Abstract
This paper is focused on researching a class of mixed fractional differential system with p-Laplacian operators. Based on the properties of the corresponding Green’s function, different combinations of superlinearity or sublinearity for the nonlinearities and other appropriate conditions, the existence of multiple positive solutions are derived via the Guo–Krasnosel’skii fixed point theorem. An example is then given to illustrate the usability of the main results.
Similar content being viewed by others
1 Introduction
In this paper, we investigate the following mixed fractional differential system:
where \(1<\beta _{i}\leq 2 \), \(1\leq n-1<\alpha _{1} \leq n \), \(1\leq m-1<\alpha _{2} \leq m \), \(n, m\geq 2\), \(D^{\beta _{i}}\) is the Riemann–Liouville derivative operator, \({}^{c}D^{\alpha _{i}}\) is the Caputo derivative. \(\mu _{i}>0\) is a constant, \(\eta _{i}\in (0, 1)\), \(\varepsilon _{i}>0\) and satisfies \(1-\varepsilon _{i}^{p_{i}-1} \eta ^{\beta _{i}-1}>0\), \(\varphi _{p_{i}}\) is the Laplacian operator defined by \(\varphi _{p_{i}}(s)=|s|^{p_{i}-2}s\), \((\varphi _{p_{i}})^{-1}= \varphi _{q_{i}}\), \(\frac{1}{p_{i}}+\frac{1}{q_{i}}=1\), \(p_{i}>1\), \(\int _{0}^{1}a(s)v(s)\,dA_{1}(s)\), \(\int _{0}^{1}b(s)u(s)\,dA_{2}(s)\) denote the Riemann–Stieltjes integrals with a signed measure, that is \(A_{i}: [0, 1]\rightarrow [0, +\infty )\) is the function of bounded variation. \(a, b : [0, 1]\rightarrow [0,+\infty )\) are continuous, \(f_{i}: [0,1]\times [0,+\infty )\times [0,+\infty ) \rightarrow [0, + \infty )\) is a continuous function, \(i=1, 2\).
Compared with the integer order systems, fractional differential systems are regarded as a better tool in the description of some problems in science and engineering. Arafal et al. [1] presented a fractional order model for infection of CD4+T cells:
where \(\alpha _{1}, \alpha _{2}, \alpha _{3}>0\). In the mathematical context, many mathematicians and applied scholars have studied the fractional differential equation or system in recent years [2,3,4,5,6,7,8,9,10,11,12,13,14,15]. In addition, by applying the functional analysis methods such as the lower and upper solutions, monotone iterative techniques, fractional integro-differential equations or singular equations are researched by Dumitru et al. [16], Denton et al. [17], Lyons and Neugebauer [18], Ambrosio [19], Zhou and Qiao [20]. There are also related books [21, 22].
Cabada and Wang in [23] studied the following factional differential equation:
where \(2 <\alpha \leq 3\), \(0 <\lambda \), \(\lambda \neq \alpha \), \(D^{\alpha }\) is the Caputo fractional derivative, and \(f: [0,1] \times [0,+\infty )\rightarrow [0, +\infty )\) is a continuous function. By the use of Guo–Krasnosel’skii fixed point theorem, the authors in [23] obtained the positive solution to Eq. (1.2). Cabada and Wang also discussed the solution of Eq. (1.2) when \(D^{\alpha }\) is the Riemann–Liouville fractional derivative [24].
The p-Laplacian equation is the second order quasilinear differential operator, it arises in the modeling of various physical and natural phenomena. Fractional differential equation with p-Laplacian operator can describe the nonlinear phenomena in non-Newtonian fluids and establishes complex process models; for some related articles, see [25,26,27,28,29,30,31]. Via variational methods, Li and Wei [32] dealt with fractional p-Laplacian equations, the existence and multiplicity of nontrivial solutions were obtained. Wu et al. [33] researched the following fractional differential turbulent flow model and obtained the iterative solutions of the equation:
where \(1 <\alpha \), \(\gamma \leq 2\), \(D^{\alpha }\), \(D^{\gamma }\) are the Riemann–Liouville fractional derivatives, \(h: [0,+\infty )\rightarrow [0, +\infty )\) is a continuous and increasing function.
Fractional differential systems with p-Laplacian operators have also attracted tremendous attention [34,35,36,37,38,39,40]. Among them, applying the monotone iterative approach, the authors in [34] got the extremal solutions of the following system:
where \(0<\alpha _{i}\), \(\gamma _{i}\leq 1\), \(1<\beta _{i}\leq 2\), \(D_{0^{+}}^{\alpha _{i}}\), \(D_{0^{+}}^{\beta _{i}} D_{0^{+}}^{\gamma _{i}}\) are the Riemann–Liouville fractional derivatives, \(i=1, 2\).
Inspired by the above articles, in this article we discuss the mixed fractional differential system with p-Laplacian operators under integral boundary value conditions. To the best of our knowledge, there is very little research on mixed fractional differential systems, especially if the system has p-Laplacian operators. Through the application of the Guo–Krasnosel’skii fixed point theorem, the existence of multiple positive solutions of the system is achieved.
2 Preliminaries and lemmas
Definition 2.1
The Caputo fractional order derivative of order \(\alpha >0\), \(n-1 < \alpha < n\), \(n\in \mathbb{N}\) is defined as
where \(u \in C^{n}(J, \mathbb{R})\), \(\mathbb{R}=(-\infty , +\infty )\), \(\mathbb{N}\) denotes the natural number set, \(n =[\alpha ]+1\), and \([\alpha ]\) denotes the integer part of α.
Definition 2.2
Let \(\alpha >0\) and let u be piecewise continuous on \((0, +\infty )\) and integrable on any finite subinterval of \([0, +\infty)\). Then, for \(t>0\), we call
the Riemann–Liouville fractional integral of u of order α.
Lemma 2.1
Let \(n-1<\alpha \leq n\), \(u \in C^{n}[0, 1]\). Then
where \(c_{i} \in \mathbb{R}\) (\(i=1, 2, \ldots , n-1\)), n is the smallest integer greater than or equal to α.
Let \({\varphi _{p_{1}}}({}^{c}D_{0^{+}}^{\alpha _{1}}u(t))=\overline{u}(t)\), \({\varphi _{p_{2}}}({}^{c}D_{0^{+}}^{\alpha _{2}}v(t))=\overline{v}(t)\), then \(\overline{u}(0)=0\), \(\overline{u}(1)=\varepsilon _{1}^{p_{1}-1} \overline{u}(\eta _{1})\), \(\overline{v}(0)=0\), \(\overline{v}(1)= \varepsilon _{2}^{p_{2}-1}\overline{v}(\eta _{2})\), we now consider the following system:
Similar to [43], if \(y_{i}\in C[0, 1]\), then the system (2.1) has a unique solution,
where
For \(y_{i}\in C[0, 1]\), consider the system
Through calculation, we conclude that system (2.3) is equal to
Lemma 2.2 was obtained by the author herself and her collaborator in [44]
Lemma 2.2
Assume the following condition \((\mathbf{H}_{0})\) holds.
- \((\mathbf{H}_{0})\) :
-
$$ k_{1}= \int _{0}^{1}a(s)\,dA_{1}(s)>0, \qquad k_{2}= \int _{0}^{1}b(s)\,dA_{2}(s)>0,\quad 1-\mu _{1}\mu _{2}k_{1}k_{2}>0. $$
Let \(h_{i}\in C(0, 1)\cap L(0, 1)\) (\(i=1, 2\)), then the system with the coupled boundary conditions
has a unique integral representation,
where
and
Lemma 2.3
The Green function \(\overline{H}_{i}(t, s)\), \(G_{i}(t, s)\) (\(i=1, 2\)) defined separately by (2.2), (2.7) has the following properties:
-
(i)
\(\overline{H}_{i}(t, s)\), \(G_{i}(t, s): [0, 1]\times [0, 1]\to [0,+\infty )\) are continuous,
-
(ii)
$$\begin{aligned} \frac{(1-s)^{\alpha _{i}-1}(1-t^{\alpha _{i}-1})}{\varGamma (\alpha _{i})} \leq G_{i}(t, s)\leq \frac{(1-s)^{\alpha _{i}-1}}{\varGamma (\alpha _{i})}, \quad t, s\in [0, 1]. \end{aligned}$$
Proof
Obviously, (i) holds, we only prove (ii). From the definition of \(G_{i}(t, s)\), for \(0\leq t\leq s\leq 1\), it is obvious that (ii) holds.
For \(0\leq s\leq t\leq 1\), we have \(t-ts\geq t-s\), then
so, we know \(\frac{(1-s)^{\alpha _{i}-1}(1-t^{\alpha _{i}-1})}{\varGamma ( \alpha _{i})} \leq G_{i}(t, s)\). It is also defined by \(G_{i}(t, s)\), and we obtain \(G_{i}(t, s)\leq \frac{(1-s)^{\alpha _{i}}}{\varGamma (\alpha _{i})}\). Thus, (ii) holds. The proof is completed. □
Similar to the proof in [35], Lemma 2.4 was obtained.
Lemma 2.4
For \(t, s\in [0, 1]\), the functions \(K_{i}(t, s)\) and \(H_{i}(t, s)\) (\(i=1, 2\)) defined as (2.3) satisfy
where
Remark 2.1
From Lemma 2.4, for \(t, \tau , s\in [0, 1]\), we have
where \(\omega =\frac{\varrho }{\rho }\), ϱ, ρ are defined as Lemma 2.4, \(0<\omega <1\).
Let \(X=C[0, 1]\times C[0, 1]\), then X is a Banach space with the norm
Let
where ω is defined as Remark 2.1. It is easy to see that K is a positive cone in X. For any \((u,v)\in K\), we can define an integral operator \(T: K\to X\) by
We know that \((u, v)\) is a positive solutions of system (1.1) if and only if \((u, v)\) is a fixed point of T in K.
Lemma 2.5
\(T: X\to X\) is a completely continuous operator and \(T(K)\subseteq K\).
Proof
By a routine discussion, we see that \(T: X \rightarrow X\) is well defined, so we only prove \(T(K)\subseteq K\). For any \((u, v)\in K\), \(0\leq t\), \(t'\leq 1\), by Remark 2.1, we have
So we have
i.e.,
In the same way as (2.11) and (2.12), we can prove that
Therefore, we have \(T (K )\subseteq K\).
According to the Ascoli–Arzela theorem, we see that \(T: K\rightarrow K\) is completely continuous. The proof is completed. □
Lemma 2.6
([45])
Let K be a positive cone in a Banach space E, \(\varOmega _{1}\) and \(\varOmega _{2}\) are bounded open sets in E, \(\theta \in \varOmega _{1}\), \(\overline{\varOmega }_{1}\subset \varOmega _{2}\), \(T :K\cap \overline{\varOmega }_{2}\backslash \varOmega _{1} \rightarrow K\) is a completely continuous operator. If the following conditions are satisfied:
or
then T has at least one fixed point in \(K\cap (\overline{ \varOmega }_{2}\backslash \varOmega _{1})\).
3 Main results
Denote
In what follows, we list the conditions to be used later:
- \((\mathbf{H}_{1})\) :
-
\(f_{i0}\in (\varphi _{p_{i}} (\frac{l _{i}}{\omega } ), +\infty ]\), \(f_{i\infty }\in (\varphi _{p_{i}} (\frac{l_{i}}{\omega } ), +\infty ]\).
- \((\mathbf{H}_{2})\) :
-
\(f^{0}_{i}\in [0, \varphi _{p_{i}}(L_{i}) )\), \(f^{\infty }_{i}\in [0, \varphi _{p_{i}}(L_{i}) )\).
- \((\mathbf{H}_{3})\) :
-
There exist constants \(d_{i}\in (0, L_{i})\) and \(r_{1}>0\), such that
$$ f_{i}(t, x, y)\leq \varphi _{p_{i}}(d_{i} r_{1}), \quad 0\leq t\leq 1, 0 \leq x, y\leq r_{1}. $$ - \((\mathbf{H}_{4})\) :
-
There exist constants \(d^{*}_{i}\in (l_{i}, + \infty )\) and \(R_{1}>0\), \([a, b]\subset (0, 1)\), such that
$$ f_{i}(t, x, y)\geq \varphi _{p_{i}}\bigl(d^{*}_{i} R_{1}\bigr),\quad a\leq t\leq b, \omega R_{1}\leq x, y \leq R_{1}. $$
Theorem 3.1
Assume that \((\mathbf{H}_{0})\), \((\mathbf{H}_{1})\), \((\mathbf{H}_{3})\) hold, then system (1.1) has at least two positive solutions \((u_{1}, v_{1})\) and \((u_{2}, v_{2})\) such that \(0<\|(u_{1}, v_{1})\|<r_{1}<\|(u_{2}, v_{2})\|\).
Proof
(I) By \((\mathbf{H}_{3})\), there exist constants \(d_{i}\in (0, L_{i})\) and \(r_{1}>0\), such that
Let \(K_{r_{1}}=\{(u, v)\in K: \|(u, v)\|< r_{1} \}\). For any \((u, v)\in \partial K_{r_{1}}\), by the definition of \(\|\cdot \|\), we know that
Thus, for any \((u, v)\in \partial K_{r_{1}}\), by (3.1) and (3.2), we can obtain
Hence, for any \((u, v)\in \partial K_{r_{1}}\), by Lemmas 2.3, 2.4 and (3.3), we have
Similar to (3.4), for any \((u, v)\in \partial K_{r_{1}}\), we also have
Consequently
(II) With the first inequality of \((\mathbf{H}_{1})\), \(f_{i0}\in (\varphi _{p_{i}} (\frac{l_{i}}{\omega } ), +\infty ]\), there exists a real number \(r\in (0, r_{1})\), such that
Let \(K_{r}=\{(u, v)\in K: \|(u, v)\|< r\}\). For any \((u, v)\in \partial K_{r}\),
By Lemmas 2.3, 2.4 and (3.6), (3.7), we have
Therefore, we obtain
(III) With the second inequality of \((\mathbf{H}_{1})\), \(f_{i\infty } \in (\varphi _{p_{i}} (\frac{l_{i}}{\omega } ), + \infty ]\), there exist real numbers \(r^{*}_{2}\), \(r^{**}_{2}\), such that
Choose \(r_{2}=\max \{2r_{1}, \frac{r^{*}}{\omega \theta }, \frac{r ^{**}_{2}}{\omega \theta } \}\). Let \(K_{r_{2}}=\{(u, v)\in K:\|(u, v)\|< r_{2}\}\). For any \((u, v)\in \partial K_{r_{2}} \), by the definition of \(\|\cdot \|\), we have
Thus, for any \((u, v)\in \partial K_{r_{2}}\), by (3.10), (3.11), we have
So, for any \((u, v)\in \partial K_{r_{2}} \), by Lemmas 2.3, 2.4 and (3.12), we know
Hence, we obtain
It follows from the above discussion, (3.5), (3.9), (3.14), Lemmas 2.5, 2.6, that T has fixed points \((u_{1}, v _{1})\in \overline{K}_{r_{2}}\backslash K_{r}\), \((u_{2}, v_{2})\in \overline{K}_{r}\backslash K_{r_{1}}\), that is to say, system (1.1) has at least two positive solutions \((u_{1}, v_{1})\), \((u_{2}, v_{2})\), satisfying \(0<\|(u_{1}, v_{1})\|<r_{1}<\|(u_{2}, v_{2})\|\). The proof is completed. □
Theorem 3.2
Assume that \((\mathbf{H}_{0})\), \((\mathbf{H}_{2})\), \((\mathbf{H}_{4})\) hold, then system (1.1) has at least two positive solutions \((u_{1}, v_{1})\) and \((u_{2}, v_{2})\) such that \(0<\|(u_{1}, v_{1})\|<R_{1}<\|(u_{2}, v_{2})\|\).
Proof
(I) By \((\mathbf{H}_{4})\), there exist constants \(d^{*}_{i}\in (l_{i}, +\infty )\) and \(R_{1}>0\), such that
Let \(K_{R_{1}}=\{(u, v)\in K: \|(u, v)\|< R_{1}\}\). For any \((u, v) \in \partial K_{R_{1}}\),
Thus, for any \((u, v)\in \partial K_{R_{1}} \), by Lemmas 2.3, 2.4 and (3.15), (3.16), we get
So, we have
(II) With the first inequality of \((\mathbf{H}_{2})\), \(f^{0}_{i} \in [0, \varphi _{p_{i}}(L_{i}) )\), there exists a real number \(R_{2}\in (0, R_{1})\), such that
Let \(K_{R_{2}}=\{(u, v)\in K: \|(u, v)\|< R_{2}\}\). For any \((u, v) \in \partial K_{R_{2}}\),
Therefore, for any \((u, v)\in \partial K_{R_{2}} \), by Lemmas 2.3, 2.4 and (3.19), (3.20), we have
By a similar proof to (3.21), for any \((u, v)\in \partial K_{R _{2}}\), we also have
Thus,
(III) With the second inequality of \((\mathbf{H}_{2})\), \(f^{\infty } _{i}\in [0, \varphi _{p_{i}}(L_{i}) )\), there exists \(R^{*}>0\), such that
Now there are two situations.
Case 1. \(f_{i}\) is bounded on \([0, +\infty )\), then we choose \(\overline{R}>0\), such that
Let \(R_{3}=\max \{2R_{1}, \overline{R}\}\), \(K_{R_{3}}=\{(u, v)\in K: \|(u, v)\|< R_{3}\}\). For any \((u, v)\in \partial K_{R_{3}}\), we know
Similar to (3.25), for any \((u, v)\in \partial K_{R_{3}}\), we have
Thus,
Case 2. \(f_{1}\) and \(f_{2}\) have at least one unbounded function, assume both \(f_{1}\) and \(f_{2}\) are unbounded. (If \(f_{1}\) or \(f_{2}\) is unbounded, the proof is similar.) Choose \(R_{3}=\max \{2R_{1}, \frac{R^{*}}{\omega } \}\), such that
Let \(K_{R_{3}}=\{(u, v)\in K: \|(u, v)\|< R_{3}\}\). For any \((u, v) \in \partial K_{R_{3}}\), by (3.24), (3.27), we have
Similar to (3.28), for any \((u, v)\in \partial K_{R_{3}}\), we have
Thus,
Through the above discussion, (3.18), (3.22), (3.26) (or (3.29)), Lemmas 2.5, 2.6, T has fixed points \((u_{1}, v _{1})\in \overline{K}_{R_{1}}\backslash K_{R_{2}}\), \((u_{2}, v_{2}) \in \overline{K}_{R_{3}}\backslash K_{R_{1}}\), that is to say, system (1.1) has at least two positive solutions \((u_{1}, v_{1})\), \((u_{2}, v_{2})\), satisfying \(0<\|(u_{1}, v_{1})\|<R_{1}<\|(u_{2}, v _{2})\|\). The proof is completed. □
4 An example
Consider the following fractional differential system:
where \(\beta _{1}=\beta _{2}=\frac{3}{2}\), \(\alpha _{1}=\alpha _{2}= \frac{5}{2}\), \(\mu _{1}=\frac{1}{2}\), \(\mu _{2}=1\), \(A_{1}(t)=t^{ \frac{1}{3}}\), \(A_{2}(t)=t\), \(\varepsilon _{1}=\varepsilon _{2}= \frac{1}{4}\), \(\eta _{1}=\eta _{2}=\frac{1}{2}\), \(a(s)=s^{2}\), \(b(s)=s\), \(p_{1}=p_{2}=2\). Then we have
Condition \((\textbf{H}_{0})\) holds. Through calculation, \(L_{1}=L _{2}=2.43299\), \(l_{1}=l_{2}=6.80274\), \(\omega =0.01953\). Choose
Take \(d_{1}=d_{2}=2\), \(r_{1}=180\), we have
Then, by Theorem 3.1, system (4.1) has at least two positive solutions \((u_{1}, v_{1})\) and \((u_{2}, v_{2})\) such that \(0<\|(u_{1}, v_{1})\|<180<\|(u_{2}, v_{2})\|\).
References
Arafa, A.A.M., Rida, S.Z., Khalil, M.: Fractional modeling dynamics of HIV and CD4+ T-cells during primary infection. EPJ Nonlinear Biomed. Phys. 2012, 6 (2012)
Wang, G., Agarwal, R., Cabada, A.: Existence results and monotone iterative technique for systems of nonlinear fractional differential equations. Appl. Math. Lett. 25(6), 1019–1024 (2012)
Wang, Y., Liu, L., Zhang, X., Wu, Y.: Positive solutions of a fractional semipositone differential system arising from the study of HIV infection models. Appl. Math. Comput. 258, 312–324 (2015)
Cui, Y.: Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 51, 48–54 (2016)
Wang, Y.: Positive solutions for fractional differential equation involving the Riemann–Stieltjes integral conditions with two parameters. J. Nonlinear Sci. Appl. 9, 5733–5740 (2016)
Zou, Y., He, G.: On the uniqueness of solutions for a class of fractional differential equations. Appl. Math. Lett. 74, 68–73 (2017)
Zhu, B., Liu, L., Wu, Y.: Local and global existence of mild solutions for a class of semilinear fractional integro-differential equations. Fract. Calc. Appl. Anal. 20(6), 1338–1355 (2017)
Wang, Y., Liu, L.: Uniqueness and existence of positive solutions for the fractional integro-differential equation. Bound. Value Probl. 2017, 12 (2017)
Cui, Y., Ma, W., Wang, X., Su, X.: Uniqueness theorem of differential system with coupled integral boundary conditions. Electron. J. Qual. Theory Differ. Equ. 9, 1–10 (2018)
Zhang, X., Liu, L., Wu, Y., Zou, Y.: Existence and uniqueness of solutions for systems of fractional differential equations with Riemann–Stieltjes integral boundary condition. Adv. Differ. Equ. 2018, 204 (2018)
Jiang, J., Liu, W., Wang, H.: Positive solutions to singular Dirichlet-type boundary value problems of nonlinear fractional differential equations. Adv. Differ. Equ. 2018, 169 (2018)
Cui, Y., Ma, W., Sun, Q., Su, X.: New uniqueness results for boundary value problem of fractional differential equation. Nonlinear Anal. 23(1), 31–39 (2018)
Hao, X., Zhang, L., Liu, L.: Positive solutions of higher order fractional integral boundary value problem with a parameter. Nonlinear Anal., Model. Control 24(2), 210–223 (2019)
Zhang, X., Jiang, J., Wu, Y., Cui, Y.: Existence and asymptotic properties of solutions for a nonlinear Schrödinger elliptic equation from geophysical fluid flows. Appl. Math. Lett. 90, 229–237 (2019)
Yue, Y., Tian, Y., Bai, Z.: Infinitely many nonnegative solutions for a fractional differential inclusion with oscillatory potential. Appl. Math. Lett. 88, 64–72 (2019)
Dumitru, B., Asef, M., Shahram, R.: On the existence of solutions for some infinite coefficient-symmetric Caputo–Fabrizio fractional integro-differential equations. Bound. Value Probl. 2017, 145 (2017)
Denton, Z., Ramírez, J.D.: Existence of minimal and maximal solutions to RL fractional integro-differential initial value problems. Opusc. Math. 37(5), 705–724 (2017)
Lyons, J.W., Neugebauer, J.T.: Positive solutions of a singular fractional boundary value problem with a fractional boundary condition. Opusc. Math. 37(3), 421–434 (2017)
Ambrosio, V.: Zero mass case for a fractional Berestycki–Lions-type problem. Adv. Nonlinear Anal. 7(3), 365–374 (2018)
Zhou, Z., Qiao, Y.: Solutions for a class of fractional Langevin equations with integral and anti-periodic boundary conditions. Bound. Value Probl. 2018, 152 (2018)
Giovanni, M.B., Radulescu, V.D., Servadei, R.: Variational Methods for Nonlocal Fractional Problems. Encyclopedia of Mathematics and Its Applications, vol. 162. Cambridge University Press, Cambridge (2016)
Papageorgiou, N.S., Radulescu, V.D., Repovs, D.D.: Nonlinear Analysis—Theory and Methods. Springer Monographs in Mathematics. Springer, Cham (2019)
Cabada, A., Wang, G.: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389, 403–411 (2012)
Cabada, A., Wang, G.: Nonlinear fractional differential equations with integral boundary value conditions. Appl. Math. Comput. 228, 251–257 (2014)
Ding, Y., Wei, Z., Xu, J., O’Regan, D.: Extremal solutions for nonlinear fractional boundary value problems with p-Laplacian. J. Comput. Appl. Math. 288, 151–158 (2015)
Zhang, X., Liu, L., Wu, Y., Cui, Y.: Entire blow-up solutions for a quasilinear p-Laplacian Schrodinger equation with a nonsquare diffusion term. Appl. Math. Lett. 74, 85–93 (2017)
Bai, C.: Existence and uniqueness of solutions for fractional boundary value problems with p-Laplacian operator. Adv. Differ. Equ. 2018, 4 (2018)
Wu, J., Zhang, X., Liu, L., Wu, Y., Cui, Y.: The convergence analysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular decreasing nonlinearity. Bound. Value Probl. 2018, 82 (2018)
Wang, Y.: Existence and nonexistence of positive solutions for mixed fractional boundary value problem with parameter and p-Laplacian operator. J. Funct. Spaces 2018, Article ID 1462825 (2018)
Guo, L., Liu, L., Wu, Y.: Iterative unique positive solutions for singular p-Laplacian fractional differential equation system with several parameters. Nonlinear Anal., Model. Control 23(2), 182–203 (2018)
Guo, L., Liu, L.: Maximal and minimal iterative positive solutions for singular infinite-point p-Laplacian fractional differential equations. Nonlinear Anal., Model. Control 23(6), 851–865 (2018)
Li, A., Wei, C.: On fractional p-Laplacian problems with local conditions. Adv. Nonlinear Anal. 7(4), 485–496 (2018)
Wu, J., Zhang, X., Liu, L., Wu, Y.: Twin iterative solutions for a fractional differential turbulent flow model. Bound. Value Probl. 2016, 98 (2016)
Li, S., Zhang, X., Wu, Y., Caccetta, L.: Extremal solutions for p-Laplacian differential systems via iterative computation. Appl. Math. Lett. 26, 1151–1158 (2013)
Zhang, X., Liu, L., Wiwatanapataphee, B., Wu, Y.: The eigenvalue for a class of singular p-Laplacian fractional differential equations involving the Riemann–Stieltjes integral boundary condition. Appl. Math. Comput. 235, 412–422 (2014)
Wang, Y., Jiang, J.: Existence and nonexistence of positive solutions for the fractional coupled system involving generalized p-Laplacian. Adv. Differ. Equ. 2017, 337 (2017)
Ren, T., Li, S., Zhang, X., Liu, L.: Maximum and minimum solutions for a nonlocal p-Laplacian fractional differential system from eco-economical processes. Bound. Value Probl. 2017, 118 (2017)
Liu, X., Jia, M., Ge, W.: The method of lower and upper solutions for mixed fractional four-point boundary value problem with p-Laplacian operator. Appl. Math. Lett. 65, 56–62 (2017)
Hao, X., Wang, H., Liu, L., Cui, Y.: Positive solutions for a system of nonlinear fractional nonlocal boundary value problems with parameters and p-Laplacian operator. Bound. Value Probl. 2017, 182 (2017)
Yan, F., Zuo, M., Hao, X.: Positive solution for a fractional singular boundary value problem with p-Laplacian operator. Bound. Value Probl. 2018, 51 (2018)
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Podlubny, I.: Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198. Academic Press, New York (1999)
Xu, J., Dong, W.: Existence and uniqueness of positive solutions for a fractional boundary value problem with p-Laplacian operator. Acta Math. Sinica (Chin. Ser.) 59, 385–396 (2016)
Zi, Y., Wang, Y.: Positive solutions for Caputo fractional differential system with coupled boundary conditions. Adv. Differ. Equ. 2019, 80 (2019)
Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988)
Acknowledgements
The author thanks the anonymous reviewers for carefully reading this paper and constructive comments.
Availability of data and materials
Not applicable.
Funding
This work is supported by the National Natural Science Foundation of China (11701252, 11671185, 61703194), the Science Research Foundation for Doctoral Authorities of Linyi University (LYDX2016BS080), the Natural Science Foundation of Shandong Province of China (ZR2018MA016), and the Applied Mathematics Enhancement Program of Linyi University.
Author information
Authors and Affiliations
Contributions
This entire work has been completed by the author. The author read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The author declares that she has no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Cite this article
Wang, Y. Multiple positive solutions for mixed fractional differential system with p-Laplacian operators. Bound Value Probl 2019, 144 (2019). https://doi.org/10.1186/s13661-019-1257-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-019-1257-2