Abstract
In this article, we consider the following boundary-value problem of nonlinear fractional differential equation with p-Laplacian operator:
where \(1<\alpha \leq 2\) is a real number, the time scale T is a nonempty closed subset of \(\mathbb{R}\). \(D^{\alpha }\) is the conformable fractional derivative on time scales, \(\phi_{p}(s)=\vert s \vert ^{p-2}s\), \(p>1\), \(\phi_{p}^{-1}=\phi_{q}\), \(1/p+1/q=1\), and \(f:[0, \sigma (1)]\times [0,+ \infty )\to [0,+\infty )\) is continuous. By the use of the approach method and fixed-point theorems on cone, some existence and multiplicity results of positive solutions are acquired. Some examples are presented to illustrate the main results.
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1 Introduction
In this paper, the existence and multiplicity of positive solutions for the following fractional differential boundary-value problem on time scales is studied:
where \(1<\alpha \leq 2\), \(D^{\alpha }\) is the conformable fractional derivative on time scales, \(\phi_{p}(s)=\vert s \vert ^{p-2}s\), \(p>1\), \(\phi_{p}^{-1}= \phi_{q}\), \(1/p+1/q=1\), and \(f: [0, \sigma (1)]\times [0, +\infty ) \to [0, +\infty )\) is continuous.
The existence of positive solutions for boundary-value problem on time scales has become the focus in recent years; for details, see [1–6]. Due to the wide applications, many researchers studied the existence of positive solutions for fractional derivatives boundary-value problem [7–21] and the references therein. Meanwhile, the boundary-value problem with p-Laplacian operator have also been discussed extensively in the literature; for example, see [4, 11, 22–27].
For \(\alpha = 2\), problem (1.1), (1.2) is called a fourth order p-Laplacian boundary-value problem which has been studied in [4].
Dong et al. [22] investigated the boundary-value problem for a fractional differential equation with the p-Laplacian operator
where \(1<\alpha \leq 2\) is a real number, \(D^{\alpha }\) is the conformable fractional derivative, \(\phi_{p}(s)=\vert s \vert ^{p-2}s\), \(p>1\), \(\phi _{p}^{-1}=\phi_{q}\), \(1/p+1/q=1\),\(f: [0, 1] \times [0, +\infty ) \to [0, +\infty )\) is continuous. By the use of the fixed-point theorems on cone, some existence and multiplicity results of positive solutions are obtained.
Motivated by the work mentioned above, we investigate the existence and multiplicity of positive solutions for (1.3), (1.4) on time scales. The rest of this paper is organized as follows. In Sect. 2, we recall some concepts relative to the new conformable fractional calculus and give some lemmas with respect to the corresponding Green’s function. In Sect. 3, we investigate the existence and multiplicity of positive solution for boundary-value problem (1.1), (1.2). In Sect. 4, we present some examples to illustrate our main results, respectively.
2 Preliminaries and lemmas
In this section, we introduced notations and definitions of conformable fractional derivative on time scales and some lemmas. Let T be a time scale and denote \([a,b]_{T}=: [a,b]\cap T\). These results can be found in the recent literature; see [2, 3, 6].
Definition 2.1
A time scale T is a nonempty closed subset of \(\mathbb{R}\); assume that T has the topology that it inherits from the standard topology on \(\mathbb{R}\). Define the forward and backward jump operators σ, ρ: \(T\rightarrow T\) by
In this definition we put \(\inf \emptyset =\sup T\), \(\sup \emptyset = \inf T\). Set \(\sigma^{2}(t)=\sigma (\sigma (t))\), \(\rho^{2}(t)=\rho ( \rho (t))\). The sets \(T^{k}\) and \(T_{k}\) which are derived from the time scale T are as follows:
Denote interval I on T by \(I_{T}=I\cap T\).
Definition 2.2
If \(f :T \rightarrow \mathbb{R}\) is a function and \(t\in T_{k}\), then the delta derivative of f at the point t is defined to be the number \(f^{\Delta }(t)\) (provided it exists) with the property that, for each \(\epsilon >0\), there is a neighborhood U of t such that
for all \(s\in U\). The function f is called Δ-differentiable on \(T^{k}\) if \(f^{\Delta }(t)\) exists for all \(t\in T^{k}\).
Definition 2.3
([3])
Let \(\alpha \in (1, 2]\) and \(f: T\rightarrow \mathbb{R}\), \(t\in T^{k}\). For \(t>0\), we define \(D^{\alpha }f(t)\) to be the number (provided it exists) with the property that, given any \(\varepsilon >0\), there is a δ-neighborhood \(V_{t} \subset T\) of \(t, \delta >0\), such that
We call \(D^{\alpha } f(t)\) the conformable fractional derivative of f of order α at t, and we define the conformable fractional derivative at 0 as \(D^{\alpha } f(0)=\lim_{t\rightarrow 0^{+}}D^{ \alpha } f(t)\).
Lemma 2.1
([3])
Let \(\alpha \in (1, 2]\) and f be two times delta differentiable at \(t \in T^{k} \). The following relation holds: \(D^{\alpha }f(t) = t^{2-\alpha }f^{\bigtriangleup \bigtriangleup }(t)\).
Definition 2.4
([2])
A function \(f: T \rightarrow \mathbb{R}\) is called regulated provided its right-sided limits exist (and are finite) at all right-dense points in T and its left-sided limits exist (and are finite) at all left-dense points in T.
Definition 2.5
([2])
A function \(f: T\rightarrow \mathbb{R}\) is called rd-continuous provided it is continuous at right-dense points in T and its left-sided limit exist (finite) at all left-dense points in T. The set of rd-continuous functions \(f: T\rightarrow \mathbb{R}\) will be denoted by \(C_{rd}(T, R)\).
Lemma 2.2
([2])
Assume \(f: T\rightarrow \mathbb{R}\).
-
(i)
If f is continuous, then f is rd-continuous.
-
(ii)
If f is rd-continuous, then f is regulated.
Definition 2.6
([3])
Let \(f: T\rightarrow \mathbb{R}\) be a regulated function and \(1<\alpha \leq 2\). Then the α-fractional integral of f is defined by
Lemma 2.3
Let \(t>0\), \(\alpha \in (1, 2]\), and the function \(f: [0, \infty )_{T} \rightarrow \mathbb{R}\) be rd-continuous, then \(D^{\alpha }I^{\alpha }f(t)=f(t)\).
Proof
Since \(f(t)\) is rd-continuous, then \(f(t)\) is regulated, and \(I^{\alpha }f(t)\) is twice times differentiable. In view of Lemma 2.1, one has
The proof is complete. □
Lemma 2.4
(Mean value theorem [6])
Let \(a\geq 0\) and \(f: T\rightarrow \mathbb{R}\) be a function continuous on \([a, b]_{T}\) which is conformable fractional differentiable of order with α on \([a, b]_{T}\). Then there exist \(\xi , \tau \in [a, b]_{T}\) such that
Lemma 2.5
Let \(\alpha \in (1, 2], f\) be a α-differentiable function at \(t> 0\), then \(D^{\alpha }f(t)= 0\) for \(t\in [0, 1]_{T}\) if and only if \(f(t)= a_{0}+ a_{1}t\), where \(a_{k}\in \mathbb{R}\), for \(k= 0, 1\).
Proof
The sufficiency follows by the definition of the delta derivative on time scales.
Next, given \(t_{1}, t_{2}\in [0, 1]_{T}\) with \(t_{1}< t_{2}\), by Lemma 2.4, there exists \(\xi , \tau \in (t_{1}, t_{2})_{T}\) such that
By means of \(D^{\alpha }f(\xi )= D^{\alpha }f(\tau )=0\), we have \(f^{\Delta }(t_{2})= f^{\Delta }(t_{1})\), with the arbitrariness of \(t_{1}\), \(t_{2}\), one has \(f^{\Delta }(t)\) is a constant, so \(f(t)= a _{0}+ a_{1}t\), for \(t\in [0, 1]_{T}\). □
With Lemma 2.3 and Lemma 2.5, the following lemma is immediate.
Lemma 2.6
Assume that \(u\in C(0, +\infty )_{T}\) with a fractional derivative of order \(\alpha \in (1, 2]\). Then
for some \(c_{k}\in \mathbb{R}\), \(k= 0, 1\).
We present below the Green’s function and its properties.
Lemma 2.7
Given \(y\in C[0, \sigma (1)]_{T}\), the unique solution of
is
where
Proof
By the use of the Lemma 2.6, we can deduce from equation (2.3) an equivalent integral equation,
for some \(c_{0}, c_{1} \in \mathbb{R}\). By (2.4), there are
Therefore, the unique solution of Problem (2.3), (2.4) is
The proof is complete. □
We point out here that (2.5) becomes the usual Green’s function when \(\alpha = 2\) on time scales.
Lemma 2.8
Let \(y\in C[0,\sigma (1)]\) and \(1<\alpha \leq 2\). Then the problem
has a unique solution
Proof
Applying operator \(I^{\alpha }\) on both sides of (2.6), with Lemma 2.6,
So,
for some \(C_{0}, C_{1} \in \mathbb{R}\). By the boundary conditions \(D^{\alpha }u(0)= D^{\alpha }u(\sigma (1))= 0\), as a consequence we have
Therefore, the solution \(u(t)\) of fractional differential equation boundary-value problem (2.6) and (2.7) satisfies
Thus, the fractional differential equation boundary-value problem (2.6) and (2.7) is equivalent to the problem
Lemma 2.7 implies that fractional differential equation boundary-value problem (2.6), (2.7) has a unique solution
The proof is complete. □
Lemma 2.9
The function \(G(t, s)\) defined by (2.5) satisfies:
-
(i)
\(G(t, s)\geq 0\), for \(t\in [0, \sigma (1)]\), \(s\in [0, 1]\), and \(G(t, s)> 0\), for \(t\in (0, \sigma (1))\), \(s\in (0, 1)\);
-
(ii)
\(G(t,s)\leq G(s,s)\), for \(t\in [0,\sigma (1)]\), \(s \in [0, 1]\);
-
(iii)
\(G(t,s)\geq \frac{\sigma (1)}{4}G(s,s)\), for \(t\in [\frac{\sigma (1)}{4}, \frac{3\sigma (1)}{4}]\), \(s\in [0, 1]\).
Proof
Observing the expression of \(G(t, s)\), it is clear that \(G(t, s) \geq 0\) for \(t\in [0, \sigma (1)]\), \(s\in [0, 1]\), and \(G(t, s)> 0\), for \(t\in (0, \sigma (1))\), \(s\in (0, 1)\). Moreover, \(G(t, s)\) is decreasing with respect to t for \(s\leq t\), and increasing for \(t\leq s\). By the fact
We have
for \(t\in [0,\sigma (1)]\), \(s\in [0,1]\). Furthermore, if \(t\in [\frac{\sigma (1)}{4}, \frac{3\sigma (1)}{4}]\), \(s\in [0, 1]\), one has
which implies the desired results. □
Lemma 2.10
([27])
The following relations hold:
-
(1)
If \(1< q \leq 2\), then \(\vert \phi_{q}(u+ v)- \phi_{q}(u) \vert \leq 2^{2-q}\vert v \vert ^{q-1}\) for \(u, v\in \mathbb{R}\).
-
(2)
If \(q> 2\), then \(\vert \phi_{q}(u+ v)-\phi_{q}(u) \vert \leq (q-1)( \vert u \vert + \vert v \vert )^{q-2} \vert v \vert \) for \(u, v\in \mathbb{R}\).
Lemma 2.11
([28])
Suppose E is a Banach space and \(T_{n}: E\rightarrow E\), \(n= 3, 4, \ldots \) are completely continuous operators, \(T: E\rightarrow E\). If \(\Vert T_{n}u- Tu \Vert \) uniformly to zero when \(n\rightarrow \infty \) for all bounded set \(\Omega \subseteq E\), then \(T: E\rightarrow E\) is completely continuous.
Definition 2.7
The map θ is said to be a nonnegative continuous concave functional on a cone P of a Banach space E provided that \(\theta : P\rightarrow [0, \infty ) \) is continuous and
for all \(x, y\in P\) and \(0< t< 1\).
The following fixed-point theorems are useful in our proofs.
Lemma 2.12
([29])
Let E be a Banach space, \(P\subseteq E\) be a cone, and \(\Omega_{1}\), \(\Omega_{2}\) be two bounded open balls of E centered at the origin with \(\overline{\Omega_{1}}\subset \Omega_{2}\). Suppose that \(\mathcal{A}: P\cap (\overline{\Omega_{2}}\backslash \Omega_{1}) \rightarrow P\) is a completely continuous operator such that either
-
(i)
\(\Vert \mathcal{A} x \Vert \leq \Vert x \Vert \), \(x\in P\cap \partial \Omega_{1}\), and \(\Vert \mathcal{A} x \Vert \geq \Vert x \Vert \), \(x\in P\cap \partial \Omega_{2}\), or
-
(ii)
\(\Vert \mathcal{A} x \Vert \geq \Vert x \Vert \), \(x\in P\cap \partial \Omega_{1}\), and \(\Vert \mathcal{A} x \Vert \leq \Vert x \Vert \), \(x\in P\cap \partial \Omega_{2}\),
holds. Then \(\mathcal{A}\) has a fixed point in \(P\cap (\overline{ \Omega_{2}}\backslash \Omega_{1})\).
Lemma 2.13
([30])
Let P be a cone in a real Banach space E, \(P_{c}= \{x\in P \mid \Vert x \Vert \leq c\}\), θ be a nonnegative continuous concave functional on P such that \(\theta (x)\leq \Vert x \Vert \), for all \(x\in \overline{P_{c}}\), and \(P(\theta , b, d)= \{x\in P \mid b\leq \theta (x), \Vert x \Vert \leq d\}\). Suppose \(\mathcal{A}: \overline{P_{c}} \rightarrow \overline{P_{c}}\) is a completely continuous and there exist constants \(0< a< b< d\leq c\) such that
-
(C1)
\(\{x\in P(\theta , b, d) \mid \theta (x)> b\}\) is nonempty, and \(\theta (\mathcal{A}x)> b\), for \(x\in P(\theta , b, d)\);
-
(C2)
\(\Vert \mathcal{A}x \Vert < a\), for \(x\leq a\);
-
(C3)
\(\theta (\mathcal{A}x)> b\), for \(x\in P(\theta , b, c)\) with \(\Vert \mathcal{A}x \Vert > d\).
Then \(\mathcal{A}\) has at least three fixed points \(x_{1}\), \(x_{2}\), \(x _{3}\) with
Remark 2.1
([30])
If we have \(d= c\), then condition (C1) of Lemma 2.13 implies condition (C3) of Lemma 2.13.
3 Existence results
Let \(E= \{u: [0,\sigma (1)]\rightarrow \mathbb{R} \}\) be endowed with the ordering \(u\leq v\) if \(u(t)\leq v(t)\) for all \(t\in [0, \sigma (1)]\), and the norm \(\Vert u \Vert = \max_{0\leq t\leq \sigma (1)}\vert u(t) \vert \). Define
Given a function \(f\in C([0, \sigma (1)]\times [0, \infty ), [0, \infty ))\), define \(T, T_{n}: P\rightarrow E\) as
Lemma 3.1
\(T: P\rightarrow P\) is completely continuous.
Proof
Firstly, take the constant in the second member to be independent on n, Hence, we show that \(T_{n}: P\rightarrow P\) are completely continuous for \(n= 3, 4, \ldots \) . Given \(u\in P\), with Lemma 2.9 and the nonnegativity of \(f(t, u)\), one has
so
For \(u\in P\),
It follows that
Hence, \(T_{n}u\in P\), and so \(T_{n}: P\rightarrow P\). Let \(\Omega \subset P\) be bounded, i.e., there exists a positive constant \(M> 0\) such that \(\Vert u \Vert \leq M\) for all \(u\in \Omega \). Let
then, for \(u\in \Omega \), we have
Hence, \(T_{n}(\Omega )\) is bounded for \(n= 3, 4, \ldots \) .
On the other hand, given \(\epsilon > 0\), let
then, for each \(u\in \Omega \), \(t_{1}, t_{2}\in [0, \sigma (1)]\), \(t_{1} \leq t_{2}\), and \(t_{2}- t_{1}< \delta \), one has
That is to say \(T_{n}(\Omega )\) has equicontinuity. In fact, we consider three situations.
(1) \(0< t_{1} \leq t_{2}< \frac{1}{n}\).
(2) \(0< t_{1}\leq \frac{1}{n} \leq t_{2}< 1\).
(3) \(\frac{1}{n}< t_{1} \leq t_{2}< 1\).
By the means of the Arzela–Ascoli theorem, we see that \(T_{n}: P \rightarrow P\) are completely continuous operators.
Secondly, it is clear that \(T: P\rightarrow P\). We prove that \(T_{n}: P\rightarrow P\) have uniform convergence to T and \(T: P\rightarrow P\) is completely continuous too.
With the use of Lemma 2.10,
Given \(\epsilon > 0\), let
then \(\Vert T_{n}u- Tu \Vert < \epsilon \), for all \(n> N\). In fact,
By the use of Lemma 2.11, \(T: P\rightarrow P\) is completely continuous. □
We take into account that the Green’s function satisfy \(G(t, s)\geq 0\) for \(t\in [0, \sigma (1)]\), \(s\in [0, 1]\), and \(G(t, s)> 0\), for \(t\in (0, \sigma (1))\), \(s\in (0, 1)\). The following constants are well defined:
Theorem 3.1
Let \(f\in C([0, \sigma (1)]\times [0, \infty ), [0, \infty ))\). Assume that there exist two different positive constants \(r_{2}, r_{1}\), and \(r_{2}\neq r_{1}\) such that
-
(H1)
\(f(t, u)\leq \phi_{p}(Mr_{1})\), for \((t, u)\in [0,\sigma (1)]\times [0, r_{1}]\);
-
(H2)
\(f(t, u)\geq \phi_{p}(Nr_{2})\), for \((t, u)\in [\frac{ \sigma (1)}{4}, \frac{3\sigma (1)}{4}]\times [\frac{\sigma (1)}{4}r _{2}, r_{2}]\).
Then Problem (1.1), (1.2) has at least one positive solution u such that \(\min \{r_{2}, r_{1}\}\leq \Vert u \Vert \leq \max \{r_{2}, r _{1}\}\).
Proof
By Lemma 3.1, \(T:P\rightarrow P\) is completely continuous. Without loss of generality, suppose \(0< r_{1}< r_{2}\), and let
For \(u\in \partial \Omega_{1}\), we have \(0\leq u(t)\leq r_{1}\) for all \(t\in [0, \sigma (1)]\). It follows from (H1) that
So,
For \(u\in \partial \Omega_{2}\), by the definition of P, we have
By assumption (H2), for \(t\in [\frac{\sigma (1)}{4}, \frac{3\sigma (1)}{4}]\), we have
So,
Therefore, by Lemma 2.12, we complete the proof. □
Theorem 3.2
Suppose \(f\in C([0, \sigma (1)]\times [0, \infty ), [0, \infty ))\) and there exist constants \(0< a< b<c\) such that the following assumptions hold:
-
(A1)
\(f(t, u)\leq \phi_{p}(Ma)\), for \((t, u)\in [0, \sigma(1)]\times [0, a]\);
-
(A2)
\(f(t, u)\geq \phi_{p}(Nb)\), for \((t, u)\in [\frac{\sigma (1)}{4}, \frac{3\sigma (1)}{4}]\times [b,c]\);
-
(A3)
\(f(t, u)\leq \phi_{p}(Mc)\), for \((t, u)\in [0, \sigma(1)]\times [0, c]\).
Then the boundary-value problem (1.1), (1.2) has at least three positive solutions \(u_{1}\), \(u_{2}\), \(u_{3}\) with
Proof
We show that all the conditions of Lemma 2.13 are satisfied. If \(u\in \overline{P}_{c}\), then \(\Vert u \Vert \leq c\). Assumption (A3) implies \(f(t, u(t))\leq \phi_{p}(Mc)\) for \(0\leq t\leq \sigma (1)\), consequently,
Hence, \(T: \overline{P}_{c}\rightarrow \overline{P}_{c}\). Similarly, if \(u\in \overline{P}_{a}\), then assumption (A1) yields \(f(t, u(t)) \leq \phi_{p}(Ma)\), \(0\leq t\leq \sigma (1)\). Therefore, condition (C2) of Lemma 2.13 is satisfied.
Choose
Then \(u(t)\in P(\theta , b, c)\), \(\theta (u)= \theta (\frac{b+ c}{2})> b\), consequently,
Hence, if \(u\in P(\theta , b, c)\), then \(b\leq u(t)\leq \ c \) for \(\frac{\sigma (1)}{4}\leq t\leq \frac{3\sigma (1)}{4}\). From assumption (A3), we have \(f(t, u(t))\geq \phi_{p}( Nb)\) for \(\frac{\sigma (1)}{4}\leq t\leq \frac{3\sigma (1)}{4}\). So
i.e.,
This shows that condition (C1) of Lemma 2.13 is satisfied.
By Lemma 2.13 and Remark 2.1, Problem (1.1), (1.2) has at least three positive solutions \(u_{1}\), \(u_{2}\), \(u_{3}\), satisfying
The proof is complete. □
4 Examples
Example 4.1
Let \(T= \mathbb{R}\), \(\alpha =\frac{3}{2}\), \(p=3\), consider the following fractional differential equation boundary-value problem:
By a simple computation, we obtain \(M= 3.75\), \(N \approx 5.987\). Choose \(r_{1}= 1\), \(r_{2}= \frac{1}{5}\), then
With the use of Theorem 3.1, the fractional differential equation boundary-value problem (4.1) and (4.2) has at least one positive solution u such that \(\frac{1}{5}\leq \Vert u \Vert \leq 1\).
Example 4.2
Let \(T= \mathbb{R}\), consider the following fractional differential equation boundary-value problem:
where
We obtain \(M= 3.75\), \(N\approx 5.987\). Choose \(a= 0.1\), \(b= 1\), \(c=4\), then
With the use of Theorem 3.2, the fractional differential equation boundary-value problem (4.3) and (4.4) has at least three positive solutions \(u_{1}\), \(u_{2}\) and \(u_{3}\) with
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The corresponding author is a professor. He has worked on nonlinear functional analysis and fractional boundary-value problems for many years. The first author and the second author are doctorate candidates. Their research field is the solvability of fractional boundary-value problems.
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This work is supported by NSFC (11571207), the Taishan Scholar project and SDUST graduate innovation project SDKDYC170343.
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Sheng, K., Zhang, W. & Bai, Z. Positive solutions to fractional boundary-value problems with p-Laplacian on time scales. Bound Value Probl 2018, 70 (2018). https://doi.org/10.1186/s13661-018-0990-2
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DOI: https://doi.org/10.1186/s13661-018-0990-2