarwal RP, Lakshmikantham V:
*Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations*. World Scientific, Singapore; 1993.View ArticleGoogle Scholar - Agrawal OP, Defterli O, Băleanu D: Fractional optimal control problems with several state and control variables.
*J. Vib. Control*2010, 16: 1967–1976. 10.1177/1077546309353361MathSciNetView ArticleGoogle Scholar - Băleanu D, Diethelm K, Scalas E, Trujillo JJ Series on Complexity, Nonlinearity and Chaos. In
*Fractional Calculus Models and Numerical Methods*. World Scientific, Boston; 2012.Google Scholar - Băleanu D, Avkar T: Lagrangians with linear velocities within Riemann-Liouville fractional derivatives.
*Nuovo Cimento B*2004, 119: 73–79.Google Scholar - Băleanu, D, Mustafa, OG, O’Regan, D: A uniqueness criterion for fractional differential equations with Caputo derivative. Nonlinear Dyn. doi:10.1007/s11071–012–0449–4 (in press)Google Scholar
- Bhalekar S, Daftardar-Gejji V, Băleanu D, Magin RL: Transient chaos in fractional Bloch equations.
*Comput. Math. Appl.*2012. doi:10.1016/j.camwa.2012.01.069Google Scholar - Delavari H, Băleanu D, Sadati J: Stability analysis of Caputo fractional-order nonlinear systems revisited.
*Nonlinear Dyn.*2012, 67: 2433–2439. 10.1007/s11071-011-0157-5View ArticleGoogle Scholar - Hartman P:
*Ordinary Differential Equations*. Wiley, New York; 1964.Google Scholar - Herzallah MAE, Băleanu D: Fractional Euler-Lagrange equations revisited.
*Nonlinear Dyn.*2012. doi:10.1007/s11071–011–0319–5Google Scholar - Kamke E:
*Differentialgleichungen, Lösungsmethoden und Lösungen I*. Akademie Verlag, Leipzig; 1961.Google Scholar - Kilbas AA, Srivastava HM, Trujillo JJ:
*Theory and Applications of Fractional Differential Equations*. North-Holland, New York; 2006.Google Scholar - Miller KS, Ross B:
*An Introduction to the Fractional Calculus and Fractional Differential Equations*. Wiley, New York; 1993.Google Scholar - Mustafa OG: Initial value problem with infinitely many linear-like solutions for a second-order differential equation.
*Appl. Math. Lett.*2005, 18: 931–934. 10.1016/j.aml.2004.07.036MathSciNetView ArticleGoogle Scholar - Mustafa OG: On smooth traveling waves of an integrable two-component Camassa-Holm shallow water system.
*Wave Motion*2009, 46: 397–402. 10.1016/j.wavemoti.2009.06.011MathSciNetView ArticleGoogle Scholar - Mustafa OG: On the uniqueness of flow in a recent tsunami model.
*Appl. Anal.*2012. doi:10.1080/00036811.2011.569499Google Scholar - Podlubny I:
*Fractional Differential Equations*. Academic Press, San Diego; 1999.Google Scholar - Rudin W:
*Real and Complex Analysis*. 3rd edition. McGraw-Hill, New York; 1987.Google Scholar - Samko SG, Kilbas AA, Marichev OI:
*Fractional Integrals and Derivatives, Theory and Applications*. Gordon & Breach, Switzerland; 1993.Google Scholar

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