- Research
- Open Access
Sequential evolution conformable differential equations of second order with nonlocal condition
- Mohamed Bouaouid1Email author,
- Khalid Hilal1 and
- Said Melliani1
https://doi.org/10.1186/s13662-019-1954-2
© The Author(s) 2019
- Received: 20 August 2018
- Accepted: 10 January 2019
- Published: 23 January 2019
Abstract
In this paper, we investigate second order evolution differential equation in the frame of sequential conformable derivatives with nonlocal condition. First, we establish Duhamel’s formula in terms of a standard cosine family of linear operators. Then, we prove some results concerning the existence, uniqueness, stability, and regularity of mild solution concept. Moreover, we present a concrete application of the main results.
Keywords
- Fractional differential equations
- Cosine family of linear operators
- Conformable derivative and nonlocal condition
MSC
- 34A08
- 47D09
1 Introduction
Differential equations with nonlocal conditions play a crucial role in numerous fields of science, physics, engineering, and so on. The theory of such equations with respect to different types of derivatives has been investigated by many authors. For the well-known classical derivative, the second order Cauchy problem with nonlocal condition was studied by Hernández [11]. In recent years, fractional differential equations have been increasingly used to formulate many problems in biology, chemistry, and other areas of applications [13–15, 17]. For Caputo’s fractional derivative, a fractional Cauchy problem of order \(\beta \in (1,2)\) with nonlocal condition was treated in [18] by Shur et al. Mainly, they studied the existence and uniqueness of the corresponding mild solution. For physical interpretations of nonlocal condition, we refer to [8, 9, 16].
The conformable derivative was introduced by Khalil et al. [12]. It is well commented in a nice paper of Al-Refai et al. [5] in which they study the Sturm–Liouville eigenvalue problems with respect to the conformable derivative. Moreover, many interesting problems, associated with the conformable derivative, have been investigated. For more details, we refer to the works [1–4, 7, 10].
The notion of sequential fractional derivative was considered in the famous book [14, p. 209] in which a complete study of some special class of sequential differential equations with respect to Caputo’s derivative was given. Attracted by this type of problem, many authors have been interested in the sequential differential equations with respect to various fractional derivative types [6, 22, 23].
Based on the fact that the sequential problem (1.1) is well adapted with the fractional Laplace transform [1], we will be interested in the mild solutions of the above nonlocal Cauchy problem. Our method shares similarities with the standard techniques used in the classical cases [11, 21]. Precisely, we use the classical cosine family to elaborate a formula of Duhamel type. This formula leads us to treating our problem by using fixed point theory. Concretely, under the compactness of the cosine family associated with the operator A and the boundedness condition for the function \(f(t,x)\), we prove that problem (1.1) admits at least one solution. Furthermore, by adding some contraction conditions, we prove the uniqueness of the mild solution and its continuous dependance with respect to initial data. Moreover, under some regularity conditions for the function \(f(t,x)\) combined with a suitable condition on the domain \(D(A)\), we obtain the differentiability of the mild solution with respect to the conformable derivative.
This paper is summarized as follows. In Sect. 2, we review some tools related to the conformable derivative as well as some needed results. Section 3 will be devoted to the statements and the proof of the main results. In Sect. 4, as application, we study a concrete sequential conformable second order partial differential equation with nonlocal condition. In Sect. 5, we tried to discuss the problem of a definition for α-cosine family.
2 Preliminaries
We start this by recalling some concepts on conformable calculus [12].
Definition 2.1
Theorem 2.1
The following definition gives us the adapted Laplace transform to the conformable derivative [1].
Definition 2.2
The action of the fractional Laplace transform on the conformable derivative is given by the following proposition.
Proposition 2.1
Now, we recall some results concerning the cosine family theory [21].
Definition 2.3
- 1.
\(C(0)=I\);
- 2.
\(C(s+t)+C(s-t)=2C(s)C(t)\) for all \(t,s\in \mathbb{R}\);
- 3.
\(t\longmapsto C(t)x\) is continuous for each fixed \(x\in X\).
Proposition 2.2
- 1.There exist constants \(K\geq 1\) and \(\omega \geq 0\) such that$$ \bigl\vert S(t)-S(s) \bigr\vert \leq K \biggl\vert \int _{s}^{t}\exp \bigl(\omega \vert r \vert \bigr) \biggr\vert \,dr \quad \textit{for all } t,s\in \mathbb{R}. $$
- 2.If \(x\in X\) and \(t,s\in \mathbb{R}\), then \(\int _{s}^{t}S(r)x\,dr\in D(A)\) and$$ A \int _{s}^{t}S(r)x\,dr=C(t)x-C(s)x. $$
- 3.
If \(t\longmapsto C(t)x\) is differentiable, then \(S(t)x\in D(A)\) and \(\frac{d C(t)}{dt}x=AS(t)x\).
- 4.For λ such that \(\operatorname{Re}(\lambda )>\omega \), we have$$\begin{aligned}& \lambda ^{2}\in \rho (A), \quad \bigl(\rho (A): \textit{is the resolvent set of } A \bigr), \\& \lambda \bigl(\lambda ^{2}I-A \bigr)^{-1}x= \int _{0}^{+\infty }e^{-\lambda t}C(t)x\,dt, \quad x\in X, \\& \bigl(\lambda ^{2}I-A \bigr)^{-1}x= \int _{0}^{+\infty }e^{-\lambda t}S(t)x\,dt, \quad x\in X. \end{aligned}$$
3 Main results
- \((H_{1})\) :
-
The function \(f(t,\cdot ): X\longrightarrow X\) is continuous, and for all \(r>0\), there exists a function \(\mu _{r}\in L ^{\infty }([0,\tau ],\mathbb{R}^{+})\) such that \({\displaystyle{\sup_{\Vert x\Vert \leq r}}}\Vert f(t,x)\Vert \leq \mu _{r}(t)\) for all \(t\in [0,\tau ]\);
- \((H_{2})\) :
-
The function \(f(\cdot ,x):[0,\tau ] \longrightarrow X\) is continuous for all \(x\in X\);
- \((H_{3})\) :
-
There exists a constant \(l_{1}>0\) such that \(\Vert g(y)-g(x)\Vert \leq l_{1}\vert y-x\vert \) for all \(x,y\in \mathcal{C}\);
- \((H_{4})\) :
-
There exists a constant \(l_{2}>0\) such that \(\Vert h(y)-h(x)\Vert \leq l_{2}\vert y-x\vert \) for all \(x,y\in \mathcal{C}\).
3.1 Existence and uniqueness of the mild solution
Definition 3.1
Theorem 3.1
Proof
Now, we will show that \(\varGamma _{2}\) is continuous and compact.
Continuity of \(\varGamma _{2}\). Let \((x_{n})\subset B_{r}\) such that \(x_{n}\longrightarrow x\) in \(B_{r}\). Then, by using assumption \((H_{1})\), we obtain \(\Vert s^{\alpha -1}[f(s,x_{n}(s))-f(s,x(s))]\Vert \leq 2\mu _{r}(s)s^{\alpha -1}\) and \(f(s,x_{n}(s))\longrightarrow f(s,x(s))\) as \(n\longrightarrow +\infty \).
Compactness of \(\varGamma _{2}\). Claim 1: We prove that \(\{\varGamma _{2}(x)(t), x\in B_{r}\}\) is relatively compact in X.
Claim 2: We show that \(\varGamma _{2}(B_{r})\) is equicontinuous.
- \((H_{5})\) :
-
There exists a constant \(l_{3}>0\) such that \(\Vert f(t,y)-f(t,x)\Vert \leq l_{3}\Vert y-x\Vert \) for all \(x,y\in X\) and \(t\in [0,\tau ]\).
Theorem 3.2
Proof
3.2 Continuous dependence of the mild solution
Now, we will give some results concerning the continuous dependence of the mild solution.
Theorem 3.3
Proof
Theorem 3.4
Proof
Remark 3.1
3.3 Special case of nonlocal conditions
Proposition 3.1
Proof
3.4 Regularity of the mild solution
- \((H_{6})\) :
-
The function f is \((\alpha )\)-differentiable of the first variable and differentiable of the second variable.
- \((H_{7})\) :
-
\((x_{0}+g(x))\in D(A)\) and \(t\longmapsto C(t)[x_{0}+g(x)]\) is \((\alpha )\)-differentiable for all \(x\in \mathcal{C}\).
Theorem 3.5
Proof
4 Application
5 Comment
By noticing the relation \(C(t)=C((t^{\frac{1}{\alpha }})^{\alpha })\) for a cosine family \((C(t))_{t\in \mathbb{R}}\), it comes to us to consider the family of functions \(t\longmapsto C_{\alpha }(t):=C(t^{\alpha })\) and to propose as in the case of semigroup [4] the following definition for α-cosine family.
6 Conclusion
We have obtained Duhamel’s formula for sequential evolution conformable differential equations of second order with nonlocal condition. Under some suitable conditions, we have also obtained an existence result for the mild solution. In the case where the contraction condition type is satisfied, we have proved the uniqueness of the mild solution as well as its continuous dependence with respect to the initial data.
In the light of the above comment and as the anonymous referee has proposed, it would be interesting to consider non-sequential conformable second order differential equations with nonlocal condition in a coming paper.
Declarations
Acknowledgements
The authors express their sincere thanks to members of the Springer nature waivers team, Springer submission support, Springer transfer desk for their strong help. Also, the authors are very thankful to the anonymous referee for carefully reading the paper and for their comments and suggestions.
Funding
Not applicable.
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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