- Research
- Open Access
- Published:
Fractional hybrid differential equations with three-point boundary hybrid conditions
Advances in Difference Equations volume 2019, Article number: 125 (2019)
Abstract
In this paper, we study the existence of solutions for hybrid fractional differential equations involving fractional Caputo derivative of order \(1 < \alpha\leq 2\). Our results rely on a hybrid fixed point theorem for a sum of three operators due to Dhage. An example is provided to illustrate the theory.
1 Introduction
Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary noninteger order. Applications of fractional differential equations can be found in various fields of science and engineering. Indeed, there are numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetism, and so on [23, 26, 28, 30]. Recent developments of fractional differential and integral equations are given in [1,2,3, 36,37,38,39,40].
Many authors have studied the existence of solutions of fractional boundary value problems under various boundary conditions and by different approaches. We refer the readers to the papers [4, 5, 7, 16, 17, 19, 22, 24, 29, 33] and references therein.
In recent years, hybrid fractional differential equations have achieved a great deal of interest and attention of several researchers. For some developments on the existence results for hybrid fractional differential equations, we refer to [6, 8,9,10,11,12,13,14,15, 20, 21, 25, 27, 31, 32, 34, 35] and es references therein.
This paper deals with the existence and uniqueness of solutions for boundary-value problem of the fractional differential equations
where \({{}^{\mathrm{c}}\!D}_{0^{+}}^{\alpha}\) and \({{}^{\mathrm{c}}\!D}_{0^{+}}^{\beta}\) denote the Caputo fractional derivatives of orders α and β, respectively, \(0<\beta \leq1\), \(a_{i}\), \(b_{i}\), \(c_{i}\), \(i=1,2 \), are real constants such that \(a_{1}+b_{1}\neq0\), \(a_{2}\eta^{1-\beta}+b_{2}T^{1-\beta }\neq0\), \(g \in C (J\times\mathbb {R}, \mathbb {R} \setminus\{0\} )\), and \(f, h \in C (J\times\mathbb {R}, \mathbb {R} )\).
This paper can be considered as a generalization of [19]. For example, if we choose \(f(t, x(t)) = 0\) and \(g(t, x(t)) = 1\) as constant functions, then our problem (1.1) reduces to the boundary value problem
The paper is organized as follows. In Sect. 2, we introduce some notations, definitions. and lemmas. Then, in Sect. 3, we prove existence results for problems (1.1) by employing the hybrid fixed point theorem for three operators in a in Banach algebra due to Dhage. Finally, we illustrate the obtained results by an example.
2 Preliminaries
In this section, we recall some basic definitions of fractional calculus [23, 30] and present some auxiliary lemmas.
Definition 2.1
The Riemann–Liouville fractional integral of order \(\alpha>0\) for a continuous function \(f:[0,\infty) \rightarrow\mathbb{R}\) is defined as
where Γ is the Euler gamma function.
Definition 2.2
Let \(\alpha> 0\) and \(n = [\alpha] + 1\). If \(f \in C^{n}([a, b])\), then the Caputo fractional derivative of order α defined by
exists almost everywhere on \([a, b]\) (\([\alpha]\) is the integer part of α).
Lemma 2.3
Let \(\alpha>\beta>0\) and \(f \in L^{1}([a, b])\). Then for all \(t \in[a, b]\), we have:
-
\(I_{0^{+}}^{\alpha} I_{0^{+}}^{\beta}f(t)= I_{0^{+}}^{\alpha+\beta}f(t)\),
-
\({{}^{\mathrm{c}}\!}D_{0^{+}}^{\alpha}I_{0^{+}}^{\alpha}f(t)=f(t)\),
-
\({{}^{\mathrm{c}}\!}D_{0^{+}}^{\beta}I_{0^{+}}^{\alpha}f(t)=I_{0^{+}}^{\alpha- \beta}f(t)\).
Lemma 2.4
Let \(\alpha>0 \). Then the differential equation
has a solution
where \(m-1 <\alpha< m \).
Lemma 2.5
Let \(\alpha>0 \). Then
for some \(c_{j}\in\mathbb {R}\), \(j= 0, 1, 2,\ldots, m -1\), where \(m = [\alpha] + 1\).
Define the supremum norm \(\|\cdot\|\) in \(E=C (J,\mathbb {R} )\) by
and the multiplication in E by
Clearly, E is a Banach algebra with respect to the supremum norm and multiplication in it.
To prove the existence result for the nonlocal boundary value problem (1.1), we will use the following hybrid fixed point theorem for three operators in a Banach algebra E due to Dhage [18].
Lemma 2.6
Let S be a closed convex bounded nonempty subset of a Banach algebra E, and let \(A, C : E\longrightarrow E\) and \(B :S \longrightarrow E\) be three operators such that:
-
(a)
A and C are Lipschitzian with a Lipschitz constants δ and ρ, respectively;
-
(b)
B is compact and continuous;
-
(c)
\(x=AxBy +Cx\Rightarrow x \in S \) for all \(y \in S\),
-
(d)
\(\delta M+\rho<1\), where \(M= \|B(S) \|\).
Then the operator equation \(AxBx+Cx=x\) has a solution in S.
3 Main results
In this section, we prove the existence results for the boundary value problems for hybrid differential equations with fractional order on the closed bounded interval \(J = [0, T]\).
Lemma 3.1
Let h be continuous function on \(J:=[0,T]\). Then the solution of the boundary value problem
with boundary conditions
satisfies the equation
Proof
Applying the Riemann–Liouville fractional integral operator of order α to both sides of (3.1) and using Lemma 2.5, we have
Consequently, the general solution of (3.1) is
Applying the boundary conditions (3.2) in (3.4), we find that
Therefore we have
Substituting the values of \(c_{0}\), \(c_{1}\) into (3.5), we get (3.3). □
Now we list the following hypotheses.
-
(H1)
The functions \(g:J\times\mathbb {R} \longrightarrow \mathbb {R} \setminus\{0\}\) and \(h, f: J\times\mathbb {R} \longrightarrow \mathbb {R} \) are continuous.
-
(H2)
There exist two positive functions \(\phi_{0}\), \(\phi_{1}\) with bounds \(\|\phi_{0}\|\) and \(\|\phi_{0}\|\), respectively, such that
$$ \bigl\vert f(t, x)-f(t,y) \bigr\vert \leq\phi_{0}(t)|x-y| $$(3.6)and
$$ \bigl\vert g(t, x)-g(t,y) \bigr\vert \leq \phi_{1}(t)|x-y|, $$(3.7)for all \((t,x, y) \in J\times\mathbb {R}\times\mathbb {R}\).
-
(H3)
There exist a function \(p\in L^{\infty}(J,\mathbb{R_{+}})\) and a continuous nondecreasing function \(\psi:[0, \infty) \longrightarrow (0, \infty)\) such that
$$ \bigl\vert h (t, x ) \bigr\vert \leq p(t)\psi\bigl( \vert x \vert \bigr) $$(3.8)for all \(t \in J\) and \(x \in\mathbb {R}\).
-
(H4)
There exists \(r > 0 \) such that
$$ r\geq\frac{g_{0}\varLambda+f_{0}}{1-\|\phi_{0}\|\varLambda-\|\phi_{1}\|} $$(3.9)and
$$ \|\phi_{0}\|\varLambda+\|\phi_{1}\|< 1, $$(3.10)where \(f_{0} = \sup_{t\in J }|f (t, 0)|\), \(g_{0} = \sup_{t\in J }|g (t, 0)|\), and
$$\begin{aligned} \varLambda =&\psi(r)\|p\| \biggl(\frac{T^{\alpha}}{\varGamma(\alpha +1)}+\frac{|b_{1}|T^{\alpha}}{|a_{1}+b_{1}|\varGamma(\alpha+1)} \\ &{}+\frac{ (|b_{1}|T+(|a_{1}|+|b_{1}|)T )\varGamma(2-\beta)}{|a_{1}+b_{1}||a_{2}\eta ^{1-\beta}+b_{2}T^{1-\beta}|\varGamma(\alpha-\beta+1)} \\ &{}\times \bigl( \bigl(|a_{2}|\eta^{\alpha-\beta}+|b_{2}|T^{\alpha-\beta } \bigr) +|\lambda_{2}| \bigr)+\frac{|\lambda_{1}|}{|a_{1}+b_{1}|} \biggr). \end{aligned}$$(3.11)
Theorem 3.2
Assume that conditions (H1)–(H4) hold. Then problem (1.1) has at least one solution defined on J.
Proof
Define the set
Clearly, S is a closed convex bounded subset of the Banach space E. By Lemma 3.1 the boundary value problem (1.1) is equivalent to the equation
Define three operators \(A, C : E\longrightarrow E\) and \(B :S \longrightarrow E\) by
and
Then the integral equation (3.12) can be written in the operator form as
We will show that the operators A, B, and C satisfy all the conditions of Lemma 2.6. This will be achieved in the following series of steps.
Step 1: First, we show that A and C are Lipschitzian on E. Let \(x, y \in E\). Then by (H2), for \(t \in J\), we have
for all \(t\in J\). Taking the supremum over t, we obtain
for all \(x, y \in E\). Therefore A is Lipschitzian on E with Lipschitz constant \(\|\phi_{0}\|\).
Now, for \(C : E\longrightarrow E\), \(x, y \in E\), we have
for all \(t\in J\). Taking the supremum over t, we obtain
Hence \(C : E\longrightarrow E\) is Lipschitzian on E with Lipschitz constant \(\|\phi_{1}\|\).
Step 2: We show that B is is a completely continuous operator from S into E. First, we show that B is continuous on S. Let \(\{x_{n}\}\) be a sequence in S converging to a point \(x \in S\). Then by the Lebesgue dominated convergence theorem we have
for all \(t \in J\). This shows that B is a continuous operator on S.
Next, we will prove that the set \(B(S)\) is a uniformly bounded in S. For any \(x \in S\), we have
Using (3.8), we can write
Thus \(\|Bx \|\leq\varLambda \) for all \(x \in S\) with Λ given in (3.11). This shows that B is uniformly bounded on S.
Now, we will show that \(B(S)\) is an equicontinuous set in E.
Let \(t_{1}, t_{2} \in J\). Then for any \(x \in S\), by (3.8) we get
Obviously, the right-hand side of inequality (3.13) tends to zero independently of \(x \in S\) as \(t_{2} \to t_{1}\). As a consequence of the Ascoli–Arzelà theorem, B is a completely continuous operator on S.
Step 3: Hypothesis (c) of Lemma 2.6 is satisfied.
Let \(x \in E\) and \(y \in S\) be arbitrary elements such that \(x = AxBy + Cx\). Then we have
Thus
Taking the supremum over t, we get
Step 4: Finally, we show that \(\delta M + \rho< 1\), that is, (d) of Lemma 2.6 holds.
Since
we have
with \(\delta=\|\phi_{0}\|\) and \(\rho= \|\phi_{1}\|\) Thus all the conditions of Lemma 2.6 are satisfied, and hence the operator equation \(x =AxBx + Cx\) has a solution in S. As a result, problem (1.1) has a solution on J. □
Example 3.3
Let us consider the following boundary value problem:
We take
We can show that
where
Hence we have
Then
and
Using the Matlab program, it follows by (3.9) and (3.10) that the constant r satisfies the inequality \(0.0146< r< 21.8589\). As all the conditions of Theorem 3.2 are satisfied, problem (3.14) has at least one solution on J.
4 Concluding remarks
In this paper, we have provided some sufficient conditions guaranteeing the existence of solutions for a class of hybrid fractional differential equations involving fractional Caputo derivative of order \(1 < \alpha\leq 2\). Our results rely on a hybrid fixed point theorem for a sum of three operators due to Dhage. Our results extend and complete those in the literature.
References
Abbas, S., Benchohra, M., Graef, J., Henderson, J.: Implicit Fractional Differential and Integral Equations: Existence and Stability. de Gruyter, Berlin (2018)
Abbas, S., Benchohra, M., N’Guérékata, G.M.: Topics in Fractional Differential Equations. Springer, New York (2012)
Abbas, S., Benchohra, M., N’Guérékata, G.M.: Advanced Fractional Differential and Integral Equations. Nova Science Publishers, New York (2015)
Ahmad, B.: Nonlinear fractional differential equations with anti-periodic type fractional boundary conditions. Differ. Equ. Dyn. Syst. 21(4), 387–401 (2013)
Ahmad, B., Ntouyas, S.K.: Fractional differential inclusions with fractional separated boundary conditions. Fract. Calc. Appl. Anal. 15(3), 362–382 (2012)
Ahmad, B., Ntouyas, S.K., Tariboon, J.: A nonlocal hybrid boundary value problem of Caputo fractional integro-differential equations. Acta Math. Sci. 36(6), 1631–1640 (2016)
Akman Yildiz, T., Khodabakhshi, N., Baleanu, D.: Analysis of mixed-order Caputo fractional system with nonlocal integral boundary condition. Turk. J. Math. 42, 1328–1337 (2018)
Aydogan, M.S., Baleanu, D., Mousalou, A., Rezapour, S.: On approximate solutions for two higher-order Caputo–Fabrizio fractional integro-differential equations. Adv. Differ. Equ. 2017, Article ID 221 (2017)
Aydogan, M.S., Baleanu, D., Mousalou, A., Rezapour, S.: On high order fractional integro-differential equations including the Caputo–Fabrizio derivative. Bound. Value Probl. 2018, Article ID 90 (2018)
Baleanu, D., Ghafarnezhad, K., Rezapour, S., Shabibi, M.: On the existence of solutions of a three steps crisis integro-differential equation. Adv. Differ. Equ. 2018, Article ID 135 (2018)
Baleanu, D., Jafari, H., Khan, H., Johnston, S.J.: Results for mild solution of fractional coupled hybrid boundary value problems. Open Math. 13, 601–608 (2015)
Baleanu, D., Khan, H., Jafari, H., Khan, R.A., Alipour, M.: On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions. Adv. Differ. Equ. 2015, Article ID 318 (2015)
Baleanu, D., Mousalou, A., Rezapour, S.: On the existence of solutions for some infinite coefficient-symmetric Caputo–Fabrizio fractional integro-differential equations. Bound. Value Probl. 2017, Article ID 145 (2017)
Baleanu, D., Mousalou, A., Rezapour, S.: A new method for investigating approximate solutions of some fractional integro-differential equations involving the Caputo–Fabrizio derivative. Adv. Differ. Equ. 2017, Article ID 51 (2017)
Baleanu, D., Mousalou, A., Rezapour, S.: The extended fractional Caputo–Fabrizio derivative of order \(0\leq \sigma<1\) on \(C_{\Bbb {R}}[0,1]\) and the existence of solutions for two higher-order series-type differential equations. Adv. Differ. Equ. 2018, Article ID 255 (2018)
Benchohra, M., Hamani, S., Ntouyas, S.K.: Boundary value problems for differential equations with fractional order. Surv. Math. Appl. 3, 1–12 (2008)
Delbosco, D., Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204, 609–625 (1996)
Dhage, B.C.: A fixed point theorem in Banach algebras with applications to functional integral equations. Kyungpook Math. J. 44, 145–155 (2004)
Fu, X.: Existence results for fractional differential equations with three-point boundary conditions. Adv. Differ. Equ. 2013, Article ID 257 (2013)
Herzallah, A.E.M., Baleanu, D.: On fractional order hybrid differential equations. Abstr. Appl. Anal. 2014, Article ID 389386 (2014)
Hilal, K., Kajouni, A.: Boundary value problems for hybrid differential equations with fractional order. Adv. Differ. Equ. 2015, Article ID 183 (2015)
Houas, M., Dahmani, Z., Benbachir, M.: New results for a boundary value problem for differential equations of arbitrary order. Int. J. Mod. Math. Sci. 7(2), 195–211 (2013)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Sudies, vol. 204. Elsevier, Amsterdam (2006)
Liu, X., Liu, Z.: Existence results for fractional differential inclusions with multivalued term depending on lower-order derivative. Abstr. Appl. Anal. 2012, Article ID 423796 (2012). https://doi.org/10.1155/2012/423796
Mahmudov, N., Matar, M.: Existence of mild solutions for hybrid differential equations with arbitrary fractional order. TWMS J. Pure Appl. Math. 8(2), 160–169 (2017)
Miller, K.S., Ross, B.: An Introduction to Fractional Calculus and Fractional Differential Equations. Wiley, New YorK (1993)
Nouri, K., Baleanu, D., Torkzadeh, L.: Study on application of hybrid functions to fractional differential equations. Iran. J. Sci. Technol., Trans. A, Sci. 42(3), 1343–1350 (2018)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1993)
Sabri Thabet, T.M., Dhakne, M.B.: On boundary value problems of higher order abstract fractional integro-differential equations. Int. J. Nonlinear Anal. Appl. 7, 165–184 (2016)
Samko, G., Kilbas, A.A., Marichev, O.I.: Fractional Integral and Derivative: Theory and Applications. Gordon & Breach, Yverdon (1993)
Sitho, S., Ntouyas, S.K., Tariboon, J.: Existence results for hybrid fractional integro-differential equations. Bound. Value Probl. 2015, Article ID 113 (2015)
Sun, S., Zhao, Y., Han, Z., Lin, Y.: The existence of solutions for boundary value problem of fractional hybrid differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 4961–4967 (2012)
Ugurlu, E., Baleanu, D., Tas, K.: On the solutions of a fractional boundary value problem. Turk. J. Math. 42, 1307–1311 (2018)
Ullah, Z., Ali, A., Khan, R.A., Iqbal, M.: Existence results to a class of hybrid fractional differential equations. Matrix Sci. Math. 1(1), 13–17 (2018)
Zhao, Y., Wang, Y.: Existence of solutions to boundary value problem of a class of nonlinear fractional differential equations. Adv. Differ. Equ. 2014, Article ID 174 (2014)
Zhou, Y.: Attractivity for fractional differential equations in Banach space. Appl. Math. Lett. 75, 1–6 (2018)
Zhou, Y.: Attractivity for fractional evolution equations with almost sectorial operators. Fract. Calc. Appl. Anal. 21(3), 786–800 (2018)
Zhou, Y., Peng, L., Huang, Y.Q.: Duhamel’s formula for time-fractional Schrödinger equations. Math. Methods Appl. Sci. 41, 8345–8349 (2018)
Zhou, Y., Peng, L., Huang, Y.Q.: Existence and Hölder continuity of solutions for time-fractional Navier–Stokes equations. Math. Methods Appl. Sci. 41, 7830–7838 (2018)
Zhou, Y., Shangerganesh, L., Manimaran, J., Debbouche, A.: A class of time-fractional reaction–diffusion equation with nonlocal boundary condition. Math. Methods Appl. Sci. 41, 2987–2999 (2018)
Availability of data and materials
Not applicable.
Funding
The work was supported by the National Natural Science Foundation of China (No. 11671339).
Author information
Authors and Affiliations
Contributions
All authors contributed equally to each part of this work. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Cite this article
Derbazi, C., Hammouche, H., Benchohra, M. et al. Fractional hybrid differential equations with three-point boundary hybrid conditions. Adv Differ Equ 2019, 125 (2019). https://doi.org/10.1186/s13662-019-2067-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-019-2067-7
MSC
- 34A08
- 34A12
Keywords
- Fractional differential equation
- Caputo fractional derivative
- Hybrid
- Fixed point theorem