Abstract
In this paper we study existence results for initial value problems for hybrid fractional integro-differential equations. A couple of hybrid fixed point theorems for the sum of three operators are used for proving the main results. Examples illustrating the results are also presented.
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1 Introduction
Fractional differential equations arise in the mathematical modeling of systems and processes occurring in many engineering and scientific disciplines such as physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, economics, control theory, signal and image processing, biophysics, blood flow phenomena, etc. [1–6]. For some recent developments on the topic, see [7–17] and the references therein.
Hybrid fractional differential equations have also been studied by several researchers. This class of equations involves the fractional derivative of an unknown function hybrid with the nonlinearity depending on it. Some recent results on hybrid differential equations can be found in a series of papers [18–26].
In this paper we study existence results for initial value problems for hybrid fractional integro-differential equations. In Section 3 we consider the following initial value problem for hybrid fractional integro-differential equations:
where \(D^{\alpha}\) denotes the Riemann-Liouville fractional derivative of order α, \(0<\alpha\leq1\), \(I^{\phi}\) is the Riemann-Liouville fractional integral of order \(\phi>0\), \(\phi\in\{ \beta_{1},\beta_{2},\ldots,\beta_{m}\}\), \(f\in C( J\times{\mathbb{R}},{\mathbb{R}}\setminus\{0\})\), \(g\in C(J\times\mathbb{R}, \mathbb {R})\) and \(h_{i}\in C(J\times\mathbb{R}, \mathbb{R})\) with \(h_{i}(0,0)=0\), \(i=1,2,\ldots,m\). An existence result is obtained for the initial value problem (1.1) by using a hybrid fixed point theorem for three operators in a Banach algebra due to Dhage [27].
As a second problem we discuss in Section 4 an initial value problem for hybrid fractional sequential integro-differential equations,
where \(0<\alpha,\omega\leq1\), \(1<\alpha+\omega\leq2\), functions f, h, and constants \(\beta_{1},\beta_{2},\ldots,\beta_{m}\) are defined as in problem (1.1), \(g\in C(J\times\mathbb{R}^{2}, \mathbb {R})\), and \(I^{\gamma}\) is the Riemann-Liouville fractional integral of order γ. By using a useful generalization of Krasnoselskii’s fixed point theorem due to Dhage [28], we prove an existence result for the initial value problem (1.2). Note that if \(m=1\), \(\beta_{1}=\gamma=0\), and \(h_{1}(t,x(t))=-\lambda x(t)\), λ is a constant, then the first equation of (1.2) is reduced to the hybrid fractional Langevin equation (first formulated by Langevin in 1908) of the form
which is a generalization of the well-known classical results in [29].
The rest of the paper is organized as follows: In Section 2 we recall some useful preliminaries. In Section 3 we study the existence of the initial value problem (1.1), while in Section 4 we deal with the initial value problem (1.2). Examples illustrating the obtained results are presented in Section 5.
2 Preliminaries
In this section, we introduce some notations and definitions of fractional calculus [1, 5] and present preliminary results needed in our proofs later.
Definition 2.1
The Riemann-Liouville fractional derivative of order \(q > 0\) of a continuous function \(f:(0,\infty)\rightarrow\mathbb{R}\) is defined by
where \(n=[q]+1\), \([q]\) denotes the integer part of a real number q, provided the right-hand side is point-wise defined on \((0,\infty)\), where Γ is the gamma function defined by \(\Gamma(q)=\int_{0}^{\infty} e^{-s}s^{q-1}\,ds\).
Definition 2.2
The Riemann-Liouville fractional integral of order \(p > 0\) of a continuous function \(f:(0,\infty)\rightarrow \mathbb{R}\) is defined by
provided the right-hand side is point-wise defined on \((0,\infty)\).
Lemma 2.1
[1]
Let \(q > 0\) and \(x\in C(0,T)\cap L(0,T)\). Then the fractional differential equation
has a unique solution
where \(k_{i} \in\mathbb{R}\), \(i=1,2,\ldots,n\), and \(n-1< q< n\).
Lemma 2.2
[1]
Let \(q > 0\). Then for \(x\in C(0,T)\cap L(0,T)\) we have
where \(n-1< q< n\).
Let \(E=C(J,\mathbb{R})\) be the space of continuous real-valued functions defined on \(J=[0,T]\). Define a norm \(\|\cdot\|\) and a multiplication in E by
Clearly E is a Banach algebra with respect to above supremum norm and the multiplication in it.
3 Hybrid fractional integro-differential equations
In this section we consider the initial value problem (1.1). The following hybrid fixed point theorem for three operators in a Banach algebra E, due to Dhage [27], will be used to prove the existence result for the initial value problem (1.1).
Lemma 3.1
Let S be a nonempty, closed convex and bounded subset of a Banach algebra E and let \(A,C:E\to E\) and \(B:S\to E\) be three operators satisfying:
- (a1):
-
A and C are Lipschitzian with Lipschitz constants δ and ρ, respectively,
- (b1):
-
B is compact and continuous,
- (c1):
-
\(x=AxBy+Cx\Rightarrow x \in S\) for all \(y\in S\),
- (d1):
-
\(\delta M+\rho<1\), where \(M=\|B(S)\|\).
Then the operator equation \(x=AxBx+Cx\) has a solution.
Lemma 3.2
Suppose that \(0<\alpha\leq1\) and functions f, g, \(h_{i}\), \(i=1,2,\ldots,m\) satisfy problem (1.1). Then the unique solution of the hybrid fractional integro-differential problem (1.1) is given by
Proof
Applying the Riemann-Liouville fractional integral of order α to both sides of (1.1) and using Lemma 2.2, we have
Since \(x(0)=0\), \(h(0,0)=0\), and \(f(0,0)\neq0\), it follows that
Thus (3.1) holds. The proof is completed. □
Theorem 3.1
Assume that:
- (H1):
-
The functions \(f:J\times\mathbb{R}\to\mathbb {R}\setminus\{0\}\) and \(h_{i}:J\times\mathbb{R}\to\mathbb{R}\), \(h_{i}(0,0)=0\), \(i=1,2,\ldots,m\), are continuous and there exist two positive functions ϕ, \(\psi_{i}\), \(i=1,2,\ldots,m\) with bound \(\| \phi\|\) and \(\|\psi_{i}\|\), \(i=1,2,\ldots,m\), respectively, such that
$$ \bigl\vert f\bigl(t,x(t)\bigr)-f\bigl(t,y(t)\bigr)\bigr\vert \leq\phi(t)\bigl\vert x(t)-y(t)\bigr\vert $$(3.2)and
$$ \bigl\vert h_{i}\bigl(t,x(t)\bigr)-h_{i} \bigl(t,y(t)\bigr)\bigr\vert \leq\psi_{i}(t)\bigl\vert x(t)-y(t) \bigr\vert ,\quad i=1,2,\ldots,m, $$(3.3)for \(t\in J\) and \(x,y\in\mathbb{R}\).
- (H2):
-
There exist a function \(p\in C(J,\mathbb{R}^{+})\) and a continuous nondecreasing function \(\Psi:[0,\infty)\to(0,\infty)\) such that
$$ \bigl\vert g\bigl(t,x(t)\bigr)\bigr\vert \leq p(t)\Psi\bigl( \vert x\vert \bigr),\quad (t,x)\in J\times\mathbb{R}. $$(3.4) - (H3):
-
There exists a number \(r>0\) such that
$$ r\ge\frac{ F_{0}\|p\|\Psi(r)\frac{T^{\alpha}}{\Gamma (\alpha+1)}+K_{0}\sum_{i=1}^{m}\frac{T^{\beta_{i}}}{\Gamma(\beta _{i}+1)}}{ 1-\|\phi\|\|p\|\Psi(r)\frac{T^{\alpha }}{\Gamma(\alpha+1)}-\sum_{i=1}^{m}\frac{\|\psi_{i}\|T^{\beta _{i}}}{\Gamma(\beta_{i}+1)}}, $$(3.5)where \(F_{0}=\sup_{t\in J}|f(t,0)|\) and \(K_{0}=\sup_{t\in J}|h_{i}(t,0)|\), \(i=1,2,\ldots,m\), and
$$ \|\phi\| \|p\|\Psi(r)\frac{T^{\alpha}}{\Gamma(\alpha+1)} +\sum _{i=1}^{m}\frac{\|\psi_{i}\|T^{\beta_{i}}}{\Gamma(\beta_{i}+1)}< 1. $$(3.6)
Then problem (1.1) has at least one solution on J.
Proof
Set \(E=C(J,\mathbb{R})\) and define a subset S of E as
where r satisfies inequality (3.5).
Clearly S is closed, convex, and bounded subset of the Banach space E. By Lemma 3.2, problem (1.1) is equivalent to the integral equation (3.1). Now we define three operators; \(\mathcal{A}:E\to E\) by
\(\mathcal{B}:S\to E\) by
and \(\mathcal{C}: E\to E\) by
We shall show that the operators \(\mathcal{A}\), \(\mathcal{B}\), and \(\mathcal{C}\) satisfy all the conditions of Lemma 3.1. This will be achieved in the following series of steps.
Step 1. We first show that \(\mathcal{A}\) and \(\mathcal{C}\) are Lipschitzian on E.
Let \(x,y\in E\). Then by (H1), for \(t\in J\) we have
which implies \(\|\mathcal{A}x-\mathcal{A}y\|\leq\|\phi\| \|x-y\|\) for all \(x,y\in E\). Therefore, \(\mathcal{A}\) is a Lipschitzian on E with Lipschitz constant \(\|\phi\|\).
Analogously, for any \(x,y\in E\), we have
This means that
Thus, \(\mathcal{C}\) is a Lipschitzian on E with Lipschitz constant \(\sum_{i=1}^{m}\frac{\|\psi_{i}\|T^{\beta_{i}}}{\Gamma (\beta_{i}+1)}\).
Step 2. The operator \(\mathcal{B}\) is completely continuous on S.
We first show that the operator \(\mathcal{B}\) is continuous on E. Let \(\{x_{n}\}\) be a sequence in S converging to a point \(x\in S\). Then by the Lebesgue dominated convergence theorem, for all \(t\in J\), we obtain
This implies that \(\mathcal{B}\) is continuous on S.
Next we will prove that the set \(\mathcal{B}(S)\) is a uniformly bounded in S. For any \(x\in S\), we have
for all \(t\in J\). Therefore, \(\|\mathcal{B}\|\leq K_{1}\), which shows that \(\mathcal{B}\) is uniformly bounded on S.
Now, we will show that \(\mathcal{B}(S)\) is an equicontinuous set in E. Let \(\tau_{1},\tau_{2}\in J\) with \(\tau_{1}<\tau_{2}\) and \(x\in S\). Then we have
which is independent of \(x\in S\). As \(\tau_{1}\to\tau_{2}\), the right-hand side of the above inequality tends to zero. Therefore, it follows from the Arzelá-Ascoli theorem that \(\mathcal{B}\) is a completely continuous operator on S.
Step 3. The hypothesis (c1) of Lemma 3.1 is satisfied.
Let \(x\in E\) and \(y\in S\) be arbitrary elements such that \(x=\mathcal {A}x\mathcal{B}y+\mathcal{C}x\). Then we have
which leads to
Therefore, \(x\in S\).
Step 4. Finally we show that \(\delta M+\rho<1\), that is, (d1) of Lemma 3.1 holds.
Since
and by (H3) we have
with \(\delta=\|\phi\|\) and \(\rho=\sum_{i=1}^{m}\frac {\|\psi_{i}\|T^{\beta_{i}}}{\Gamma(\beta_{i}+1)}\).
Thus all the conditions of Lemma 3.1 are satisfied and hence the operator equation \(x=\mathcal{A}x\mathcal {B}x+\mathcal{C}x\) has a solution in S. In consequence, problem (1.1) has a solution on J. This completes the proof. □
4 Hybrid fractional sequential integro-differential equations
In this section we consider the initial value problem (1.2). An existence result will be proved by using the following fixed point theorem due to Dhage.
Lemma 4.1
[28]
Let M be a nonempty, closed, convex and bounded subset of the Banach space X and let \(A: X\to X\) and \(B:M\to X\) be two operators such that
-
(i)
A is a contraction,
-
(ii)
B is completely continuous, and
-
(iii)
\(x=Ax+By\) for all \(y\in M \Rightarrow x\in M\).
Then the operator equation \(Ax+Bx=x\) has a solution.
Lemma 4.2
Suppose that \(0<\alpha,\omega\leq 1\), \(1<\alpha+\omega\leq2\), \(\gamma>0\), and the functions f, g, \(h_{i}\), \(i=1,2,\ldots,m\) satisfy problem (1.2). Then the unique solution of the hybrid fractional sequential integro-differential problem (1.2) is given by
Proof
By using the method of proving of Lemma 3.2 and applying the semigroup property, i.e., \(I^{\omega}I^{\beta_{i}}h_{i}=I^{\beta_{i}+\omega}h_{i}\), \(i=1,2,\ldots ,m\), we obtain the desired integral equation in (4.1). □
Theorem 4.1
Assume that:
- (A1):
-
The functions \(f:J\times\mathbb{R}\to\mathbb {R}\setminus\{0\}\) and \(g:J\times\mathbb{R}^{2}\to\mathbb{R}\), are continuous and there exist two positive functions ϕ, χ with bound \(\|\phi\|\) and \(\|\chi\|\), respectively, such that
$$ \bigl\vert f\bigl(t,x(t)\bigr)-f\bigl(t,y(t)\bigr)\bigr\vert \leq\phi(t)\bigl\vert x(t)-y(t)\bigr\vert $$(4.2)and
$$ \bigl\vert g\bigl(t,x(t),y(t)\bigr)-g\bigl(t,\bar{x}(t), \bar{y}(t)\bigr)\bigr\vert \leq\chi (t) \bigl(\bigl\vert x(t)-y(t)\bigr\vert +\bigl\vert \bar{x}(t)-\bar{y}(t)\bigr\vert \bigr), $$(4.3)for \(t\in J\) and \(x,y, \bar{x}, \bar{y}\in\mathbb{R}\).
- (A2):
-
\(|f(t,x)|\le\mu(t)\), \(\forall(t,x)\in J\times {\mathbb{R}}\), \(\mu\in C(J,{\mathbb{R}}^{+})\), \(|g(t, x, y)|\le\nu(t)\), \(\forall(t,x, y)\in J\times{\mathbb{R}}\times{\mathbb{R}}\), \(\nu\in C(J,{\mathbb{R}}^{+})\), and \(|h_{i}(t, x)|\le\theta_{i}(t)\), \(\forall(t,x)\in J\times{\mathbb{R}}\), \(\theta_{i}\in C(J,{\mathbb{R}}^{+})\), \(i=1,2, \ldots, m\).
If
then problem (1.2) has at least one solution on J.
Proof
Setting \(\sup_{t\in J}|\mu(t)|=\|\mu\|\), \(\sup_{t\in J}|\nu(t)|=\|\nu\|\), \(\sup_{t\in J}|\theta_{i}(t)|=\|\theta_{i}\|\), \(i=1,2,\ldots, m\), and choosing
we consider \(B_{R}=\{x\in C(J, {\mathbb{R}}): \|x\|\le R\}\). We define the operators \(\mathcal{A}:E\to E\) as in (3.7), \(\mathcal{D}: B_{R}\to E\) by
and
and
For any \(y\in B_{R}\), we have
and therefore \(\|x\|\le R\), which means that \(x\in B_{R}\). Hence, the condition (iii) of Lemma 4.1 holds.
Next we will show that \(\mathcal{Q}\) satisfy the condition (ii) of Lemma 4.1. The operator \(\mathcal{Q}\) is obviously continuous. Also, \(\mathcal{Q}\) is uniformly bounded on \(B_{R}\) as
Let \(\tau_{1},\tau_{2}\in J\) with \(\tau_{1}<\tau_{2}\) and \(x\in B_{R}\). We define \(\sup_{(t,x)\in J\times B_{R}}|h_{i}(t,x)|=\bar{h}_{i}<\infty\), \(i=1,2,\ldots, m\). Then we have
which is independent of x and tends to zero as \(\tau_{2}-\tau_{1}\to 0\). Thus, \(\mathcal{Q}\) is equicontinuous. So \(\mathcal{Q}\) is relatively compact on \(B_{R}\). Hence, by the Arzelá-Ascoli theorem, \(\mathcal{Q}\) is compact on \(B_{R}\).
Now we show that \({\mathcal{T}}\) is a contraction mapping. Let \(x, y\in B_{R}\). Then for \(t\in J\) we have
Hence, by (4.4), \(\mathcal{T}\) is a contraction mapping, and thus the condition (i) of Lemma 4.1 is satisfied.
Thus all the assumptions of Lemma 4.1 are satisfied. Therefore, the conclusion of Lemma 4.1 implies that problem (1.2) has at least one solution on J. □
5 Examples
In this section, we present two examples to illustrate our results.
Example 5.1
Consider the following hybrid fractional integro-differential equation:
where
and
Here \(\alpha=3/4\), \(T=1\), \(m=4\), \(\beta_{1}=1/3\), \(\beta_{2}=7/4\), \(\beta_{3}=10/3\), and \(\beta_{4}=29/6\). We can show that
and
It follows that \(\phi(t)=(1+2t)(2+10t)^{-2}\), \(\psi _{1}(t)=18t(3+t)^{-1}/75\), \(\psi_{2}(t)=4t(4+e^{t})^{-1}/21\), \(\psi _{3}(t)=16(45+5(5+t)^{2})^{-1}\), and \(\psi_{4}(t)=5t(4-t)^{-2}/8\), which give norms \(\|\phi\|=1/4\), \(\|\psi_{1}\|=3/50\), \(\|\psi_{2}\| =4(28+7e)^{-1}/3\), \(\|\psi_{3}\|=8/85\), and \(\|\psi_{4}\|=5/72\).
Since
we set \(p(t)=((t-1)^{2}+3)/(13-t^{2})\) and also the function Ψ as
It is easy to verify that \(\|p\|=4/13\), \(F_{0}=\sup_{t\in[0,2]}|f(t,0)|=4/27\), and \(K_{0}=\sup_{t\in[0,2]}|h_{i}(t, 0)|=2/25\), \(i=1,2,3,4\). We see that condition (H3) is followed with a number \(r\in[0.187454,131.292851]\). Consequently all conditions in Theorem 3.1 are satisfied. Therefore, problem (5.1) has at least one solution on \([0,1]\).
Example 5.2
Consider the following hybrid fractional sequential integro-differential equation:
where
and
Here \(\alpha=4/5\), \(\omega=2/3\), \(m=5\), \(T=1\), \(\beta_{1}=1/6\), \(\beta _{2}=3/5\), \(\beta_{3}=5/4\), \(\beta_{4}=7/3\), \(\beta_{5}=9/2\), \(\gamma=8/3\). We can show that
and
Therefore, we choose
It is easy to see that
The functions \(h_{i}(t,x(t))\), \(i=1,2,\ldots,5\), are bounded by the corresponding positive functions \(\theta_{i}(t)\), \(i=1,2,\ldots,5\), as follows:
Hence the conditions (A1) and (A2) are satisfied. Putting \(\|\phi\| =3/20\), \(\|\chi\|=1/3\), \(\|\mu\|=4/5\), and \(\|\nu\|=2\), we obtain
6 Conclusions
In this paper, we have studied the existence of solutions for initial value problems of first and second order for hybrid fractional integro-differential equations. The first result has been obtained by using a hybrid fixed point theorem for three operators in a Banach algebra due to Dhage [27], while the second has been obtained by using a useful generalization of Krasnoselskii’s fixed point theorem due to Dhage [28]. The main results are well illustrated with the help of examples.
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This research was funded by College of Industrial Technology, King Mongkut’s University of Technology North Bangkok, Thailand.
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Sitho, S., Ntouyas, S.K. & Tariboon, J. Existence results for hybrid fractional integro-differential equations. Bound Value Probl 2015, 113 (2015). https://doi.org/10.1186/s13661-015-0376-7
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DOI: https://doi.org/10.1186/s13661-015-0376-7