Fractional Navier boundary value problems

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.

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.

1. (i)

   \(B^{+}((0,1))\) is the set of nonnegative measurable functions in \((0,1)\).

2. (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.

3. (iii)

   For \(\gamma\in(1,2]\), \(C_{2-\gamma }([0,1])=\{w\in C((0,1]): x\rightarrow x^{2-\gamma}w(x)\in C ( [ 0,1 ] ) \}\).

4. (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

1. (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 \(H(x,s):=x^{\gamma-2} ( 1-s ) ^{\gamma -2}(1-\max(x,s))\).

2. (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 \(\int_{0}^{1}(1-s)^{\gamma -1}\varphi(s)\,ds<\infty\).

3. (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.

We recall the definition of the Riemann-Liouville derivative.

Definition 2.1

(see [17, 22, 23])

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 \(B^{+}((0,1))\).

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)
2. (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 \(B^{+}((0,1))\).

1. (i)

   Inequalities in (2.3) follow immediately from (2.2) and (1.7).

2. (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

Combining (2.7) and (2.8), we obtain the required result. □

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

1. (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 \((0,1]\times[0,1]\).

2. (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

   $$ l_{n}=m^{n+1}\biggl( \int_{0}^{1}t^{\beta-2}(1-t)^{\alpha}p(t)\,dt \biggr)^{n}\quad \textit{and}\quad r_{n}=M^{n+1} \biggl( \int_{0}^{1}t^{\beta -2}(1-t)^{\alpha}p(t)\,dt \biggr)^{n}. $$
3. (iii)

   \(H_{n+1}(x,s)=\int_{0}^{1}H_{n}(x,t)G(t,s)p(t)\,dt\) for each \(n\in\mathbb{N}\).

4. (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

By (3.10) and (2.5) we have

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

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.

Affiliations

1. Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia
   • Imed Bachar
2. College of Sciences and Arts, Department of Mathematics, King Abdulaziz University, Rabigh Campus, P.O. Box 344, Rabigh, 21911, Saudi Arabia
   • Habib Mâagli
3. Institute of Mathematics ‘Simion Stoilow’ of the Romanian Academy, P.O. Box 1-764, Bucharest, 014700, Romania
   • Vicenţiu D. Rădulescu
4. Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, Craiova, 200585, Romania
   • Vicenţiu D. Rădulescu

Corresponding author

Correspondence to Vicenţiu D. Rădulescu.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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