- Research
- Open Access
A coupled fixed point theorem and application to fractional hybrid differential problems
- Tahereh Bashiri^{1},
- Seyed Mansour Vaezpour^{1} and
- Choonkil Park^{2}Email author
https://doi.org/10.1186/s13663-016-0511-x
© Bashiri et al. 2016
- Received: 12 November 2015
- Accepted: 28 February 2016
- Published: 5 March 2016
Abstract
The proof of the existence theorem is based on a coupled fixed point theorem of Krasnoselskii type, which extends a fixed point theorem of Burton (Appl. Math. Lett. 11:85-88, 1998). Finally, our results are illustrated by a concrete example.
Keywords
- hybrid initial value problem
- Banach space
- coupled fixed point theorem
- Riemann-Liouville fractional derivative
1 Introduction
Nonlinear differential equations are crucial tools in the modeling of nonlinear real phenomena corresponding to a great variety of events, in relation with several fields of the physical sciences and technology. For instance, they appear in the study of the air motion or the fluids dynamics, electricity, electromagnetism, or the control of nonlinear processes, among others (see [2]). The resolution of nonlinear differential equations requires, in general, the development of different techniques in order to deduce the existence and other essential properties of the solutions. There are still many open problems related the solvability of nonlinear systems, apart form the fact that this is a field where advances are continuously taking place.
Perturbation techniques are useful in the nonlinear analysis for studying the dynamical systems represented by nonlinear differential and integral equations. Evidently, some differential equations representing a certain dynamical system have no analytical solution, so the perturbation of such problems can be helpful. The perturbed differential equations are categorized into various types. An important type of these such perturbations is called a hybrid differential equation (i.e. quadratic perturbation of a nonlinear differential equation). See [3] and the references therein.
Indeed, the fractional differential equations have recently been intensively used in modeling several physical phenomena and have been studied by many researchers in recent years [14–22]; therefore they seem to deserve an independent study of their theory parallel to the theory of ordinary differential equations.
On top of that, the study of coupled systems involving fractional differential equations is also important as such systems occur in various problems of applied nature, for instance, see [24–29]. Lately, Su [30] discussed a two-point boundary value problem for a coupled system of fractional differential equations. Gafiychuk et al. [31] analyzed the solutions of coupled nonlinear fractional reaction-diffusion equations.
2 Preliminaries
- (i)
the map \(t \rightarrow g(t,x,y)\) is measurable for each \(x,y \in\mathbb{R}\),
- (ii)
the map \(x \rightarrow g(t,x,y)\) is continuous for each \(x \in \mathbb{R}\),
- (iii)
the map \(y \rightarrow g(t,x,y)\) is continuous for each \(y \in\mathbb{R}\).
The class \(\mathcal{C}(J\times\mathbb{R}\times \mathbb{R},\mathbb{R})\) is called the Carathéodory class of functions on \(J\times\mathbb{R}\times\mathbb{R,}\) which are Lebesgue integrable when bounded by a Lebesgue integrable function on J.
We need some precise definitions of the basic concepts. The following is a discussion of some of the concepts we will need.
Definition 1
[32]
Definition 2
[32]
Definition 3
[32]
Lemma 1
- (K1)
The equality \(D^{\alpha}I^{\alpha}f(t) = f(t)\) holds.
- (K2)The equalityholds almost everywhere on J.$$ I^{\alpha}D^{\alpha}f (t) = f (t) - \frac{[D^{\alpha-1}f(t)]_{t=0}}{\Gamma(\alpha)} t^{\alpha- 1} $$
The following is a fixed point theorem in Banach spaces due to Burton [1].
Lemma 2
[1]
- (i)
A is a contraction with constant \(\alpha<1\),
- (ii)
B is completely continuous,
- (iii)
\(x=Ax+By \Rightarrow x \in S\) for all \(y \in S\).
Now, we recall the definition of a coupled fixed point for a bivariate mapping.
Definition 4
[34]
An element \((x,y)\in X\times X\) is called a coupled fixed point of a mapping \(T:X\times X \rightarrow X\) if \(T(x,y)=x\) and \(T(y,x)=y\).
Let us denote by Φ the family of all functions \(\varphi:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) fulfilling \(\varphi(r)< r\) for \(r>0\) and \(\varphi(0)=0\).
- (i)
the function \(t\rightarrow x- f(t,x)\) is absolutely continuous for each \(x \in\mathbb{R}\), and
- (ii)
\((x,y)\) satisfies the system of equations in (1),
3 Main result
Throughout this section, let \(X= C(J, \mathbb{R})\) equipped with the supremum norm. Clearly it is a Banach space with respect to pointwise operations and the supremum norm.
In the following lemma we introduce a certain Banach space which is used in our results.
Lemma 3
Proof
Clearly X̃ is a Banach space and \(\|\cdot\|\) is a norm on X̃. □
Now, we prove a coupled fixed point theorem which is a generalization of Lemma 2 of Dhage.
Theorem 1
- (C1)there exists \(\varphi_{A} \in\Phi\) such that for all \(x,y \in X\), we havefor some constant \(\sigma>0\),$$\|Ax-Ay\|\leq\sigma\varphi_{A}\bigl(\Vert x-y\Vert \bigr), $$
- (C2)
B is completely continuous,
- (C3)
\(x=Ax+By \Rightarrow x \in S\) for all \(y \in S\).
Proof
- (H0)
The function \(x \rightarrow x - f (t, x)\) is increasing in \(\mathbb {R}\) for all \(t \in J\).
- (H1)There exists a constant \(M\geq L>0\) such thatfor all \(t \in J\) and \(x, y \in\mathbb{R}\).$$\bigl\vert f\bigl(t,x(t)\bigr)-f\bigl(t,y(t)\bigr)\bigr\vert \leq \frac{L(\vert x(t)-y(t)\vert )}{2(M+\vert x(t)-y(t)\vert )}, $$
- (H2)Fix$$F_{0}=\max_{t\in J} \bigl\vert f(t,0)\bigr\vert . $$
- (H3)There exists a continuous function \(h \in C(J,\mathbb{R})\) such that$$g\bigl(t, x(t), y(t)\bigr)\leq h(t),\quad x, y \in\mathbb{R}, t\in J. $$
As a consequence of Lemma 1 we have the following lemma which is useful in the existence results.
Lemma 4
[23]
Now we are going to prove the following existence theorem for the FHDEs of system (1).
Theorem 2
Assume that hypotheses (H1)-(H3) hold. Then the FHDEs of system (1) has a solution defined on J.
Proof
Next we show that B is compact and continuous operator on S.
4 Illustrative example
Example 1
Declarations
Acknowledgements
The authors are grateful to the reviewers for their valuable comments and suggestions.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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