- Research
- Open Access
Existence results for fractional-order differential equations with nonlocal multi-point-strip conditions involving Caputo derivative
- Bashir Ahmad1Email author,
- Ahmed Alsaedi1 and
- Alaa Alsharif1
https://doi.org/10.1186/s13662-015-0684-3
© Ahmad et al. 2015
- Received: 10 October 2015
- Accepted: 2 November 2015
- Published: 11 November 2015
Abstract
Keywords
- fractional order derivative
- nonlocal conditions
- strip
- existence
- fixed point
MSC
- 34A08
- 34B15
1 Introduction
The study of fractional-order differential equations supplemented with a variety of initial and boundary conditions, such as classical, nonlocal, multi-point, periodic/anti-periodic, and integral boundary conditions, has attracted significant attention in recent years. In consequence, the literature on the topic is now much enriched and covers theoretical aspects as well as analytic/numerical methods for solving fractional-order initial and boundary value problems. The widespread applications of fractional calculus modeling techniques in several disciplines of applied and technical sciences have played a key role in the popularity of the subject. Examples include viscoelasticity, control theory, biological sciences, ecology, aerodynamics, electro-dynamics of complex medium, environmental issues, etc. An important and useful feature characterizing fractional-order differential and integral operators (in contrast to integer-order operators) is their nonlocal nature that accounts for the past and hereditary behavior of materials and processes involved in the real world problems. For examples and details, we refer the reader to the works [1–5].
Nonlocal conditions, introduced by Bitsadze and Samarskii [6], are regarded as more plausible than the classical initial/boundary conditions in view of their ability to describe certain peculiarities of chemical, physical or other processes happening inside the domain. Computational fluid dynamics (CFD) studies of blood flow indicate that it is not always possible to assume circular cross-section of blood arteries. Several approaches have been proposed to resolve this issue. However, the idea of introducing integral boundary conditions [7] is found to be quite a productive one. Also, integral boundary conditions are applied to regularize ill-posed parabolic backward problems in time partial differential equations, see, for example, mathematical models for bacterial self-regularization [8]. Some recent results on fractional-order boundary value problems involving nonlocal and integral boundary conditions can be found in [9–20] and the references cited therein.
The content of the paper is organized as follows. Section 2 is devoted to some basic concepts and a lemma concerning the unique solution of a linear variant of problem (1.1)-(1.2). In Section 3, we present our main results which are obtained via Krasnoselskii’s fixed point theorem, Schauder type fixed point theorem, nonlinear alternative for single-valued maps and Banach’s theorem. It is imperative to note that the exposition of indicated tools of fixed point theory is new in the context of problem (1.1)-(1.2). Finally, we discuss some examples for illustration of the main results.
2 Preliminaries
First of all, we recall some basic definitions.
Definition 2.1
[3]
Definition 2.2
[3]
Next we present an auxiliary lemma to define the solution for problem (1.1)-(1.2).
Lemma 2.3
Proof
Substituting the values of \(c_{0}\), \(c_{1}\) in (2.4), we get (2.2). This completes the proof. □
3 Main results
Now we state the known fixed point results which we need in the forthcoming analysis.
Lemma 3.1
(Krasnoselskii [21])
Let \(\mathcal{Q}\) be a closed, convex, bounded and nonempty subset of a Banach space Y. Let \(\mathcal{\phi}_{1}\), \(\mathcal{\phi}_{2}\) be operators such that (a) \(\mathcal{\phi}_{1}\nu_{1}+\mathcal{\phi}_{2}\nu_{2} \in\mathcal{Q}\) whenever \(\nu_{1},\nu_{2} \in\mathcal{Q}\); (b) \(\mathcal{\phi}_{1}\) is compact and continuous; and (c) \(\mathcal{\phi}_{2}\) is a contraction mapping. Then there exists \(\nu\in\mathcal{Q}\) such that \(\nu=\mathcal{\phi}_{1}\nu+\mathcal{\phi}_{2}\nu\).
Lemma 3.2
[21]
Let X be a Banach space. Assume that \(T:X\longrightarrow X\) is a completely continuous operator and the set \(V=\{u\in X|u=\epsilon Tu,0<\epsilon<1\} \) is bounded. Then T has a fixed point in X.
Lemma 3.3
(Nonlinear alternative for single-valued maps [22])
- (i)
\(\mathcal{U}\) has a fixed point in V̅, or
- (ii)
there is \(x\in\partial V \) (the boundary of V in \(E_{1} \)) and \(\kappa\in(0,1) \) with \(x=\kappa\) \(\mathcal{U} (x) \).
Our first existence result is based on Lemma 3.1.
Theorem 3.4
- (\({\mathcal{H}}_{1}\)):
-
\(|f(t,x) -f(t,y)|\leq \ell|x-y|\), \(\ell >0\), \(\forall t\in[0,1]\), \(x,y\in\mathbb{R}\);
- (\({\mathcal{H}}_{2}\)):
-
\(|f(t,x)|\leq \omega(t) \), \(\forall(t,x)\in[0,1] \times\mathbb{R}\), \(\omega\in C([0,1], \mathbb{R^{+}})\).
Proof
In the next result, we make use of Lemma 3.2.
Theorem 3.5
Assume that there exists a positive constant \(L_{1}\) such that \(|f(t,x)|\leq L_{1} \) for all \(t\in[0,1] \), \(x\in\mathcal{C} \). Then there exists at least one solution for problem (1.1)-(1.2).
Proof
The following theorem deals with the uniqueness of solutions for problem (1.1)-(1.2).
Theorem 3.6
Let \(f:[0,1]\times\mathbb {R}\longrightarrow \mathbb{R}\) be a continuous function satisfying the condition (\({\mathcal{H}}_{1}\)) and that \(\ell\vartheta_{1}<1\), where \(\vartheta_{1}\) is given by (3.2). Then there exists a unique solution for problem (1.1)-(1.2) on \([0, 1]\).
Proof
Our final result relies on Lemma 3.3 (nonlinear alternative for single-valued maps).
Theorem 3.7
- (\({\mathcal{H}}_{3}\)):
-
there exist a function \(p\in\mathcal{C} ([0,1], \mathbb{R^{+}})\) and a nondecreasing function \(\varphi:\mathbb{R^{+}}\longrightarrow\mathbb{R^{+}}\) such that \(|f(t,x)|\leq p(t)\varphi(\|x\|) \), \(\forall(t,x) \in[0,1]\times\mathbb{R}\);
- (\({\mathcal{H}}_{4}\)):
-
there exists a constant \(M>0 \) such that$$\begin{aligned} &M \biggl[\varphi(M)\|p\| \biggl\{ \frac{1}{\Gamma(q+1)}+\frac{|\delta |\sigma ^{q}}{|1-\delta|\Gamma(q+1)} + \biggl(\frac{|\delta|\sigma}{|\mathcal{A}(1-\delta)|}+\frac {1}{|\mathcal {A}|} \biggr) \\ &\quad{}\times \biggl(|a|\frac{\varrho_{1}^{q-\mu}}{\Gamma(q-\mu+1)}+|b|\frac {\varrho _{2}^{q-\mu}}{\Gamma(q-\mu+1)}+|c| \frac{({\beta_{2}}^{q-\mu+1} -{\beta_{1}}^{q-\mu+1})}{\Gamma(q-\mu+2)} \biggr) \biggr\} \biggr]^{-1}>1. \end{aligned}$$
Proof
Example 3.8
Example 3.9
Declarations
Acknowledgements
The paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University. The authors, therefore, acknowledge with thanks DSR technical and financial support.
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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