Abstract
This paper is concerned with an integral boundary value problem of fractional differential equations with p-Laplacian operator. Sufficient conditions ensuring the existence of extremal solutions for the given problem are obtained. Our results are based on the method of upper and lower solutions and monotone iterative technique.
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1 Introduction
This paper studies the existence of extremal solutions for the boundary value problem of a fractional p-Laplacian equation with the following form:
where \(D^{\tau }\) and \(D^{\sigma }\) are the standard Riemann–Liouville fractional derivatives, \(I^{\gamma }\), \(I^{\epsilon }\) are the Riemann–Liouville fractional integral, and \(1<\sigma \), \(\tau <2\), \(\gamma ,\epsilon >1\), \(0<\theta \), \(\zeta <1\), \(d, e\in \mathbb{R}\), \(h\in C([0,1]\times \mathbb{R}\times \mathbb{R}, \mathbb{R})\), \(k\in C([0,1]\times \mathbb{R},\mathbb{R})\). The p-Laplacian operator is defined as \(\phi _{p}(t)=|t|^{p-2}t\), \(p>1\), and \((\phi _{p})^{-1}=\phi _{q}\), \(\frac{1}{p}+\frac{1}{q}=1\).
Recently, much attention has been paid to the study of the existence of extremal solutions, for fractional differential equations with corresponding initial or boundary conditions; see [1,2,3,4,5,6,7,8,9]. The monotone iterative technique, combined with the method of upper and lower solutions, provides an effective mechanism to prove constructive existence results for nonlinear differential equations, the advantage and importance of the technique needs no special emphasis [10, 11]. By using the monotone iterative technique, Ahmad [12] and Alsaedi [13] successfully investigated initial value problems for nonlinear fractional differential equations with fractional derivatives. Han [14] considered the existence of positive solutions for the following problem:
where \(2<\alpha \leq 3\), \(1<\beta \leq 2\) are real numbers, \(\lambda >0\) is a parameter, and \(f: (0, +\infty )\rightarrow (0, +\infty )\) is continuous. By using the properties of Green function and the Guo–Krasnosel’skii fixed-point theorem on cones, several existence results of at least one or two positive solutions in terms of different eigenvalue interval are obtained. By means of the monotone iterative method, Wang [15] investigated the fractional integral boundary problem
where \(\lambda >0\), and f, g are continuous functions. However, the existence results in [10] mainly depend upon a restrictive condition, i.e.,
It is a critical condition in order to discuss the monotone iterative sequences. Therefore, it is natural to ask whether similar results can be obtained if
Being directly inspired by Wang [15], the purpose of this paper is to study the nonlinear integral boundary value problem for p-Laplacian differential equations. The nonlinear terms h, k are not required to satisfy monotonicity conditions on the unknown function u or their derivatives. The monotone iterative technique combined with the method of upper and lower solutions is applied. In particular, we construct two well-defined monotone iterative sequences of upper and lower solutions and prove that they converge uniformly to the actual solution of the problem.
2 Preliminaries
In this section, we deduce some preliminary results which will be used in the next section.
Denote \(C_{\tau }[0,1]=\{u :u \in C[0,1], D^{\tau }u(t)\in C[0,1]\}\) and endow it with the norm \(\|u\|_{\tau }=\|u\|+\|D^{\tau }u\|\), where \(\|u\|=\max_{0\leq t\leq 1}|u(t)|\) and \(\|D^{\tau }u\|=\max_{0\leq t\leq 1}|D^{\tau }u(t)|\). Then \((C_{\tau }[0,1],\| \cdot \|_{\tau })\) is a Banach space (see [16]).
Definition 2.1
A function \(u(t)\in C_{ \tau }[0,1]\) satisfying \(D^{\sigma }(\phi _{p}(-D^{\tau }u(t)))\in C[0,1]\) is called a lower solution of problem (1.1) if
A function \(v(t)\in C_{\tau }[0,1]\) satisfying \(D^{\sigma }(\phi _{p}(-D ^{\tau }v(t)))\in C[0,1]\) is called an upper solution of problem (1.1) if the above inequalities are reversed.
For the sake of convenience, we now present some assumptions as follows.
- (H1):
-
Assume that \(u_{0},v_{0}\in C_{\tau }[0,1]\) satisfying \(D^{\sigma }(\phi _{p}(-D^{\tau }u_{0}(t))) \), \(D^{\sigma }(\phi _{p}(-D ^{\tau }v_{0}(t)))\in C[0,1]\) are lower and upper solutions of problem (1.1), respectively, and \(u_{0}(t)\leq v_{0}(t)\), \(D^{\tau }v_{0}(t) \leq D^{\tau }u_{0}(t)\), \(t\in [0,1]\).
- (H2):
-
There exists a constant \(L\in \mathbb{R}\) such that
$$ h\bigl(t,u(t),D^{\tau }u(t)\bigr)-h\bigl(t,v(t),D^{\tau }v(t) \bigr)\leq L\bigl[\phi _{p}\bigl(-D^{ \tau }v(t)\bigr)-\phi _{p}\bigl(-D^{\tau }u(t)\bigr)\bigr], $$for \(u_{0}(t)\leq u(t)\leq v(t) \leq v_{0}(t)\), \(D^{\tau } v_{0}(t) \leq D^{\tau }v(t)\leq D^{\tau } u(t)\leq D^{\tau } u_{0}(t)\), \(t\in [0,1]\).
- (H3):
-
There exists a constant \(\mu \geq 0\), such that
$$ k\bigl(t,\phi _{p}\bigl(-D^{\tau }v(t)\bigr)\bigr)-k \bigl(t,\phi _{p}\bigl(-D^{\tau }u(t)\bigr)\bigr)\geq \mu \bigl[\phi _{p}\bigl(-D^{\tau }v(t)\bigr)-\phi _{p}\bigl(-D^{\tau }u(t)\bigr)\bigr], $$for \(u_{0}(t)\leq u(t)\leq v(t) \leq v_{0}(t)\), \(D^{\tau } v_{0}(t) \leq D^{\tau }v(t)\leq D^{\tau } u(t)\leq D^{\tau } u_{0}(t)\), \(t\in [0,1]\).
- (H4):
-
\(\varGamma (\sigma +\gamma )>\mu \theta ^{\sigma +\gamma -1}\).
- (H5):
-
\(2\varGamma (\sigma +\gamma ) |L|<\varGamma (\sigma )[\varGamma ( \sigma +\gamma )-\mu \theta ^{\sigma +\gamma -1}]\).
- (H6):
-
For any \(t\in (0,1)\), we have
$$ \varGamma (2-\sigma )t^{\sigma }L>1-\sigma $$and
$$ \varGamma (2-\sigma )\mu \theta ^{\gamma }< \varGamma (\gamma ). $$
Lemma 2.1
([17])
Let \(f(t)\in C[0,1]\), \(a \in \mathbb{R}\), and \(\varGamma (\sigma +\gamma )\neq \mu \theta ^{\sigma +\gamma -1} \), then the fractional boundary value problem
is equivalent to
where
and \(\Delta =\varGamma (\sigma )[\varGamma (\sigma +\gamma )-\mu \theta ^{\sigma +\gamma -1}]\).
Lemma 2.2
([17])
If (H4) holds, then the Green’s function \(J(t,s)\) satisfies
Lemma 2.3
Let L, \(a\in \mathbb{R}\), \(w(t)\), \(f(t) \in C[0,1]\) and (H4), (H5) hold, then the boundary value problem
has a unique solution \(w(t)\in C[0,1]\).
Proof
It follows from Lemma 2.1 that problem (2.2) is equivalent to the following integral equation:
Let
For any \(u,w\in C[0,1]\), by (H4) and Lemma 2.2, we have
Noting that (H5) holds, which implies \(\frac{2\varGamma (\sigma + \gamma )| L| }{\varGamma (\sigma )[\varGamma (\sigma +\gamma )- \mu \theta ^{\sigma +\gamma -1}]}<1\), we have
By the Banach fixed point theorem, the operator A has a unique fixed point. That is, (2.2) has a unique solution. □
Lemma 2.4
Assume that \(z(t)\in C[0,1]\), \(k \in \mathbb{R}\). Then the fractional boundary value problem
is equivalent to
where
Proof
We can transform the equation \(-D^{\tau }v(t)=z(t)\) to an equivalent integral equation
Note that \(u(0)=0\), we have \(C_{2}=0\). Consequently, we have the following form:
and
On the other hand \(D^{\tau -1}u(1)=k\), and we obtain
Therefore, the solution of problem (2.3) is
□
Lemma 2.5
Assume that \(k,a\in \mathbb{R}\), \(u(t)\in C_{\tau }[0,1]\), \(f(t)\in C[0,1]\) and (H4), (H5) hold. Then the boundary value problem
has a unique solution \(u(t)\).
Proof
Let \(\phi _{p}(-D^{\tau }u(t))=w(t)\), and consider the boundary value problem:
From Lemma 2.3, we know that (2.6) has a unique solution \(w(t) \in C[0,1]\). Note that \(\phi _{p}(-D^{\tau }u(t)) =w(t)\in C[0,1]\), and \(-D^{\tau }u(t)=\phi _{q}(w(t))\in C[0,1]\). Then the problem (2.5) is transformed into to the following problem:
By Lemma 2.4, the solution of (2.7) can be written
Combining with (2.6) and (2.7), we assert that the boundary problem (2.5) has a unique solution \(u(t)\). □
Lemma 2.6
([18, Lemma 2.6])
Assume that (H6) holds, \(w(t)\in C[0,1]\), satisfying \(D^{\sigma }w(t)\in C[0,1]\) and
then \(w(t)\geq 0\), \(\forall t\in [0,1]\).
Lemma 2.7
If \(u(t)\in C[0,1]\) satisfies
then \(u(t)\geq 0\), \(\forall t \in [0,1]\).
Proof
By Lemma 2.4, we know that (2.3) has a unique solution
It is easy to verify that the Green’s function \(H(t,s)\geq 0\), t, \(s \in [0,1]\). Let \(z(t)\geq 0\) and \(k\geq 0\). Then we obtain (2.9) and \(u(t)\geq 0\), \(\forall t \in [0,1]\). □
3 Main results
Theorem 3.1
Suppose that (H1)–(H6) hold. Then problem (1.1) has extremal solution \(u^{*}, v^{*}\in [u _{0},v_{0}]\). Moreover,
and
Proof
For \(n=0,1,2,\ldots \) , we define
and
In view of Lemma 2.5, the functions \(u_{1}\) and \(v_{1}\) are well defined. First, we show that \(u_{0}(t)\leq u_{1}(t)\leq v_{1}(t) \leq v_{0}(t)\), and \(D^{\tau } v_{0}(t)\leq D^{\tau }v_{1}(t) \leq D^{\tau } u_{1}(t)\leq D^{\tau } u_{0}(t)\), \(t\in [0,1]\). Let \(\delta (t)=\phi _{p}(-D^{\tau }u_{1}(t))-\phi _{p}(-D^{\tau }u_{0}(t))\). From (3.1) and (H1), we obtain
Also \(\delta (0)=0\) and
In view of Lemma 2.6, we have \(\phi _{p}(-D^{\tau }u_{1}(t))\geq \phi _{p}(-D^{\tau }u_{0}(t))\), \(t\in [0,1]\), since \(\phi _{p}(x)\) is nondecreasing, thus
Let \(\alpha (t)=u_{1}(t)-u_{0}(t)\), it follows from (3.1) and (3.3) that
According to Lemma 2.7, we have \(u_{1}(t)\geq u_{0}(t)\), \(\forall t \in [0,1]\).
By a similar way, we can show that \(v_{0}(t)\geq v_{1}(t)\), and \(D^{\tau }v_{0}(t)\leq D^{\tau }v_{1}(t)\). Now, we put \(p(t)=\phi _{p}(-D ^{\tau }v_{1}(t))-\phi _{p}(-D^{\tau }u_{1}(t))\). From (H2) and (H3), we have
also \(p(0)=0\), and
In view of Lemma 2.6, we have \(p(t)\geq 0\), \(\forall t\in [0,1]\). Thus we have \(\phi _{p}(-D^{\tau }v_{1}(t))\geq \phi _{p}(-D^{\tau }u_{1}(t))\), that is, \(D^{\tau }v_{1}(t)\leq D^{\tau }u_{1}(t)\), since \(\phi _{p}\) is nondecreasing. Therefore \(D^{\tau } v_{0}(t)\leq D^{\tau }v_{1}(t) \leq D^{\tau } u_{1}(t)\leq D^{\tau } u_{0}(t)\) \(\forall t\in [0,1]\) holds.
Let \(\theta (t)=v_{1}(t)-u_{1}(t)\). From (H1), we have
Moreover, we get \(v_{1}(t)\geq u_{1}(t)\), from Lemma 2.7. Hence, we have the relation \(u_{0}(t)\leq u_{1}(t)\leq v_{1}(t)\leq v_{0}(t)\), \(\forall t\in [0,1] \).
In the following, we show that \(u_{1}(t)\), \(v_{1}(t)\) are lower and upper solutions of problem (1.1), respectively. From (3.1)–(3.2) and (H2), (H3), one gets
Also \(D^{\tau }u_{1}(0)=0\), \(u_{1}(0)=0\), and
This proves that \(u_{1}(t)\) is a lower solution of the problem (1.1). Similarly, we find that \(v_{1}(t)\) is an upper solution of (1.1).
Using mathematical induction, we see that
and
for \(t\in [0,1]\) and \(n=1,2,3,\ldots \) .
Since the space of solutions is \(C_{\tau }[0,1]\), the sequences \(\{u_{n}\}\) and \(\{v_{n}\}\) are uniformly bounded and equi-continuous. The Arzela–Ascoli theorem guarantees that they are relatively compact sets in the space \(C_{\tau }[0,1]\). Therefore, \(\{u_{n}\}\) and \(\{v_{n}\}\) converge, say to \(u^{*}(t)\) and \(v^{*}(t)\), uniformly on \([0,1]\), respectively. That is,
and
uniformly on \(t\in [0,1]\). Moreover, from (3.1) and (3.2), we find that \(u^{*}(t)\) and \(v^{*}(t)\) are solutions of problem of (1.1).
Finally, we prove that \(u^{*}(t)\), \(v^{*}(t)\) are the minimal and maximal solutions of problem (1.1), respectively. Let \(u(t)\in [u_{0},v_{0}]\) be any solution of the problem (1.1). We suppose that \(u_{n}(t) \leq u(t)\leq v_{n}(t)\), \(D^{\tau }v_{n}(t)\leq D^{\tau }u(t)\leq D^{ \tau }u_{n}(t)\) \(\forall t\in [0,1]\) for some n. Let \(y(t)=\phi _{p}(-D ^{\tau }u(t))-\phi _{p}(-D^{\tau }u_{n+1}(t))\), \(x(t)=\phi _{p}(-D^{ \tau }v_{n+1}(t))-\phi _{p}(-D^{\tau }u(t))\). Then, by assumptions (H2) and (H3), we see that
also \(y(0)=0\), and
On the other hand
Using Lemma 2.6, we have
Let \(m(t)=u(t)-u_{n+1}(t)\), \(\xi (t)=v_{n+1}(t)-u(t)\), by (3.4), we get
and
These results and Lemma 2.7 imply that \(u_{n+1}(t)\leq u(t)\leq v_{n+1}(t)\), \(t \in [0,1]\), so by induction \(u^{*}(t)\leq u(t)\leq v^{*}(t)\), \(D^{ \tau }v^{*}(t)\leq D^{\tau }u(t)\leq D^{\tau }u^{*}(t)\), \(t\in [0,1]\) by taking as \(n\longrightarrow \infty \). The proof is complete. □
4 Iteration procedure and a numerical example
In this section, a numerical procedure is introduced to obtain an appropriate solution of (1.1). For a given accuracy δ, we take \(u_{n}\) and \(v_{n}\) as δ-accurate approximations of x and y, respectively, according to the stopping criterion \(E(N)<\delta \), where for each n, \(E(n)\) is defined by
For the iteration equation (3.1), let \(\phi _{p}(-D^{\tau }u_{n+1}(t))=x _{n+1}\), and with the boundary conditions \(u_{n+1}(0)=0\), \(D^{\tau -1}u _{n+1}(1)=k\), and by Lemma 2.4,
where \(k=I^{\epsilon }u_{n}(\zeta )+e=\frac{1}{\varGamma (\epsilon )}\int _{0}^{\zeta }(\zeta -s)^{\epsilon -1}u_{n}(s)\,ds+e\) and
Thus the iteration equation (3.1) can be rewritten as
Applying Lemma 2.1 to (4.2), we obtain
where \(a= I^{\gamma }k(\theta ,x_{n}(\theta )) -\mu I^{\gamma }x_{n}( \theta )+d\), and
Discretize the interval \([0,1]\) with the nodes \(t_{i}=ih\), \(b= \frac{1}{K}\), \(K=\mathbb{N}\). Let \(u_{n+1}^{(i)}\approx u_{n+1}(t_{i})\), \(x _{n+1}^{(i)}\approx x_{n+1}(t_{i})\), \(H(i,j)=H(t_{i},s_{j})\), \(J(i,j)=J(t _{i},s_{j})\) and \(h_{n}^{(j)}=h(s_{j},Bx_{n}(s_{j}),-\phi _{q}(x_{n}(s _{j})))+Lx_{n}(s_{j})\). Using the trapezoidal quadrature rule to approximate the integral in the right hand sides of (4.3) and (4.1), we obtain the following linear systems of equations:
and
for the unknown \(x_{n+1}^{(i)}\), \(u_{n+1}^{(i)}\), \(0\leq i\leq K\), where \(\{d_{j}\}\) are the coefficients in the rule\(,d_{0}=d_{K}=1\), \(d_{j}=2\) for \(0\leq i\leq K-1\).
Setting \(J_{ij}=\frac{b}{2}J(i,j)d_{j}\), \(H_{ij}=\frac{b}{2}H(i,j)d _{j}\), the matrix \(\varPhi =(J_{ij})\), \(A=\mathbb{I}+L\varPhi \), and \(G=(H_{(ij)})\) with \(\mathbb{I}\) the identity matrix. The systems (4.4) and (4.5) can be written as a system of matrix-vector equations
where \(\overrightarrow{x}_{n+1}=[{x}_{n+1}^{(0)},{x}_{n+1}^{(1)}, \ldots ,{x}_{n+1}^{(K)}] \), \(\overrightarrow{U}_{n+1}=[{u}_{n+1}^{(0)}, {u}_{n+1}^{(1)},\ldots ,{u}_{n+1}^{(K)}]\), \(S=[t_{0},t_{1},\ldots ,t _{K}]^{T}\) and \(\overrightarrow{F}_{n}\) is a column vector of its component \(F_{n}^{i}=\frac{b}{2}\sum_{j=0}^{K}J(i,j)d_{j}h_{n} ^{(j)}\).
Example 4.1
Consider the following fractional boundary value problem:
where \(\sigma =\frac{5}{4}\), \(\tau =\frac{3}{2}\), \(\gamma =\frac{5}{4}\), \(\epsilon =\frac{3}{2}\), \(\theta =\frac{1}{4} \), \(\zeta =\frac{1}{2}\), \(d=0.1\), \(e=0.2\), \(p=4\), and
Take \(u_{0}(t)=\frac{1}{2}t^{\frac{1}{2}}\), \(v_{0}(t)=2t^{\frac{1}{2}}-\frac{\sqrt{ \varPi }}{4}t^{2}+\frac{8}{15\sqrt{\varPi }}t^{\frac{5}{2}}\), then \(-1\leq -t^{\frac{1}{2}}+t=D^{\frac{3}{2}}v_{0}(t)\leq D^{\frac{3}{2}}u _{0}(t)=0\). It is not difficult to verify that \(u_{0}\), \(v_{0}\) are lower and upper solutions of problem (4.6), respectively. So (H1) holds. In addition, we have
and
where \(u_{0}(t)\leq u(t)\leq v(t)\leq v_{0}(t)\). Therefore (H2) and (H3) hold.
From (4.7) and (4.8), we have \(L=\frac{1}{16}\), \(\mu =1\). Then
It appears that (H4), (H5) and (H6) hold. By Theorem 3.1, the boundary value problem (4.6) has extremal solutions in \([u_{0}(t),v_{0}(t)]\).
Applying the numerical scheme to Example 4.1, we obtain approximate solutions to \({u_{n}}\) and \({v_{n}}\) for \(1\leq n\leq N\), with N some integer. The graphs of to \({u_{n}}\) and \({v_{n}}\) for \(n=0,1,2,3,4,34\) are shown in Fig. 1. Computed error values \(E(n)\) are displayed in Table 1. We found that, for \(\delta =10^{-10}\), it took \(N=34 \) iterations for \(E(N)<\delta \).
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The authors sincerely thank the editor and reviewers for their valuable suggestions and useful comments to improve the manuscript.
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This work is supported by Fostering Foundation of Northeast Petroleum University (No. 2017PYYL-08).
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He, Y., Bi, B. Existence and iteration of positive solution for fractional integral boundary value problems with p-Laplacian operator. Adv Differ Equ 2019, 415 (2019). https://doi.org/10.1186/s13662-019-2317-8
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DOI: https://doi.org/10.1186/s13662-019-2317-8