- Research
- Open Access
Basic theory of initial value problems of conformable fractional differential equations
- Wenyong Zhong^{1}Email author and
- Lanfang Wang^{1}
https://doi.org/10.1186/s13662-018-1778-5
© The Author(s) 2018
- Received: 16 February 2018
- Accepted: 24 August 2018
- Published: 12 September 2018
Abstract
In this paper, we discuss the basic theory of the conformable fractional differential equation \(T^{a}_{\alpha}x(t)=f(t,x(t)), t\in[a,\infty)\), subject to the local initial condition \(x(a)=x_{a} \) or the nonlocal initial condition \(x(a)+g(x)=x_{a}\), where \(0<\alpha<1\), \(T^{a}_{\alpha}x(t)\) denotes the conformable fractional derivative of a function \(x(t)\) of order α, \(f:[a,\infty)\times\mathbb{R}\mapsto\mathbb{R}\) is continuous and g is a given functional defined on an appropriate space of functions. The theory of global existence, extension, boundedness, and stability of solutions is considered; by virtue of the theory of the conformable fractional calculus and by the use of fixed point theorems, some criteria are established. Several concrete examples are given to illustrate the possible application of our analytical results.
Keywords
- Conformable fractional derivatives
- Fractional differential equations
- Initial value problems
- Fixed point theorems
1 Introduction
Fractional derivative is a generalization of the classical one to an arbitrary order, and it is as old as calculus. It has been applied to almost every field of science, engineering, and mathematics in the last three decades [1–10]. At present, there exist a number of definitions of fractional derivatives in the literature, each depending on a given set of assumptions, the most popular of which are the Riemann–Liouville and Caputo fractional derivatives. But it is worth noting that these two kinds of derivatives do not satisfy the classical chain rule.
Recently, Khalil et al. in [11] introduced a new well-behaved definition of a fractional derivative, called the conformable fractional derivative, which satisfies the chain rule. The new definition has attracted a great deal of attention from many researchers. And for the basic properties of the conformable fractional derivative, some results have been obtained [11–13]; its several applications and generalizations were also discussed [14–18]. But the investigation of the theory of conformable fractional differential equations has only been started quite recently.
There is a vast literature concerning the existence of solutions to an initial value problem for the differential equations with the Riemann–Liouville or Caputo fractional derivatives [29–32]. While in the setting of the conformable fractional derivatives, as far as we know, the global existence and extension and boundedness of solutions have not been discussed in the literature. It is worth pointing out that the global existence and boundedness of solutions play a prerequisite role in the discussion of the stabilities of solutions.
The rest of paper is organized as follows. Section 2 preliminarily provides some definitions and lemmas which are crucial to the following discussion. In Sect. 3, we first establish some criteria for the global existence, extension, and boundedness of solutions to the local initial value problem by means of some fixed point theorems and by the use of the conformable fractional calculus, and further discuss the stabilities of solutions; and then we investigate the existence of solutions to the nonlocal initial value problem. Finally, we give several concrete examples to illustrate the possible application of our analytical results.
2 Preliminaries
In this section, we preliminarily provide some definitions and lemmas which are useful in the following discussion. It is always assumed that \(\alpha\in(0,1]\) throughout this paper.
Definition 2.1
Definition 2.2
Lemma 2.1
Lemma 2.2
([22])
If \(T^{a}_{\alpha}f(t)\) is continuous on \([a,b]\), then \(I^{a}_{\alpha }T^{a}_{\alpha}f(t)=f(t)-f(a)\).
Lemma 2.3
If f is differentiable at t in \([a,b]\), then it is also α-differentiable at t and \(T^{a}_{\alpha}f(t)=(t-a)^{1-\alpha}\frac{\mathrm{ d}f(t)}{\mathrm{ d} t}\).
Lemma 2.5
(Chain rule [13])
Lemma 2.6
([11])
By an argument similar to the one used in [11], a general version of the mean value theorem for the conformable fractional derivative is yielded as follows [20]. It plays a crucial role in the study of the extension of solutions.
Lemma 2.7
The following lemma is a direct consequence of the application of the mean value theorem [13].
Lemma 2.8
If \(T^{a}_{\alpha}f(t)\leq0\) on \([a,b]\), then f is decreasing on \([a,b]\).
We next present an extended Gronwall’s inequality, which generalizes the result in [13]; and it plays a key role in the discussion of the extension and stabilities of solutions.
Lemma 2.9
Proof
3 Main results
- (H1)
The function \(f:\mathfrak{D}\mapsto\mathbb{R}\) is continuous.
- (H2)There exists a positive constant L such that, for any \((t,u), (t,v)\) in \(\mathfrak{D}\),$$\bigl\vert f(t,u)-f(t,v) \bigr\vert \leq L \vert u-v \vert . $$
- (H3)There exists a nonnegative function h such that, for any \((t,u)\) in \(\mathfrak{D}\),for which \(I_{\alpha}^{a}h(t)\) is bounded on \([a,\infty)\).$$\bigl\vert f(t,u) \bigr\vert \leq h(t) \vert u \vert $$
- (H4)There exist a nonnegative function l and a positive constant L such that, for any \((t,u), (t,v)\) in \(\mathfrak{D}\),for which \(I_{\alpha}^{a} l(t)\) is bounded on \([a,\infty)\).$$\bigl\vert f(t,u)-f(t,v) \bigr\vert \leq l(t) \vert u-v \vert \leq L \vert u-v \vert $$
3.1 Local initial value problems
In this subsection, we establish some criteria for the global existence, extension, boundedness, and stabilities of solutions to the local initial value problem. By Lemmas 2.1 and 2.2, the initial value problem (1.1)–(1.2) is easily transformed into an equivalent integral equation.
Lemma 3.1
Now, we are in a position to present a result of existence and uniqueness of the solution to the initial value problem (1.1)–(1.2).
Theorem 3.1
If (H1)–(H2) hold, then the initial value problem (1.1)–(1.2) has exactly one solution defined on \([a,b]\).
Proof
We next discuss the extension to the right of the solutions of Eq. (1.1) with initial condition (1.2).
Lemma 3.2
If (H1) holds. Let \(x(t)\) be a solution of the initial value problem (1.1)–(1.2) defined on \([a,t^{+})\) with \(t^{+}\neq\infty\). If the limit of \(x(t)\) exists as t tends to \(t^{+}\), then the solution \(x(t)\) can be extended to the closed interval \([a,t^{+}]\).
Proof
Definition 3.1
Let J be the maximal existence interval of the solution \(x(t)\) of the initial value problem (1.1)–(1.2), then the solutions \(x(t)\) is called to come arbitrarily close to the boundary of \(\mathfrak{D}=[a,\infty)\times\mathbb{R}\) to the right if for any closed and bounded domain \(\mathfrak{D}_{0}\) in \(\mathfrak{D}\), it is impossible that the point \((t,x(t))\) always remains in \(\mathfrak{D}_{0}\) for every t in J.
Theorem 3.2
If (H1)–(H2) hold, then the solution of the initial value problem (1.1)–(1.2) comes arbitrarily close to the boundary of \(\mathfrak{D}=[a,\infty)\times\mathbb{R}\) to the right.
Proof
According to Theorem 3.1, the initial value problem (1.1)–(1.2) has a unique solution, and denote the solution by \(x(t)\). Let J stand for the maximal existence interval of \(x(t)\). Again, using Theorem 3.1, we infer that \(J=[a,\infty)\) or \([a,t^{+})\) with \(t^{+}\neq\infty\).
The desired result is obvious if \(J=[a,\infty)\).
Using Theorems 3.1 and 3.2, we now give a result guaranteeing that the solution of Eq. (1.1) with the initial condition (1.2) is defined and bounded on \([a,\infty)\).
Theorem 3.3
If (H1)–(H3) hold, then the solution of the initial value problem (1.1)–(1.2) is defined and bounded on \([a,\infty)\).
Proof
By Theorem 3.1, Eq. (1.1) with the initial condition (1.2) has a unique solution. Denote the solution by \(x(t)\) for which its maximal existence interval is \([a,t^{+})\). It remains to show that \(t^{+}=\infty\) and that \(x(t)\) is bounded on \([a,\infty)\).
If \(t^{+}\neq\infty\), then Theorem 3.2 immediately implies \(\lim_{t\rightarrow t^{+}}x(t)=\infty\), which contradicts the boundedness of \(x(t)\) on \([a,t^{+})\). Thus \(t^{+}=\infty\), and therefore the desired result follows. □
By Gronwall’s inequality, we next investigate the stabilities of the solutions to the problem (1.1)–(1.2).
Definition 3.2
Theorem 3.4
If (H1), (H3) and (H4) hold, then every solution to Eq. (1.1) with the local initial condition is always stable.
Proof
3.2 Nonlocal initial value problems
In this subsection, the existence of solutions to the nonlocal initial value problem is discussed. We next introduce a fixed theorem to be adopted to prove the main result in this subsection.
Lemma 3.3
([33])
- (C1)
\(\mathcal{A}\) has a fixed point \(u\in\bar{\mathcal{U}}\); or
- (C2)
there exist a point \(u\in\partial{\mathcal{U}}\) and \(\lambda \in(0,1)\) with \(u=\lambda\mathcal{A}(u)\), where \(\bar{\mathcal{U}}\) and \(\partial{\mathcal{U}}\), respectively, represent the closure and boundary of \(\mathcal{U}\).
- (H5)
f is a continuous function defined on \([a,b]\times\mathbb{R}\).
- (H6)
There exist a positive constant γ in \((0,1)\) and a nonnegative and nondecreasing function ϕ in \(C([0,\infty))\) such that \(\phi(z)<\gamma z\) for \(z>0\) and \(|g(u)-g(v)|\leq\phi(\| u-v\|)\) for any \(u, v\) in \(C([a,b])\).
- (H7)There exist a nonnegative function φ in \(C([a,b])\) for which \(\varphi>0\) on a subinterval of \([a,b]\) and a nonnegative and nondecreasing function ψ in \(C([0,\infty ))\) such thatfor any \((t,u)\) in \([a,b]\times\mathbb{R}\) and$$\bigl\vert f(t,u) \bigr\vert \leq \varphi(t)\psi\bigl( \vert u \vert \bigr) $$By Lemmas 2.1 and 2.2, it is easy to verify the following lemma.$$\sup_{r\in(0,\infty)}\frac{r}{ \vert x_{a} \vert +\psi(r)I_{\alpha}^{a}\varphi(b)}>\frac{1}{1-\gamma}. $$
Lemma 3.4
In order to utilize the fixed point theorem to discuss the existence of solutions to the nonlocal initial value problem, we first define some sets of functions in \(C([a,b])\) and operators.
Using the standard arguments, the complete continuity of the operator \(\mathcal{A}_{1}:\bar{\mathcal{U}}_{r}\mapsto C([a,b])\) can be verified, and it is also easy to check that the operator \(\mathcal{A}_{2}:\bar{\mathcal{U}}_{r}\mapsto C([a,b])\) is a nonlinear contraction under the condition (H6). Here we omit their proofs.
Lemma 3.5
If (H5) holds, then the operator \(\mathcal{A}_{1}:\bar{\mathcal {U}}_{r}\mapsto C([a,b])\) is completely continuous.
Lemma 3.6
If (H6) holds, then the operator \(\mathcal{A}_{2}:\bar{\mathcal {U}}_{r}\mapsto C([a,b])\) is a nonlinear contraction.
We now present the main result in this subsection.
Theorem 3.5
If (H5)–(H7) hold, then the nonlocal initial value problem (1.1), (1.3) exists at least one solution defined on \([a,b]\).
Proof
3.3 Illustrative examples
Let \(\mathfrak{D}=[a,\infty)\times\mathbb{R}\), \(f(t,x)=\mathrm{ e}^{{-\frac{(t-a)^{\alpha}}{\alpha}}}(x+\sin x), h(t)=l(t)=2\mathrm{ e}^{{-\frac{(t-a)^{\alpha}}{\alpha}}}\), and \(L=2\).
(I) Local initial value problems
(II) Stabilities
(III) Nonlocal initial value problems
4 Conclusion
By the use of the conformable fractional calculus and by means of fixed point theorems, some criteria are established for the global existence, extension, boundedness, and stabilities of solutions to the local initial value problem; and the existence result of solutions to the nonlocal initial value problem is also obtained. The obtained conditions are easy to satisfy and check. For \(\alpha=1\), the classical results corresponding to ordinary differential equations will be yielded.
Declarations
Acknowledgements
The authors are grateful to the referees for carefully reading the paper and for their comments and suggestions.
Funding
The paper is supported by the Natural Science Foundation of Hunan Province of China (Grant no. 11JJ3007).
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Oldham, K.B., Spanier, J.: Fractional Calculus: Theory and Applications, Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974) MATHGoogle Scholar
- Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integral and Derivatives: Theory and Applications. Gordon and Breach, Switzerland (1993) MATHGoogle Scholar
- Kilbas, A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) MATHGoogle Scholar
- Diethelm, K.: The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics, vol. 204. Springer, Berlin (2010) View ArticleMATHGoogle Scholar
- Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) MATHGoogle Scholar
- Lundstrom, B.N., Higgs, M.H., et al.: Fractional differentiation by neocortical pyramidal neurons. Nat. Neurosci. 11, 1335–1342 (2008) View ArticleGoogle Scholar
- West, B.J.: Colloquium: fractional calculus view of complexity: a tutorial. Rev. Mod. Phys. 86, 1169–1184 (2014) View ArticleGoogle Scholar
- Zhong, W.Y., Lin, W.: Nonlocal and multiple-point boundary value problem for fractional differential equations. Comput. Math. Appl. 59, 1345–1351 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Zhong, W.Y., Wang, L.F.: Monotone and concave positive solutions to three-point boundary value problems of higher-order fractional differential equations. Abstr. Appl. Anal. 2015, Article ID 728491 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Zhong, W.Y.: Positive solutions for multipoint boundary value problem of fractional differential equations. Abstr. Appl. Anal. 2010, Article ID 601492 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Khalil, R., Horani, M.A., et al.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Jarad, F., Uğurlu, E., et al.: On a new class of fractional operators. Adv. Differ. Equ. (2017). https://doi.org/10.1186/s13662-017-1306-z MathSciNetGoogle Scholar
- Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Eslami, M.: Exact traveling wave solutions to the fractional coupled nonlinear Schrodinger equations. Appl. Math. Comput. 285, 141–148 (2016) MathSciNetGoogle Scholar
- Ekici, M., Mirzazadeh, M.: Optical soliton perturbation with fractional-temporal evolution by first integral method with conformable fractional derivatives. Optik 127, 10659–10669 (2016) View ArticleGoogle Scholar
- Weberszpil, J., Helaël-Neto, J.A.: Variational approach and deformed derivatives. Physica A 450, 217–227 (2016) MathSciNetView ArticleGoogle Scholar
- Anderson, D.R., Ulness, D.J.: Properties of the Katugampola fractional derivative with potential application in quantum mechanics. J. Math. Phys. 56, 063502 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Katugampola, U.N.: A new fractional derivative with classical properties. arXiv:1410.6535
- Al-Rifae, M., Abdeljawad, T.: Fundamental results of conformable Sturm–Liouville eigenvalue problems. Complexity 2017, Article ID 3720471 (2017) MathSciNetMATHGoogle Scholar
- Asawasamrit, S., Ntouyas, S.K., Thiramanus, P., Tariboon, J.: Periodic boundary value problems for impulsive conformable fractional integrodifferential equations. Bound. Value Probl. 2016, Article ID 122 (2016) View ArticleMATHGoogle Scholar
- Abdeljawad, T., Alzabut, J.: A generalized Lyapunov-type inequality in the frame of conformable derivatives. Adv. Differ. Equ. (2017). https://doi.org/10.1186/s13662-017-1383-z MathSciNetGoogle Scholar
- Dong, X., Bai, Z., Zhang, W.: Positive solutions for nonlinear eigenvalue problems with conformable fractional differential derivatives. J. Shandong Univ. Sci. Technol. Nat. Sci. 35, 85–90 (2016) Google Scholar
- Song, Q.L., Dong, X.Y., et al.: Existence for fractional Dirichlet boundary value problem under barrier strip conditions. J. Nonlinear Sci. Appl. 10, 3592–3598 (2017) MathSciNetView ArticleGoogle Scholar
- He, L.M., Dong, X.Y., et al.: Solvability of some two-point fractional boundary value problems under barrier strip conditions. J. Funct. Spaces 2017, Article ID 1465623 (2017) MathSciNetMATHGoogle Scholar
- Batarfi, H., Losada, J., et al.: Three-point boundary value problems for conformable fractional differential equations. J. Funct. Spaces 2015, Article ID 706383 (2015) MathSciNetMATHGoogle Scholar
- Souahi, A., Makhlouf, A.B., et al.: Stability analysis of conformable fractional-order nonlinear systems. Indag. Math. 28, 1265–1274 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Abdeljawad, T., AL Horani, M., Khalil, R.: Conformable fractional semigroups of operators. J. Semigroup Theory Appl. 2015, Article ID 7 (2015) Google Scholar
- Bayour, B., Torres, D.F.M.: Existence of solution to a local fractional nonlinear differential equation. J. Comput. Appl. Math. 312, 127–133 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Lin, W.: Global existence theory and chaos control of fractional differential. J. Math. Anal. Appl. 332, 709–726 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Lakshmikantham, V., Vatsala, A.S.: Basic theory of fractional differential equations. Nonlinear Anal. TMA 69, 2677–2682 (2008) MathSciNetView ArticleMATHGoogle Scholar
- N’Guérékata, G.M.: A Cauchy problem for some fractional abstract differential equation with non local conditions. Nonlinear Anal. TMA 70, 1873–1876 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Balachandran, K., Park, J.Y.: Nonlocal Cauchy problem for abstract fractional semilinear evolution equations. Nonlinear Anal. TMA 71, 4471–4475 (2009) MathSciNetView ArticleMATHGoogle Scholar
- O’Regan, D.: Fixed-point theory for the sum of two operators. Appl. Math. Lett. 9, 1–8 (1996) MathSciNetView ArticleMATHGoogle Scholar