Abstract
Using fixed point results of \(\alpha-\psi\)-Geraghty contractive type mappings, we examine the existence of solutions for some fractional differential equations in b-metric spaces. By some concrete examples we illustrate the obtained results.
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1 Introduction
In 2012, Samet et al. [11] presented the concepet of α-admissible mappings, which was expanded by several authors (see [5, 6, 9]). Baleanu, Rezapour, and Mohammadi [3] studied the existence of a solution for problem \(D^{\nu}w(\xi)=h(\xi, w(\xi))\) \((\xi\in[0,1],1<\nu\leq2)\). Afshari, Aydi, and Karapinar [1, 2] considered generalized \(\alpha-\psi\)-Geraghty contractive mappings in b-metric spaces.
We investigate the existence of solutions for some fractional differential equations in b-metric spaces. We denote \(I=[0,1]\).
Definition 1.1
The Caputo derivative of order ν of a continuous function \(h:[0,\infty)\rightarrow\mathbb{R}\) is defined by
where \(n-1<\nu<n\), \(n=[\nu]+1\), \([\nu]\) is the integer part of ν, and
Definition 1.2
The Riemann–Liouville derivative of a continuous function h is defined by
where the right-hand side is defined on \((0,\infty)\).
Let Ψ be the set of all increasing continuous functions \(\psi: [0,\infty) \to\mathbb{[}0,\infty)\) such that \(\psi(\lambda x)\leq \lambda\psi(x)\leq\lambda x\) for \(\lambda>1\), and let \(\mathcal{B}\) be the family of nondecreasing functions \(\gamma: [0,\infty) \to[0,\frac{1}{s^{2}})\) for some \(s\geq1\).
Definition 1.3
([1])
Let \((X,d)\) be a b-metric space (with constant s). A function \(g:X\rightarrow X\) is a generalized \(\alpha-\psi\)-Geraghty contraction if there exists \(\alpha:X\times X\to [0,\infty)\) such that
for all \(z,t \in X\), where \(\gamma\in\mathcal{B}\) and \(\psi\in\Psi\).
Definition 1.4
([11])
Let \(g: X\rightarrow X\) and \(\alpha: X\times X\rightarrow[0,\infty)\) be given. Then g is called α-admissible if for \(z,t\in X\),
Theorem 1.5
([1])
Let \((X,d)\) be a complete b-metric space, and let \(f:X\rightarrow X\) be a generalized \(\alpha-\psi\)-Geraghty contraction such that
-
(i)
f is α-admissible;
-
(ii)
there exists \(u_{0}\in X\) such that \(\alpha(u_{0},fu_{0})\geq1\);
-
(iii)
if \(\{u_{n}\}\subseteq X\), \(u_{n}\rightarrow u\) in X, and \(\alpha(u_{n},u_{n+1})\geq1\), then \(\alpha(u_{n},u)\geq 1\).
Then f has a fixed point.
2 Main result
By \(X=C(I)\) we denote the set of continuous functions. Let \(d:X\times X\to[0,\infty)\) be given by
Evidently, \((X,d)\) is a complete b-metric space with \(s=2\) but is not a metric space.
Now we study the problem
under the conditions
where \(D^{\nu}\) is the Riemann–Liouville derivative, and \(h:I\times X\rightarrow\mathbb{R}\) is continuous.
Lemma 2.1
([13])
Given \(h\in C(I\times X,\mathbb{R})\) and \(3 <\nu\leq4\), the unique solution of
where
is given by \(w(\xi)=\int_{0}^{1} G(\xi,\zeta)h(s,w(s))\,ds\), where
If \(h(\xi,w(\xi))=1\), then the unique solution of (7)–(8) is given by
Lemma 2.2
([13])
In Lemma 2.1, \(G(\xi,\zeta)\) given in (9) satisfies the following conditions:
-
(1)
\(G(\xi, \zeta) > 0\), and \(G(\xi,\zeta)\) is continuous for \(\xi,\zeta\in I\);
-
(2)
\(\frac{(\nu-2)\sigma(\xi)\rho(\zeta)}{\Gamma(\nu)}\leq G(\xi,\zeta)\leq\frac{r_{0}\rho(\zeta)}{\Gamma(\nu)}\),
where
Theorem 2.3
Suppose
-
(i)
there exist \(\theta:\mathbb{R}^{2}\rightarrow\mathbb{R}\) and \(\psi\in\Psi\) such that
$$\bigl\vert h(\xi,c)-h(\xi,d) \bigr\vert \leq \frac{1}{2\sqrt{2}} \frac{\Gamma(\nu+1)}{4\nu} \frac{\psi( \vert c-d \vert ^{2})}{\sqrt{4\| (c-d)^{2}\|_{\infty}+1}} $$for \(\xi\in I\) and \(c,d\in\mathbb{R}\) with \(\theta(c,d)\geq0\);
-
(ii)
there exists \(y_{0}\in C(I)\) such that \(\theta(y_{0}(\xi),\int _{0}^{1}G(\xi,\zeta)h(\zeta,y_{0}(\xi))\,d\zeta)\geq0\), \(\xi\in I\);
-
(iii)
for \(\xi\in I\) and \(y,z\in C(I)\), \(\theta(y(\xi),z(\xi))\geq0\) implies
$$\theta \biggl( \int_{0}^{1}G(\xi,\zeta)h\bigl(\zeta,y(\zeta)\bigr)d \zeta, \int _{0}^{1}G(\xi,\zeta)h\bigl(\zeta,z(\xi)\bigr)d \zeta \biggr)\geq 0; $$ -
(iv)
if \(\{y_{n}\}\subseteq C(I)\), \(y_{n}\rightarrow y\) in \(C(I)\), and \(\theta(y_{n},y_{n+1})\geq0\), then \(\theta(y_{n},y)\geq0\).
Then problem (7) has at least one solution.
Proof
By Lemma 2.1 \(y\in C(I)\) is a solution of (7) if and only if it is a solution of \(y(\xi)=\int_{0}^{1} G(\xi,\zeta)h(\zeta,y(\zeta))\,d\zeta\), and we define \(A : C(I)\rightarrow C(I)\) by \(Ay(\xi)=\int_{0}^{1} G(\xi,\zeta)h(\zeta,y(\zeta))\,d\zeta\) for \(\xi\in I\). For this purpose, we find a fixed point of A. Let \(y,z\in C(I)\) be such that \(\theta(y(\xi),z(\xi))\geq0\) for \(\xi\in I\). Using (i), we get
Hence, for \(y,z\in C(I)\) and \(\xi\in I\) with \(\theta(y(\xi),z(\xi ))\geq 0\), we have
Let \(\alpha: C(I)\times C(I)\rightarrow[0,\infty)\) be defined by
Define \(\gamma: [0,\infty)\rightarrow[0,\frac{1}{4})\) by \(\gamma (q)=\frac{q}{4q+1}\) and \(s=2\).
So
Then A is an \(\alpha-\psi-\)contractive mapping. From (iii) and the definition of α we have
for \(y,z\in C(I)\). Thus, A is α-admissible. By (ii) there exists \(y_{0}\) \(\in C(I)\) such that \(\alpha(y_{0},Ay_{0})\geq1\). By (iv) and Theorem 1.5 there is \(y^{*}\in C(I)\) such that \(y^{*}=Ay^{*}\). Hence \(y^{*}\) is a solution of the problem. □
Corollary 2.4
Suppose that there exist \(\theta:\mathbb{R}^{2}\rightarrow\mathbb{R}\) and \(\psi\in \Psi\) such that
for \(\xi\in I\) and \(c,d\in\mathbb{R}\) with \(\theta(c,d)\geq0\). Also, suppose that conditions (ii)–(iv) from Theorem 2.3 hold for h, where \(G(\xi,\zeta)\) is given in (9). Then the problem
where
has at least one solution.
Proof
By Lemma 2.2
Using (10) and (12), by Theorem 2.3 we obtain
The rest of the proof is according to Theorem 2.3. □
Lemma 2.5
([8])
If \(h\in C(I\times X,\mathbb{R})\) and \(h(\xi,w(\xi))\leq0\), then the problem
has a unique positive solution
where \(G(\xi,\zeta)\) is given by
Lemma 2.6
([12])
The function \(G(\xi,\zeta)\) in Lemma 2.5 has the following property:
where \(\xi,\zeta\in I\) and \(3<\nu\leq4\).
Based on Theorem 2.3, we get the following result.
Corollary 2.7
Assume that there exist \(\theta:\mathbb {R}^{2}\rightarrow\mathbb{R}\) and \(\psi\in\Psi\) such that
where \(M=\sup_{\xi\in I} \int_{0}^{1}G(\xi,\zeta)\,d\zeta\). Also, suppose that conditions (ii)–(iv) from Theorem 2.3 are satisfied, where \(G(\xi,\zeta)\) is given in (14). Then problem (13) has at least one solution.
Proof
By Lemma 2.5 \(y\in C(I)\) is a solution of (13) if and only if a solution of \(y(\xi)=\int_{0}^{1} G(\xi,\zeta)h(\zeta,y(\zeta))\,d\zeta\). Define \(A : C(I)\rightarrow C(I)\) by \(Ay(\xi)=\int_{0}^{1} G(\xi,\zeta)h(\zeta,y(\zeta))\,d\zeta\) for \(\xi\in I\). We find a fixed point of A. Let \(y,z\in C(I)\) be such that \(\theta(y(\xi),z(\xi))\geq0\) for \(\xi\in I\). By (i) and Lemma 2.6 we get
Suppose that conditions (ii)–(iv) from Theorem 2.3 are satisfied, where \(G(\xi,\zeta)\) is given in (14). By Theorem 2.3 problem (13) has at least one solution.
Let \((X,d)\) be given in (4). For the equation
via
where \(h:I\times X\rightarrow\mathbb{R}\) is continuous, we have the following result. □
Theorem 2.8
Assume that there exist \(\theta:\mathbb{R}^{2}\rightarrow\mathbb{R}\), \(\gamma\in \mathcal{B}\), and \(\psi\in\Psi\) such that
Suppose conditions (ii)–(iv) from Theorem 2.3 hold, where \(A:C(I)\rightarrow C(I)\) is defined by
Then (15) has at least one solution.
Proof
A function \(y\in C(I)\) is a solution of (15) if and only if it is a solution of
Then (15) is equivalent to finding \(y^{*}\in C(I)\) that is a fixed point of A. Let \(y,z\in C(I)\) with \(\theta(y(\xi),z(\xi))\geq0\), \(\xi\in I\). By (i) we have
for all \(y,z\in C(I)\) with \(\theta(y(\xi),z(\xi))\geq 0\), \(\xi\in I\), so that
Let \(\alpha: C(I)\times C(I)\rightarrow[0,\infty)\) be defined by
Then
for all \(y,z\in C(I)\), and thus A is an \(\alpha-\psi-\)contractive mapping. From Theorem 1.5, based on the proof of Theorem 2.3, we can deduce the proof of Theorem 2.8. □
Here we find a positive solution for
where
Note that \(^{c}D^{\nu}\) is the Caputo derivative of order ν. We consider the Banach space of continuous functions on I endowed with the sup norm. We have the following lemma.
Lemma 2.9
([4])
Let \(0<\nu\leq1\) and \(h\in C([0,T]\times X,\mathbb{R})\) be given. Then the equation
with
has a unique solution given by
where \(G(\xi,\zeta)\) is defined by
By Lemma 2.9 and Theorem 2.4 we get the following conclusion.
Corollary 2.10
Assume that there exist \(\theta:\mathbb{R}^{2}\rightarrow\mathbb{R}\) and \(\psi\in \Psi\) such that
for \(\xi\in I\) and \(c,d\in\mathbb{R}\) with \(\theta(c,d)\geq0\). Suppose conditions (ii)–(iv) from Theorem 2.3 are satisfied, where \(G(\xi,\zeta)\) is given in (17). Then the following problem has at least one solution:
Proof
It is easily that \(\min_{t\in[0,1]}\int_{0}^{1} G(t,s)\,ds=\frac{1}{3}\) and \(\max_{t\in[0,1]}\int_{0}^{1} G(t,s)\,ds=\frac{80}{51}\). By Theorem 2.3 we conclude the desired result. □
Example 2.11
Let \(\psi(r)=r\), \(\theta(x,z)=xz\), and \(y_{n}(\xi)=\frac{\xi}{n^{2}+1}\). We consider \(h:I\times[-2,2]\to[-2,2]\) and the periodic boundary value problem
with
Then
for \(\xi\in I\) and \(c,d\in[-2,2]\) with \(\theta(c,d)\geq0\). Because \(y_{0}(\xi)=\xi\), thus
for all \(\xi\in I\). Also, \(\theta(y(\xi),z(\xi))=y(\xi)z(\xi)\geq 0\) implies that
It is obvious that condition (iv) in Corollary (2.4) holds. Hence by Corollary 2.4 problem (18) has at least one solution.
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Afshari, H., Kalantari, S. & Baleanu, D. Solution of fractional differential equations via \(\alpha-\psi\)-Geraghty type mappings. Adv Differ Equ 2018, 347 (2018). https://doi.org/10.1186/s13662-018-1807-4
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DOI: https://doi.org/10.1186/s13662-018-1807-4