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Stability of Ulam–Hyers and Ulam–Hyers–Rassias for a class of fractional differential equations
Advances in Difference Equations volume 2020, Article number: 103 (2020)
Abstract
In this paper, we investigate a class of nonlinear fractional differential equations with integral boundary condition. By means of Krasnosel’skiĭ fixed point theorem and contraction mapping principle we prove the existence and uniqueness of solutions for a nonlinear system. By means of Bielecki-type metric and the Banach fixed point theorem we investigate the Ulam–Hyers and Ulam–Hyers–Rassias stability of nonlinear fractional differential equations. Besides, we discuss an example for illustration of the main work.
1 Introduction
Fractional derivatives provide an effective instrument in the modeling of many physical phenomena. Fractional differential equations and fractional integral equations appeared in various fields such as polymer rheology, blood flow phenomena, electrodynamics of complex medium, modeling and control theory, signal processing, and so on; see [1, 2]. In recent years, many researchers proved the existence and uniqueness of solutions to fractional differential equations [3–8]. Moreover, integral boundary problems had a variety of applications in real-life problems such as blood flow, underground water flow, population dynamics, thermoplasticity, chemical engineering, and so on; see [9–11].
On the other hand, S.M. Ulam presented the stability problem of the solutions of functional equations (of group homomorphisms) in 1940 in a talk given at Wisconsin University [12]. In 1941, Hyers [13] gave the first answer to the question in Banach spaces. Since then, many researchers were interested in Ulam-type stability. With a wide expansion of the fractional calculus, the study of stability for fractional differential equations also attracted the attention of researchers [14, 15].
In 2011, Wang et al. [16] investigated the Ulam stability and data dependence for fractional differential equations with Caputo derivative:
where \({}^{c}D^{\alpha }(\cdot )\) is the Caputo fractional derivative, \(\alpha \in (0,1)\).
Abbas, Benchohra et al. [17] researched the existence and Ulam stability for the fractional differential equation
where \(\alpha \in (0,1)\), \(\beta \in [0,1]\), \(\gamma =\alpha +\beta -\alpha \beta \), \(T>1\), \(\phi \in \mathbb{R}\), \({}^{H}D^{\alpha ,\beta }_{1+}(\cdot )\) is the Hilfer–Hadamard fractional derivative, and \({}^{H}I^{\alpha , 1-\gamma }_{1+}(\cdot )\) is the Hadamard integral.
Chalishajar et al. [18] discussed the existence, uniqueness, and Ulam–Hyers stability of solutions for the following coupled system of fractional differential equations with integral boundary conditions:
where \({}^{c}D^{\alpha }_{0+}(\cdot )\) and \({}^{c}D^{\beta }_{0+}(\cdot )\) are the Caputo fractional derivatives, \(p,\widetilde{p}>0\), \(q, \widetilde{q}\geq 0\) are real numbers, and \(a_{1}\), \(a_{2}\), \(\widetilde{a}_{1}\), \(\widetilde{a}_{2}\) are continuous functions.
Vanterler da C. Sousa et al. [19–21] studied the ψ-Hilfer fractional derivative and the stability of Hyers–Ulam–Rassias and Hyers–Ulam of the following Volterra integro-differential equation [14]:
where \(f(t,u)\) is a continuous function with respect to the variables t and u on \(I\times \mathbb{R}\), \(K(t,s,u)\) is continuous with respect to t, s, and u on \(I\times I\times \mathbb{R}\), σ is a given constant, \({}^{H}D^{\alpha ,\beta ;\psi }_{0+}(\cdot )\) is the right-sided ψ-Hilfer fractional derivative with \(\alpha \in (0,1)\) and \(\beta \in [0,1]\), and \(I^{1-\gamma }_{0+}(\cdot )\) is the ψ-Riemann–Liouville fractional integral with \(\gamma \in [0,1)\).
In this paper, we consider the following class of fractional differential equations with integral boundary condition:
where \({}^{c}D^{\alpha }_{0+}(\cdot )\) is the Caputo derivative with \(0<\alpha <1\), \(I^{\beta }_{0+}(\cdot )\) is the Riemann–Liouville fractional integral with \(\beta >0\), \(\eta \in (0,1]\) is a fixed real number, \(u\in C^{1}[0,1]\), and \(f:[0,1]\times \mathbb{R}\rightarrow \mathbb{R}\) is a continuous function.
Equation (1.1) represents a constitutive relation for viscoelastic model of fractional differential equation. Equation (1.1) [22] can also be used to describe macroscopic models II for electrodiffusion of ions in nerve cells when molecular diffusion is anomalous subdiffusion due to binding, crowding, or trapping.
This paper is organized as follows. In the second section, we recall some basic definitions of fractional calculus, the concepts of Ulam–Hyers and Ulam–Hyers–Rassias stability for Eq. (1.1) and fixed point theorems. In the third section, we investigate the existence and uniqueness of solutions for problem (1.1)–(1.2). Moreover, we discuss the Ulam–Hyers and Ulam–Hyers–Rassias stability for Eq. (1.1). In the last section, we provide an illustrative example.
2 Preliminaries
In this section, we recall some useful definitions, notations, and the fundamental results about fractional derivatives (refer to [23, 24] and [25]). Also, we present the concepts of the Ulam–Hyers and Ulam–Hyers–Rassias stability for Eq. (1.1).
Let \(C^{1}[0,1]=\{u | u\mbox{ is a differentiable function on }[0,1]\mbox{ and its derivative is continuous}\}\) with the norm
Definition 2.1
For a real-valued integrable function \(f:(0,+\infty )\rightarrow \mathbb{R}\), the Riemann–Liouville fractional integral of order \(0<\alpha <1\) is defined as
where Γ is the gamma function.
Definition 2.2
The Caputo fractional derivative \({}^{c}D_{0+}^{\alpha }\) of an absolutely continuous (or differentiable) function \(f(t)\) of order \(0<\alpha <1\) is defined as
Definition 2.3
The two-parametric Mittag-Leffler function is defined as
The Laplace transform of the Caputo derivative \({}^{c}D_{0+}^{\alpha }f(t)\) is
The Laplace transform of the two-parametric Mittag-Leffler function is
where \(E_{\alpha ,\beta }^{(k)}(y)=\frac{d^{k}}{dy^{k}}E_{\alpha ,\beta }(y)= \sum_{j=0}^{\infty } \frac{(j+k)!y^{j}}{j!\varGamma (\alpha j+\alpha k+\beta )}\), \(k=0,1,2, \ldots \) .
Definition 2.4
([26])
A function f is of exponential order λ if there exist constants \(M>0\) and λ such that for some \(t_{0}>0\),
Next, we present the concepts of the Ulam– and Ulam–Hyers–Rassias stability for Eq. (1.1). The following Definitions 2.5 and 2.6 are adapted from [14].
Definition 2.5
If \(x(t)\) is a continuously differentiable function satisfying
where \(\theta > 0\), and there are a solution \(u(t)\) of Eq. (1.1) and a constant \(C>0\) independent of \(x(t)\) and \(u(t)\) such that
then we say that the Eq. (1.1) has the Ulam–Hyers stability.
Definition 2.6
If \(x(t)\) is a continuously differentiable function satisfying
where \(\sigma :[0,1]\rightarrow [0,+\infty )\) is a continuous function, and there exist a solution \(u(t)\) of Eq. (1.1) and a constant \(C>0\) independent of \(x(t)\) and \(u(t)\) such that
then we say that the Eq. (1.1) has the Ulam–Hyers–Rassias stability.
Theorem 2.1
([27] (Krasnosel’skiĭ fixed point theorem))
LetMbe a closed convex and nonempty subset of a Banach spaceX. LetA, Bbe operators such that
- (i)
\(Ax+By\in M\)whenever\(x,y\in M\),
- (ii)
Ais compact and continuous,
- (iii)
Bis a contraction mapping.
Then there exists\(z\in M\)such that\(z=Az+Bz\).
Theorem 2.2
([28] (Banach))
Let\((X,d)\)be a generalized complete metric space. Assume that\(\varLambda :X\rightarrow X\)is a strictly contractive operator with Lipschitz constant\(K<1\). If there exists a nonnegative integerksuch that\(d(\varLambda ^{k+1}x,\varLambda ^{k}x)<\infty \)for some\(x\in X\), then the following three propositions hold:
- (1)
The sequence\(\{\varLambda ^{n}x\}\)converges to a fixed point\(x^{*}\)ofΛ;
- (2)
\(x^{*}\)is a unique fixed point ofΛin\(X^{*}=\{y\in X: d(\varLambda ^{k}x,y)<\infty \}\);
- (3)
If\(y\in X^{*}\), then\(d(y,x^{*})\leq \frac{1}{1-K}d(\varLambda y,y)\).
Theorem 2.3
([26])
Iffis piecewise continuous function on\([0,\infty )\)of exponential order λ, then the Laplace transform\(\mathcal{L}(f(t))\)exists for\(\operatorname{Re}(s)>\lambda \)and converges absolutely.
3 Main results
In this section, we derive the existence and uniqueness of solutions for the integral boundary problem (1.1)–(1.2). Moreover, we study the Ulam–Hyers and Ulam–Hyers–Rassias stability for Eq. (1.1).
3.1 Existence and uniqueness results
In this subsection, by means of the Krasnosel’skiĭ fixed point theorem and contraction mapping principle, we investigate the existence and uniqueness of solutions for problem (1.1)–(1.2) in \(C^{1}[0,1]\).
Lemma 3.1
Let\(u(t)\in C^{1}[0,1]\), \(0<\alpha <1\), \(\beta >0\). For any\(g \in C[0,1]\)and\(\eta \in (0,1]\), the solution of the boundary value problem
is given by
where\(G(t,s)\)is called the Green’s function of problem (3.1)–(3.2)and is given by
Proof
Since \(u(t)\in C^{1}[0,1]\), \(u(t)\) and \({}^{c}D^{\alpha }_{0+}u(t)\) are bounded. For any \(t\in [0,1]\), we have that \(u'\) and \({}^{c}D^{\alpha }_{0+}\) are of exponential order. By Theorem 2.3 and Definition 2.4 we have that the Laplace transform of both \(u'(t)\) and \({}^{c}D^{\alpha }_{0+}u(t)\) exist for \(u(t)\in C^{1}[0,1]\).
Taking the Laplace transform on both sides of Eq. (3.1), by Eq. (2.1) we obtain
Using the inverse Laplace transform, by Eq. (2.2) we have
Equation (3.4) is the equivalent fractional integral equation of Eq. (3.1), so we have
From Eqs. (3.2) and (3.5) we have
Therefore the unique solution of problem (3.1)–(3.2) is
where \(G(t,s)\) is given by (3.3). This completes the proof. □
Remark 3.1
By the definition of the two-parameter Mittag-Leffler function we get
which is a convergent series of real numbers. Therefore there exists a constant \(E_{1-\alpha ,2}>0\) such that
Moreover, by Eq. (3.3) and the continuity of the two-parameter Mittag-Leffler function there exists a constant \(M>0\) such that
Remark 3.2
For a continuous function \(f:[0,1]\times \mathbb{R}\rightarrow \mathbb{R}\), there exists a constant \(N>0\) such that
Theorem 3.1
Let\(0<\alpha <1\), \(\beta >0\), and\(\eta \in [0,1]\)be fixed real numbers. Let\(f:[0,1]\times \mathbb{R}\rightarrow \mathbb{R}\)be a continuous function satisfying the Lipschitz condition with respect to second argument, that is,
for all\(t\in [0,1]\)and\(h_{1},h_{2}\in \mathbb{R}\), where\(L>0\)is a Lipschitz constant. Then problem (1.1)–(1.2) has unique solution in\(C^{1}[0,1]\), provided that
Proof
By Lemma 3.1 the equivalent fractional integral of problem (1.1)–(1.2) is given by
where \(G(t,s)\) is given by Eq. (3.3).
Consider the operator T defined on \(C^{1}[0,1]\) by
By the continuity of the functions \(G(t,s)\) and \(f(t,u(t))\) we have \(Tu\in C^{1}[0,1]\) for any \(u\in C^{1}[0,1]\). This proves that T maps \(C^{1}[0,1]\) into itself.
We define the set \(\mathcal{B}=\{u\in C^{1}[0,1]:\|u\|<\delta \}\) and choose \(\delta > \frac{MN \varGamma (\beta +1)}{\varGamma (\beta +1)-\eta ^{\beta }}\).
From Eq. (3.6), for \(u\in \mathcal{B}\), we obtain
Hence \(T\mathcal{B}\subseteq \mathcal{B}\).
Next, we show that T is a contraction operator. For \(u_{1}, u_{2}\in C^{1}[0,1]\) and \(t\in [0,1]\), from Eq. (3.6), using the Lipschitz condition on f, we have
As \(\frac{\eta ^{\beta }}{\varGamma (\beta +1)}+LM<1\), T is a contraction mapping. By the contraction mapping principle it has a unique fixed point, which is the unique solution of problem (1.1)–(1.2). □
Theorem 3.2
Let\(f:[0,1]\times \mathbb{R}\rightarrow \mathbb{R}\)be a continuous function, and letfsatisfy the Lipschitz condition with respect to second argument:
for all\(t\in [0,1]\)and\(h_{1},h_{2}\in \mathbb{R}\), where\(L>0\)is a Lipschitz constant, and
Then problem (1.1)–(1.2) has at least one solution in\(C^{1}[0,1]\).
Proof
We consider the operators A and B on \(C^{1}[0,1]\) defined by
Consider \(W_{r}=\{u\in C^{1}[0,1]:\|u\|\leq r\}\) and choose
For any \(u,v\in W_{r}\), having in mind Remark 3.1, Remark 3.2, and the definitions of the operators A and B, we conclude that
Therefore we obtain \(Au+Bv\in W_{r}\).
By Theorem 3.1 the operator B is a contraction mapping if
It follows from the proof of the operator T.
By the continuity of the two-parameter Mittag-Leffler function and \(f(t.u(t))\), for any continuous function \(u\in W_{r}\), the operator A is continuous.
For any \(u\in W_{r}\), from Remarks 3.1 and 3.2 we have
Hence A is uniformly bounded on \(W_{r}\).
For any \(u\in W_{r}\) and \(t_{1}, t_{2}\in [0,1]\) such that \(t_{1}< t_{2}\),
The constant \(NE_{1-\alpha ,1}(-(1-\theta )^{1-\alpha })\) is independent of u, so A is relatively compact on \(W_{r}\). Therefore by the Arzelà–Ascoli theorem the operator A is compact on \(W_{r}\). By Theorem 2.1 problem (1.1)–(1.2) has at least one solution on \([0,1]\). □
3.2 Ulam–Hyers–Rassias and Ulam–Hyers stability
In this subsection, by means of the Bielecki metric and Banach fixed-point theorem we investigate the Ulam–Hyers–Rassias and Ulam–Hyers stability in \(C^{1}[0,1]\) for Eq. (1.1).
Consider the space \(C^{1}[0,1]\) endowed with the Bielecki-type metric
where \(\sigma :[0,1]\rightarrow (0,\infty )\) is a nondecreasing continuous function. Obviously, \((C^{1}[0,1],d)\) is a complete metric space.
Theorem 3.3
Let\(f:[0,1]\times \mathbb{R}\rightarrow \mathbb{R}\)be a continuous function satisfying the Lipschitz condition
with\(L>0\). Moreover, let\(\sigma :[0,1]\rightarrow (0,\infty )\)be a nondecreasing continuous function, and suppose that there exists a constant\(\xi \in [0,1)\)such that
If\(x\in C^{1}[0,1]\)satisfies
and\(L\xi <1\), then there exists a solution\(u(t)\)of Eq. (1.1) in\(C^{1}[0,1]\)such that
This means that under the above conditions, the fractional differential Eq. (1.1) has the Ulam–Hyers–Rassias stability.
Proof
By Lemma 3.1 the equivalent fractional integral equation of Eq. (1.1) is given by
which follows from the proof of Eq. (3.4).
We conclude that \(u(t)\) satisfies Eq. (1.1) if and only if \(u(t)\) satisfies the integral Eq. (3.10).
We consider the operator \(\varLambda :C^{1}[0,1]\rightarrow C^{1}[0,1]\) defined by
By the continuity of the two-parameter Mittag-Leffler function and f the operator Λ is continuous.
First, we prove that the operator Λ is strictly contractive in \((C^{1}[0,1],d)\). From Eq. (3.7), for any \(v,w\in C^{1}[0,1]\), we obtain
Since \(L\xi <1\), the operator Λ is strictly contractive.
On the other hand, let \(x\in C^{1}[0,1]\) satisfy Eq. (3.8). By the Laplace transform and the inverse Laplace transform we obtain that x satisfies
which follows from the proof of Lemma 3.1.
By Eq. (3.7), Eq. (3.11), and the definition of the operator Λ we get
Therefore we conclude that
By means of item 2 of Theorem 2.2 there exists a unique element
such that \(\varLambda u=u\) or, equivalently,
Since Eq. (3.10) is the equivalent integral equation of Eq. (1.1), we conclude that \(u(t)\) is a solution of Eq. (1.1). Also, from item 3 of Theorem 2.2 and Eq. (3.12) we have
By the definition of d we obtain that inequality (3.9) holds. □
Theorem 3.4
Let\(f:[0,1]\times \mathbb{R}\rightarrow \mathbb{R}\)be a continuous function satisfying the Lipschitz condition
with\(L>0\). Moreover, let\(\sigma :[0,1]\rightarrow (0,\infty )\)be a nondecreasing continuous function, and suppose that there exists a constant\(\xi \in [0,1)\)such that
and\(L\xi <1\). If\(x\in C^{1}[0,1]\)satisfies
where\(\theta >0\), then there exists a solution\(u(t)\)of Eq. (1.1) in\(C^{1}[0,1]\)such that
This means that under the above conditions, the fractional differential Eq. (1.1) has the Ulam–Hyers stability.
Proof: The first part of the proof follows the same steps as in the proof of Theorem 3.3. Consider the operator \(\varLambda :C^{1}[0,1]\rightarrow C^{1}[0,1]\) defined by
For any \(v,w\in C^{1}[0,1]\), we have
Since \(L\xi <1\), we conclude that the operator Λ is strictly contractive in \((C^{1}[0,1],d)\), which follows from the proof of Theorem 3.3.
Suppose that \(x\in C^{1}[0,1]\) satisfies Eq. (3.14). By means of the Laplace transform, the inverse Laplace transform, and Remark 3.1 we obtain
Now by the definition of the operator Λ we get
Since σ is a positive nondecreasing function, we have
Having in mind item 2 of Theorem 2.2, there exists a unique element
such that \(\varLambda u=u\), which means that \(u(t)\) is a solution of Eq. (1.1).
Thus from item 3 of Theorem 2.2 and Eq. (3.16) it follows that
By the definition of the Bielecki-type metric d we obtain
4 Example
Example 4.1
Consider the following fractional differential equation with integral boundary condition:
where \(u(t)\in C^{1}[0,1]\).
Comparing with problem (1.1)–(1.2), we have
Clearly, we obtain
Here we get \(L=N=\frac{1}{8}\).
Further, by Remark 3.1 we have
Therefore by Theorems 3.1 and 3.2 problem (4.1)–(4.2) has a unique solution.
Next, we investigate the Ulam–Hyers and Ulam–Hyers–Rassias stability for Eq. (4.1).
Letting \(\sigma (t)=e^{t}\), by Remark 3.1 we obtain
Thus \(\sigma (t)=e^{t}\) satisfies Eq. (3.7) with \(\xi =\frac{3}{4}\), and \(L\xi =\frac{3}{32}<1\).
Hence Theorem 3.3 guarantees that Eq. (4.1) has the Ulam–Hyers–Rassias stability. Further, Theorem 3.4 guarantees that Eq. (4.1) has the Ulam–Hyers stability.
The Ulam–Hyers and Ulam–Hyers–Rassias stability for Eq. (4.1) is independent of the initial value condition. Using MATLAB, the solution \(u(t)\) of Eq. (4.1) with initial value condition \(u(0)=0\) is computed and depicted in Fig. 1.
Now consider \(x\in C^{1}[0,1]\), the solution of the following fractional differential equation:
We conclude that x satisfies Eq. (3.8). Therefore we have
see Fig. 2.
On the other hand, consider \(y\in C^{1}[0,1]\), the solution of the following fractional differential equation:
Then y satisfies Eq. (3.8), and we have
see Fig. 2.
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The authors are grateful to anonymous referees for their constructive comments and suggestions which have greatly improved this paper.
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This work was supported by China Scholarship Council (No. 201807585008), the Natural Science Foundation of China (No. 11501051), Science and Technology Development Foundation of Jilin Province (No. 20180520025JH), Thirteen Five-year Science and Technology Research Plan Project of Jilin Province Education Department (No. JJKH20190547KJ).
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Dai, Q., Gao, R., Li, Z. et al. Stability of Ulam–Hyers and Ulam–Hyers–Rassias for a class of fractional differential equations. Adv Differ Equ 2020, 103 (2020). https://doi.org/10.1186/s13662-020-02558-4
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DOI: https://doi.org/10.1186/s13662-020-02558-4