Abstract
In this paper, we discuss the existence and multiplicity of positive solutions for a system of fractional differential equations with boundary condition and advanced arguments. The existence result is proved via Leray–Schauder’s fixed point theorem type in a vector Banach space. Further, by using a new fixed point theorem in order Banach space, we study the multiplicity of positive solutions. Finally, some examples are given to illustrate our results.
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1 Introduction
Fractional calculus and differential equations have now proved to be important tools modeling many real world phenomena like chemistry and physics [11, 22, 23, 25]). For the description of hereditary properties of fractional calculus, see [20, 24, 32, 37] and the references therein.
Recently, there have been some papers dealing with the existence and multiplicity of solution (or positive solution) of nonlinear initial fractional differential equation by the use of techniques of nonlinear analysis, see [2–7, 9, 33, 35, 38].
For example, Chai obtained in [10] the existence of at least one nonnegative solution and two positive solutions by using fixed point theorem on cone for the following problem:
Su et al. [31] studied the existence of one and two positive solutions by using the fixed point index theory of the following boundary values problems:
Tang et al. [34] studied the existence of positive solutions of fractional differential equation with p-Laplacian of the following type (1.3) by using the coincidence degree theory.
In this work, we study the existence and multiplicity of positive solutions of the following problem:
where \(\eta \in (0,1), \gamma \in (0, \frac{1}{\eta ^{\alpha -\beta -1}} ), D_{0^{+}}^{\alpha }, D_{0^{+}}^{ \beta }\) are the standard Riemann–Liouville fractional derivatives with \(\alpha \in (2,3), \beta \in (1,2)\) such that \(\alpha \geq \beta + 1\), p-Laplacian operator is defined as \(\varphi _{p}(s) = |s|^{p-2}s, p > 1\), and the functions \(f,g \in C({\mathbb{R}}^{2},{\mathbb{R}})\).
In recent years, many authors studied the existence of solutions for systems of difference and differential equations with and without fractional derivative by using the vector version of the fixed point theorem (see [1, 8, 13, 15–19, 21, 26–28], the monograph of Graef et al. [12], and the references therein).
For establishing the existence and multiplicity of positive solutions of problem (1.4), let us list the following assumptions:
- \((H_{1})\):
-
\(a_{i} \in L^{1}[0,1]\) is nonnegative and \(a_{i}(t) \not \equiv 0\) on any subinterval of \([0,1]\) for \(i =1,2\).
- \((H_{2})\):
-
The advanced argument \(\theta \in C((0,1),(0,1])\) and \(0 \leq \theta (t) \leq 1, \forall t \in (0,1)\).
This work is organized as follows: In Sect. 2, we introduce all the background material used in this paper such as fractional calculus analysis and some results from fixed point theory. In Sects. 3, 4, the existence and multiplicity results of solutions for a system of fractional p-Laplace differential equations (1.4) are discussed by using the fixed point theorems in the generalized Banach space. We end the paper with two examples to illustrate our main results.
2 Preliminaries
In this section, we introduce some preliminary facts which are used throughout this paper.
Definition 2.1
([14])
Let X be a real Banach space. A nonempty closed convex set \(P \subset X\) is called cone if
-
(1)
\(x \in P,\lambda \geq 0\), then \(\lambda x \in P\);
-
(2)
\(x \in P, -x \in P\), then \(x =0\).
If \(x,y\in \mathbb{R}^{n}\) with \(x=(x_{1},\ldots ,x_{n})\) and \(y=(y_{1},\ldots ,y_{n})\), then by \(x\leq y\) we mean \(x_{i}\leq y_{i}\) for all \(i=1,\ldots ,n\). Also we set \(|x|=(|x_{1}|,\ldots ,|x_{n}|)\), \(\max (x,y)=(\max (x_{1},y_{1}),\ldots ,\max (x_{n},y_{n}))\), and \(\mathbb{R}_{+}^{n}=\{x\in \mathbb{R}^{n}: x_{i}>0\}\). If \(c\in \mathbb{R}\), then \(x\leq c\) means \(x_{i}\leq c\) for each \(i=1,\ldots ,n\).
Definition 2.2
Let X be a nonempty set, and consider the space \(\mathbb{R}^{m}_{+}\) endowed with the usual component-wise partial order. The mapping \(d: X\times X\rightarrow \mathbb{R}^{m}_{+}\), which satisfies all the usual axioms of the metric, is called a generalized metric in Perov’s sense and \((X,d)\) is called a generalized metric space.
Let \((X,d)\) be a generalized metric space in Perov’s sense. For \(r:=(r_{1},\ldots ,r_{m})\in \mathbb{R}_{+}^{m}\), we will denote by
the open ball centered in \(x_{0}\) with radius r, and by
the closed ball centered at \(x_{0}\) with radius r.
Theorem 2.1
Let X be a generalized Banach space, and let \(N: X\to X\) be a completely continuous operator. Then either
-
(i)
the equation \(N(x) = x\) has at least one solution, or
-
(ii)
the set \(\mathcal{M} =\{x\in X| \mu N(x) = x, \mu \in (0,1)\}\) is unbounded.
Theorem 2.2
([30])
Let \((X, \|\cdot \|)\) be a normed space, \(P_{1},P_{2} \subset X\) be two cones; \(P:= P_{1} \times P_{2};r,R \in {\mathbb{R}}^{2}_{+}, P_{r,R}:= \{u \in P_{i}: r_{i} \leq \|u_{i}\| \leq R_{i}\}\) with \(0 < r < R\); and let \(N:P_{r,R} \rightarrow P, N = (N_{1},N_{2})\) be a compact map. Assume that, for each \(i \in \{1, 2\}\), one of the following conditions is satisfied in \(P_{r,R}\):
-
(1)
\(N_{i}(u_{i}) \nprec u_{i}\) if \(\|u_{i}\| = r_{i}\), and \(N_{i}(u_{i}) \nsucc u_{i}\) if \(\|u_{i}\| = R_{i}\);
-
(2)
\(N_{i}(u_{i}) \nsucc u_{i}\) if \(\|u_{i}\| = r_{i}\), and \(N_{i}(u_{i}) \nprec u_{i}\) if \(\|u_{i}\| = R_{i}\).
Then N has a fixed point u in P with \(r_{i} \leq \|u_{i}\| \leq R_{i}\) for \(i \in \{1, 2\}\), where ⪯, namely \(u \preceq v\) if and only if \(v - u \in P\). We shall say that \(u \prec v\) if \(v - u \in P \setminus \{0\}\).
Remark 2.1
([30])
In Theorem (2.2) four cases are possible for \(u \in p_{r,R}\):
- (\(c_{1}\)):
-
\(N_{1}(u) \nprec u_{1}\) if \(\|u_{1}\| = r_{1}\), and \(N_{1}(u) \nsucc u_{1}\) if \(\|u_{1}\| = R_{1}\), \(N_{2}(u) \nprec u_{2}\) if \(\|u_{2}\| = r_{2}\), and \(N_{2}(u) \nsucc u_{2}\) if \(\|u_{2}\| = R_{2}\);
- (\(c_{2}\)):
-
\(N_{1}(u) \nprec u_{1}\) if \(\|u_{1}\| = r_{1}\), and \(N_{1}(u) \nsucc u_{1}\) if \(\|u_{1}\| = R_{1}\), \(N_{2}(u) \nsucc u_{2}\) if \(\|u_{2}\| = r_{2}\), and \(N_{2}(u) \nprec u_{2}\) if \(\|u_{2}\| = R_{2}\);
- (\(c_{3}\)):
-
\(N_{1}(u) \nsucc u_{1}\) if \(\|u_{1}\| = r_{1}\), and \(N_{1}(u) \nprec u_{1}\) if \(\|u_{1}\| = R_{1}\), \(N_{2}(u) \nprec u_{2}\) if \(\|u_{2}\| = r_{2}\), and \(N_{2}(u) \nsucc u_{2}\) if \(\|u_{2}\| = R_{2}\);
- (\(c_{4}\)):
-
\(N_{1}(u) \nsucc u_{1}\) if \(\|u_{1}\| = r_{1}\), and \(N_{1}(u)\nprec u_{1}\) if \(\|u_{1}\| = R_{1}\), \(N_{2}(u) \nsucc u_{2}\) if \(\|u_{2}\| = r_{2}\), and \(N_{2}(u) \nprec u_{2}\) if \(\|u_{2}\| = R_{2}\).
Theorem 2.3
([29])
Let \((X, \|\cdot \|)\) be a Banach space, \(P_{1},P_{2} \subset X\) be two cones, and \(P:= P_{1} \times P_{2}\) be the corresponding cone of \(X^{2} = X \times X\), and let \(\alpha _{i}, \beta _{i} > 0\). We denote
with \(\alpha _{i}\neq \beta _{i} ,r_{i} = \min \{\alpha _{i}, \beta _{i}\}\) and \(R_{i} = \max \{\alpha _{i}, \beta _{i}\}\) for \(i = 1, 2\). Assume that \(N:\overline{ W_{1} \times W_{2}} \rightarrow P, N = (N_{1},N_{2})\) is a compact map (where \(W_{i} = U_{\alpha _{i}}\cup V_{\beta _{i}}\textit{ for }i = 1,2\)) and there exist \(h_{i} \in P_{i} \setminus \{0\}, i = 1, 2\), such that for each \(i \in \{1,2\}\) the following condition is satisfied in \(\overline{ W_{1} \times W_{2}}\):
Then
-
(1)
N has at least one fixed point \(u = (u_{1},u_{2})\) in P such that \(u_{i} \in U_{\alpha _{i}} \setminus \overline{V_{\beta _{i}}}\) for \(i=1,2\) if \(\alpha _{i} > \beta _{i}\) for \(i = 1, 2\);
-
(2)
N has at least two fixed points located in \((U_{\alpha _{1}} \setminus \overline{V_{\beta _{1}}}) \times U_{ \alpha _{2}}\) and \((U_{\alpha _{1}} \setminus \overline{V_{\beta _{1}}}) \times (V_{ \beta _{2}} \setminus U_{\alpha _{2}})\) if \(\beta _{1} < \alpha _{1}\) and \(\beta _{2} > \alpha _{2}\);
-
(3)
N has at least two fixed points located in \(U_{\alpha _{1}}\times (U_{\alpha _{2}} \setminus \overline{V_{\beta _{2}}}) \) and \((V_{\beta _{1}} \setminus \overline{U_{\alpha _{1}}}) \times (U_{ \alpha _{2}} \setminus \overline{V_{\beta _{2}}})\) if \(\beta _{1} > \alpha _{1}\) and \(\beta _{2} < \alpha _{2}\);
-
(4)
N has at least four (three nontrivial) fixed points in \(U_{\alpha _{1}} \times U_{\alpha _{2}}, U_{\alpha _{1}}\times (V_{ \beta _{2}} \setminus \overline{U_{\alpha _{2}}}), (V_{\beta _{1}} \setminus \overline{U_{\alpha _{1}}}) \times U_{\alpha _{2}} \), and \((V_{\beta _{1}} \setminus \overline{U_{\alpha _{1}}}) \times (V_{ \beta _{2}} \setminus \overline{U_{\alpha _{2}}})\) if \(\alpha _{i} < \beta _{i}\) for \(i= 1,2\).
Remark 2.2
([29])
Our previous results can be easily generalized to systems of n operator equations.
Definition 2.3
([7])
The fractional integral of Riemann–Liouville of the function \(h \in L^{1}((0,\infty ),{\mathbb{R}})\) of order \(\alpha > 0\) is defined by
where \(\Gamma (\alpha )\) is the Euler gamma function defined by
Definition 2.4
For a function \(h \in AC^{n}(J)\), the Riemann–Liouville fractional order derivative of order \(\alpha > 0\) of h is defined by
where \(n = [\alpha ] + 1\) and \([\alpha ]\) denotes the integer part of the real number α.
Remark 2.3
([7])
-
(1)
If \(\lambda > -1\)
$$\begin{aligned} D_{0^{+}}^{\alpha } t^{\lambda } = \frac{\Gamma (\lambda +1)}{\Gamma (\lambda -\alpha +1} t^{\lambda - \alpha }, \end{aligned}$$and \(D_{0^{+}}^{\alpha } t^{\alpha - m} = 0, m = 1,2,\ldots ,n\), where \(n = [\alpha ] + 1\).
-
(2)
\(D_{0^{+}}^{\alpha } I_{0^{+}}^{\alpha } u(t) = u(t) \) for all \(u \in C(0,1)\cap L^{1}(0,1)\).
-
(3)
If \(u \in L^{1}(0,1), \alpha > \beta > 0\), then
$$\begin{aligned} D_{0^{+}}^{\beta }I_{0^{+}}^{\alpha } u(t) = I_{0^{+}}^{\alpha -\beta } u(t). \end{aligned}$$
Lemma 2.1
([7])
If we assume that \(u \in C(0, 1) \cap L^{1}(0, 1)\), then the fractional differential equation
has \(u(t) = C_{1} t^{\alpha -1} + C_{2} t^{\alpha -2} + \cdots + C_{n} t^{ \alpha -n}, C_{i} \in \mathbb{R}, i = 1,2,\ldots ,n\), as a unique solution, where \(n = [\alpha ] + 1\).
Lemma 2.2
([7])
Suppose that \(u \in C(0, 1) \cap L^{1}(0, 1)\) is such that \(D_{0^{+}}^{\alpha } u \in C(0, 1) \cap L^{1}(0, 1)\). Then
for some \(C_{i} \in \mathbb{R}, i=1,2,\ldots ,n\), where \(n = [\alpha ] + 1\).
Lemma 2.3
([10])
If \(x,y \geq 0,\gamma > 0\), then
Lemma 2.4
([10])
Let \(c > 0, \gamma > 0\). For any \(x, y \in [0, c]\), we have that
-
(1)
if \(\gamma > 1\), then \(|x^{\gamma } - y^{\gamma }| \leq \gamma c^{\gamma -1} |x-y|\);
-
(2)
if \(0 < \gamma \leq 1\), then \(|x^{\gamma } - y^{\gamma }| \leq |x-y|^{\gamma }\).
3 Existence result
Denote by \(C([0,1])\) the Banach space of all continuous functions from \([0,1]\) into \({\mathbb{R}}\) with the norm
Define the cone P in \(C([0,1]^{2})\) as \(P = \{ u \in C([0,1]): u(t) \geq 0, t \in [0,1]\}\). Let \(q > 1\) and \(\tilde{q} > 1\) satisfy the relation \(\frac{1}{p}+\frac{1}{q}=1, \frac{1}{\tilde{p}}+\frac{1}{\tilde{q}}=1\), where \(p,\tilde{p}\) are given by (1.4).
To prove the existence of solutions to (1.4), we need the following auxiliary lemma.
Lemma 3.1
Given \(h_{1} , h_{2} \in C [0,1], \eta \in (0,1), \gamma \in (0, \frac{1}{\eta ^{\alpha -\beta -1}} )\), and \(\alpha \geq \beta + 1\), the unique solution of C boundary value problem for a coupled system
is
and
where
Proof
Integrating equation (3.1) from 0 to t, we have
and so,
From Lemma 2.2,
From (3.3), \(B = C = 0\), and so
Now, from Remark 2.3
Therefore
By boundary condition (3.3), we have
and replacing in (3.7), we obtain
Splitting the second integral in two parts of the form
we have \(k = \gamma \eta ^{\alpha -\beta -1} t^{\alpha -1}\), and thus
This completes the proof. □
Lemma 3.2
([35])
Let \(\rho \in (0, 1)\) be fixed. The kernel \(G_{1}(t, s)\) satisfies the following properties:
-
(1)
\(G_{1}(t,s)\in C([0,1]\times [0,1])\) and \(G_{1}(t,s) > 0\) for all \(s,t \in (0,1)\);
-
(2)
\(G_{1}(t,s) \leq G_{1}(1,s)\) for all \(s\in (0,1)\);
-
(3)
\(\min_{\rho \leq t \leq 1} G_{1}(t,s) \geq \rho ^{\alpha -1} G_{1}(1,s) \) for all \(s \in [0,1]\).
We are now ready to present our main result. In this section we give an existence result based on the nonlinear alternative of Leray–Schauder type.
Theorem 3.1
Assume (\(H_{1}\))–(\(H_{2}\)) and that the following condition holds:
- (\(H_{3}\)):
-
There exist functions \(p, q, h, \breve{p}, \breve{q}\), and \(\bar{h} \in L^{1}([0,1],{\mathbb{R}}_{+})\) and constants \(\alpha _{1}, \alpha _{2}, \alpha _{3}\), and \(\alpha _{4} \in [0, 1)\) such that
$$\begin{aligned} \bigl\vert f(u,v) \bigr\vert \leq p(t) \vert u \vert ^{\alpha _{1}} + q(t) \vert v \vert ^{\alpha _{2}} + h(t)\quad \textit{for each } t \in [0,1] \textit{ and } u,v \in {\mathbb{R}} \end{aligned}$$and
$$\begin{aligned} \bigl\vert g(u,v) \bigr\vert \leq \breve{p}(t) \vert u \vert ^{\alpha _{3}} + \breve{q}(t) \vert v \vert ^{ \alpha _{4}} + \breve{h}(t)\quad \textit{for each } t \in [0,1] \textit{ and } u,v \in {\mathbb{R}}. \end{aligned}$$If \(\alpha _{1}p, \alpha _{2}p, \alpha _{3}q\), and \(\alpha _{4}q \in [0, 1)\), then system (1.4) has at least one solution.
Proof
Let N be the operator
defined by
where
and
We shall use the Leray–Schauder fixed point theorem to prove that N has a fixed point. The proof will be given in several steps.
Step 1. To show that N is continuous, let \((u_{n},v_{n})\) be a sequence such that \((u_{n},v_{n}) \rightarrow (u,v) \in C[0,1] \times C[0,1]\) as \(n\rightarrow \infty \). Then we have
By Lemma 3.2 and \(t \in [0,1]\),
On the other hand, since f is a continuous function combined with the fact that
then there exists \(N \geq 1\) such that for all \(\tau \in [0,1]\) the following estimate
holds for \(n \geq N\). By the Lebesgue dominated convergence theorem, we have
Similarly,
Consequently, N is continuous.
Step 2. N maps bounded sets into bounded sets in \(C[0,1] \times C[0,1]\), it suffices to show that for any \(r> 0\) there exists a positive constant vector \(l = (l_{1}, l_{2})\) such that, for each \((u,v) \in B_{r}= \{(u,v) \in C[0,1] \times C[0,1]: \|u\| \leq r, \|v \| \leq r\}\), we have
For each \(t\in [0,1]\), we have
Hence
Similarly, we have
Step 3. N maps bounded sets into equicontinuous. Let \(u \in B_{r}\) be a bounded set as in Step 2, \(t_{1},t_{2} \in [0, 1]\) with \(t_{1} < t_{2}\), from (3.5) and Lemma 2.3, we have
By Lemma 2.4 we obtain
Similarly, we have
The continuity of \(G_{1}\) implies that the right-hand side of the above inequality tends to zero if \(t_{2}\rightarrow t_{1}\). Therefore, by Arzela–Ascoli N is completely continuous.
Step 4. A priori bounds. Now it remains to show that the set
is bounded. Let \((u, v) \in {\mathcal{M}}\), then there exists \(0 < \lambda < 1\) such that \(u = \lambda N_{1}(u,v)\) and \(v= \lambda N_{2}(u, v)\). Thus, for \(t \in [0, 1]\), we have
Hence,
Similarly, we obtain
Notice that if \(\epsilon \leq \delta \) and \(\|u\| > 1\), then \(\|u\|^{\epsilon } \leq \|u\|^{\delta }\) Thus, \(\|u\|^{\epsilon } \leq 1+ \|u\|^{\delta }\) for all u. We then have
where
If \(\|u\| + \|v\| > 1\), then
or
This implies that
then
where
and
As a consequence of Theorem 2.1, the operator N has a fixed point that is a solution of system (1.4). This completes the proof of the theorem. □
4 Multiplicity of positive solutions
In this section, our goal is to establish positive solutions and multiplicity of solutions for the problem to system (1.4). To this end, first in this section we assume the functions \(f,g \in C({\mathbb{R}}^{2},{\mathbb{R}_{+}})\) and define the operator on P as \(N: P^{2} \rightarrow P^{2}\) to be the completely continuous map \(N = (N_{1},N_{2})\) given in the proof of Theorem 3.1. Then (3.5) and (3.6) are equivalent to the fixed point problem
If \(v \in P\) and
and \(u_{i}(t_{i}) = \|u_{i}\|\), by Lemma 3.2 this implies that, for any \(t \in [\rho , 1]\),
Hence
Define the cone \(P_{i}\) for \(i = 1, 2\) in P by
and the product cone \(P = P_{1} \times P_{2}\) in \(P^{2}\), then \(N(P) \subset P\). Before we state our main result, we introduce the following notations: \(\alpha _{i}, \beta _{i} > 0\) with \(\alpha _{i} \neq \beta _{i}\), we let \(r_{i} = \min \{\alpha _{i}, \beta _{i}\}\), \(R_{i} = \max \{\alpha _{i}, \beta _{i}\} i =1,2\).
Also, let
and
Theorem 4.1
Assume that there exist \(\alpha _{i}, \beta _{i} > 0\) with \(\alpha _{i} \neq \beta _{i}, i = 1, 2\), such that
Then (1.4) has a positive solution \(u = (u_{1},u_{2})\) with \(r_{i} \leq \|u_{i}\| \leq R_{i},i = 1, 2\), where \(r_{i} = \min \{\alpha _{i}, \beta _{i}\}\), \(R_{i} = \max \{\alpha _{i}, \beta _{i}\}\). Moreover, the corresponding orbit of u is included in the rectangle \([\rho r_{1},R_{1}] \times [\rho r_{2},R_{2}]\).
Proof
First note that if \(u \in P_{r,R}\), then \(r_{1} \leq \||u_{1}\| \leq R_{1}\) and \(r_{2} \leq \|u_{2}\| \leq R_{2}\), and by the definition of P,
for all t, showing that the orbit of u for \(t \in [\rho ,1]\) is included in the rectangle \([\rho r_{1},R_{1}] \times [\rho r_{2},R_{2}]\).
Also, if we know for example that \(\|u_{1}\| = \alpha _{1}\), then
We now prove that, for every \(u \in P_{r,R}\) and \(i \in \{1,2\}\), the following properties hold:
guaranteeing the applicability of Theorem 2.2. Indeed, if \(\|u_{1}\|= \alpha _{1}\) and we would have \(u_{1}\prec N_{1}(u)\), then
for all t. This yields the contradiction \(\alpha _{1} < \alpha _{1}\).
Now, if \(\|u_{1}\|= \beta _{1}\) and \(u_{1} \succ N_{1}(u)\), then for \(t\in [\rho ,1]\) we obtain
Then we deduce that \(\beta _{1} > \beta _{1}\), which is a contradiction. Hence (4.2) holds for \(i = 1\). Similarly, (4.2) is true for \(i = 2\). By Theorem 2.2, we see that N has at least one fixed point in P. Therefore, system (1.4) has at least one positive solution. □
Now we study the existence of multiple positive solutions for the systems of fractional boundary value problem with p-Laplacian boundary conditions.
- \((H_{4})\):
-
\(f,g \) are positive and increasing, i.e.,
$$\begin{aligned} 0 \leq u \leq x , 0 \leq v \leq y\quad \text{imply } 0 \leq f(u,v) \leq f(x,y), 0 \leq g(u,v) \leq g(x,y). \end{aligned}$$
We present the following general existence, multiplicity, and localization result.
Theorem 4.2
Let conditions \((H_{1})-(H_{2})-(H_{4})\) hold, and assume that the norm \(\|\cdot \|\) is monotone with respect to each cone \(P_{i} (i = 1, 2)\). Moreover, suppose that there exist \(\alpha _{i}, \beta _{i} > 0\), with \(\alpha _{i} \neq \beta _{i}, i = 1, 2\), such that
where \(R_{i} = \max \{\alpha _{i}, \beta _{i}\} (i = 1, 2)\).
Then problem (1.4) has at least
-
(1)
one solution \(u = (u_{1}, u_{2})\) such that \(\beta _{i} < \|u_{i}\| < \alpha _{i}\) for \(i = 1, 2\), if \(\alpha _{i} > \beta _{i}\) for \(i = 1, 2\);
-
(2)
two solutions \((u_{1}, u_{2})\) and \((v_{1}, v_{2})\) such that \(\beta _{1} < \|u_{1}\| < \alpha _{1}\), \(\beta _{2} < \|u_{2}\| < \alpha _{2}\), \(\beta _{1} < \|v_{1}\| < \alpha _{1}\), and \(\|v_{2}\| < \alpha _{2}\) if \(\alpha _{1} > \beta _{1}\) and \(\alpha _{2} < \beta _{2}\);
-
(3)
two solutions \((u_{1}, u_{2})\) and \((v_{1}, v_{2})\) such that \(\alpha _{1} < \|u_{1}\| < \beta _{1}\), \(\alpha _{2} < \|u_{2}\| < \beta _{2}\), \(\|v_{1}\| < \alpha _{1}\), and \(\beta _{2} < \|v_{2}\| < \alpha _{2}\) if \(\alpha _{1} < \beta _{1}\) and \(\alpha _{2} > \beta _{2}\);
-
(4)
four solutions \((u_{1}, u_{2}),(v_{1}, v_{2}),(w_{1}, w_{2})\), and \((z_{1}, z_{2})\) such that \(\beta _{i} < \|u_{i}\| < \alpha _{i}, \alpha _{1}< \|v_{1}\| <\beta _{1}\), and \(\|v_{2}\| < \alpha _{2} , \|w_{1}\| < \alpha _{1}, \alpha _{2}< \|w_{1} \| <\beta _{2}\), and \(\|z_{i}\| < \alpha _{i}\), if \(\alpha _{i} < \beta _{i}\) for \(i = 1, 2\).
Proof
We shall apply Theorem 2.3 to the operator \(N = (N_{1},N_{2})\) defined as in (3.8) and (3.9). Let us see that it satisfies conditions (2.1)(2.2).
First we prove that
Indeed, if not,
From \(0 \leq u_{1} \leq \alpha _{1} \rho ^{\alpha -1}\) and \(0 \leq u_{2} \leq R_{2} \rho ^{\alpha -1}\), by \((H_{1}),(H_{4})\) it follows that
By Lemma (3.2) we obtain
and the norm of X being monotone,
By assumption (4.3),
so we obtain the contradiction
Hence (4.5) holds.
Now, we prove that \(u_{1} \neq N_{1}(u) + \mu \rho ^{\alpha -1} \) for every \(u \in P\) with \(\|u_{1}\|=\beta _{1}, \|u_{2}\| \leq R_{2}\) and all \(\mu \geq 0\).
Assume the contrary, i.e., \(u_{1} = N_{1}(u) + \mu \rho ^{\alpha -1}\) for some \(u \in P\) with \(\|u_{1}\|=\beta _{1}, \|u_{2}\| \leq R_{2}\) and some \(\mu \geq 0\). Then \(u_{1} - N_{1}(u)\in P_{1}\), so \(0 \leq N_{1}(u) \leq u_{1}\), and the norm of X being monotone
Also, from condition \((H_{4})\), \(0 \leq \beta _{1}\rho ^{\alpha -1} \leq u_{1}\) and \(0 \leq u_{2}\), so we obtain
then by \((H_{1})\) we obtain
and by Lemma 3.2 we conclude \(0 \leq N_{1}(\beta _{1}\rho ^{\alpha -1},0) \leq N_{1}(u_{1},u_{2})\). Hence, by monotonicity of the norm,
Now, from (4.6) we have
which contradicts assumption (4.4). Therefore, conditions (2.1)–(2.2) hold for i = 1. Similarly, they can be verified for \(i = 2\). □
5 Applications
Example 5.1
Consider the fractional differential equation with advanced argument for p-Laplacian:
where \(\alpha =\frac{5}{2}, \beta =\frac{7}{6}, \eta =\frac{7}{10}, p= \tilde{p}=\frac{3}{2}, q=\tilde{q}= 3 , a_{1}(t) =\frac{t^{-1/2}}{4}, a_{2}(t)= \frac{{7t^{-1/2}}}{2}, \varphi _{3} (\int _{0}^{1} a_{1}(t) \,dt )= \frac{1}{4}, \varphi _{3} (\int _{0}^{1} a_{2}(t) \,dt )=\frac{\sqrt{7}}{2}, \alpha _{1}p=\alpha _{3}\tilde{p}= \frac{3}{8}\in (0,1)\), \(\alpha _{2}p=\frac{3}{10}\in (0,1), \alpha _{4} \tilde{p}=\frac{3}{12}\in (0,1)\)
and
It is clear that, for all \((t,u,v)\in [0,1]\times \mathbb{R}^{2}\),
Hence all the conditions of Theorem 3.1 hold, this implies that problem (5.1) has at least one solution.
Example 5.2
Consider the fractional differential equation with advanced argument for p-Laplacian:
where \(f,g\in C(\mathbb{R}^{2},\mathbb{R}_{+})\) are nondecreasing in u and v, \(\theta (t)=t^{\gamma }, \gamma \in (0,1)\). Assume that
and
From conditions (5.3) and (5.4), we can prove that there exist \(\alpha _{1},\alpha _{2},\beta _{1},\beta _{2}>0 , \alpha _{1} < \beta _{1}, \alpha _{2}= \beta _{1}\), and \(\beta _{2}=\alpha _{1}\) such that
and
Then we set
and
We concluded that (5.5) and (5.6) guarantee (4.1). Hence, by Theorem 4.2, problem (5.2) has a positive solution.
Since \(f,g\) are positive and increasing, we can easily show that
Thus conditions (4.3) and (4.4) hold. Then, by Theorem 4.2, problem (5.2) has multiplicity of solutions.
6 Conclusions
In this present work, we discussed some existence multiplicity results for system of fractional differential equations, under various assumptions on the right-hand side nonlinearity. The main assumptions on the nonlinearity are the continuity and some Nagumo–Bernstein type growth conditions. We have used fixed point theory in vector metric spaces with general properties from functional analysis. Also the positivity result for a fractional system of differential equations was considered. We hope that this paper can provide contributions to the questions of existence, positivity, and multiplicity of solutions for fractional differential equations on bounded domains.
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Funding
The research of J.J. Nieto has been partially supported by the Agencia Estatal de Investigación (AEI) of Spain under Grant MTM2016-75140-P, co-financed by the European Community fund FEDER, and by Xunta de Galicia under grant ED431C 2019/02.
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Mahdjouba, A., Nieto, J.J. & Ouahab, A. System of fractional boundary value problem with p-Laplacian and advanced arguments. Adv Differ Equ 2021, 352 (2021). https://doi.org/10.1186/s13662-021-03508-4
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DOI: https://doi.org/10.1186/s13662-021-03508-4