1 Introduction

As is well known Sturm-Liouville problems play an important role in many scientific problems. Second order Sturm-Liouville equations are of the form

$$ -\bigl(p(x)y^{\prime}\bigr)^{\prime}+q(x)y=\lambda w(x)y,\quad -\infty < a< b< \infty, $$

where p, q, and w are real-valued functions such that \(p^{-1}\), q, and w are locally integrable functions on a given interval and \(w>0\) for almost all x. However, this classical equation can be generalized to fractional differential equation [1] with the help of left- and right-sided Riemann-Liouville integrals of order α (\(\operatorname{Re}\alpha>0\)) as follows [2–4]:

$$\begin{aligned}& \bigl(I_{a^{+}}^{\alpha}y\bigr) (x) = \frac{1}{\Gamma(\alpha)} \int_{a}^{x}(x-s)^{\alpha-1}y(s)\,ds,\quad x>a, \\ & \bigl(I_{b^{-}}^{\alpha}y\bigr) (x)=\frac{1}{\Gamma(\alpha)} \int_{x}^{b}(s-x)^{\alpha-1}y(s)\,ds,\quad x< b, \end{aligned}$$

where Γ denotes the gamma function. Then the left-sided and right-sided Riemann-Liouville derivatives of order α are defined as

$$\begin{aligned}& \bigl(D_{a^{+}}^{\alpha}y\bigr) (x)=D^{m} \bigl(I_{a^{+}}^{m-\alpha}y\bigr) (x),\quad x>a, \\ & \bigl(D_{b^{-}}^{\alpha}y\bigr) (x)=(-D)^{m} \bigl(I_{b^{-}}^{m-\alpha}y\bigr) (x),\quad x< b, \end{aligned}$$

where \(\operatorname{Re}\alpha\in(m-1,m)\), and

$$\begin{aligned}& \bigl(^{c}D_{a^{+}}^{\alpha}y\bigr) (x)= \bigl(I_{a^{+}}^{m-\alpha}D^{m}y\bigr) (x),\quad x>a, \\ & \bigl(^{c}D_{b^{-}}^{\alpha}y\bigr) (x)= \bigl(I_{b^{-}}^{m-\alpha}(-D)^{m}y\bigr) (x),\quad x< b. \end{aligned}$$

Here, secondary ones are called the left- and right-sided Caputo derivatives of order α.

Fractional calculus is one of the most useful tools to investigate the hidden properties of dynamical systems. During the last few decades a lot of developments were done in both theoretical and the applied view points [4–14], namely, some important results from classical analysis were generalized to the fractional case and fractional calculus was successfully applied to real problems which appear in science and engineering. One of the most challenging hot topics in fractional calculus is to find real world application in physics. We recall that an one-dimensional dissipative Schrödinger-type operators together with their dilations and eigenfunction expansions was discussed in [15] and it was motivated by the problems appearing in semiconductor physics (also see [16, 17]).

It is well known that [18] a linear operator A acting on a Hilbert space H is said to be dissipative if the imaginary part of the corresponding inner product is nonnegative, that is,

$$ \operatorname{Im}(Af,f)\geq0,\quad f\in D(A), $$

where \(D(A)\) is the domain of the operator A. To investigate the spectral properties of the boundary value problems it is useful to understand the nature of the associated operator. To be more precise, we should note that if an operator associated with the boundary value problem is Hermitian (selfadjoint) then all eigenvalues of the problem are real. On the other side, an important class of nonselfadjoint operators consists of dissipative operators. For ordinary differential operator generated by an ordinary differential equation and associated boundary conditions (singular) dissipative operators have been studied in many works [16–29]. However, in fractional calculus it seems that there is no work in this field. In this paper our main aim is to construct a regular dissipative fractional operator associated with a fractional boundary value problem.

Finally, in this paper we denote by \(L_{w_{\alpha}}^{2}[a,b]\) the Hilbert space consisting of all functions y such that

$$ \int_{a}^{b}\vert y\vert^{2}w_{c}\,dx< \infty $$

with the inner product

$$ (y,z)_{L_{w_{c}}^{2}}= \int_{a}^{b}y\overline{z}w_{c}\,dx. $$

2 Boundary value problem

As in [1], in this paper we consider the following fractional Sturm-Liouville differential expression:

$$ \mathcal{L}_{\alpha,x}:=^{C}D_{b^{-}}^{\alpha} \bigl( p(x)D_{a^{+}}^{\alpha } \bigr) +q(x),\quad 0 < \alpha \leq 1, $$

on the interval \([a,b]\). Here we assume that p and q are real-valued functions having finite values at each point on \([a,b]\) and \(p(x)\neq0\). Now we shall handle the following boundary value problem:

$$\begin{aligned}& \mathcal{L}_{\alpha,x}y=\lambda w_{\alpha}(x), \end{aligned}$$
(2.1)
$$\begin{aligned}& \cos\beta I_{a^{+}}^{1-\alpha}y(a)+\sin\beta p(a)D_{a^{+}}^{\alpha }y(a)=0, \end{aligned}$$
(2.2)
$$\begin{aligned}& I_{a^{+}}^{1-\alpha}y(b)-hp(b)D_{a^{+}}^{\alpha}y(b)=0, \end{aligned}$$
(2.3)

where \(w_{\alpha}(x)\) is the real-valued function such that \(w_{\alpha }(x)>0\) on \([a,b]\), β is a real number and h is a complex number such that \(h=h_{1}+ih_{2}\) with \(h_{2}>0\).

Let \(\mathbb{L}\) be an operator with domain \(D(\mathbb{L})\), which consists of those functions y such that \({}^{c}D_{b^{-}}^{\alpha} (p(x)D_{a^{+}}^{\alpha}y ) \) is meaningful satisfying (2.2), (2.3) and

$$ \frac{1}{w_{\alpha}(x)} ( \mathcal{L}_{\alpha,x}y ) \in L_{w_{\alpha}}^{2}[a,b] $$

with the rule

$$ \mathbb{L}y=\frac{1}{w_{\alpha}(x)} ( \mathcal{L}_{\alpha,x}y ), \quad y\in D( \mathbb{L}). $$

Then

$$ \mathbb{L}y=\lambda y,\quad y\in D(\mathbb{L}) $$

coincides with the problem (2.1)-(2.3). Then we have the following.

Theorem 2.1

The operator \(\mathbb{L}\) is dissipative in \(L_{w_{\alpha}}^{2}[a,b]\).

Proof

For \(y\in D(\mathbb{L})\) we have

$$ ( \lambda-\overline{\lambda} ) w_{\alpha}(x)y\overline{y}=\mathcal{L}_{\alpha,x}y\overline{y}-\overline{\mathcal{L}_{\alpha,x}y}y $$

and using (2.2) and (2.3) we obtain

$$\begin{aligned} ( \mathbb{L}y,y ) _{L_{w_{c}}^{2}}- ( y,\mathbb{L}y ) _{L_{w_{c}}^{2}} =& I_{a^{+}}^{1-\alpha}y(b)\overline{p(b)D_{a^{+}}^{\alpha }y(b)}-p(b)D_{a^{+}}^{\alpha}y(b) \overline{I_{a^{+}}^{1-\alpha}y(b)} \\ & {}-I_{a^{+}}^{1-\alpha}y(a)\overline{p(a)D_{a^{+}}^{\alpha}y(a)}+p(a)D_{a^{+}}^{\alpha}y(a) \overline{I_{a^{+}}^{1-\alpha}y(a)} \\ =& 2i\operatorname{Im}h\bigl\vert p(b)D_{a^{+}}^{\alpha}y(b) \bigr\vert ^{2}, \end{aligned}$$

which implies that \(\mathbb{L}\) is dissipative in \(L_{w_{\alpha}}^{2}[a,b]\). □

Then we arrive at the following corollary ([18], p.176).

Corollary 2.1

Let λ be an eigenvalue of the operator \(\mathbb{L}\). Then \(\operatorname{Im}\lambda\geq0\).

For the special case of α we have additional results [19, 20, 25–29].

Corollary 2.2

For \(\alpha=1\) the boundary value problem (2.1)-(2.3) reduces to

$$\begin{aligned}& -\bigl(p(x)y^{\prime}\bigr)^{\prime}+q(x)y=\lambda w(x)y, \end{aligned}$$
(2.4)
$$\begin{aligned}& \cos\beta y(a)+\sin\beta p(a)y^{\prime}(a)=0, \end{aligned}$$
(2.5)
$$\begin{aligned}& y(b)-hp(b)y^{\prime}(b)=0. \end{aligned}$$
(2.6)

Let \(\lambda=\lambda_{0}\) be an eigenvalue of the problem (2.4)-(2.6). Then \(\operatorname{Im}\lambda_{0}>0\). Moreover, the multiplicity of \(\lambda_{0}\) is finite. The set of all eigenvalues are countable. All root vectors (eigen- and associated vectors) of the problem (2.4)-(2.6) span the Hilbert space \(L_{w}^{2}[a,b]\).

3 Eigenparameter dependent boundary value problem

The second problem is as follows:

$$\begin{aligned}& \mathcal{L}_{\alpha,x}y=\lambda w_{\alpha}(x)y, \end{aligned}$$
(3.1)
$$\begin{aligned}& \gamma_{1}I_{a^{+}}^{1-\alpha}y(a)-\gamma _{2}p(a)D_{a^{+}}^{\alpha }y(a)=\lambda\bigl( \gamma _{1}^{\prime}I_{a^{+}}^{1-\alpha}y(a)-\gamma _{2}^{\prime}p(a)D_{a^{+}}^{\alpha}y(a) \bigr) , \end{aligned}$$
(3.2)
$$\begin{aligned}& I_{a^{+}}^{1-\alpha}y(b)-hp(b)D_{a^{+}}^{\alpha}y(b)=0, \end{aligned}$$
(3.3)

where \(w_{\alpha}(x)\) is the real-valued function such that \(w_{\alpha }(x)>0\) on \([a,b]\), \(\gamma_{1}\), \(\gamma_{2}\), \(\gamma_{1}^{\prime }\), \(\gamma_{2}^{\prime}\) are real numbers such that \(\gamma:=\gamma _{1}^{\prime }\gamma_{2}-\gamma_{1}\gamma_{2}^{\prime}>0\) and h is a complex number such that \(h=h_{1}+ih_{2}\) with \(h_{2}>0\).

Let \(H=L_{w}^{2}[a,b]\oplus \mathbb{C}\) be the Hilbert space with the inner product

$$ \langle Y,Z \rangle_{H}=(y,z)_{L_{w_{c}}^{2}}+\frac{1}{\gamma}y_{1} \overline{z}_{1}, $$

where

$$ Y=\left( \textstyle\begin{array}{c} y(x) \\ y_{1}\end{array}\displaystyle \right) ,\qquad Z=\left( \textstyle\begin{array}{c} z(x) \\ z_{1}\end{array}\displaystyle \right) \in H, $$

where \(y_{1}=\gamma^{\prime}_{1}I_{a^{+}}^{1-\alpha}y(a)- \gamma^{\prime}_{2} p(a)D_{a^{+}}^{\alpha}y(a)\).

We construct the set \(D(\mathbb{L}_{\lambda})\) consisting of all functions \(Y=\binom{y(x)}{y_{1}}\) such that \(y\in L_{w}^{2}[a,b]\) satisfies the condition (3.3) with the rule

$$ \mathbb{L}_{\lambda}Y=\left( \textstyle\begin{array}{c} \frac{1}{w_{\alpha}(x)} ( \mathcal{L}_{\alpha,x}y ) \\ \gamma_{1}I_{a^{+}}^{1-\alpha}y(a)-\gamma_{2}p(a)D_{a^{+}}^{\alpha }y(a)\end{array}\displaystyle \right) . $$

Then the problem (3.1)-(3.3) coincides with the problem

$$ \mathbb{L}_{\lambda}Y=\lambda Y. $$

We have the following.

Theorem 3.1

The operator \(\mathbb{L}_{\lambda}\) is dissipative in H.

Proof

For \(Y\in D(\mathbb{L}_{\lambda})\) we obtain

$$ \langle\mathbb{L}_{\lambda}Y,Y \rangle_{H}- \langle Y,\mathbb{L}_{\lambda}Y \rangle_{H}=2i \operatorname{Im}h\bigl\vert p(b)D_{a^{+}}^{\alpha}y(b)\bigr\vert ^{2}. $$

Since \(Y\in D(\mathbb{L}_{\lambda})\) we see that \(\mathbb {L}_{\lambda}\) is dissipative in H. □

Corollary 3.1

Let λ be an eigenvalue of the operator \(\mathbb{L}_{\lambda}\). Then \(\operatorname{Im}\lambda \geq 0\). For the special case of α we have additional results [21–24].

Corollary 3.2

For \(\alpha=1\) the boundary value problem (3.1)-(3.3) reduces to

$$\begin{aligned}& -\bigl(p(x)y^{\prime}\bigr)^{\prime}+q(x)y=\lambda w(x)y, \end{aligned}$$
(3.4)
$$\begin{aligned}& \gamma_{1}y(a)-\gamma_{2}p(a)y^{\prime}(a)= \lambda\bigl( \gamma_{1}^{\prime}y(a)-\gamma _{2}^{\prime}p(a)y^{\prime}(a) \bigr) , \end{aligned}$$
(3.5)
$$\begin{aligned}& y(b)-hp(b)y^{\prime}(b)=0. \end{aligned}$$
(3.6)

Let \(\lambda=\lambda_{0}\) be an eigenvalue of the problem (3.4)-(3.6). Then \(\operatorname{Im}\lambda_{0}>0\). Moreover, the multiplicity of \(\lambda_{0}\) is finite. The set of all eigenvalues are countable. All root vectors (eigen- and associated vectors) of the problem (2.4)-(2.6) span the Hilbert space \(L_{w}^{2}[a,b]\oplus\mathbb{C}\).

4 Conclusion

It is well known that the dissipative operators arise in several real world applications and even naturally in mathematics. In this manuscript we considered new operators, namely they are both dissipative and of fractional calculus type. The generalization proposed in this manuscript will extend considerably the possibility to extract new features from the dynamics of complex systems involving non-local effects. Bearing this in mind we discussed first of all the boundary value problem (2.1)-(2.3) and we showed that the corresponding fractional operator \(\mathbb{L}\) is dissipative in \(L_{w}^{2}[a,b]\). After that we investigated the boundary value problem (3.1)-(3.3) and we proved that the corresponding operator \(\mathbb {L}_{\lambda}\) is dissipative in \(L_{w}^{2}[a,b]\oplus \mathbb{C}\).