Existence results for sequential fractional integro-differential equations with nonlocal multi-point and strip conditions
Received: 24 September 2016
Accepted: 8 November 2016
Published: 22 November 2016
Abstract
In this paper we investigate a new kind of nonlocal multi-point boundary value problem of Caputo type sequential fractional integro-differential equations involving Riemann-Liouville integral boundary conditions. Several existence and uniqueness results are obtained via suitable fixed point theorems. Some illustrative examples are also presented. The paper concludes with some interesting observations.
Keywords
MSC
1 Introduction
Fractional differential equations arise in many engineering and scientific disciplines such as physics, chemistry, aerodynamics, electrodynamics of complex medium or polymer rheology. In fact, the tools of fractional calculus have considerably improved the mathematical modeling of many real world problems. It has been mainly due to the fact that fractional-order operators provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. For theoretical development and applications of the subject, we refer the reader to the books [1–3] and a series of papers [4–14], and the references cited therein.
Sequential fractional differential equations are also found to be of much interest [15, 16]. In fact, the concept of sequential fractional derivative is closely related to the non-sequential Riemann-Liouville derivatives; for details, see [17]. In [18], the authors studied different kinds of boundary value problems involving sequential fractional differential equations. In a recent article [19], the existence of solutions for higher-order sequential fractional differential inclusions with nonlocal three-point boundary conditions was discussed.
Here we emphasize that the coupling of nonlocal multi-point and Riemann-Liouville type strip condition considered on an arbitrary segment \((0, \eta) \subset[0, 1]\) can be interpreted as the linear combination of the values of the unknown function at nonlocal points \(\zeta_{i} \in(0, 1)\) is proportional to the strip contribution of the unknown function. The consideration of the sequential fractional integro-differential equation (1.1) together with multi-point cum strip condition makes the problem (1.1)-(1.2) new.
The rest of the paper is arranged as follows. In Section 2, we establish a basic result that lays the foundation for defining a fixed point problem equivalent to the given problem (1.1)-(1.2). The main results, based on Banach’s contraction mapping principle, Krasnoselskii’s fixed point theorem and nonlinear alternative of Leray-Schauder type, are obtained in Section 3. Illustrating examples are discussed in Section 4, while Section 5 contains some interesting observations.
2 Background material
This section is devoted to some fundamental concepts of fractional calculus [20] and a basic lemma related to the linear variant of the given problem.
Definition 2.1
Definition 2.2
Definition 2.3
Remark 2.4
To define the solution for problem (1.1)-(1.2), we consider the following lemma dealing with the linear variant of (1.1)-(1.2).
Lemma 2.1
Proof
In the next lemma, we present some estimates that we need in the sequel.
Lemma 2.2
- (i)
\(|\int_{0}^{\eta}\frac{(\eta-s)^{\delta-1}}{\Gamma(\delta)} (\int_{0}^{s} e^{-k(s-\tau)} (\int_{0}^{\tau}\frac{(\tau- \omega)^{q-2}}{\Gamma(q-1)}y(\omega) \,d\omega )\,d\tau )\,ds |\leq\frac{\eta^{q+\delta-2}}{k^{2}\Gamma(q)\Gamma(\delta)}( \eta k +e^{-k \eta}-1)\Vert y\Vert \).
- (ii)
\(|\sum_{i=1}^{m}a_{i}\int_{0}^{\zeta_{i}}e^{-k(\zeta_{i}-s)} (\int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}y(\tau)\,d\tau )\,ds |\leq\sum_{i=1}^{m}\vert a_{i}\vert\zeta_{i}^{q-1}(1-e ^{-k\zeta_{i}})\frac{\Vert y\Vert }{k\Gamma(q)}\).
- (iii)
\(|\int_{0}^{t}e^{-k(t-s)} (\int_{0}^{s}\frac{(s-\tau)^{q-2}}{ \Gamma(q-1)}y(\tau)\,d\tau )\,ds |\leq\frac{1}{k\Gamma(q)}(1-e ^{-k})\Vert y\Vert \).
Proof
3 Existence and uniqueness results
This section is devoted to the main results concerning the existence and uniqueness of solutions for the problem (1.1)-(1.2). First of all, we fix our terminology.
Let \(X= \{ x:x\in C([0,1],\mathbb{R}) \text{ and } {}^{c}D^{\beta } x\in C([0,1],\mathbb{R}) \} \) denotes the space equipped with the norm \(\Vert x\Vert _{X}=\Vert x\Vert +\Vert {}^{c}D^{\beta}x \Vert =\sup_{t\in[0,1]} |x(t)|+ \sup_{t\in[0,1]}| {}^{c}D^{\beta}x(t)|\). Observe that \((X,\|\cdot \|_{X})\) is a Banach space.
Observe that problem (1.1)-(1.2) has solutions if the operator defined by (3.1) has fixed points.
Theorem 3.1
- \((H_{1})\) :
-
\(|f(t,x,y,z)-f(t,x_{1},y_{1},z_{1})| \le L[\|x-x _{1}\|+\|y-y_{1}\|+\|z-z_{1}\|]\),
Proof
As \(LL_{1} (\Lambda+\frac{\Lambda_{1}}{\Gamma(2-\delta)} )<1\), F is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle. This completes the proof. □
Now, we state a known result due to Krasnoselskii [21] which is needed to prove the existence of at least one solution of (1.1)-(1.2).
Theorem 3.2
Let M be a closed, convex, bounded, and nonempty subset of a Banach space X. Let \(\mathcal{G}_{1}\), \(\mathcal{G}_{2}\) be the operators such that: (i) \(\mathcal{G}_{1}x+\mathcal{G}_{2} y \in M\) whenever \(x, y \in M\); (ii) \(\mathcal{G}_{1}\) is compact and continuous; (iii) \(\mathcal{G}_{2}\) is a contraction mapping. Then there exists \(z \in M\) such that \(z=\mathcal{G}_{1} z+\mathcal{G}_{2} z\).
Theorem 3.3
- (\(H_{2}\)):
-
\(|f(t,x_{1}, x_{2}, x_{3})|\le\mu(t)\), \(\forall(t,x _{1}, x_{2}, x_{3}) \in[0,1] \times\mathbb{R}^{3}\) with \(\mu \in C([0,1], \mathbb{R}^{+})\).
Proof
Remark 3.4
In the next theorem, we prove the existence of solutions for the problem (1.1)-(1.2) via the Leray-Schauder nonlinear alternative.
Lemma 3.1
- (i)
F has a fixed point in U̅, or
- (ii)
there is a \(u\in\partial U\) (the boundary of U in C) and \(\lambda\in(0,1)\) with \(u=\lambda F(u)\).
Theorem 3.5
- (\(H_{3}\)):
-
there exist a function \(\phi\in C([0,1], \mathbb{R} ^{+})\), and a nondecreasing, subhomogeneous (that is, \(\Omega(kx) \le k\Omega(x)\) for all \(k\ge1\) and \(x\in{\mathbb{R}}^{+}\)) function \(\Omega: {\mathbb{R}}^{+}\to{ \mathbb{R}}^{+}\) such that \(|f(t,x _{1}, x_{2}, x_{3})|\le\phi(t)\Omega(\Vert x_{1}\Vert +\Vert x_{2}\Vert +\Vert x_{3}\Vert )\), for all \((t,x_{1}, x_{2}, x_{3}) \in[0,1] \times\mathbb{R}^{3}\);
- (\(H_{4}\)):
-
there exists a constant \(M>0\) such thatwhere Λ, \(\Lambda_{1}\) and \(L_{1}\) are given by (3.3).$$\frac{M}{ ( \Lambda+\frac{\Lambda_{1}}{\Gamma(2-\beta)} ) \Vert \phi \Vert L _{1}\Omega(M)} > 1, $$
Proof
The result will follow from the Leray-Schauder nonlinear alternative (Lemma 3.1) once we have proved the boundedness of the set of all solutions to equations \(x=\theta F x\) for \(\theta\in[0,1]\).
4 Examples
5 Conclusions
We have discussed the existence and uniqueness of solutions for sequential fractional integro-differential equations involving the Caputo (Liouville-Caputo) derivative supplemented with nonlocal multi-point boundary conditions coupled with Riemann-Liouville type strip condition. Our results are not only new in the given configuration but also correspond to some new situations associated with the specific values of the parameters involved in the given problem. For example, our results correspond to the multi-point boundary conditions with classical nonlocal strip condition: \(\sum_{i=1}^{m}a_{i} x(\zeta_{i})=\lambda \int_{0}^{\eta} x(s)\,ds\) if we take \(\delta=1\) in (1.2). In the case we choose \(a_{i}=0\), \(i=1, \dots, (m-1)\), \(a_{m}=1\), and \(\zeta_{m} \to1\), our results correspond to the condition \(x(1)=\lambda\int _{0}^{\eta} \frac{(\eta-s)^{\delta-1}}{\Gamma(\delta)}x(s)\,ds\).
Declarations
Acknowledgements
We thank the reviewers for their useful remarks that led to the improvement of our work.
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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