Abstract
We study an initial value problem for an implicit fractional differential equation with the Liouville–Caputo fractional derivative. By using fixed point theory and an approximation method, we obtain some existence and uniqueness results.
1. Introduction
Differential equations of fractional order have recently been proven to be valuable tools in the modeling of many physical phenomena [1–3]. There has also been a significant theoretical development in fractional differential equations in recent years; see the monographs of Abbas et al. [4], Kilbas et al. [5], Miller and Ross [6], Podlubny [7] and Samko et al. [8].
The basic theory for initial value problems for fractional differential equations involving the Riemann–Liouville and Liouville–Caputo differential operator was discussed by Diethelm [9].
Recently, fractional functional differential equations, fractional differential inclusions and impulsive fractional differential equations with different conditions were studied, for example, by Aghajani et al. [10], Ahmad and Nieto [11], Benchohra et al. [12], Chalishajar and Karthikeyan [13,14], Henderson and Ouahab [15,16], Ouahab [17,18] and in the references therein.
In this paper, we study the existence of solutions of the following implicit fractional differential equation with initial condition:
where J = [0, b], 0 < α < 1 and f : J × ℝ × ℝ → ℝ is a continuous function.
This problem is motivated by the importance of implicit ordinary differential equations of the form:
under various initial and boundary conditions. Implicit equations have been considered by many authors [19–26]. Furthermore, our intention is to extend the results to implicit differential equations of fractional order.
Very recently, some existence results for an implicit fractional differential equation on compact intervals were investigated [27–29].
Our goal in this work is to give some existence and uniqueness results for implicit fractional differential equations.
2. Fractional Calculus
According to the Riemann–Liouville approach to fractional calculus, the notation of the fractional integral of order α (α > 0) is a natural consequence of the well-known formula (usually attributed to Cauchy) that reduces the calculation of the n−fold primitive of a function f to a single integral of the convolution type. The Cauchy formula reads:
Definition 1. The fractional integral of order α > 0 of a function f ∈ L1(a, b) is defined by:
where Γ is the classical gamma function. When a = 0, we write Iαf(t) = f(t) * ϕα(t), where for t > 0, ϕα(t) = 0 for t ≤ 0 and ϕα → δ as α → 0+ where δ is the Dirac delta function. For consistency, I0 = Id (Identity operator), i.e., I0f(t) = f(t). Furthermore, by Iαf(0+), we mean the limit (if it exists) of Iαf(t) for t → 0+; this limit may be infinite.
After the notion of the fractional integral, that of the fractional derivative of order α (α > 0) becomes a natural requirement, and one is attempted to substitute α with −α in the above formulas. However, this generalization needs some care in order to guarantee the convergence of the integral and to preserve the well-known properties of the ordinary derivative of integer order. Denoting by Dn, with n ∈ ℕ, the operator of the derivative of order n, we first note that:
i.e., Dn is the left inverse (and not the right inverse) to the corresponding integral operator In. We can easily prove that:
As a consequence, we expect that Dα is defined as the left inverse to Iα. For the fractional derivative of order α > 0 with integer n, such that n − 1 < α ≤ n, we have:
Definition 2. For a function f given on interval [a, b], the Riemann–Liouville fractional-order derivative of order α of f is defined by:
provided the right-hand side is defined.
Defining for consistency, D0 = I0 = Id, then we easily recognize that:
and
Of course, Properties (3) and (4) are a natural generalization of those known when the order is a positive integer.
Note the remarkable fact that the fractional derivative Dαf is not zero for the constant function f(t) = 1, if α ∉ ℕ. In fact, Equation (4) with γ = 0 illustrates that:
It is clear that Dα1 = 0, for α ∈ ℕ, due to the poles of the gamma function at the points 0, −1, −2,.….
We now observe an alternative definition of fractional derivative, introduced by Caputo [30,31] in the late 1960s and adopted by Caputo and Mainardi [32] in the framework of the theory of linear viscoelasticity (see a review in [2]).
Definition 3. Let f ∈ ACn([a, b]). The Liouville–Caputo fractional-order derivative of f is defined by:
This definition is of course more restrictive than the Riemann–Liouville definition, in that it requires the absolute integrability of the derivative of order n. Whenever we use the operator cDα, we (tacitly) assume that this condition is met. We easily recognize that in general:
unless the function f(t), along with its first n − 1 derivatives, vanishes at t = 0+. In fact, assuming that the passage of the n−derivative under the integral is legitimate, we recognize that, for n − 1 < α < n and t > 0,
and therefore, recalling the fractional derivative of the power Function (4):
The alternative definition, that is Definition 3, for the fractional derivative thus incorporates the initial values of the function and of lower order. The subtraction of the Taylor polynomial of degree n − 1 at t = 0+ from f(t) means a sort of regularization of the fractional derivative. In particular, according to this definition, the relevant property for which the fractional derivative of a constant is still zero is satisfied, i.e.,
At the end of this section, we present some properties of a special function. Denote Eα,β the generalized Mittag–Leffler special function defined by:
Also,
where ϒ is a contour, which starts and ends at −∞ and encircles the disc
counterclockwise.
Lemma 1. [9] Let α > 0, n = [α] and λ ∈ ℝ. The solution of the initial value problem:
where q ∈ C[0, b] is a given function, can be expressed in the form:
with:
where, k = 0, 1, …, n − 1 and.
Lemma 2. Let v : [0, b] → [0, ∞) be a real function, and w is a nonnegative, locally-integrable function on [0, b]. Assume that there are constants a > 0 and 0 < β < 1, such that:
then there exists a constant K = K(β), such that:
for every t ∈ [0, b].
Proof. From Gronwall’s lemma for singular kernels, whose proof can be found in Lemma 7.1.1 on page 188 of [33], we know that:
where c ∈ ℝ is a constant dependent on β. Now,
is bounded for s ∈ [0, b] and for s → t−:
where a > 0 is a constant. This implies that:
with constant K = K(β).
Lemma 3. Let (X, d) be a non-empty complete metric space with a contraction mapping T : X → X. Then, T admits a unique fixed-point x∗ in X(i.e., T (x∗) = x∗).
Lemma 4. If K is a convex subset of a topological vector space V and T is a continuous mapping of K into itself, so that T (K) is contained in a compact subset of K, then T has a fixed point.
Lemma 5. Consider a sequence of real-valued continuous functions {fn}n∈ℕ defined on a closed and bounded interval [0, b]. If this sequence is uniformly bounded and equicontinuous, then there exists a subsequence that converges uniformly.
For further readings and details on fractional calculus, we refer to the books and papers by Kilbas et al. [5], Podlubny [7] and Samko et al. [8].
3. Existence and Uniqueness
In this section, we prove some existence results and describe the structure of the solution set. We first note that if y ∈ C[0, b] is an absolutely continuous function on [0, b] satisfying Equation (1), then:
Theorem 1. Assume that there exist K1, K2 > 0, such that:
If, then there exists unique y ∈ C(J, ℝ), which satisfy Equation (1).
Proof. Consider N : C(J, ℝ) → C(J, ℝ) defined by:
Let z1, z2 ∈ C(J, ℝ), then:
Hence:
From the Banach fixed point theorem, Lemma 3, there exists a unique z ∈ C(J, ℝ), such that z = N(z).
Therefore:
Set
This implies that cDαy(t) = z(t), and hence:
The goal of the second result of this section is to apply the Schauder fixed point theorem, Lemma 4. For the study of this problem, we present some auxiliary lemmas.
Lemma 6. Let ϵ, ϵ′ ∈ (0, 1) and be an operator defined by:
Then, is a linear, continuous and invertible operator.
Proof. It is clear that is a linear operator. For every z ∈ C1(J, ℝ), we have:
Hence, is continuous. Now, we show that if (z) = 0, then z = 0, that is is injective. Indeed, let z ∈ C(J, ℝ), such that (z) = 0, then:
From Lemma 1, we have: z = 0.
Let h ∈ C(J, ℝ); we consider the fractional Cauchy problem:
Again from Lemma 1, we obtain that:
This implies that is bijective, and from the Banach isomorphism theorem, −1 is a continuous operator.
Lemma 7. Let F : C1(J, ℝ) → C(J, ℝ) be an operator defined by:
We note
Assume:
(H1) there exist M1, M3 > 0, M2 ∈ (0, 1), such that:
Then, F is continuous and compact.
Proof. The proof will be given in three steps.
Step 1: F is continuous.
Let (zn)n∈ℕ ⊂ C1(J, ℝ) be a sequence, such that (zn)n∈ℕ → z ∈ C1(J, ℝ).
Then:
Thus:
Furthermore:
and hence:
where:
Hence:
Step 2: F sends bounded sets in C1(J, ℝ) into bounded sets in C1(J, ℝ).
For each t ϵ J and
, we have:
Then:
Step 3: F maps bounded sets into equicontinuous sets.
Let t1; t2 ϵ J, such that t1 < t2 and
; we have:
where:
Then, the right-hand side tends to zero as t2 − t1 → 0. By the Arzelá-Ascoli theorem, Lemma 5, we conclude that F : C1(J, ℝ) → C(J, ℝ) is a compact continuous operator.
Now, we present a result without Lipschitz condition.
Theorem 2. Suppose that there is a constant M > 0, such that for each λ ∈ [0, 1) and z ∈ C1(J, ℝ) with, we have:
. Then, the implicit Problem (1) has at least one solution. Moreover, if 0 < M2 < 1, then is compact.
Proof. From Lemmas 6 and 7, the operators
, F : C1(J, ℝ) → C(J, ℝ) are linear, continuous, invertible and continuous and compact, respectively. Hence,
is a compact continuous operator. Set:
Assume that there exists λ ∈ [0, 1) and z ∈ ∂U, such that
Hence:
which contradicts with (H2). As a consequence of the nonlinear alternative of Leray–Schauder, we deduce that
has a fixed point z in U, which is a solution to Problem (1). Now, we show that
is compact. Let
, then:
Thus,
and:
Let:
Thus:
where:
and:
By Lemma 2, there exists K3 > 0, such that:
Therefore,
Additionally, since
:
and:
This implies that there exists K4 > 0, such that:
Therefore,
is bounded. Hence,
is compact. Now, since
is a continuous operator, the set
is a closed set. It is clear that
and thus, we conclude that
is compact.
By the approximation method, we present the existence of the solution for Problem (1):
Theorem 3. Assume that:
(H3) for each ϵ > 0, there exists δ > 0 and k = k(ϵ), with:
such that, if |t1 − t2| < δ,
,
, then we have:
If the approximation sequence (zn)n∈ℕ defined by:
is bounded, then there exist at least one solution of Problem (1).
Proof. Since (zn)n∈ℕ is bounded, we only need to show that (zn)n∈ℕ is equicontinuous. We show this fact by induction. We consider n = 1; let t1, t2 ∈ J. Then, by (H3) we have, for > 0; we consider r > 0, such that k(r) ≤ ϵ there exists δ > 0, such that for
, |t1 − t2| < η;
we have:
We suppose that
, |t1 − t2| < η;
for each p ≤ n: Let p = n + 1:
Set:
Without loss of generality, we can assume that t1 ≥ t2:
where M = ||z||∞.
By (H3), we have:
Hence, by the Arzelá-Ascoli theorem, Lemma 5, we conclude that (zn)n∈ℕ is relatively compact in C(J, ℝ). Then, there exists a subsequence
, which converges to some limit z. Since f is continuous, this implies that:
Example 1. Consider the Cauchy problem:
where f : J × ℝ → ℝ, h : ℝ → ℝ are continuous functions. Assume that there exist K1, K2 > 0, such that:
If, then, from Theorem 1, Problem (11) has a unique solution.
For instance, take J = [0, 1] and with and t ϵ J and y ϵ ℝ. Then, Problem (11) becomes:
It is clear that the functions f and h in Problem (12) are continuous and:
We have:
This implies that Problem (12) has a unique solution.
4. Conclusions
We have proven an existence result for implicit fractional differential equations. In the future, we will extend the results to other fractional derivatives and boundary value problems.
Acknowledgments
The research has been partially supported by the Ministerio de Economía y Competitividad of Spain under Grants MTM2010-15314 and MTM2013–43014–P, XUNTA de Galicia, the local government, under Grant R2014/002 and co-financed by the European Community fund FEDER.
Author Contributions
Each of the authors, Juan J. Nieto, Abelghani Ouahab and V. Venktesh, contributed to each part of this study equally and read and approved the final version of this manuscript.
Conflicts of Interest
The authors declare no conflict of interest.
References
- Gaul, L.; Klein, P.; Kempfle, S. Damping description involving fractional operators. Mech. Syst. Signal Process. 1991, 5, 81–88. [Google Scholar]
- Mainardi, F. Fractional calculus: Some basic problems in continuum and statistical mechanis. In Fractals and Fractional Calculus in Continuum Mechanics; Carpinteri, A., Mainardi, F., Eds.; Springer-Verlag: Wien, Austria, 1997; pp. 291–348. [Google Scholar]
- Metzler, F.; Schick, W.; Kilian, H.G.; Nonnenmacher, T.F. Relaxation in filled polymers: A fractional calculus approach. J. Chem. Phys. 1995, 103, 7180–7186. [Google Scholar]
- Abbas, S.; Benchohra, M.; N’Guérékata, G.M. Topics in Fractional Differential Equations; Springer: New York, NY, USA, 2012. [Google Scholar]
- Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006; Volume 204, North-Holland Mathematics Studies. [Google Scholar]
- Miller, K. S.; Ross, B. An Introduction to the Fractional Calculus and Differential Equations; John Wiley: New York, NY, USA, 1993. [Google Scholar]
- Podlubny, I. Fractional Differential Equations; Academic Press: San Diego, CA, USA, 1999. [Google Scholar]
- Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives, Theory and Applications; Gordon and Breach: Yverdon, Switzerland, 1993. [Google Scholar]
- Diethelm, K. The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type; Springer-Verlag: Berlin, Germany, 2010; Volume 2004, Lecture Notes in Mathematics. [Google Scholar]
- Aghajani, A.; Banaś, J.; Jalilian, Y. Existence of solutions for a class of nonlinear Volterra singular integral equations. Comput. Math. Appl. 2011, 62, 1215–1227. [Google Scholar]
- Ahmad, B.; Nieto, J.J. Anti-periodic fractional boundary value problems with nonlinear term depending on lower order derivative. Fract. Calc. Appl. Anal. 2012, 15, 451–462. [Google Scholar]
- Benchohra, M.; Henderson, J.; Ntouyas, S.K.; Ouahab, A. Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 2008, 338, 1340–1350. [Google Scholar]
- Chalishajar, D.N.; Karthikeyan, K. Existence and uniqueness results for boundary value problems of higher order fractional integro-differential equations involving Gronwall’s inequality in Banach spaces. Acta Math. Sci. 2013, 33, 758–772. [Google Scholar]
- Chalishajar, D.N.; Karthikeyan, K. Boundary value problems for impulsive fractional evolution integro-differential equations with Gronwall’s inequality in Banach spaces. Discontin. Nonlinearity Complex. 2014, 3, 33–48. [Google Scholar]
- Henderson, J.; Ouahab, A. Fractional functional differential inclusions with finite delay. Nonlinear Anal. 2009, 70, 2091–2105. [Google Scholar]
- Henderson, J.; Ouahab, A. Impulsive differential inclusions with fractional order. Comput. Math. Appl. 2010, 59, 1191–1226. [Google Scholar]
- Ouahab, A. Some results for fractional boundary value problem of differential inclusions. Nonlinear Anal. 2008, 69, 3877–3896. [Google Scholar]
- Ouahab, A. Semilinear fractional differential inclsions. Comput. Math. Appl. 2012, 64, 3235–3252. [Google Scholar]
- Benavides, T.D. An existence theorem for implicit differential equations in a Banach space. Ann. Mat. Pura Appl. 1978, 4, 119–130. [Google Scholar]
- Emmanuele, G. Convergence of successive approximations for implicit ordinary differential equations in Banach spaces. Funkc. Ekvac. 1981, 24, 325–330. [Google Scholar]
- Emmanuele, G.; Ricceri, B. On the existence of solutions of ordinary differential equations in implicit form in Banach spaces. Ann. Mat. Pura Appl. 1981, 129, 367–382, In Italian. [Google Scholar]
- Hokkanen, V.M. Continuous dependence for an implicit nonlinear equation. J. Differ. Equ. 1994, 110, 67–85. [Google Scholar]
- Hokkanen, V.M. Existence of a periodic solution for implicit nonlinear equations. Differ. Integral Equ. 1996, 9, 745–760. [Google Scholar]
- Li, D. Peano’s theorem for implicit differential equations. J. Math. Anal. Appl. 2001, 258, 591–616. [Google Scholar]
- Ricceri, B. Lipschitz solutions of the implicit Cauchy problem g(x′) = f(t, x), x(0) = 0, with f discontinuous in x. Rend. Circ. Mat. Palermo 1985, 34, 127–135. [Google Scholar]
- Seikkala, S.; Heikkilä, S. Uniqueness, comparison and existence results for discontinuous implicit differential equations. Nonlinear Anal. 1997, 30, 1771–1780. [Google Scholar]
- Benchohra, M.; Lazreg, J.E. Nonlinear fractional implicit differential equations. Commun. Appl. Anal. 2013, 17, 471–482. [Google Scholar]
- Benchohra, M.; Lazreg, J.E. Existence and uniqueness results for nonlinear implicit fractional differential equations with boundary conditions. Rom. J. Math. Comput. Sci. 2014, 4, 60–72. [Google Scholar]
- Vityuk, A.N.; Mikhailenko, A.V. The Darboux problem for an implicit differential equation of fractional order. Ukr. Mat. Visn. 2010, 7. [Google Scholar]J. Math. Sci. (N. Y.) 2011, 175, 391–401, In Russian.
- Caputo, M. Elasticità e Dissipazione; Zanichelli: Bologna, Italy, 1969. [Google Scholar]
- Caputo, M. Linear models of dissipation whose Q is almost frequency independent, Part II. Geophys. J. R. Astr. Soc. 1967, 13, 529–529. [Google Scholar]
- Caputo, M.; Mainardi, F. Linear models of dissipation in anelastic solid. Riv. Nuovo Cimento (Ser. II) 1971, 1, 161–198. [Google Scholar]
- Henry, D. Geometric Theory of Semilinear Parabolic Partial differential Equations; Springer-Verlag: Berlin, Gremany; New York, NY, USA, 1989. [Google Scholar]
© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).