Abstract
We study the following fractional Navier boundary value problem:
where \(\alpha,\beta\in(1,2]\), \(D^{\alpha}\) and \(D^{\beta}\) stand for the standard Riemann-Liouville fractional derivatives, and \(\xi ,\zeta\geq0\) are such that \(\xi+\zeta>0\).
Our purpose is to prove the existence, uniqueness, and global asymptotic behavior of a positive continuous solution, where \(f:(0,1)\times[0,\infty)\rightarrow{}[0,\infty)\) is continuous and dominated by a function p satisfying appropriate integrability condition.
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1 Introduction
The existence, uniqueness, and global asymptotic behavior of positive continuous solutions related to fractional differential equations have been studied by many researchers. Many fractional differential equations subject to various boundary conditions have been addressed; see, for instance, [1–12] and the references therein. It is known that fractional differential equations serve as a good tool to model many phenomena in various fields of science and engineering (see [13–24] and references therein for discussions of various applications).
In [2], the authors proved the existence and uniqueness of a positive solution to the following fractional boundary value problem:
where \(1<\alpha\leq2 \), \(\xi,\zeta\geq0\) are such that \(\xi+\zeta >0\), and \(\varphi(x,s)\in C^{+} ( (0,1)\times[0,\infty ) ) \) satisfies appropriate conditions. Inspired by the above-mentioned paper, we aim at studying similar problem in the case of fractional Navier boundary value problem. More precisely, we are concerned with the problem
where \(\alpha,\beta\in(1,2]\), and \(\xi,\zeta\geq0\) are such that \(\xi+\zeta>0\). The nonlinear term \(f(x,s)\) is required to be a nonnegative continuous function in \((0,1)\times[0,\infty)\) dominated by a function p belonging to the class \(\mathcal{J}_{\alpha,\beta}\) defined as follows.
Definition 1.1
Let \(\alpha,\beta\in(1,2]\). A nonnegative measurable function p on \((0,1)\) belongs to the class \(\mathcal{J}_{\alpha,\beta}\) iff
Next, we introduce the following notation.
-
(i)
\(B^{+}((0,1))\) is the set of nonnegative measurable functions in \((0,1)\).
-
(ii)
Let X be a metric space, we denote by \(C(X)\) (resp. \(C^{+}(X)\)) the set of continuous (resp. nonnegative continuous) functions in X.
-
(iii)
For \(\gamma\in(1,2]\), \(C_{2-\gamma }([0,1])=\{w\in C((0,1]): x\rightarrow x^{2-\gamma}w(x)\in C ( [ 0,1 ] ) \}\).
-
(iv)
For \(\gamma\in(1,2]\), \(G_{\gamma}(x,s)\) is the Green function of the operator \(u\rightarrow-D^{\gamma}u\), with boundary data \(\lim_{x\rightarrow0^{+}}D^{\gamma -1}u(x)=u(1)=0\). From [8], Lemma 7, we have
$$ G_{\gamma} ( x,s ) =\frac{1}{\Gamma ( \gamma ) } \bigl( x^{\gamma-2} ( 1-s ) ^{\gamma-1}- \bigl( ( x-s ) ^{+} \bigr) ^{\gamma-1} \bigr) , $$(1.4)where \(x^{+}=\max(x,0)\).
Proposition 1.2
(see [8])
Let \(1<\gamma\leq2\) and \(\varphi\in B^{+}((0,1))\). Then we have
-
(i)
For \((x,s)\in(0,1]\times[0,1]\),
$$ \frac{(\gamma-1)}{\Gamma ( \gamma ) }H(x,s)\leq G_{\gamma} ( x,s ) \leq\frac{1}{\Gamma ( \gamma ) }H(x,s), $$(1.5)where \(H(x,s):=x^{\gamma-2} ( 1-s ) ^{\gamma -2}(1-\max(x,s))\).
-
(ii)
The function \(x\rightarrow G_{\gamma}\varphi (x):=\int_{0}^{1}G_{\gamma} ( x,s ) \varphi(s)\,ds\) belongs to \(C_{2-\gamma}([0,1])\) if and only if \(\int_{0}^{1}(1-s)^{\gamma -1}\varphi(s)\,ds<\infty\).
-
(iii)
If the map \(s\rightarrow(1-s)^{\gamma-1}\varphi (s)\in C((0,1))\cap L^{1}((0,1))\), then \(G_{\gamma}\varphi\) belongs to \(C_{2-\gamma}([0,1])\), and it is the unique solution of the problem
$$ \textstyle\begin{cases} D^{\gamma}u(x)=-\varphi(x),\quad 0< x< 1, \\ \lim_{x\rightarrow0^{+}}D^{\gamma-1}u(x)=u(1)=0.\end{cases} $$
Throughout this paper, for \(\alpha,\beta\in(1,2]\), let \(G(x,s)\) be the Green function of the operator \(u\rightarrow D^{\alpha}(D^{\beta }u)\) with Navier boundary conditions \(\lim_{x\rightarrow 0^{+}}D^{\beta-1}u(x)=\lim_{x\rightarrow0^{+}}D^{\alpha -1}(D^{\beta }u)(x)=u(1)=D^{\beta}u(1)=0\). Then we have
For a given function p in \(B^{+}((0,1))\), we put
and we will prove that \(\kappa_{p}<\infty\) if and only if \(p\in\mathcal{J}_{\alpha,\beta}\).
From here on, let ξ, ζ be two nonnegative constants such that \(\xi+\zeta>0\), and \(\theta(x)\) be the unique solution of the problem
We can easily verify that, for \(x\in(0,1]\), \(\theta(x)=\xi h_{1}(x)+\zeta h_{2}(x)\), where
and
Note that from (1.5), (1.9), and (1.10) it follows that there exists a constant \(c>0\) such that, for each \(x\in(0,1]\),
To state our existence results, a combination of the following hypotheses are required.
- (A1):
-
f is in \(C^{+}((0,1)\times[0,\infty))\).
- (A2):
-
There exists \(p\in\mathcal {J}_{\alpha,\beta} \cap C^{+}((0,1))\) with \(\kappa_{p}\leq \frac{1}{2}\) such that, for each \(x\in(0,1)\), the map \(s\rightarrow s ( p ( x ) -f ( x,s\theta ( x ) ) ) \) is nondecreasing on \([ 0,1 ] \).
- (A3):
-
For each \(x\in(0,1)\), the function \(s\rightarrow sf ( x,s ) \) is nondecreasing on \([0,\infty)\).
Our main results are the following.
Theorem 1.3
Under conditions (\(\mathrm{A}_{1}\))-(\(\mathrm{A}_{2}\)), problem (1.2) admits a solution \(u\in C_{2-\beta}([0,1])\) such that
where \(c_{0}\in(0,1)\).
Moreover, this solution is unique if hypothesis (\(\mathrm{A}_{3}\)) is also satisfied.
Corollary 1.4
Let \(\alpha,\beta\in(1,2]\), and h be a nonnegative function in \(C^{1}([0,\infty))\) such that the map \(s\rightarrow \varrho(s)=sh(s)\) is nondecreasing on \([0,\infty)\). Let \(q\in C^{+}((0,1))\) and assume that the function \(\tilde {q}(x):=q(x)\max_{0\leq t\leq\theta ( x ) }\varrho^{\prime}(t)\) belongs to \(\mathcal{J}_{\alpha,\beta}\). Then for \(\lambda\in{}[0,\frac{1}{2\kappa_{\tilde {q}}})\), the problem
admits a unique solution \(u\in C_{2-\beta}([0,1])\) such that
Our paper is organized as follows. In Section 2, we establish some properties of \(G(x,s)\). In particular, we prove the existence of a constant \(c>0\) such that, for all \(x,t,s\in(0,1)\),
This implies that \(\kappa_{p}<\infty\) if and only if \(p\in\mathcal{J}_{\alpha,\beta}\). In Section 3, for a given function \(p\in\mathcal{J}_{\alpha,\beta}\) with \(\kappa_{p}\leq \frac{1}{2}\), we construct the Green function \(\mathcal{H} ( x,s ) \) of the operator \(u\rightarrow D^{\alpha}(D^{\beta }u)+p(x)u\) with boundary conditions \(\lim_{x\rightarrow0^{+}}D^{\beta -1}u(x)=\lim_{x\rightarrow0^{+}}D^{\alpha-1}(D^{\beta }u)(x)=u(1)=D^{\beta}u(1)=0 \), and we derive some of its properties including the following:
and
where W and \(W_{p}\) are defined by
Exploiting these results, we prove our main results by means of a perturbation argument.
2 Estimates on the Green function
We recall the definition of the Riemann-Liouville derivative.
Definition 2.1
The Riemann-Liouville derivative of fractional order \({\gamma>0}\) of a function g is defined as
where \(n-1\leq\gamma< n\in\mathbb{N}\).
Next, we prove some properties of \(G ( x,s ) \).
Proposition 2.2
Let \(\alpha,\beta\in(1,2]\). Then there exist two constants \(m>0\) and \(M>0\) such that, for all \((x,s)\in(0,1]\times[0,1]\), we have
Proof
Using (1.6) and (1.5), we have
On the other hand, using again (1.6), (1.5), and the inequality \(1-\max(x,s)\geq(1-x)(1-s)\), we get
□
Using Proposition 2.2, we deduce the following.
Corollary 2.3
Let \(\alpha,\beta\in(1,2]\). Then there exists a constant \(c>0\) such that, for all \(x,t,s\in(0,1)\), we have
Proposition 2.4
Let \(\alpha,\beta\in(1,2]\), and p be a function in \(B^{+}((0,1))\).
-
(i)
There exists a constant \(c>0\) such that
$$ \frac{1}{c} \int_{0}^{1}t^{\beta-2}(1-t)^{\alpha }p(t)\,dt \leq\kappa_{p}\leq c \int_{0}^{1}t^{\beta -2}(1-t)^{\alpha}p(t)\,dt, $$(2.3)where \(\kappa_{p}\) is given by (1.7).
In particular,
$$ \kappa_{p}< \infty\quad\textit{if and only if}\quad p\in\mathcal{J}_{\alpha ,\beta}. $$(2.4) -
(ii)
For \(x\in(0,1]\), we have
$$ W(\theta p) (x)\leq\kappa_{p}\theta(x), $$(2.5)where \(\theta(x):=\xi h_{1}(x)+\zeta h_{2}(x)\), and \(h_{1}\) and \(h_{2} \) are given respectively in (1.9) and (1.10).
Proof
Let p be a function in \(B^{+}((0,1))\).
-
(i)
Inequalities in (2.3) follow immediately from (2.2) and (1.7).
-
(ii)
Since \(\theta(x):=\xi h_{1}(x)+\zeta h_{2}(x)\), it suffices to prove (2.5) for \(h_{1}\) and \(h_{2}\). To this end, observe that from (1.6) it follows that, for each \(x,s\in(0,1)\),
$$ \lim_{r\rightarrow0} \frac{G(s,r)}{G(x,r)}=\frac {G(s,0)}{G(x,0)}= \frac{h_{1}(s)}{h_{1}(x)}. $$
So by Fatou’s lemma and (1.7) we deduce that
that is,
Similarly, we prove that \(W(h_{2}p)(x)\leq\kappa _{p}h_{2}(x)\) by observing that
This ends the proof. □
Corollary 2.5
Let \(\alpha,\beta\in(1,2]\) and \(\varphi\in\mathcal {B}^{+} ((0,1)) \). Then \(x\rightarrow W\varphi(x)\in C_{2-\beta} ( [0,1] ) \) if and only if \(\int_{0}^{1} ( 1-s ) ^{\alpha-1}\varphi ( s ) \,ds<\infty\).
Proof
The assertion follows from (2.1) and the dominated convergence theorem. □
Proposition 2.6
Let \(\alpha,\beta\in(1,2]\) and \(\varphi\in\mathcal {B}^{+} ( (0,1) ) \) be such that \(s\rightarrow(1-s)^{\alpha -1}\varphi(s)\in C((0,1))\cap L^{1}((0,1))\). Then Wφ is the unique nonnegative solution in \(C_{2-\beta}([0,1])\) of
Proof
Let \(\varphi\in\mathcal{B}^{+} ( (0,1) ) \). From (1.6) and the Fubini-Tonelli theorem we obtain
where \(G_{\alpha}\varphi(t)=\int_{0}^{1}G_{\alpha } ( t,s ) \varphi(s)\,ds\).
Since the function \(s\rightarrow(1-s)^{\alpha-1}\varphi (s)\in C((0,1))\cap L^{1}((0,1))\), we deduce by Proposition 1.2 that \(G_{\alpha}\varphi\) is the unique solution in \(C_{2-\alpha}([0,1])\) of
On the other hand, by using (1.5) we deduce that
Hence, the function \(t\rightarrow(1-t)^{\beta-1}G_{\alpha }\varphi(t)\in C((0,1))\cap L^{1}((0,1))\). Therefore, using (2.6) and Proposition 1.2, we deduce that Wφ is the unique solution in \(C_{2-\beta}([0,1])\) of
3 Proofs of main results
Let \(\alpha,\beta\in(1,2]\). For \(( x,s ) \in (0,1]\times[0,1]\), put \(H_{0}(x,s)=G(x,s)\) and
Now, let \(\mathcal{H}:(0,1]\times[0,1]\rightarrow \mathbb{R}\) be defined by
provided that the series converges.
Lemma 3.1
Let \(\alpha,\beta\in(1,2]\) and \(m,M>0\) be as in (2.1). Let \(p\in\mathcal{J}_{\alpha,\beta}\) with \(\kappa_{p}<1\). Then on \((0,1]\times[0,1]\), we have
-
(i)
\(H_{n}(x,s)\leq\kappa_{p}^{n}G(x,s)\) for each \(n\in\mathbb{N}\).
So, \(\mathcal{H} ( x,s ) \) is well defined in \((0,1]\times[0,1]\).
-
(ii)
For each \(n\in\mathbb{N}\),
$$ l_{n}x^{\beta-2}(1-x) ( 1-s ) ^{\alpha-1}\leq H_{n}(x,s)\leq r_{n}x^{\beta-2}(1-x) ( 1-s ) ^{\alpha-1}, $$(3.3)where
$$ l_{n}=m^{n+1}\biggl( \int_{0}^{1}t^{\beta-2}(1-t)^{\alpha}p(t)\,dt \biggr)^{n}\quad \textit{and}\quad r_{n}=M^{n+1} \biggl( \int_{0}^{1}t^{\beta -2}(1-t)^{\alpha}p(t)\,dt \biggr)^{n}. $$ -
(iii)
\(H_{n+1}(x,s)=\int_{0}^{1}H_{n}(x,t)G(t,s)p(t)\,dt\) for each \(n\in\mathbb{N}\).
-
(iv)
\(\int_{0}^{1}\mathcal{H} ( x,t ) G(t,s)p(t)\,dt=\int_{0}^{1}G ( x,t ) \mathcal{H}(t,s)p(t)\,dt\).
Proof
By simple induction we prove (i), (ii), and (iii).
(iv) By Lemma 3.1(i) we have
Therefore, the series \(\sum_{n\geq0}\int_{0}^{1}H_{n}(x,t)G(t,s)p(t)\,dt\) converges.
So by applying the dominated convergence theorem we deduce that
□
Proposition 3.2
Let \(\alpha,\beta\in(1,2]\) and \(p\in \mathcal{J}_{\alpha,\beta}\) with \(\kappa_{p}<1\). Then the function \((x,s)\rightarrow x^{2-\beta}\mathcal{H} ( x,s ) \in C([0,1]\times[0,1])\).
Proof
Clearly, the function \((x,s)\rightarrow x^{2-\beta}H_{0}(x,s)\in C([0,1]\times[0,1])\).
Assume that the function \((x,s)\rightarrow x^{2-\beta }H_{n-1}(x,s)\in C([0,1]\times[0,1])\).
Using Lemma 3.1(i) and (2.1), we have, for all \((x,s,t)\in{}[0,1]\times[0,1]\times(0,1]\),
So by (3.1) and the dominated convergence theorem we conclude that the function \((x,s)\rightarrow x^{2-\beta }H_{n}(x,s)\in C([0,1]\times[0,1])\).
From Lemma 3.1(i) and (2.1) we deduce that
Therefore, the series \(\sum_{n\geq0} ( -1 ) ^{n}x^{2-\beta}H_{n}(x,s)\) is uniformly convergent on \([0,1]\times[0,1]\), and so the function \((x,s)\rightarrow x^{2-\beta}\mathcal{H} ( x,s ) \in C([0,1]\times[ 0,1])\). □
Lemma 3.3
Let \(\alpha,\beta\in(1,2]\) and \(p\in \mathcal{J}_{\alpha,\beta}\) with \(\kappa_{p}\leq\frac{1}{2}\). Then for \(( x,s ) \in(0,1]\times[0,1]\), we have
Proof
Let \(p\in\mathcal{J}_{\alpha,\beta}\) with \(\kappa_{p}\leq \frac{1}{2}\). By Lemma 3.1(i) we deduce that
Now, from the expression of \(\mathcal{H}\) we have
Since the series \(\sum_{n\geq0}\int_{0}^{1}G(x,t)H_{n}(t,s)p(t)\,dt\) converges, we conclude by (3.7) and (3.1) that
namely,
On the other hand, since
we deduce that
Hence, \(\mathcal{H} ( x,s ) \leq G ( x,s ) \), and by (3.8) we have
□
Corollary 3.4
Let \(\alpha,\beta\in(1,2]\) and \(p\in \mathcal{J}_{\alpha,\beta}\) with \(\kappa_{p}\leq\frac{1}{2}\).
Let \(\varphi\in\mathcal{B}^{+} ( (0,1) ) \). Then
Proof
The assertion follows from Proposition 3.2, (3.5), and (2.1). □
Lemma 3.5
Let \(\alpha,\beta\in(1,2]\) and \(p\in \mathcal{J}_{\alpha,\beta}\) with \(\kappa_{p}\leq\frac{1}{2}\).
Let \(h \in\mathcal{B}^{+} ( (0,1) ) \). Then we have, for \(x\in(0,1]\),
In particular, if \(W(ph )<\infty\), then
Proof
Let \((x,s)\in(0,1]\times[0,1]\). Then by (3.8) we have
Let \(h\in\mathcal{B}^{+} ( (0,1) ) \). Using the Fubini theorem, we obtain
Using Lemma 3.1(iv) and again the Fubini theorem, we have
that is,
So
□
Proposition 3.6
Let \(\alpha,\beta\in(1,2]\) and \(p\in \mathcal{J}_{\alpha,\beta}\cap C(\mathcal{(}0,1\mathcal{))}\) with \(\kappa_{p}\leq\frac{1}{2}\). Let \(\varphi\in\mathcal{B}^{+} ( (0,1) ) \) be such that \(s\rightarrow(1-s)^{\alpha-1}\varphi(s)\in C((0,1))\cap L^{1}((0,1))\). Then \(W_{p}\varphi\in C_{2-\beta}([0,1])\), and it is the unique nonnegative solution of the problem
satisfying
Proof
By Corollary 3.4 the function \(x\rightarrow p(x)W_{p}\varphi ( x ) \in C((0,1))\).
Using (3.10) and (2.1), we have that there exists \(c\geq 0\) such that
Therefore,
Hence, by Proposition 2.6 the function \(u=W_{p}\varphi=W\varphi-W ( pW_{p}\varphi ) \) satisfies the equation
By integration of inequalities (3.5) we obtain (3.13).
Let us prove the uniqueness. Let \(v\in C_{2-\beta }([0,1])\) be another solution of problem (3.12) satisfying \(v\leq W\varphi\).
Put \(\tilde {v}:=v+W(pv)\). Since the function \(s\rightarrow (1-s)^{\alpha-1}p(s)v(s)\in C((0,1))\cap L^{1}((0,1))\), then by Proposition 2.6 it follows that
From the uniqueness in Proposition 2.6 we conclude that
So
where \((v-u)^{+}=\max(v-u,0)\) and \((v-u)^{-}=\max (u-v,0)\).
From (3.13), (3.14), (1.11), and (2.5), there exists a constant \(\tilde {c}>0\), such that
Therefore, \(u=v\) by Lemma 3.5. □
Proof of Theorem 1.3
Consider \(\xi\geq0\) and \(\zeta\geq0\) with \(\xi+\zeta >0\). Let \(\alpha,\beta\in(1,2]\) and \(p\in\mathcal{J}_{\alpha,\beta }\cap C((0,1))\) be such that \((A_{2})\) is satisfied.
Let
where \(\theta(x):=\xi h_{1}(x)+\zeta h_{2}(x)\), and \(h_{1}\) and \(h_{2} \) are defined respectively by (1.9) and (1.10).
Define the operator \(\mathcal{F}\) on \(\mathcal{S}\) by
Using (\(\mathrm{A}_{2}\)), we get
Next, we prove that \(\mathcal{FS}\subseteq\mathcal{S}\). Indeed, using (3.16) and (3.15), we have, for \(u\in\mathcal {S}\),
and
Observe that, by (\(\mathrm{A}_{2}\)), \(\mathcal{F}\) becomes nondecreasing on \(\mathcal{S}\).
Define the sequence \(\{v_{n}\}\) by \(v_{0}= ( 1-\kappa _{p} ) \theta\) and \(v_{n+1}=\mathcal{F}v_{n}\) for \(n\in \mathbb{N}\). Since \(\mathcal{FS}\subseteq\mathcal{S}\), we have \(v_{1}=\mathcal{F}v_{0}\geq\) \(v_{0}\), and by the monotonicity of \(\mathcal{F}\) we deduce that
Using (\(\mathrm{A}_{1}\))-(\(\mathrm{A}_{2}\)) and the dominated convergence theorem, we deduce that the sequence \(\{v_{n}\}\) converges to a function \(u\in\mathcal{S}\) satisfying
that is,
and by (3.15) we have \(W ( pu ) \leq W ( p\theta ) \leq\theta<\infty\). Therefore, by Lemma 3.5 we deduce that
We claim that u is a solution.
Indeed, from (3.16) and (1.11), there exists a constant \(c>0\) such that
So, by Proposition 2.6 the function \(W ( uf ( \cdot,u ) ) \in C_{2-\beta}([0,1])\). This implies by (3.17) that \(u\in C_{2-\beta}([0,1])\).
Now, since the function \(s\rightarrow(1-s)^{\alpha -1}u(s)f ( s,u(s) ) \in C((0,1))\cap L^{1}((0,1))\), we deduce by Proposition 2.6 that u is a solution.
It remains to prove the uniqueness. Let v be another solution in \(C_{2-\beta}([0,1])\) to problem (1.2) satisfying (1.12). Since \(v\leq\theta\), we deduce by (3.18) that
This implies that \(s\rightarrow(1-s)^{\alpha -1}v(s)f ( s,v(s) ) \in C((0,1))\cap L^{1}((0,1))\). Let \(\tilde {v}:=v+W ( vf (\cdot,v ) ) \). By Proposition 2.6, we have
Hence,
Let \(\omega:(0,1)\rightarrow\mathbb{R}\) be defined by
By (\(\mathrm{A}_{3}\)), \(\omega\in\mathcal{B}^{+} ((0,1)) \) and from (3.17) and (3.19) we deduce that
where \((v-u)^{+}=\max(v-u,0)\) and \((v-u)^{-}=\max (u-v,0)\).
From (\(\mathrm{A}_{2}\)) we have \(\omega\leq p\). So by using (1.12) and (2.5) we obtain
Hence, \(u=v\) by (3.11). □
Proof of Corollary 1.4
The statement follows from Theorem 1.3 with \(f ( x,t ) =\lambda q(x)h(t)\), \(\varrho(t)=th(t)\) and \(p(x):=\lambda q(x)\max_{0\leq t\leq\theta ( x ) }\varrho ^{\prime}(t)\). □
Example 3.7
Let \(\sigma\geq0\), \(\nu\geq0\), and \(q\in C^{+}((0,1))\) be such that
Let \(\varrho(t)=t^{\sigma+1}\ln(1+t^{\nu})\) and \(\tilde {q}(t):=q(t)\max_{0\leq s\leq\theta ( t ) }\varrho^{\prime}(s)\). Since \(\tilde {q}\in\mathcal{J}_{\alpha ,\beta}\), then for \(\xi\geq0\), \(\zeta\geq0\) with \(\xi+\zeta >0\) and \(\lambda\in[0,\frac{1}{2\kappa_{\tilde {q}}})\), the problem
has a unique solution \(u\in C_{2-\beta}([0,1])\) such that
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Acknowledgements
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding this Research group NO (RG-1435-043). The authors would like to thank the anonymous referees for their careful reading of the paper.
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Bachar, I., Mâagli, H. & Rădulescu, V.D. Fractional Navier boundary value problems. Bound Value Probl 2016, 79 (2016). https://doi.org/10.1186/s13661-016-0586-7
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DOI: https://doi.org/10.1186/s13661-016-0586-7