Abstract
A four-point boundary problem for a fractional p-Laplacian differential equation is studied. The existence of two positive solutions is established by means of the monotone iterative method. An example supporting the abstract result is given.
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1 Introduction
Fractional differential equations (FrDEs) are widely used in many fields: physical chemistry, financial mathematics, diffusion theory, transportation theory, chaos and turbulence, viscoelastic mechanics, non-newtonian fluid mechanics, seismic analysis. Therefore, many scholars have studied fractional differential equations, to mention a few (see, for example, [1–15]). The standard approach to study boundary value problems (BVPs) for FrDEs is based on the passage to equivalent integral equations and further application of the methods and techniques of modern nonlinear analysis. In particular, to study (multiple) positive solutions, one can combine the classical Green function methods with fixed point theorems in cones (see, for example, [1–3, 5, 7–9, 11]).
On the other hand, BVPs involving p-Laplacian have attracted a lot of attention during the last decades (see, for example, [16–19]). Also, we refer to [20–28], where BVPs for FrDE involving the p-Laplacian were considered. References [22, 25, 28], where the four-point BVPs were considered, are of our special interest. In [25], Wang et al. considered the BVP of the form
with \(\alpha ,\beta \in R\); \(1<\alpha ,\beta \leq 2\); \(0\leq a,b\leq 1\); \(0< \xi ,\eta <1\). The authors imposed certain monotonicity conditions and applied the upper and lower solutions method.
In [22] and [28], Tian et al. studied the differential system
with the boundary conditions
and
respectively (here \(1<\alpha \), \(\beta \leq 2\); \(\gamma >0\); \(1+\gamma \leq \beta \); \(a, b>0\); \(\xi , \eta \in (0,1)\)). To be more specific, in [22], the existence of multiple positive solutions was established by means of the Leggett–Williams fixed-point theorem, while in [28], some existence results were obtained using a monotone iterative method.
In this paper, we study the BVP
where \(h\in C([0,1]\times [0,+\infty ),[0,+\infty ))\), \(D_{0+}^{\alpha }\), \(D_{0+}^{\beta }\), and \(D_{0+}^{\gamma }\) stand for the standard Riemann–Liouville differentiations, \(\phi _{p}(z)=|z|^{p-2}z\), \(p>1\); \(1<\alpha ,\beta \leq 2\), and \(\gamma =\frac{\beta -1}{2}\); \(0<\xi \leq \frac{1}{2}\); \(0<\eta < 1\); \(a, b \in [0,+\infty )\) and \(a\Gamma (\beta )\xi ^{\frac{\beta -1}{2}}<\Gamma (\frac{\beta +1}{2})\); \(b^{p-1}\eta ^{\alpha -1}< 1\). By applying a monotone iterative method, we establish the existence of two positive solutions of (1.4) (see Theorem 3.1) and support the general result by an example (see Sect. 4).
Our paper is distinguished from [22, 25, 28] in the following three aspects. Firstly, the boundary condition \(x(1)=a D_{0+}^{\gamma }x(\xi )\) in (1.4) is different from the condition \(x(1)=ax(\xi )\) in (1.1). Next, the condition \(D_{0+}^{\gamma }x(1)=a D_{0+}^{\gamma }x(\xi )\) in (1.3) links the values of derivatives of the same order. At the same time, condition \(x(1)=a D_{0+}^{\gamma }x(\xi )\) in (1.4) links the derivatives of different order (as usual, we regard \(x(1)\) as the derivative of order 0 of x at \(t = 1\)). Finally, in (1.2), the authors imposed the boundary condition \(D_{0+}^{\gamma }x(1)=a x(\xi )\), where \(1+\gamma \leq \beta \), \(\xi \in (0,1)\), and applied the Leggett–Williams fixed-point theorem. This is in a sharp contrast with the boundary condition \(x(1)=a D_{0+}^{\gamma }x(\xi )\), \(\gamma =\frac{\beta -1}{2}\), \(0<\xi \leq \frac{1}{2}\), in (1.4), which allows us to use a monotone iterative method. Summing up: although the methodology that we use rests on the one developed in [28], our setting is different from the ones considered in [22, 25, 28].
2 Preliminaries
In this section, we present some preliminary results including estimates for Green functions and solvability of non-homogeneous fractional p-Laplacian BVPs. These results constitute key ingredients of the proof of our main result (Theorem 3.1). All the function spaces considered below consist of scalar functions. The two lemmas following below are well known.
Lemma 2.1
([29])
\((1)\) If \(F\in L(0,1)\) and \(\mu > \nu >0\), then
\((2)\) If \(\mu , \nu >0\), then
Lemma 2.2
([11])
Let \(A_{i}\in R\), \(i=1,2,\ldots ,N\), and \(N=[\alpha ]+1\). Then
where \(\alpha >0\), \(F \in C(0,1)\cap L(0,1)\), \(D_{0+}^{\alpha }F \in C(0,1)\cap L(0,1)\).
Let \(B_{1}= b^{p-1}\eta ^{\alpha -1}\neq1\), \(B_{2}=a\Gamma (\beta )\xi ^{\frac{\beta -1}{2}}\neq\Gamma (\frac{\beta +1}{2})\), and denote
The following technical statement plays an important role in studying Green functions relevant to our considerations.
Lemma 2.3
Let G and M be defined by (2.1) and (2.2), respectively. If \(a \Gamma (\beta )\xi ^{\frac{\beta -1}{2}}< \Gamma (\frac{\beta +1}{2})\) and \(b^{p-1}\eta ^{\alpha -1}<1\), then:
(a) \(G, M \in C([0,1]\times [0,1])\);
(b) \(G(t,z)>0\), \(M(t,z)>0\) for all \(t,z\in (0,1)\);
(c) there exist two positive functions \(\mu ,\nu \in C((0,1),(0,+\infty ))\) such that, for all \(z \in (0,1)\), one has
Proof
(i) This statement follows immediately from (2.1) and (2.2).
(ii) In order to prove that \(G(t,z)>0\) for all \(t,z\in (0,1)\), consider, first, the case
Put
Obviously, in the considered case, \(g(t,z)> 0\).
On the other hand, \(\forall \xi \in (0,\frac{1}{2}]\), \(z\in [0,\xi ]\), we have
Then
which implies that
Obviously,
Therefore,
The remaining three cases \(0< \xi \leq z\leq t < 1 \) or \(0 < t \leq z \leq \xi < 1 \) or \(0 \leq t\leq z <1 \), \(\xi \leq z\), can be treated using the similar method, so that we omit the obvious modifications. Thus, \(G(t,z)>0\) for all \(t,z\in (0,1)\).
Similarly, to prove that \(M(t,z)>0\), for all \(t,z\in (0,1)\), consider, first, the case
Put
Obviously, \(m(t,z)> 0 \) for \(0\leq z\leq t\leq 1\) in the considered case. So
One can apply a similar argument in order to treat the remaining three cases \(0< \eta \leq z\leq t < 1 \) or \(0 < t \leq z \leq \eta < 1 \) or \(0 \leq t\leq z <1 \), \(\eta \leq z\). Thus, \(M(t,z)> 0 \) for \(t,z \in (0,1)\).
(iii) Obviously, for a fixed z, the functions g and m, given by (2.3) and (2.4), respectively, are increasing in t for \(t\leq z\) and decreasing in t for \(t\geq z\). Therefore,
Put
It is clear that \(\mu ,\nu \in C((0,1),(0,+\infty ))\).
Consider four cases.
If \(0\leq z\leq t\leq 1\), \(z\leq \xi \), then
If \(0<\xi \leq z\leq t\leq 1\), then
If \(0\leq t\leq z\leq \xi < 1\), then
If \(0\leq t\leq z\leq 1\), \(\xi \leq z\), then
Thus,
Similarly, consider four cases for the function ν.
If \(0\leq z\leq t\leq 1\), \(z\leq \eta \), then
If \(0<\eta \leq z\leq t\leq 1\), then
If \(0\leq t\leq z\leq \eta < 1\), then
If \(0\leq t\leq z\leq 1\), \(\eta \leq z\), then
Thus,
The proof of Lemma 2.3 is complete. □
The next statement provides the existence and uniqueness result for the non- homogeneous problems of our interest.
Lemma 2.4
Assume that
(i) \(\phi _{p}(z)=|z|^{p-2}z\), \(p > 1\);
(ii) \(\phi _{q}=(\phi _{p})^{-1}\), \(\frac{1}{p}+\frac{1}{q}=1\);
(iii) \(1<\alpha ,\beta \leq 2\) and \(\gamma =\frac{\beta -1}{2}\);
(iv) \(0<\xi \leq \frac{1}{2}\), \(0<\eta < 1\), \(a,b\in [0,+\infty )\).
Then, for any \(y\in C[0,1]\), the problem
admits the unique solution
Proof
By Lemma 2.2, one has
where \(A_{1},A_{2}\in R\). Combining (2.7) with \(D_{0+}^{\beta }x(0)=0\) (cf. (2.5)), we have \(A_{2}=0\). Then
from which it follows that
Hence,
and
Next, combining (2.10) and (2.11) with \(D_{0+}^{\beta }x(1)=bD_{0+}^{\beta }x(\eta )\) (cf. once again (2.5)), we obtain
So
Then
Applying now Lemma 2.2 to (2.12), we have
where \(C_{1},C_{2}\in R\). Since \(x(0)=0\) (see (2.5), we have \(C_{2}=0\). Therefore, (2.13) reduces to
Applying \(D_{0+}^{\gamma }\) to both sides of (2.14), and by Lemma 2.1, we have
So
Combining (2.15) and (2.16) with \(x(1)=a D_{0+}^{\gamma }x(\xi )\) (see again (2.5)), we have
Thus, we obtain the unique solution of problem (2.5):
The proof of Lemma 2.4 is complete. □
We complete this section with the following simple observation.
Lemma 2.5
Let \(E=C[0,1]\) be the space of continuous functions equipped with the standard sup-norm \(\|x\|=\max_{0\leq t\leq 1}|x(t)|\) and denote by \(P=\{x\in E\mid x(t)\geq 0,0\leq t\leq 1\}\) the corresponding cone. Let \(T : P\rightarrow E\) be given by
where \(h\in C([0,1]\times [0,+\infty ),[0,+\infty ))\) and G and M are defined by (2.1) and (2.2), respectively. Then T takes P into itself, and as such is completely continuous.
Proof
Since G, M and h are nonnegative and continuous, one has \(T(P) \subset P\) and T is continuous. To prove the complete continuity of T, one needs to use the standard argument based on the Arzela–Ascoli theorem and Lebesgue dominated convergence theorem (see, for example, [23]). □
3 Main result
We are now in a position to formulate our main result. To this end, denote
where μ and ν are provided by Lemma 2.3(iii).
Theorem 3.1
Let \(h\in C([0,1]\times [0,+\infty ),[0,+\infty ))\) and assume that there exists a positive constant k satisfying the following conditions:
\((S_{1})\) if \(0\leq t\leq 1\) and \(0\leq s_{1}\leq s_{2}\leq k\), then \(h(t, s_{1})\leq h(t, s_{2})\);
\((S_{2})\) \(\max_{0\leq t\leq 1}h(t, k)\leq \phi _{p}(kJ)\);
\((S_{3})\) \(h(t,0)\neq0\) for all \(0\leq t\leq 1\).
Then problem (1.4) admits two positive solutions \(x^{*}\) and \(y^{*}\) such that:
(i) \(0<\|x^{*}\|\leq k\) and \(\lim_{n\rightarrow \infty }T^{n} x_{0}=x^{*}\), where \(x_{0}(t)=k\) for all \(0\leq t\leq 1\);
(ii) \(0<\|y^{*}\|\leq k\) and \(\lim_{n\rightarrow \infty }T^{n} y_{0}=y^{*}\), where \(y_{0}(t)=0\) for all \(0\leq t\leq 1\).
Proof
Let \(\Omega =\{x\in P\mid \|x\|\leq k\}\). Assume \(x\in \Omega \). Obviously, \(0\leq x(t)\leq \|x\|\leq k\). From \((S_{1})\) and \((S_{2})\) it follows immediately that
We claim that \(T(\Omega )\subseteq \Omega \). In fact, for any \(x\in \Omega \), we have \(Tx\in P\), and by Lemma 2.3, one has
Hence, \(Tx\in \Omega \) and the claim follows.
Let us show the existence of the required \(x^{*}\). Take the function \(x_{0}\) equal to k identically on \(0\leq t\leq 1\). Clearly, \(\|x_{0}\|=k\) (in particular, \(x_{0}\in \Omega \)). Also, \(x_{1}(t)=Tx_{0}(t) \in \Omega \). Define
Then, for all \(n=0,1,2,\ldots \) , one has \(x_{n}\in \Omega \).
Also, using \((S_{2})\) and the formula for T, and Lemma 2.3, one obtains for any \(t \in [0, 1]\):
Hence,
By induction, one has
Moreover, by Lemma 2.5, T is completely continuous, we know that \(\overline{T(\Omega )}\) is a compact set.
Hence, there exists a subsequence \(\{{x_{n_{i}}}\}_{i=1}^{\infty }\) of \(\{{x_{n}}\}_{n=1}^{\infty }\) convergent to \(x^{*} \in \Omega \). Since \(\{{x_{n}}\}_{n=1}^{\infty }\) is monotone, one has \(x_{n} \to x^{*}\). Combining the continuity of T with \(Tx_{n}=x_{n+1}\rightarrow x^{*}\) yields \(Tx^{*}=x^{*}\).
Below, using a similar approach, we prove \(Ty^{*} = y^{*}\). Take the function \(y_{0}\) equal to 0 identically on \(0\leq t\leq 1\). Clearly, \(\|y_{0}\|=0\), and \(y_{0} \in \Omega \). Also, \(y_{1} = Ty_{0} \in \Omega \). Define
Then, for all \(n=0,1,2,\ldots\) , one has \(y_{n}\in \Omega \). By the same computation as above,
Hence, there exists a subsequence \(\{{y_{n_{i}}}\}_{i=1}^{\infty }\) of \(\{{y_{n}}\}_{n=1}^{\infty }\) convergent to \(y^{*} \in \Omega \). Since \(\{{y_{n}}\}_{n=1}^{\infty }\) is monotone, one has \(y_{n} \to x^{*}\). Combining the continuity of T with \(Ty_{n}=y_{n+1}\rightarrow y^{*}\) yields \(Ty^{*}=y^{*}\). It remains to observe that, by assumption \((S_{3})\), the zero function is not a solution of problem (1.4). So \(\|x^{*}\|> 0\), and \(\|y^{*}\|> 0\). The proof is completed. □
4 Example
Consider the following BVP:
A simple computation gives
Take \(k=8\). Then:
\((1)\) For any \(0\leq t\leq 1\), \(0\leq s_{1}\leq s_{2}\leq 8\), \(h(t, s_{1})\leq h(t, s_{2})\);
\((2)\) \(\max_{0\leq t\leq 1}h(t, k)=h(1, 8)\approx 4.9834< \phi _{p}(kJ) \approx 5.0768\);
\((3)\) \(h(t,0)=0.05\neq0\), for \(0\leq t\leq 1\).
Then problem (1.4) has two positive solutions, \(x^{*}\) and \(y^{*}\), such that
\(0<\|x^{*}\|\leq 8\) and \(\lim_{n\rightarrow \infty }T^{n} x_{0}=x^{*}\), where \(x_{0}(t)=8\),
\(0<\|y^{*}\|\leq 8\) and \(\lim_{n\rightarrow \infty }T^{n} y_{0}=y^{*}\), where \(y_{0}(t)=0\).
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Acknowledgements
The authors would like to thank Professor Zalman Balanov and Y. Tian for their help with this research.
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Xiaoping Li, Associate professor, Her main research field is fractional differential equations and boundary value problems. Minyuan He, Lecturer, Her main research field is Probabilistic statistics.
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Project was supported by Hunan Provincial Natural Science Foundation of China(2018JJ2370), supported by the Scientific Research Foundation of Hunan Provincial Education Department (16A198, 18C1019), was also supported by Hunan Province of Chenzhou Science and Technology Planning Project.
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Li, X., He, M. Monotone iterative method for fractional p-Laplacian differential equations with four-point boundary conditions. Adv Differ Equ 2020, 686 (2020). https://doi.org/10.1186/s13662-020-03066-1
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DOI: https://doi.org/10.1186/s13662-020-03066-1