Abstract
In this paper, by means of the Krasnoselskii fixed point theorem, the existence of solutions for a boundary value problem of nonlinear sequential fractional integro-differential equations are investigated. Two examples are given to illustrate our results.
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1 Introduction
Fractional differential equations have attracted much attention and have been the focus of many studies due mainly to their varied applications in many fields of science and engineering. In other words, fractional differential equations are widely used to describe many important phenomena in various fields such as physics, biophysics, chemistry, biology, control theory, economy and so on; see [14, 19, 23, 29, 33]. For an extensive literature in the study of fractional differential equations, we refer the reader to [2, 11, 15, 16, 18, 20, 21, 24, 26, 30, 32]. However, it should be noted that in recent years, there have been many works related to fractional integro-differential equations, see [1, 3, 4, 6, 8, 12, 17, 22, 28, 29] and the references therein. For some interesting and considerable applied works, we refer to [5, 7, 9, 10].
In [13], Baleanu et al. studied the existence and uniqueness of solutions for the multiterm nonlinear fractional integro-differential equation
where \(1<\alpha <2\), \(0<\beta _{i}<1\), \(\alpha -\beta _{i}\geq 1\), \(a,b\neq -1\), \(f: [0,1] \times \mathbb{R} ^{n+3}\rightarrow \mathbb{R} \) is continuous, and for the mappings \(\gamma ,\lambda : [0,1] \times [0,1] \rightarrow [ 0,\infty ) \) with the property
the maps φ and ψ being defined by \(( \varphi u ) ( t ) =\int _{0}^{t}\gamma ( t,s ) \,ds\) and \(( \psi u ) ( t )= \int _{0}^{t} \lambda ( t,s ) \,ds \).
In [31], Wang et al. proved the existence and uniqueness of positive solutions for the following fractional integro-differential equation:
where \(n-1<\alpha \leq n\), −1, \(n\geq 3\), \(b_{i}\geq 0\)\(( i=1,2,\dots ,n-3,n-1 )\), \({}^{c}D^{\alpha }\) is the Caputo fractional derivative, \(f: [0,1] \times \mathbb{R} _{+}\times \mathbb{R} _{+}\times \mathbb{R} _{+}\rightarrow \mathbb{R} _{+}\) is continuous and \(( \varphi u ) ( t ) =\int _{0}^{t}K ( t,s ) u ( s ) \,ds\), \(( \psi u ) ( t ) =\int _{0}^{t}H ( t,s ) u ( s ) \,ds\).
Motivated by the previous results, we discuss in this paper the existence of solutions for the following nonlinear sequential fractional boundary value problem:
where \(1<\alpha \leq 2\), \(0<\beta \leq 1\), \(f: [0,1] \times \mathbb{R} ^{3}\rightarrow \mathbb{R} \) is continuous and
where \(\gamma ,\lambda : [0,1] \times [0,1] \rightarrow [ 0,+\infty ) \) are such that \(\sup_{t\in [0,1] } ( \int _{0}^{1} \lambda ( t,s ) \,ds ) <\infty \) and \(\sup_{t\in [0,1] } ( \int _{0}^{1} \gamma ( t,s ) \,ds ) <\infty \).
2 Preliminaries
For convenience, in this section we recall some basic definitions and properties of the fractional calculus theory and auxiliary lemmas which will be used throughout this paper, see [23, 25, 27].
Definition 2.1
The Caputo fractional derivative of order \(\alpha >0\) of a continuous function \(u: ( 0,\infty ) \rightarrow \mathbb{R} \) is defined by
provided the right-hand side is pointwise defined on \(( 0,\infty ) \).
Definition 2.2
The Riemann–Liouville fractional integral of order \(\alpha >0\) of a continuous function \(u: ( 0,\infty ) \rightarrow \mathbb{R} \) is defined by
provided the right-hand side is pointwise defined on \(( 0,\infty ) \).
Lemma 2.1
If \(\alpha >0\), then the differential equation \({}^{c}D^{\alpha }u ( t ) =0\)has a unique solution given by
where \(c_{i}\in \mathbb{R} \), \(i=0,1,\dots ,n-1\) (n is the smallest integer such that \(n\geq \alpha \)).
Lemma 2.2
For \(\alpha >0\), let \(u\in C^{n} [0,1] \). Then
where \(c_{i}\in \mathbb{R} \), \(i=0,1,\dots ,n-1\) (n is the smallest integer such that \(n\geq \alpha \)).
Lemma 2.3
If \(y\in C [0,1] \), then the boundary value problem
has the unique solution given by
Proof
In view of Lemma 2.2, FBVP (2.1) is equivalent to the following integral equation:
Differentiating both sides of (2.3), we get
Using the boundary conditions \(u ( 1 ) =u ( 0 ) =u^{\prime } ( 1 ) =0\), we obtain
Substituting the values of \(c_{0}\), \(c_{1}\), \(c_{2}\) in (2.3) we obtain (2.2). This completes the proof. □
3 Main results
Theorem 3.1
(Krasnoselskii fixed point theorem)
Let X be a closed convex and nonempty subset of a Banach space E. Let A and B be two operators such that
-
1.
\(Ax+By\in X\), whenever \(x,y\in X\);
-
2.
A is compact and continuous;
-
3.
B is a contraction.
Then there exists \(z\in X\)such that \(z=Az+Bz\).
Let \(X=C ( I ) \) be the space of all continuous real-valued functions on \(I= [0,1] \) endowed with the norm \(\Vert u \Vert =\max_{t\in I} \vert u ( t ) \vert \).
Theorem 3.2
Assume that \(\alpha +\beta -2\geq 0\)and there exists a nonnegative function \(\theta ( t ) \in L^{1} ( 0,1 ) \)such that
for all \(t\in [0,1] \)and \(t,x,y,z,t^{\prime },y^{\prime },z^{\prime }\in \mathbb{R} \). Then problem (1.1) has at least one solution on X whenever
where \(\gamma _{0}=\sup_{t\in I} \vert \int _{0}^{t}\gamma ( t,s ) \,ds \vert \), \(\lambda _{0}=\sup_{t\in I} \vert \int _{0}^{t}\lambda ( t,s ) \,ds \vert \), and \(\theta ^{\ast }=\int _{0}^{1}\theta ( s ) \,ds\).
Proof
Choose
and set \(\varpi =\max \{ f ( t,0,0,0 ) : t\in I \} \). Consider the set \(B_{R}= \{ u\in X: \Vert u \Vert \leq R \} \), then \(B_{R}\) is a closed, bounded, and convex set of X. We define the operators A and B on X as
For any \(u\in B_{R}\) and \(t\in I\), we get with the help of inequality (3.1)
Hence, we get
Similarly, we estimate \(\Vert Bv \Vert \). Let \(v\in B_{R}\) and \(t\in I\), then
Hence, we get
Taking estimates (3.3) and (3.4) into account, we get for any u, \(v\in B_{R}\) and \(t\in I\),
since if
then \(\Vert Au+Bv \Vert \leq R\).
Now, we prove that B is a contraction. Let \(v,u\in B_{R}\) and \(t\in I\). Then, thanks to (3.1), it yields
thus
so by (3.2) we conclude that B is a contraction.
Let us prove that A is compact and continuous. The continuity of f implies that A is continuous. Also A is uniformly bounded on \(B_{R}\), indeed, from (3.3) we have
Set \(L=\max_{0\leq s\leq 1} \{ \vert f ( s,u ( s ) ,\varphi u ( s ) ,\psi u ( s ) ) \vert ,u\in B_{R} \} \). Let \(u\in B_{R}\), \(t_{1}, t_{2}\in I\), with \(t_{1}\leq t_{2}\). We have
Hence, if \(t_{2}\rightarrow t_{1}\), then \(\vert Au ( t_{2} ) -Au ( t_{1} ) \vert \rightarrow 0\). Then A is equicontinuous and so, by Arzela–Ascoli theorem, we deduce that A is compact on \(B_{R}\). So the operator A is completely continuous. Thus, by Theorem 3.1, problem (1.1) has at least one solution in X. The proof is complete. □
Example 3.1
We consider the boundary value problem (1.1) with \(f ( t,x_{1},x_{2},x_{3} ) = \frac{t^{\frac{1}{2}}e^{-t}}{4}\times\sum_{i=1}^{3}\frac{1}{1+x_{i}^{2} ( t ) }\), \(\alpha =\frac{9}{5}\), \(\beta =\frac{3}{5}\). Also we have \(f ( t,0,0,0 ) =\frac{3t^{\frac{1}{2}}e^{-t}}{4}\) thus \(\varpi =0.31 \). Let \(\lambda ( t,s ) =\gamma ( t,s ) =ts\), so that \(\gamma _{0}=\lambda _{0}=\frac{1}{2}\).
Moreover, we can verify that condition (3.1) is satisfied
so \(\theta ( t ) =\frac{t^{\frac{1}{2}}e^{-t}}{4}\) and \(\theta ^{\ast }=\frac{0.37894}{4}\). Also, condition (3.2) holds:
Therefore, by Theorem 3.2, the problem has at least one solution in \(B_{R}\) with
Example 3.2
Consider the boundary value problem (1.1) with
\(\alpha =1.3\), \(\beta =0.4\). Then \(f ( t,0,0,0 ) =10^{-2} ( 1+t^{2} ) \), thus \(\varpi =0.02\). Let \(\lambda ( t,s ) =e^{t-s}\), \(\gamma ( t,s ) = ( t-s ) ^{\beta }\), thus \(\gamma _{0}=1.7183\), \(\lambda _{0}=0.71429\). Condition (3.1) is satisfied, in fact,
We choose \(\theta ( t ) =0.02e^{t}\) then \(\theta ^{\ast }=3.4366\times 10^{-2}\). We check condition (3.2):
We conclude, by Theorem 3.2, that the problem has at least one solution in \(B_{R}\) with
Conclusion. In the present work, we have studied the existence of solutions for a fractional sequential boundary value problem. To demonstrate the existence results, we transformed the posed problem into a sum of a contraction and a compact operator, then we applied the Krasnoselskii’s fixed point theorem. We ended the article with some numerical examples illustrating the obtain results.
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Bragdi, A., Frioui, A. & Guezane Lakoud, A. Existence of solutions for nonlinear fractional integro-differential equations. Adv Differ Equ 2020, 418 (2020). https://doi.org/10.1186/s13662-020-02874-9
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DOI: https://doi.org/10.1186/s13662-020-02874-9