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Solvability of fractional boundary value problems with pLaplacian operator
Advances in Difference Equations volumeÂ 2015, ArticleÂ number:Â 352 (2015)
Abstract
This paper studies the existence of solutions for two boundary value problems for the fractional pLaplacian equation. Under certain nonlinear growth conditions of the nonlinearity, two new existence results are obtained by using Schaeferâ€™s fixed point theorem.
1 Introduction
The fractional calculus is a generalization of the ordinary differentiation and integration on an arbitrary order that can be noninteger. In the last two decades, the theory of fractional calculus has gained importance and popularity due to its wide range of applications in varied fields of sciences and engineering. In [1â€“8], the applications are mentioned to fluid flow, rheology, dynamical processes in selfsimilar and porous structures, electrical networks, control theory of dynamical systems, viscoelasticity, electrochemistry of corrosion, chemical physics, optics and signal processing, and so on. Recently, many important results about the fractional differential equations have been given. For example, for fractional initial value problems, the existence and multiplicity of solutions were discussed in [4, 9â€“11]. In addition, for fractional boundary value problems (BVPs for short), Agarwal et al. (see [12]) considered a twopoint BVP at nonresonance, and Bai (see [13]) considered an mpoint BVP at resonance. For more papers on the fractional BVPs, see [14â€“22] and the references therein.
The turbulent flow in a porous medium is a fundamental mechanics problem. For studying this type of problems, Leibenson (see [23]) introduced the pLaplacian equation as follows:
where \(\phi_{p}(s)=s^{p2}s\), \(p>1\). Obviously, \(\phi_{p}\) is invertible and its inverse operator is \(\phi_{q}\), where \(q>1\) is a constant such that \(1/p+1/q=1\). In the past few decades, many important results about equation (1.1) with certain boundary value conditions have been obtained. We refer the readers to [24â€“31] and the references cited therein.
Motivated by the works mentioned previously, in this paper, we investigate the existence of solutions for the fractional pLaplacian equation of the form
subject to the boundary value conditions either
or
where \(0<\alpha,\beta\leq1\), \(D_{0^{+}}^{\alpha}\) is a Caputo fractional derivative, and \(f:[0,1]\times\mathbb{R}^{2}\rightarrow\mathbb{R}\) is continuous. Note that the nonlinear operator \(D_{0^{+}}^{\beta}\phi _{p}(D_{0^{+}}^{\alpha})\) reduces to the linear operator \(D_{0^{+}}^{\beta}D_{0^{+}}^{\alpha}\) when \(p=2\) and the additive index law
holds under some reasonable constraints on the function u (see [32]).
The rest of this paper is organized as follows. SectionÂ 2 contains some necessary notations, definitions and lemmas. In SectionÂ 3, based on Schaeferâ€™s fixed point theorem, we establish two theorems on the existence of solutions for BVP (1.2) (1.3) (TheoremÂ 3.1) and BVP (1.2) (1.4) (TheoremÂ 3.2). Finally, in SectionÂ 4, an explicit example is given to illustrate the main results.
2 Preliminaries
For convenience of the readers, we present here some necessary basic knowledge and definitions as regards the fractional calculus theory, which can be found, for instance, in [33, 34].
Definition 2.1
The RiemannLiouville fractional integral operator of order \(\alpha>0\) of a function \(u:(0,+\infty )\rightarrow\mathbb{R}\) is given by
provided that the righthand side integral is pointwise defined on \((0,+\infty)\).
Definition 2.2
The Caputo fractional derivative of order \(\alpha>0\) of a continuous function \(u:(0,+\infty)\rightarrow \mathbb{R}\) is given by
where n is the smallest integer greater than or equal to Î±, provided that the righthand side integral is pointwise defined on \((0,+\infty)\).
Lemma 2.1
(see [8])
Let \(\alpha>0\). Assume that \(u,D_{0^{+}}^{\alpha}u\in L([0,1],\mathbb{R})\). Then the following equality holds:
where \(c_{i}\in{\mathbb{R}}\), \(i=0,1,\ldots,n1\), and n is the smallest integer greater than or equal to Î±.
Lemma 2.2
For fixed \(l\in C([0,1],\mathbb{R})\), let us define
then the equation \(G_{l}(a)=0\) has a unique solution \(\widetilde{a}(l)\), and the function \(\tilde{a}:C[0,1]\rightarrow\mathbb{R}\) is continuous and sends bounded sets into bounded sets.
Proof
The proof is similar to the proof of PropositionÂ 2.2 in [35], so we omit the details.â€ƒâ–¡
In this paper, we take \(Y=C([0,1],\mathbb{R})\) with the norm \(\y\ _{0}=\max_{t\in[0,1]}y(t)\), and \(X=\{xx,D_{0^{+}}^{\alpha}x\in Y\}\) with the norm \(\x\_{X}=\max\{\x\_{0},\D_{0^{+}}^{\alpha}x\_{0}\}\). By means of the linear functional analysis theory, we can prove that X is a Banach space.
3 Main results
In this section, two theorems on the existence of solutions for BVP (1.2) (1.3) and BVP (1.2) (1.4) will be given under nonlinear growth restrictions of f.
As a consequence of LemmaÂ 2.1, we have the following results that are useful in what follows.
Lemma 3.1
Given \(h\in Y\), the unique solution of
is
where
and \(\phi_{q}\) is understood as the operator \(\phi_{q}:Y\rightarrow Y\) defined by \(\phi_{q}(x)(t)=\phi_{q}(x(t))\).
Proof
Assume that \(x(t)\) satisfies the equation of BVP (3.1), then LemmaÂ 2.1 implies
From the boundary value condition \(D_{0^{+}}^{\alpha}x(0)=0\), one has \(c_{1}=0\). Thus, we have
By the condition \(x(1)=0\), we get
The proof is completed.â€ƒâ–¡
Define the operator \(\mathcal{K}_{1}:X\rightarrow X\) by
where \(N_{f}:X\rightarrow Y\) is the Nemytskii operator defined by
Clearly, the fixed points of \(\mathcal{K}_{1}\) are the solutions of BVP (1.2) (1.3).
Lemma 3.2
Given \(h\in Y\), the unique solution of
is
where
and \(\eta(h)\in(0,1)\) is a unique constant dependent on \(h(t)\).
Proof
Assume that \(x(t)\) satisfies the equation of BVP (3.3), by LemmaÂ 2.1, we get
From the boundary value conditions \(x(0)=x(1)=0\), one has \(c=0\) and
Then, basing on LemmaÂ 2.2, we know equation (3.4) has a unique solution \(\widetilde{a}(I_{0^{+}}^{\beta}h)\). Moreover, by the integral mean value theorem, there exists a unique constant \(\eta(h)\in(0,1)\) dependent on \(h(t)\) such that \(\phi_{q}(\widetilde{a}(I_{0^{+}}^{\beta}h)+I_{0^{+}}^{\beta}h)(\eta(h))=0\), which implies \((\widetilde {a}(I_{0^{+}}^{\beta}h)+I_{0^{+}}^{\beta}h)(\eta(h))=0\). Thus, we have
The proof is completed.â€ƒâ–¡
Define the operator \(\mathcal{K}_{2}:X\rightarrow X\) by
where \(\eta(x)\in(0,1)\) is a unique constant dependent on \(x(t)\) and \(N_{f}\) is the Nemytskii operator defined by (3.2). Clearly, the fixed points of \(\mathcal{K}_{2}\) are the solutions of BVP (1.2) (1.4).
Our first result, based on Schaeferâ€™s fixed point theorem and LemmaÂ 3.1, is stated as follows.
Theorem 3.1
Let \(f:[0,1]\times\mathbb{R}^{2}\rightarrow \mathbb{R}\) be continuous. Assume that

(H)
there exist nonnegative functions \(a,b,c\in Y\) such that
$$\begin{aligned} \biglf(t,u,v)\bigr\leq a(t)+b(t)u^{p1}+c(t)v^{p1},\quad \forall t\in [0,1],(u,v)\in\mathbb{R}^{2}. \end{aligned}$$
Then BVP (1.2) (1.3) has at least one solution provided that
Proof
The proof will be given in the following two steps.
Step 1: \(\mathcal{K}_{1}:X\rightarrow X\) is completely continuous.
By the definition of \(\mathcal{K}_{1}\), we obtain
Obviously, the operators \(\mathcal{K}_{1}\) and \(D_{0^{+}}^{\alpha}\mathcal {K}_{1}\) are compositions of the continuous operators. So \(\mathcal{K}_{1}\) and \(D_{0^{+}}^{\alpha}\mathcal{K}_{1}\) are continuous in Y. Hence, \(\mathcal{K}_{1}\) is a continuous operator.
Let \(\Omega\subset X\) be an open bounded set, then \(\mathcal {K}_{1}(\overline{\Omega})\) and \(D_{0^{+}}^{\alpha}\mathcal{K}_{1}(\overline {\Omega})\) are bounded. Moreover, for \(\forall x\in\overline{\Omega }\), \(t\in[0,1]\), there exists a constant \(T>0\) such that \(I_{0^{+}}^{\beta}N_{f}x(t)\leq T\). Thus, in view of the ArzelÃ Ascoli theorem, we need only to prove that \(\mathcal{K}_{1}(\overline{\Omega})\subset X\) is equicontinuous.
For \(0\leq t_{1}< t_{2}\leq1\), \(x\in\overline{\Omega}\), we have
Since \(t^{\alpha}\) is uniformly continuous in \([0,1]\), we can obtain that \(\mathcal{K}_{1}(\overline{\Omega})\subset Y\) is equicontinuous. A similar proof can show that \(I_{0^{+}}^{\beta}N_{f}(\overline{\Omega })\subset Y\) is equicontinuous. This, together with the uniform continuity of \(\phi_{q}(s)\) on \([T,T]\), yields that \(D_{0^{+}}^{\alpha }\mathcal{K}_{1}(\overline{\Omega})\subset Y\) is also equicontinuous. Thus, we find that \(\mathcal{K}_{1}:X\rightarrow X\) is compact.
Step 2: A priori bounds.
Set
Now it remains to show that the set \(\Omega_{1}\) is bounded.
By LemmaÂ 2.1 and the boundary value condition \(x(1)=0\), one has
which together with
yields
For \(x\in\Omega_{1}\), we have
So, from (H), we obtain
which, together with \(\phi_{p}(D_{0^{+}}^{\alpha}x(t))=D_{0^{+}}^{\alpha}x(t)^{p1}\) and (3.7), yields
Hence, in view of (3.5), we can get
Thus, from (3.7), one has
Therefore, combining (3.8) with (3.9), we have
As a consequence of Schaeferâ€™s fixed point theorem, we deduce that \(\mathcal{K}_{1}\) has a fixed point which is the solution of BVP (1.2) (1.3). The proof is completed.â€ƒâ–¡
Our second result, based on Schaeferâ€™s fixed point theorem and LemmaÂ 3.2, is stated as follows.
Theorem 3.2
Let \(f:[0,1]\times\mathbb{R}^{2}\rightarrow \mathbb{R}\) be continuous. Suppose (H) holds, then BVP (1.2) (1.4) has at least one solution provided that
Proof
The proof of complete continuity of \(\mathcal {K}_{2}\) is similar to the proof of complete continuity of \(\mathcal {K}_{1}\), so we omit the details.
Set
From LemmaÂ 2.1 and \(x(0)=0\), one has
which together with (3.6) yields
Then, by the hypothesis (H) and (3.11), we obtain
For \(x\in\Omega_{2}\), we have
which, together with \(\phi_{p}(D_{0^{+}}^{\alpha}x(t))=D_{0^{+}}^{\alpha}x(t)^{p1}\) and (3.12), yields
Thus, in view of (3.10), we can get
So, from (3.11), one has
Hence, combining (3.13) with (3.14), we obtain
Therefore, \(\Omega_{2}\) is bounded.
As a consequence of Schaeferâ€™s fixed point theorem, we deduce that \(\mathcal{K}_{2}\) has a fixed point which is the solution of BVP (1.2) (1.4). The proof is completed.â€ƒâ–¡
4 An example
In this section, we give an example to illustrate our main results.
Example 4.1
Consider the following fractional pLaplacian equation:
Corresponding to equation (1.2), we get \(p=3\), \(\alpha=1/2\), \(\beta=3/4\) and
Choose \(a(t)=2\), \(b(t)=1/6\), \(c(t)=0\). By a simple calculation, we obtain \(\ b\_{0}=1/6\), \(\c\_{0}=0\) and
Obviously, equation (4.1) subject to the boundary value conditions (1.3) (or (1.4)) satisfies all assumptions of TheoremÂ 3.1 (or TheoremÂ 3.2). Hence, BVP (4.1) (1.3) (or BVP (4.1) (1.4)) has at least one solution.
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Zhang, B. Solvability of fractional boundary value problems with pLaplacian operator. Adv Differ Equ 2015, 352 (2015). https://doi.org/10.1186/s1366201506487
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DOI: https://doi.org/10.1186/s1366201506487