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 Open Access
Study of a class of arbitrary order differential equations by a coincidence degree method
 Nigar Ali^{1},
 Kamal Shah^{1},
 Dumitru Baleanu^{2},
 Muhammad Arif^{3}Email authorView ORCID ID profile and
 Rahmat Ali Khan^{1}
 Received: 8 May 2017
 Accepted: 18 July 2017
 Published: 3 August 2017
Abstract
Keywords
 fractional order differential equations
 Caputo derivative
 condensing operator
 Grönwall’s inequality
 topological degree method
MSC
 34A08
 35R11
1 Introduction
Section 2 is concerned with some background material and lemmas required for the main results. In Section 3, the problem under consideration of FDEs is transformed to its equivalent Fredholm integral equation. Then the required theory devoted to the aims of this paper is developed via using coincidence degree of condensing maps and using the standard singular Grönwall inequality. At the end, an example is provided for justification of the established results.
2 Background material
This section contains basics materials and preliminaries related to of noninteger order calculus and degree theory of topological type. For further details, we refer to [2–5, 35–38].
The space consisting of all continuous functions \(\mathfrak{J}\rightarrow \mathbf{R}\) is a Banach space endowed with a norm \(\lVert z \rVert_{\mathbf{Z}}=\sup \{\lvert z(t)\rvert :t\in \mathfrak{J} \}\). For simplicity, we denote the defined space by \(\mathbf{Z}=C(J, \mathbf{R})\).
Definition 2.1
Definition 2.2
For further details on fractional derivatives and integrals; see [2–5].
Lemma 2.1
[35]
Theorem 2.3
[35]
Next, we present some important definitions, propositions and theorems from [36]. For the Banach space Z, with \(\mathbf{C} \in P(\mathbf{Z})\) represents the collection of all bounded sets.
Definition 2.4
Proposition 2.1
 (i)
the set B is relatively compact if and only if \(\mathbf{B}\in \mathbf{C}\) has Kuratowski measure zero;
 (ii)
β is a seminorm, because it satisfies \(\beta ( \lambda \mathbf{B})=\lvert \lambda \rvert \beta (\mathbf{B})\), \(\lambda \in \mathbf{R}\) and \(\beta (\mathbf{B}_{1}+\mathbf{B}_{2})\leq \beta (\mathbf{B}_{1})+\beta (\mathbf{B}_{2})\);
 (iii)
\(\beta (\mathbf{B}_{1})\leq \beta (\mathbf{B}_{2})\) for \(\mathbf{B}_{1}\subset \mathbf{B}_{2}\) and \(\beta (\mathbf{B}_{1} \cup \mathbf{B}_{2})=\sup \{\beta (\mathbf{B}_{1}),\beta (\mathbf{B} _{2})\}\);
 (iv)
\(\beta (\operatorname{conv}\mathbf{B})=\beta (\mathbf{B})\);
 (v)
\(\beta (\bar{\mathbf{B}})=\beta (\mathbf{B})\).
Definition 2.5
Definition 2.6
Here, we represent the family of all strict βcontraction mappings \(\mathcal{F}:\Omega \rightarrow \mathbf{Z}\) by \(\Theta \mathbf{C}_{\beta }(\Omega )\). Further, denoting the family of all βcondensing mappings \(\mathcal{F}:\Omega \rightarrow \mathbf{Z}\) by \(\mathbf{C}_{\beta }(\Omega )\).
Remark 1
Each \(\mathcal{F}\in \mathbf{C}_{\beta }(\Omega )\) is βLipschitz with constant \(\mathcal{K}=1\), where \(\Theta \mathbf{C}_{\beta }( \Omega )\subset \mathbf{C}_{\beta }(\Omega )\).
The provided propositions are necessarily required for our analysis throughout this paper.
Proposition 2.2
Consider \(\mathcal{F},\mathcal{G}:\Omega \rightarrow \mathbf{Z} \) to be βLipschitz operators and there exist two constants \(\mathcal{K}\) and \(\mathcal{K}^{\prime}\), respectively, then their sum \(\mathcal{F}+\mathcal{G}:\Omega \rightarrow \mathbf{Z}\) is also βLipschitz with constant \(\hat{\mathcal{K}}=\mathcal{K}+ \mathcal{K}^{\prime}\).
Proposition 2.3
The operator \(\mathcal{F}\) is βLipschitz with constant \(\mathcal{K}=0\). Then the same operator \(\mathcal{F}:\Omega \rightarrow \mathbf{Z}\) is compact.
Proposition 2.4
If an operator \(\mathcal{F}:\Omega \rightarrow \mathbf{Z}\) is Lipschitz with constant \(\mathcal{K}\). Then the same operator \(\mathcal{F}\) will also be βLipschitz with the same constant \(\mathcal{K}\).
Theorem 2.7
We recall some basic properties of proposed degree theory from Isaia [37].
 (D1)
Normalization: \(\deg (I,\Omega ,z)=1\) at each \(z\in \Omega \);
 (D2)additivity on domain: For each pair of disjoint open sets \(\Omega_{1}, \Omega_{2}\subset \Omega \) and each \(z\notin (I \mathcal{F})(\bar{(\Omega )}\setminus (\Omega_{1} \cup \Omega_{2}))\), we have$$\deg (\mathfrak{I}\mathcal{F},\Omega ,z)=\deg (\mathcal{I} \mathcal{F}\Omega , \Omega_{1},z)+\deg (\mathcal{I}\mathcal{F},\Omega _{2},z); $$
 (D3)invariance property under homoptopy: \(\deg (\mathcal{I}H(t,z), \Omega , z)\) is independent of \(t\in \mathfrak{J}\) for each continuous and bounded mapping \(H:\mathfrak{J}\times \bar{\Omega }\rightarrow \mathbf{Z}\) which satisfiesand every continuous function \(z:\mathfrak{J}\rightarrow \mathbf{Z}\) which satisfies$$\beta \bigl(H(\mathfrak{J}\times \mathbf{B})\bigr)< \beta (\mathbf{B}),\quad \textit{for all } \mathbf{B}\subset \bar{\Omega } \textit{ with } \beta (\mathbf{B})>0 $$$$z\neq zH(t,z), \quad \textit{for all } t\in \mathfrak{J}, \textit{for every } z\in \partial \Omega ; $$
 (D4)existence: \(\deg (\mathcal{I}\mathcal{F},\Omega ,z) \neq 0\) yields$$z\in (\mathcal{I}\mathcal{F}) (\Omega ); $$
 (D5)
excision: \(\deg (\mathcal{I}\mathcal{F},\Omega ,z)= \deg (\mathcal{I}\mathcal{F},\Omega_{1},z)\) for each open set \(\Omega_{1} \subset \Omega \) and for all \(z\notin (\mathcal{I} \mathcal{F})(\bar{\Omega } \setminus \Omega_{1})\).
Theorem 2.8
Theorem 2.9
[38]
3 Existence of at least one solution to BVP (1)
The purpose of this section is concerning to establish the required theory for existence of at least one solutions to the BVP (1).
Lemma 3.1
Proof
Lemma 3.2
A function \(z\in \mathbf{Z}\) will be the solution of the fractional integral equation (6) if and only if z is a solution of (1).
Proof
The proof is obvious. □
 (\(A_{1}\)):

For arbitrary \(u,v\in \mathbf{Z}\) and if there exists a constant \(\mathcal{K}_{\phi } \in [0,1)\), then one has$$\bigl\lvert \phi (z)\phi (\bar{z})\bigr\rvert \leq \mathcal{K}_{\phi } \lVert z \bar{z} \rVert_{\mathbf{Z}}; $$
 (\(A_{2}\)):

for constants \(\mathcal{C}_{\phi }\), \(q_{1} \in [0,1)\) and \(\mathcal{M}_{\phi }>0\), we have the following growth condition:$$\bigl\lvert \phi (z)\bigr\rvert \leq \mathcal{C}_{\phi } \lVert z \rVert_{\mathbf{Z}} ^{q_{1}}+\mathcal{M}_{\phi },\quad \mbox{for each } z\in \mathbf{Z}; $$
 (\(A_{3}\)):

in the same fashion, for constants \(\mathcal{C}_{ \theta }\), \(q_{2} \in [0,1)\) and \(\mathcal{M}_{\theta }>0\), we have the following growth condition:$$\bigl\lvert \theta (t,z(t))\bigr\rvert \leq \mathcal{C}_{\theta } \lVert z \rVert_{\mathbf{Z}}^{q_{2}}+\mathcal{M}_{\theta }. $$
Lemma 3.3
Proof
Lemma 3.4
Proof
Lemma 3.5
The operator \(\mathcal{G}:\mathbf{Z}\rightarrow \mathbf{Z}\) is compact. Consequently, \(\mathcal{G}\) is βLipschitz with zero constant.
Proof
From now on, we will prove our main results.
Theorem 3.1
Under the hypotheses (\(A_{1}\))(\(A_{3}\)) equation (1) has at least one solution \(u\in \mathbf{Z}\). Also, the set of solutions for (1) is bounded in Z.
Proof
Therefore, we conclude that the operator T has at least one fixed point and the set of fixed points is bounded in Z. □
 (\(A_{4}\)):

There exist constants \(\mathcal{L}_{\theta }>0\), \(\lambda \in [0,1\frac{1}{r}]\) for some \(r\in (0,1\frac{1}{1q})\) such that$$\bigl\lvert \theta (t,z)\theta (t,\bar{z}) \bigr\rvert \leq \mathcal{L}_{ \theta }\lvert z\bar{z} \rvert^{\lambda }, \quad \mbox{for each } t\in \mathfrak{J}, \mbox{and for all } z, \bar{z} \in \mathbf{R}. $$
Theorem 3.2
Proof
 (\(A_{5}\)):

For a \(\mathcal{L}_{\theta }>0\), the following relation holds:$$\bigl\lvert \theta (t,z)\theta (t,\bar{z}) \bigr\rvert \leq \mathcal{L}_{ \theta }\lvert z\bar{z }\rvert ,\quad \mbox{for each } t\in \mathfrak{J}, \mbox{and for each } z, \bar{z} \in \mathbf{R}. $$
Theorem 3.3
Assume that the hypotheses (\(A_{1}\))(\(A_{5}\)) hold, then FDE (1) has a unique solution \(z\in \mathbf{Z}\) if \(\frac{ \mathcal{M}^{\ast }}{1\mathcal{K}_{\phi }}<1\).
Proof
4 Illustrative example
Example 1
Example 2
Upon computation, we get \(d=1.27739\), \(\mathcal{Q}=d\mathcal{C}_{ \phi }=0.127739\). Thus for the given boundary value problem (14) of FDEs, all the data dependence results (\(A_{1}\))(\(A_{5}\)) hold.
Further, it is easy to show by using Theorem 3.1 that there exists at least one solution for (14) which is bounded. Also, one can easily derive the assumptions of Theorems 3.2 and 3.3.
5 Concluding remarks
In this paper, we have successfully applied an a priori estimate method known as topological degree method rather than Schauder and Brouwer degree theory. Highly interesting results for the existence of at least one solution have been derived. In the future, we can extend the concerned theory to highly applicable nonlinear problems of applied analysis to investigate them for solutions.
Declarations
Acknowledgements
We are thankful to the reviewers for their useful corrections and suggestions which improved the quality of this paper. This research work has been supported financially by Abdul Wali Khan University Mardan, Pakistan and Cankaya University, Turkey.
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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