Application to the Sturm–Liouville Problem

Abstract

We study the Sturm–Liouville eigenvalue problem with Caputo fractional derivatives and show that fractional variational principles are useful for proving existence of eigenvalues and eigenfunctions.

In 1836–1837, the French mathematicians Sturm (1803–1853) and Liouville (1809–1855) published a series of articles initiating a new subtopic of mathematical analysis—the Sturm–Liouville theory. It deals with the general linear, second-order ordinary differential equation of the form

$$\begin{aligned} \frac{d}{\mathrm{d}t}\left( p(t)\frac{\mathrm{d}y}{\mathrm{d}t}\right) +q(t)y=\lambda w(t)y, \end{aligned}$$

(6.1)

where $t\in [a,b]$, and in any particular problem functions $p(t)$, $q(t)$, and $w(t)$ are known. In addition, certain boundary conditions are attached to Eq. (6.1). For specific choices of the boundary conditions, nontrivial solutions of (6.1) exist only for particular values of the parameter $\lambda =\lambda ^{(m)}$, $m=1,2,\ldots$. Constants $\lambda ^{(m)}$ are called eigenvalues and corresponding solutions $y^{(m)}(t)$ are called eigenfunctions. For a deeper discussion of the classical Sturm–Liouville theory, we refer the reader to Gelfand and Fomin (2000), van Brunt (2004). The results of this chapter can be found in the paper (Klimek et al. 2014) and are part of the PhD thesis (Odzijewicz 2013).

Recently, many researchers focused their attention on certain generalizations of the Sturm–Liouville problem. Namely, they are interested in equations of type (6.1), however, with fractional differential operators (see, e.g., (Al-Mdallal 2009, 2010; Klimek and Agrawal 2012, 2013a, b; Liu et al. 2012; Qi and Chen 2011)). In this chapter, we develop the Sturm–Liouville theory by studying the Sturm–Liouville eigenvalue problem with Caputo fractional derivatives. We show that fractional variational principles are useful for the approximation of eigenvalues and eigenfunctions. Traditional Sturm–Liouville theory does not depend upon the calculus of variations, but stems from the theory of ordinary linear differential equations. However, the Sturm–Liouville eigenvalue problem is readily formulated as a constrained variational principle, and this formulation can be used to approximate the solutions. We emphasize that it has a special importance for the fractional Sturm–Liouville equation since fractional operators are nonlocal and it can be extremely challenging to find analytical solutions. Besides, allowing convenient approximations, many general properties of the eigenvalues can be derived using the variational principle.

6.1 Useful Lemmas

In this section, we present three lemmas that are used to prove existence of solutions for the fractional Sturm–Liouville problem.

Lemma 6.1

Let $\alpha \in (0,1)$ and function $\gamma \in C([a,b];\mathbb {R})$. If

$$\begin{aligned} \int \limits _{a}^{b}\gamma (t)\frac{d}{\mathrm{d}t}\left( {^{C}_{a}} {D}_t^\alpha [h](t)\right) \,\mathrm{d}t=0 \end{aligned}$$

for each $h\in C^{1}([a,b];\mathbb {R})$ such that $\frac{d}{\mathrm{d}t}{^{C}_{a}} {D}_t^\alpha [h]\in C([a,b];\mathbb {R})$ and boundary conditions

$$ h(a)={_{a}}{ {I}}_t^{1-\alpha }h(b)=0 $$

and

$$ {^{C}_{a}} {D}_t^\alpha [h](t)|_{t=a}={^{C}_{t}} {D}_b^\alpha [h](t)|_{t=b}=0 $$

are fulfilled, then $\gamma (t)=c_{0}+c_{1}t$, where $c_{0}, c_{1}$ are some real constants.

Proof

Let us define function $h$ as follows:

$$\begin{aligned} h(t):={_{a}} {I}_t^{1+\alpha }\left[ \gamma (\tau )-c_{0}-c_{1}\tau \right] (t) \end{aligned}$$

(6.2)

with constants fixed by the conditions

$$\begin{aligned}&{_{a}} {I}_t^{2}\left[ \gamma (\tau )-c_{0}-c_{1}\tau \right] (t)|_{t=b}=0,\end{aligned}$$

(6.3)

$$\begin{aligned}&{_{a}} {I}_t^{1}\left[ \gamma (\tau )-c_{0}-c_{1}\tau \right] (t)|_{t=b}=0. \end{aligned}$$

(6.4)

Observe that function $h$ is continuous and fulfills the boundary conditions

$$ h(a)=0\qquad {_{a}}{ {I}}_t^{1-\alpha }[h](t)|_{t=b} ={_{a}} {I}_t^{2}\left[ \gamma (\tau )-c_{0}-c_{1}\tau \right] (t)|_{t=b}=0 $$

and

$$\begin{aligned} {^{C}_{a}} {D}_t^\alpha [h](t)|_{t=a}={_{a}} {D}_t^\alpha [h](t)|_{t=a}=\frac{d}{\mathrm{d}t}{_{a}} {I}_t^{2}\left[ \gamma (\tau )-c_{0}-c_{1}\tau \right] (t)|_{t=a} \end{aligned}$$

$$ ={_{a}} {I}_t^{1}\left[ \gamma (\tau )-c_{0}-c_{1}\tau \right] (t)|_{t=a}=0, $$

$$ {^{C}_{a}} {D}_t^\alpha [h](t)|_{t=b}={_{a}} {D}_t^\alpha [h](t)|_{t=b}=\frac{d}{\mathrm{d}t}{_{a}} {I}_t^{2}\left[ \gamma (\tau )-c_{0}-c_{1}\tau \right] (t)|_{t=b} $$

$$ = {_{a}} {I}_t^{1}\left[ \gamma (\tau )-c_{0}-c_{1}\tau \right] (t)|_{t=b}=0. $$

In addition,

$$ t\mapsto h'(t) = {_{a}} {I}_t^\alpha [\gamma (\tau )-c_{0}-c_{1}\tau ](t) \in C([a,b];\mathbb {R}), $$

$$ t\mapsto \frac{d}{\mathrm{d}t}{^{C}_{a}} {D}_t^\alpha [h](t)=\gamma (t)-c_{0}-c_{1}t\in C([a,b];\mathbb {R}). $$

We also have

$$\begin{aligned} \begin{aligned} \int \limits _{a}^{b}&\left( \gamma (t)-c_{0}-c_{1}t\right) \frac{d}{\mathrm{d}t}\left( {^{C}_{a}} {D}_t^\alpha [h](t)\right) \,\mathrm{d}t\\&=\int \limits _{a}^{b}\left( -c_{0}-c_{1}t\right) \frac{d}{\mathrm{d}t}\left( {^{C}_{a}} {D}_t^\alpha [h](t)\right) \,\mathrm{d}t\\&=-c_{0}\cdot {^{C}_{a}} {D}_t^\alpha [h](t)|_{t=a}^{t=b}-c_{1}t\cdot {^{C}_{a}} {D}_t^\alpha [h](t)|_{t=a}^{t=b} +c_{1}\cdot {_{a}}{ {I}}_t^{1-\alpha }[h](t)|_{t=a}^{t=b}=0. \end{aligned} \end{aligned}$$

On the other hand,

$$ \frac{d}{\mathrm{d}t}\left( {^{C}_{a}} {D}_t^\alpha [h](t)\right) =\frac{d}{\mathrm{d}t}{^{C}_{a}} {D}_t^\alpha \left[ {_{a}} {I}_t^{1+\alpha }\left[ \gamma (\tau )-c_{0}-c_{1}\tau \right] (s)\right] (t)=\gamma (t)-c_{0}-c_{1}t $$

and

$$ 0= \int \limits _{a}^{b}\left( \gamma (t)-c_{0}-c_{1}t\right) \frac{d}{\mathrm{d}t}\left( {^{C}_{a}} {D}_t^\alpha [h](t)\right) \,\mathrm{d}t = \int \limits _{a}^{b}\left( \gamma (t)-c_{0}-c_{1}t\right) ^{2}\,\mathrm{d}t. $$

Thus function $\gamma$ is

$$ \gamma (t)=c_{0}+c_{1}t. $$

The proof is complete.

Lemma 6.2

Let $\alpha \in \left( \frac{1}{2},1\right)$, $\gamma \in C([a,b];\mathbb {R})$ and ${_{a}}D^{1-\alpha }_{t}[\gamma ]\in L^2(a,b;\mathbb {R})$. If

$$\begin{aligned} \int \limits _{a}^{b}\gamma (t)\frac{d}{\mathrm{d}t}\left( {^{C}_{a}} {D}_t^\alpha [h](t)\right) \,\mathrm{d}t=0 \end{aligned}$$

for each $h\in C^{1}([a,b];\mathbb {R})$ such that $h''\in L^{2}(a,b;\mathbb {R})$, $\frac{d}{\mathrm{d}t}{^{C}_{a}} {D}_t^\alpha [h]\in C([a,b];\mathbb {R})$ and boundary conditions

$$\begin{aligned}&h(a)={_{a}}{ {I}}_t^{1-\alpha }[h](b)=0,\end{aligned}$$

(6.5)

$$\begin{aligned}&{^{C}_{a}} {D}_t^\alpha [h](t)|_{t=a}={^{C}_{a}} {D}_t^\alpha [h](t)|_{t=b}=0 \end{aligned}$$

(6.6)

are fulfilled, then $\gamma (t)=c_{0}+c_{1}t$, where $c_{0}, c_{1}$ are some real constants.

Proof

We define function $h$ as in the proof of Lemma 6.1:

$$\begin{aligned} h(t):={_{a}} {I}_t^{1+\alpha }\left[ \gamma (\tau )-c_{0}-c_{1}\tau \right] (t) \end{aligned}$$

(6.7)

with constants fixed by the conditions (6.3) and (6.4). The proof of the lemma is analogous to that of the Lemma 6.1. In addition, for the second-order derivative we have

$$\begin{aligned} \begin{aligned} h''(t)&=\frac{d}{\mathrm{d}t}{_{a}} {I}_t^\alpha \left[ \gamma (\tau )-c_{0}-c_{1}\tau \right] (t)\\&= {_{a}}D^{1-\alpha }_{t}\left[ \gamma (\tau )-c_{0}-c_{1}\tau \right] (t)\\&={_{a}}D^{1-\alpha }_{t} [\gamma ](t)-(c_{0}+c_{1}a)\frac{(t-a)^{\alpha -1}}{\varGamma (\alpha )} -c_{1}\frac{(t-a)^{\alpha }}{\varGamma (\alpha +1)}. \end{aligned} \end{aligned}$$

Let us observe that for $\alpha >1/2$:

$$ t\mapsto \frac{(t-a)^{\alpha -1}}{\varGamma (\alpha )}\in L^{2}(a,b;\mathbb {R}), $$

$$ t\mapsto \frac{(t-a)^{\alpha }}{\varGamma (\alpha +1)}\in C([a,b];\mathbb {R})\subset L^{2}(a,b;\mathbb {R}). $$

Thus, we conclude that $h''\in L^{2}(a,b;\mathbb {R})$ and function $h$ constructed in this proof fulfills all the assumptions of Lemma 6.2. The remaining part of the proof is analogous to that for Lemma 6.1.

Lemma 6.3

• (a) Let $\alpha \in \left( \frac{1}{2},1\right)$, functions $\gamma _{j}\in C([a,b];\mathbb {R})$, $j=1,2,3$ and ${_{a}}D^{1-\alpha }_{t}[\gamma _3]\in L^2(a,b;\mathbb {R})$. If

  $$\begin{aligned} \int \limits _{a}^{b}\left( \gamma _{1}(t)h(t)+\gamma _{2}(t){^{C}_{a}} {D}_t^\alpha [h](t) +\gamma _{3}(t)\frac{d}{\mathrm{d}t}\left( {^{C}_{a}} {D}_t^\alpha [h](t)\right) \right) \,\mathrm{d}t=0 \end{aligned}$$

  (6.8)

  for each $h\in C^{1}([a,b];\mathbb {R})$, such that $h''\in L^2(a,b;\mathbb {R})$ and $\frac{d}{\mathrm{d}t}{^{C}_{a}} {D}_t^\alpha [h]\in C([a,b];\mathbb {R})$, fulfilling boundary conditions

  $$\begin{aligned} h(a)={_{a}}{ {I}}_t^{1-\alpha }[h](b)=0,\end{aligned}$$

  (6.9)
  $$\begin{aligned} {^{C}_{a}} {D}_t^\alpha [h](t)|_{t=a}={^{C}_{a}} {D}_t^\alpha [h](t)|_{t=b}=0, \end{aligned}$$

  (6.10)

  then $\gamma _{3}\in C^{1}([a,b];\mathbb {R})$.

• (b) Let $\alpha \in \left( \frac{1}{2},1\right)$ and functions $\gamma _1,\gamma _2\in C([a,b];\mathbb {R})$. If

  $$\begin{aligned} \int \limits _{a}^{b}\left( \gamma _{1}(t)h(t)+\gamma _{2}(t){^{C}_{a}} {D}_t^\alpha [h](t)\right) \,\mathrm{d}t=0 \end{aligned}$$

  (6.11)

  for each $h\in C^{1}([a,b];\mathbb {R})$, such that $h''\in L^2(a,b;\mathbb {R})$ and $\frac{d}{\mathrm{d}t}{^{C}_{a}} {D}_t^\alpha [h]\in C([a,b];\mathbb {R})$, fulfilling boundary conditions (6.9) and (6.10), then

  $$\begin{aligned} -\gamma _{1}(t)-{^{C}_{t}} {D}_b^\alpha [\gamma _{2}](t)=0. \end{aligned}$$

Proof

Observe that integral (6.8) can be rewritten as

$$ \int \limits _{a}^{b}\left( \gamma _{1}(t)h(t)+\gamma _{2}(t){^{C}_{a}} {D}_t^\alpha [h](t) +\gamma _{3}(t)\frac{d}{\mathrm{d}t}{^{C}_{a}} {D}_t^\alpha [h](t)\right) \,\mathrm{d}t $$

$$ = \int \limits _{a}^{b}\left( -\left( {_{a}} {I}^{1}_{t}\circ {_{t}} {I}^{\alpha }_{b}\right) [\gamma _1](t) -{_{a}} {I}^{1}_{t}[\gamma _2](t)+\gamma _3(t)\right) \frac{d}{\mathrm{d}t}{^{C}_{a}} {D}_t^\alpha [h](t)\,\mathrm{d}x = 0 $$

due to the fact that relations

$$ \left( {_{a}} {I}^{\alpha }_{t}\circ {_{t}} {I}^{1}_{b}\circ \frac{d}{\mathrm{d}t}{^{C}_{a}} {D}_t^\alpha \right) [h](t)=h(t) $$

and

$$ \left( {_{t}} {I}_b^{1}\circ \frac{d}{\mathrm{d}\tau }{^{C}_{a}} {D}_t^\alpha \right) [h](t)=-{^{C}_{a}} {D}_t^\alpha [h](t) $$

are valid because function $h$ fulfills boundary conditions (6.9) and (6.10). Denote

$$ \gamma (t):= -\left( {_{a}} {I}^{1}_{t}\circ {_{t}} {I}^{\alpha }_{b}\right) [\gamma _{1}](t) -{_{a}} {I}^{1}_{t}[\gamma _{2}](t)+\gamma _{3}(t). $$

It is clear that $\gamma \in C([a,b];\mathbb {R})$ and ${_{a}}D^{1-\alpha }_{t}[\gamma ] \in L^{2}(a,b;\mathbb {R})$. Thus, according to Lemma 6.2, there exist constants $c_{0}$ and $c_{1}$ such that

$$ -\left( {_{a}} {I}^{1}_{t}\circ {_{t}} {I}^{\alpha }_{b}\right) [ \gamma _{1}](t)-{_{a}} {I}^{1}_{t}[\gamma _{2}](t)+\gamma _{3}(t)=c_{0}+c_{1}t. $$

Let us note that function $\gamma _{3}$ is

$$ \gamma _{3}(t)=\left( {_{a}} {I}^{1}_{t}\circ {_{t}} {I}^{\alpha }_{b}\right) [ \gamma _{1}](t)+{_{a}} {I}^{1}_{t}[\gamma _{2}](t)+c_{0}+c_{1}t. $$

Hence its first order derivative is continuous in $[a,b]$ and $\gamma _{3}\in C^{1}([a,b];\mathbb {R})$.

The proof of part (b) is similar. We write integral (6.11) as follows:

$$\begin{aligned}&\int \limits _{a}^{b}\left( \gamma _{1}(t)h(t)+\gamma _{2}(t){^{C}_{a}} {D}_t^\alpha [h](t)\right) \,\mathrm{d}t\\&\qquad = \int \limits _{a}^{b}\left( -\left( {_{a}} {I}^{1}_{t} \circ {_{t}} {I}^{\alpha }_{b}\right) [\gamma _{1}](t) -{_{a}} {I}^{1}_{t}[\gamma _{2}](t)\right) \frac{d}{\mathrm{d}t}{^{C}_{a}} {D}_t^\alpha [h](t)\,\mathrm{d}t=0. \end{aligned}$$

The function in brackets is continuous in $[a,b]$,

$$\begin{aligned}&{_{a}}D^{1-\alpha }_{t} \left[ -\left( {_{a}} {I}^{1}_{t} \circ {_{t}} {I}^{\alpha }_{b}\right) [\gamma _{1}](\tau ) -{_{a}} {I}^{1}_{t}[\gamma _{2}](\tau )\right] (t)\\&\qquad =-\left( {_{a}} {I}^{\alpha }_{t}\circ {_{t}} {I}^{\alpha }_{b}\right) [ \gamma _{1}](t)-{_{a}} {I}^{\alpha }_{t}[\gamma _{2}](t), \end{aligned}$$

and

$$\begin{aligned} -\left( {_{a}} {I}^{\alpha }_{t}\circ {_{t}} {I}^{\alpha }_{b}\right) [ \gamma _{1}]-{_{a}} {I}^{\alpha }_{t}[\gamma _{2}]\in C([a,b];{\mathbb {R}}) \subset {L^2({a,b};\mathbb {R})}, \end{aligned}$$

so we again can apply Lemma 6.2 and obtain that there exist constants $c_{0}$ and $c_{1}$ such that

$$ \left( {_{a}} {I}^{1}_{t}\circ {_{t}} {I}^{\alpha }_{b}\right) [\gamma _{1}](t) +{_{a}} {I}^{1}_{t}[\gamma _{2}](t)=c_{0}+c_{1}t. $$

Thus, functions $\gamma _{1,2}$ fulfill equation ${^{C}_{t}} {D}_b^\alpha [\gamma _{2}](t)+\gamma _{1}(t)=0$.

6.2 The Fractional Sturm–Liouville Problem

The crucial idea in the proof of main result of this chapter (Theorem 6.5) is to apply direct variational methods to the fractional Sturm–Liouville equation. Starting from the fractional Sturm–Liouville equation, the approach is to find an associated functional and to use this to find approximations to the minimizers, which are necessarily solutions to the original equation. In the case of the fractional Sturm–Liouville equation, an associated variational problem is the fractional isoperimetric problem, which is defined in the following way:

$$\begin{aligned} \min \mathcal {I}(y)=\int \limits _a^b F\left( y(t),{^{C}_{a}} {D}_t^\alpha [y](t),t\right) \,\mathrm{d}t, \end{aligned}$$

(6.12)

subject to the boundary conditions

$$\begin{aligned} y(a)=y_a,\quad y(b)=y_b \end{aligned}$$

(6.13)

and the isoperimetric constraint

$$\begin{aligned} \mathcal {J}(y)=\int \limits _a^b G\left( y(t),{^{C}_{a}} {D}_t^\alpha [y](t),t\right) \,\mathrm{d}t=\xi , \end{aligned}$$

(6.14)

where $\xi \in \mathbb {R}$ is given, and

$$\begin{aligned} \begin{array}{rcl}F :[a,b]\times \mathbb {R}^2 &{}\longrightarrow &{}\mathbb {R}\\ (y,u,t)&{} \longmapsto &{}F(y,u,t), \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{rcl}G :[a,b]\times \mathbb {R}^2 &{}\longrightarrow &{}\mathbb {R}\\ (y,u,t)&{} \longmapsto &{}G(y,u,t) \end{array} \end{aligned}$$

are functions of class $C^1$, such that $\frac{\partial F}{\partial u}$, $\frac{\partial G}{\partial u}$ have continuous ${_{t}} {D}_b^\alpha$ derivatives.

Theorem 6.4

(cf. Theorem 3.3 (Almeida and Torres 2011)) If $\bar{y}\in C[a,b]$ with ${^{C}_{a}} {D}_t^\alpha [\bar{y}]\in C([a,b];\mathbb {R})$ is a minimizer for problem (6.12)–(6.14), then there exists a real constant $\lambda$ such that, for $H=F+\lambda G$, the Euler–Lagrange equation

$$\begin{aligned} \frac{\partial H}{\partial y}(y(t),{^{C}_{a}} {D}_t^\alpha [y](t),t) +{_{t}} {D}_b^\alpha \left[ \frac{\partial H}{\partial u}(y(t),{^{C}_{a}} {D}_t^\alpha [y](t),t)\right] =0 \end{aligned}$$

(6.15)

holds for $\bar{y}$, provided $\bar{y}$ is not an Euler–Lagrange extremal for $G$, that is,

$$\begin{aligned} \frac{\partial G}{\partial y}(\bar{y}(t),{^{C}_{a}} {D}_t^\alpha [\bar{y}](t),t) +{_{t}} {D}_b^\alpha \left[ \frac{\partial G}{\partial u}(\bar{y}(t),{^{C}_{a}} {D}_t^\alpha [\bar{y}](t),t)\right] \ne 0. \end{aligned}$$

6.2.1 Existence of Discrete Spectrum

We show that, similarly to the classical case, for the fractional Sturm–Liouville problem there exists an infinite monotonic increasing sequence of eigenvalues. Moreover, apart from multiplicative factors to each eigenvalue, there corresponds precisely one eigenfunction and eigenfunctions form an orthogonal set of solutions.

We shall use the following assumptions.

(H1) Let $\frac{1}{2}<\alpha <1$ and $p,q,w_{\alpha }$ be given functions such that: $p$ is of $C^1$ class and $p(t)>0$; $q,w_{\alpha }$ are continuous, $w_{\alpha }(t)>0$ and $(\sqrt{w_{\alpha }})'$ is Hölderian of order $\beta \le \alpha -\frac{1}{2}$. Consider the fractional differential equation

$$\begin{aligned} {^{C}_{t}} {D}_b^\alpha \left[ p(\tau ){^{C}_a} {D}_{\tau }^{\alpha }[y](\tau )\right] (t)+q(t)y(t) = \lambda w_{\alpha }(t)y(t), \end{aligned}$$

(6.16)

that will be called the fractional Sturm–Liouville equation, subject to the boundary conditions

$$\begin{aligned} y(a)=y(b)=0. \end{aligned}$$

(6.17)

Theorem 6.5

Under assumptions (H1), the fractional Sturm–Liouville problem (FSLP) (6.16) and (6.17) has an infinite increasing sequence of eigenvalues $\lambda ^{(1)}, \lambda ^{(2)},\ldots$, and to each eigenvalue $\lambda ^{(n)}$ there corresponds an eigenfunction $y^{(n)}$ which is unique up to a constant factor. Furthermore, eigenfunctions $y^{(n)}$ form an orthogonal set of solutions.

Proof

The proof is similar in spirit to Gelfand and Fomin (2000) and will be divided into 6 steps. As in Gelfand and Fomin (2000), we shall derive a method for approximating both eigenvalues and eigenfunctions.

Step 1. We shall consider the problem of minimizing the functional

$$\begin{aligned} \mathcal {I}(y)= {\int \limits _{a}^{b}}\left[ p(t) ({^{C}_{a}} {D}_t^\alpha [y](t))^{2}+q(t)y^{2}(t)\right] \,\mathrm{d}t \end{aligned}$$

(6.18)

subject to an isoperimetric constraint

$$\begin{aligned} \mathcal {J}(y)= \int \limits _{a}^{b}w_{\alpha }(t)y^{2}(t)\,\mathrm{d}t=1 \end{aligned}$$

(6.19)

and boundary conditions (6.17). First, let us point out that functional (6.18) is bounded from below. Indeed, as $p(t)>0$ we have

$$\begin{aligned} \mathcal {I}(y)&= \int \limits _{a}^{b}\left[ p(t) ({^{C}_{a}} {D}_t^\alpha [y](t))^{2}+q(t)y^{2}(t)\right] \,\mathrm{d}t\\&\ge \min \limits _{t\in [a,b]}\frac{q(t)}{w_{\alpha }(t)}\cdot \int \limits _{a}^{b}w_{\alpha }(t)y^{2}(t)\,\mathrm{d}t =\min \limits _{t\in [a,b]}\frac{q(t)}{w_{\alpha }(t)}=:M_0>-\infty . \end{aligned}$$

From now on, for simplicity, we assume that $a=0$ and $b=\pi$. According to the Ritz method, we approximate solution of (6.17)–(6.19) using the following trigonometric function with coefficient depending on $w_{\alpha }$:

$$\begin{aligned} y_{m}(t)= \frac{1}{\sqrt{w_{\alpha }}}\sum _{k=1}^{m} \beta _{k}\sin (kt). \end{aligned}$$

(6.20)

Observe that $y_{m}(0)=y_{m}(\pi )=0$. Substituting (6.20) into (6.18) and (6.19) we obtain the problem of minimizing the function

$$\begin{aligned}&I(\beta _{1},\ldots ,\beta _{m})= I([\beta ])\nonumber \\&\quad = \sum _{k,j=1}^{m}\beta _{k}\beta _{j} \times \int \limits _{0}^{\pi }\left[ p(t) \left( {^{C}_{0}} {D}_t^{\alpha }\left[ \frac{\sin (k\tau )}{\sqrt{w_{\alpha }}}\right] (t) \cdot {^{C}_{0}} {D}_t^{\alpha }\left[ \frac{\sin (j\tau )}{\sqrt{w_{\alpha }}}\right] (t)\right) \right. \nonumber \\&\qquad \left. + \frac{q(t)}{w_{\alpha }(t)}\sin (kt)\sin (jt)\right] \,\mathrm{d}t \end{aligned}$$

(6.21)

subject to the condition

$$\begin{aligned} J(\beta _{1},\ldots ,\beta _{m})= J([\beta ])=\frac{\pi }{2}\sum _{k=1}^{m}(\beta _{k})^{2}=1. \end{aligned}$$

(6.22)

Since $I([\beta ])$ is continuous and the set given by (6.22) is compact, function $I([\beta ])$ attains minimum, denoted by $\lambda _m^{(1)}$, at some point $[\beta ^{(1)}]=(\beta _{1}^{(1)},\ldots ,\beta _{m}^{(1)})$. If this procedure is carried out for $m=1,2,\ldots$, we obtain a sequence of numbers $\lambda _{1}^{(1)},\lambda _2^{(1)},\ldots$. Because $\lambda _{m+1}^{(1)}\le \lambda _m^{(1)}$ and $\mathcal {I}(y)$ is bounded from below, we can find the limit

$$\begin{aligned} \lim \limits _{m\rightarrow \infty }\lambda _m^{(1)}=\lambda ^{(1)}. \end{aligned}$$

Step 2. Let

$$\begin{aligned} y_{m}^{(1)}(t)= \frac{1}{\sqrt{w_{\alpha }}}\sum _{k=1}^{m} \beta _{k}^{(1)}\sin (kt) \end{aligned}$$

denote the linear combination (6.20) achieving the minimum $\lambda _m^{(1)}$. We shall prove that sequence $(y_{m}^{(1)})_{m\in \mathbb {N}}$ contains a uniformly convergent subsequence. From now on, for simplicity, we will write $y_m$ instead of $y_m^{(1)}$. Recall that

$$\begin{aligned} \lambda _m^{(1)}=\int \limits _{0}^{\pi } \left[ p(t)\left( {^{C}_{0}} {D}_t^{\alpha }[y_m](t)\right) ^2 +q(t)y_m^2(t)\right] \,\mathrm{d}t \end{aligned}$$

is convergent, so it must be bounded, i.e., there exists a constant $M>0$ such that

$$\begin{aligned} \int \limits _{0}^{\pi } \left[ p(t)\left( {^{C}_{0}} {D}_t^{\alpha }[y_m](t)\right) ^2 +q(t)y_m^2(t)\right] \,\mathrm{d}t\le M, \quad m\in \mathbb {N}. \end{aligned}$$

Therefore, for all $m\in \mathbb {N}$ it hold the following:

$$\begin{aligned}&\int \limits _0^{\pi } p(t)\left( {^{C}_{0}} {D}_t^{\alpha }[y_m](t)\right) ^2\,\mathrm{d}t \le M+\left| \int \limits _0^{\pi } q(t)y_m^2(t)\,\mathrm{d}t\right| \\&\quad \le M+\max \limits _{t\in [0,\pi ]}\left| \frac{q(t)}{w_{\alpha }(t)}\right| \int \limits _0^{\pi } w_{\alpha }(t)y_m^2(t)\,\mathrm{d}t=M+\max \limits _{t\in [0,\pi ]}\left| \frac{q(t)}{w_{\alpha }(t)}\right| =: M_1. \end{aligned}$$

Moreover, since $p(t)>0$, one has

$$\begin{aligned} \min \limits _{t\in [0,\pi ]}p(t)\int \limits _0^{\pi } \left( {^{C}_{0}} {D}_t^{\alpha }[y_m](t)\right) ^2\,\mathrm{d}t \le \int \limits _0^{\pi } p(t)\left( {^{C}_{0}} {D}_t^{\alpha }[y_m](t)\right) ^2\,\mathrm{d}t\le M_1, \end{aligned}$$

and hence

$$\begin{aligned} \int \limits _0^{\pi } \left( {^{C}_{0}} {D}_t^{\alpha }[y_m](t)\right) ^2\,\mathrm{d}t \le \frac{M_1}{\min \limits _{t\in [0,\pi ]}p(t)}=: M_2. \end{aligned}$$

(6.23)

Using (4.67) and (6.23), condition $y_m(0)=0$ and Schwartz’s inequality, one has

$$\begin{aligned} \begin{aligned} \left| y_m(t)\right| ^2&=\left| \left( {_{0}} {I}_t^{\alpha } \circ {^{C}_{0}} {D}_t^{\alpha }\right) [y_m](t)\right| ^2 =\frac{1}{\varGamma (\alpha )^2}\left| \int \limits _0^{t} (t-\tau )^{\alpha -1}{^{C}_{0}} {D}_{\tau }^{\alpha }y_m(\tau )\,\mathrm{d}\tau \right| ^2\\&\le \frac{1}{\varGamma (\alpha )^2}\left( \int \limits _0^\pi \left| {^{C}_{0}} {D}_{\tau }^{\alpha }[y_m](\tau )\right| ^2\,\mathrm{d}\tau \right) \left( \int \limits _0^t (t-\tau )^{2(\alpha -1)}\,\mathrm{d}\tau \right) \\&\le \frac{1}{\varGamma (\alpha )^2}M_2\int \limits _0^t (t-\tau )^{2(\alpha -1)}\,\mathrm{d}\tau <\frac{1}{\varGamma (\alpha )}M_2\frac{1}{2\alpha -1}\pi ^{2\alpha -1}, \end{aligned} \end{aligned}$$

so that $(y_m)_{m\in \mathbb {N}}$ is uniformly bounded. Now, using Schwartz’s inequality, Eq. (6.23) and the fact that the following inequality holds,

$$\begin{aligned} \forall x_1\ge x_2\ge 0,~(x_1-x_2)^2\le x_1^2-x_2^2, \end{aligned}$$

we have for any $0< t_1< t_2\le \pi$ that

$$\begin{aligned} \begin{aligned}&\left| y_m(t_2)-y_m(t_1)\right| =\left| \left( {_{0}} {I}_t^{\alpha } \circ {^{C}_{0}} {D}_t^{\alpha }\right) [y_m](t_2) -\left( {_{0}} {I}_t^{\alpha }\circ {^{C}_{0}} {D}_t^{\alpha }\right) [y_m](t_1)\right| \\&=\frac{1}{\varGamma (\alpha )}\left| \int \limits _0^{t_2}(t_2-\tau )^{\alpha -1}{^{C}_{0}} {D}_t^{\alpha }[y_m](\tau )\,\mathrm{d}\tau -\int \limits _0^{t_1}(t_1-\tau )^{\alpha -1}{^{C}_{0}} {D}_t^{\alpha }[y_m](\tau )\,\mathrm{d}\tau \right| \\&=\frac{1}{\varGamma (\alpha )}\left| \int \limits _{t_1}^{t_2}(t_2-\tau )^{\alpha -1}{^{C}_{0}} {D}_t^{\alpha }[y_m](\tau )\,\mathrm{d}\tau \right. \\&\qquad \qquad \qquad \left. -\int \limits _0^{t_1}\left( (t_2-\tau )^{\alpha -1}-(t_1-\tau )^{\alpha -1}\right) {^{C}_{0}} {D}_t^{\alpha }[y_m](\tau )\,\mathrm{d}\tau \right| \\&\le \frac{1}{\varGamma (\alpha )}\left[ \left( \int \limits _{t_1}^{t_2} (t_2-\tau )^{2(\alpha -1)}\,\mathrm{d}\tau \right) ^{\frac{1}{2}}\left( \int \limits _{t_1}^{t_2}\left[ \left( {^{C}_{0}} {D}_t^{\alpha }[y_m](\tau )\right) ^2\right] \,\mathrm{d}\tau \right) ^{\frac{1}{2}}\right. \\&\qquad \qquad \qquad \left. +\left( \int \limits _0^{t_1}\left( (t_1-\tau )^{\alpha -1}-(t_2-\tau )^{\alpha -1}\right) ^2 \,\mathrm{d}\tau \right) ^{\frac{1}{2}}\right. \\&\qquad \qquad \qquad \times \left. \left( \int \limits _{0}^{t_1}\left[ \left( {^{C}_{0}} {D}_t^{\alpha }[y_m](\tau )\right) ^2\right] \,\mathrm{d}\tau \right) ^{\frac{1}{2}}\right] \\&\le \frac{\sqrt{M_2}}{\varGamma (\alpha )}\left[ \left( \int \limits _{t_1}^{t_2}(t_2-\tau )^{2(\alpha -1)}\,\mathrm{d}\tau \right) ^{\frac{1}{2}}\right. \\&\qquad \qquad \qquad \left. +\left( \int \limits _0^{t_1}\left( (t_1-\tau )^{2(\alpha -1)}-(t_2-\tau )^{2(\alpha -1)}\right) \,\mathrm{d}\tau \right) ^{\frac{1}{2}}\right] \\&=\frac{\sqrt{M_2}}{\varGamma (\alpha )\sqrt{2\alpha -1}}\left[ (t_2-t_1)^{\alpha -\frac{1}{2}}+\left[ (t_2-t_1)^{2\alpha -1} -t_2^{2\alpha -1}+t_1^{2\alpha -1}\right] ^{\frac{1}{2}}\right] \\&\le \frac{2\sqrt{M_2}}{\varGamma (\alpha )\sqrt{2\alpha -1}}(t_2-t_1)^{\alpha -\frac{1}{2}}. \end{aligned} \end{aligned}$$

Therefore, by Ascoli’s theorem, there exists a uniformly convergent subsequence $(y_{m_n})_{n\in \mathbb {N}}$ of sequence $(y_m)_{m\in \mathbb {N}}$. It means that we can find $y^{(1)}\in C([a,b];\mathbb {R})$ such that

$$\begin{aligned} y^{(1)}=\lim \limits _{n\rightarrow \infty }y_{m_n}. \end{aligned}$$

Step 3. Observe that by the Lagrange multiplier rule at $[\beta ]=[\beta ^{(1)}]$ we have

$$\begin{aligned} 0=\frac{\partial }{\partial \beta _{j}}\left[ I([\beta ]) -\lambda ^{(1)}_{m}J([\beta ])\right] |_{[\beta ]=[\beta ^{(1)}]},\quad j=1,\ldots ,m. \end{aligned}$$

Multiplying equations by an arbitrary constant $C^{j}$ and summing from 1 to $m$ we obtain

$$\begin{aligned} 0=\sum _{j=1}^{m}C^{j}\frac{\partial }{\partial \beta _{j}}\left[ I([\beta ]) -\lambda ^{(1)}_{m}J([\beta ])\right] |_{[\beta ]=[\beta ^{(1)}]}. \end{aligned}$$

(6.24)

Introducing

$$\begin{aligned} h_{m}(x)=\frac{1}{\sqrt{w_{\alpha }}}\sum _{j=1}^{m} C^{j}\sin (jt) \end{aligned}$$

we can rewrite (6.24) in the form

$$\begin{aligned} 0=\int \limits _{0}^{\pi }\left[ p(t){^{C}_{0}} {D}_t^{\alpha }[y_{m}](t){^{C}_{0}} {D}_t^{\alpha }[h_{m}](t) +[q(t)-\lambda ^{(1)}_{m}w_{\alpha }(t)]y_{m}(t)h_{m}(t)\right] \,\mathrm{d}t. \end{aligned}$$

(6.25)

Using the differentiation properties and formula ${^{C}_{0}} {D}_t^{\alpha }[y_{m}]=\frac{d}{\mathrm{d}t}{_{0}} {I}_t^{1-\alpha }[y_{m}]$ we write (6.25) as

$$\begin{aligned} 0&=\int \limits _{0}^{\pi }\left[ -p'(t){_{0}} {I}_t^{1-\alpha }[y_{m}](t){^{C}_{0}} {D}_t^{\alpha }[h_{m}](t) -p(t){_{0}} {I}_t^{1-\alpha }[y_{m}](t)\frac{d}{\mathrm{d}t}{^{C}_{0}} {D}_t^{\alpha }[h_{m}](t)\right] \,\mathrm{d}t \nonumber \\&\quad +p(t){_{0}} {I}_t^{1-\alpha }[y_{m}](t){^{C}_{0}} {D}_t^{\alpha }[h_{m}](t)|_{t=0}^{t=\pi }\nonumber \\&\quad +\int \limits _{0}^{\pi }[q(t)-\lambda ^{(1)}_{m}w_{\alpha }(t)]y_{m}(t)h_{m}(t)\,\mathrm{d}t:=I_{m}. \end{aligned}$$

(6.26)

By Lemma A.2 (with $w=1/\sqrt{w_{\alpha }}$) and Lemma A.3 (Appendix), for function $h$ fulfilling assumptions of Lemma 6.2, we shall obtain in the limit (at least for the convergent subsequence $(y_{m_n})_{n\in \mathbb {N}}$) the relation

$$\begin{aligned} 0&=\int \limits _{0}^{\pi }\left[ -p'(t){_{0}} {I}_t^{1-\alpha }[y^{(1)}](t)\;{^{C}_{0}} {D}_t^{\alpha }[h](t) -p(t){_{0}} {I}_t^{1-\alpha }[y^{(1)}](t)\frac{d}{\mathrm{d}t}{^{C}_{0}} {D}_t^{\alpha }[h](t)\right] \,\mathrm{d}t \nonumber \\&\quad + p(t){_{0}} {I}_t^{1-\alpha }[y^{(1)}](t)\;{^{C}_{0}} {D}_t^{\alpha }[h](t)|_{t=0}^{t=\pi } +\int \limits _{0}^{\pi }[q(t)-\lambda ^{(1)}w_{\alpha }(t)]y^{(1)}(t)h(t)\,\mathrm{d}t:=I. \end{aligned}$$

(6.27)

Let us check the convergence of integrals (6.26) explicitly:

$$\begin{aligned} \begin{aligned}&\left| I_{m}-I\right| \le \int \limits _{0}^{\pi }\left| -p'(t){_{0}} {I}_t^{1-\alpha }[y_{m}](t){^{C}_{0}} {D}_t^{\alpha }[h_{m}](t) +p'(t){_{0}} {I}_t^{1-\alpha }[y^{(1)}](t)\;{^{C}_{0}} {D}_t^{\alpha }[h](t)\right| \,\mathrm{d}t\\&\quad + \int \limits _{0}^{\pi }\left| p(t){_{0}} {I}_t^{1-\alpha }[y_{m}](t)\frac{d}{\mathrm{d}t}{^{C}_{0}} {D}_t^{\alpha }[h_{m}](t) -p(t){_{0}} {I}_t^{1-\alpha }[y^{(1)}](t)\;\frac{d}{\mathrm{d}t}{^{C}_{0}} {D}_t^{\alpha }[h](t)\right| \,\mathrm{d}t\\&\quad +\left| p(t){_{0}} {I}_t^{1-\alpha }[y_{m}](t){^{C}_{0}} {D}_t^{\alpha }[h_{m}](t)|_{x=0} -p(t){_{0}} {I}_t^{1-\alpha }[y^{(1)}](t)\;{^{C}_{0}} {D}_t^{\alpha }[h](t)|_{t=0}\right| \\&\quad +\left| p(t){_{0}} {I}_t^{1-\alpha }[y_{m}](t){^{C}_{0}} {D}_t^{\alpha }[h_{m}](t)|_{t=\pi } -p(t){_{0}} {I}_t^{1-\alpha }[y^{(1)}](t)\;{^{C}_{0}} {D}_t^{\alpha }[h](t)|_{t=\pi }\right| \\&\quad + \int \limits _{0}^{\pi }\left| [q(t)-\lambda ^{(1)}_{m}w_{\alpha }(t)]y_{m}(t)h_{m}(t) -\left[ q(t)-\lambda ^{(1)}w_{\alpha }(t)\right] y^{(1)}(t)\; h(t)\right| \,\mathrm{d}t. \end{aligned} \end{aligned}$$

(6.28)

For the first integral we get

$$\begin{aligned}&\int \limits _{0}^{\pi }\left| -p'(t){_{0}} {I}_t^{1-\alpha }[y_{m}](t){^{C}_{0}} {D}_t^{\alpha }[h_{m}](t) +p'(t){_{0}} {I}_t^{1-\alpha }[y^{(1)}](t)\;{^{C}_{0}} {D}_t^{\alpha }[h](t)\right| \,\mathrm{d}t\\&\quad \le ||p'||\cdot \left[ ||{^{C}_{0}} {D}_t^{\alpha }[h]||\cdot ||{_{0}} {I}_t^{1-\alpha }[y_{m} -y^{(1)}]||_{L^{1}}\right. \\&\qquad \qquad \qquad \quad \left. +M_3K_{1-\alpha }\sqrt{\pi }||{^{C}_{0}} {D}_t^{\alpha }[h_{m}-h]||_{L^{2}}\right] , \end{aligned}$$

where constant $M_3=\sup \nolimits _{m\in \mathbb {N}} ||y_{m}||$ and $||\cdot ||$ denotes the supremum norm in the $C([0,\pi ];\mathbb {R})$ space. Now, we estimate the second integral

$$\begin{aligned}&\int \limits _{0}^{\pi }\left| p(t){_{0}} {I}_t^{1-\alpha }[y_{m}](t)\frac{d}{\mathrm{d}t}{^{C}_{0}} {D}_t^{\alpha }[h_{m}](t) -p(t)I^{1-\alpha }_{0+}y^{(1)}(t)\;\frac{d}{\mathrm{d}t}{^{C}_{0}} {D}_t^{\alpha }[h](t)\right| \,\mathrm{d}t\\&\quad \le ||p||\cdot \left[ ||\frac{d}{\mathrm{d}t}{^{C}_{0}} {D}_t^{\alpha }[h]||_{L^{2}} \cdot ||{_{0}} {I}_t^{1-\alpha }[y_{m}-y^{(1)}]||_{L^{2}}\right. \\&\qquad \qquad \qquad \quad \left. +M_3K_{1-\alpha } \cdot ||\frac{d}{\mathrm{d}t}{^{C}_{0}} {D}_t^{\alpha }[h_{m}-h]||_{L^{1}}\right] . \end{aligned}$$

For the next two terms we have

$$\begin{aligned} {_{0}} {I}_t^{1-\alpha }[y_{m}](0)\xrightarrow [m\rightarrow \infty ]{\ } {_{0}} {I}_t^{1-\alpha }[y](0), \quad {_{0}} {I}_t^{1-\alpha }[y_{m}](\pi ) \xrightarrow [m\rightarrow \infty ]{\ } {_{0}} {I}_t^{1-\alpha }[y](\pi ) \end{aligned}$$

(6.29)

resulting from the convergence of sequence $y_{m} \xrightarrow [m\rightarrow \infty ]{C} y$. For the sequence $h_{m}=g_{m}/\sqrt{w_{\alpha }}$, we infer from Lemma A.3 that $h'_{m} \xrightarrow [m\rightarrow \infty ]{C} h'$. Hence, also

$$ {^{C}_{0}} {D}_t^{\alpha }[h_{m}]\xrightarrow [m\rightarrow \infty ]{C}{^{C}_{0}} {D}_t^{\alpha }[h], \quad {_{0}} {I}_t^{1-\alpha }[h'_{m}]\xrightarrow [m\rightarrow \infty ]{C}{_{0}} {I}_t^{1-\alpha }[h'] $$

and at points $t=0,\pi$ we obtain

$$\begin{aligned} {^{C}_{0}} {D}_t^{\alpha }[h_{m}](0) \xrightarrow [m\rightarrow \infty ]{\ } {^{c}_{0}} {D}_t^{\alpha }[h](0), \quad {^{C}_{0}} {D}_t^{\alpha }[h_{m}](\pi ) \xrightarrow [m\rightarrow \infty ]{\ } {^{c}_{0}} {D}_t^{\alpha }[h](\pi ). \end{aligned}$$

(6.30)

The above pointwise convergence (6.29) and (6.30) imply that

$$\begin{aligned}&\lim _{m\longrightarrow \infty } \left| p(t){_{0}} {I}_t^{1-\alpha }[y_{m}](t){^{C}_{0}} {D}_t^{\alpha }[h_{m}](t)|_{t=0} -p(t){_{0}} {I}_t^{1-\alpha }[y^{(1)}](t)\;{^{C}_{0}} {D}_t^{\alpha }[h](t)|_{t=0}\right| =0,\\&\lim _{m\longrightarrow \infty }\left| p(t){_{0}} {I}_t^{1-\alpha }[y_{m}](t) {^{C}_{0}} {D}_t^{\alpha }[h_{m}](t)|_{t=\pi }-p(t){_{0}} {I}_t^{1-\alpha }[y^{(1)}](t)\; {^{C}_{0}} {D}_t^{\alpha }[h](t)|_{t=\pi }\right| =0. \end{aligned}$$

Finally, for the last term in estimation (6.28) we get

$$\begin{aligned} \begin{aligned}&\int \limits _{0}^{\pi }|[q(t)-\lambda ^{(1)}_{m}w_{\alpha }(t)]y_{m}(t)h_{m}(t) -[q(t)-\lambda ^{(1)}w_{\alpha }(t)]y^{(1)}(t)\; h(t)|\,\mathrm{d}t\\&\quad \le \int \limits _{0}^{\pi }|q(t)(y_{m}(t)h_{m}(t)-y^{(1)}(t)\; h(t))|\,\mathrm{d}t\\&\qquad +\int _{0}^{\pi }|w_{\alpha }(t)(\lambda ^{(1)}_{m}y_{m}(t)h_{m}(t)-\lambda ^{(1)}y^{(1)}(t)\; h(t))|\,\mathrm{d}t\\&\quad \le \pi \cdot ||q||\cdot \left[ M_3 \cdot ||h_{m}-h|| + ||h||\cdot ||y_{m}-y^{(1)}||\right] \\&\qquad +\pi \cdot ||w_{\alpha }||\cdot \left[ \varLambda \left( M_3 \cdot ||h_{m}-h|| + ||h||\cdot ||y_{m}-y^{(1)}||\right) \right. \\&\qquad \left. + ||y^{(1)}h||\cdot |\lambda ^{(1)}_{m}-\lambda ^{(1)}|\right] , \end{aligned} \end{aligned}$$

where constants $M_3=\sup \nolimits _{m\in \mathbb {N}} ||y_{m}||$ and $\varLambda =\sup \nolimits _{m\in \mathbb {N}} |\lambda ^{(1)}_{m}|$. We conclude that

$$ 0= \lim _{m\longrightarrow \infty }I_{m}=I $$

and (6.27) is fulfilled for function $y^{(1)}$ being the limit of subsequence $(y_{m_n})$ of the sequence $\left( y_{m}\right) _{m\in \mathbb {N}}$.

Step 4. Let us denote in relation (6.27):

$$\begin{aligned}&\gamma _1 (t):=(q(t)-\lambda ^{(1)}w_{\alpha }(t))y^{(1)}(t),\\&\gamma _2 (t):= -p'(t){_{0}}I^{1-\alpha }_{t} [y^{(1)}](t),\\&\gamma _3 (t):= -p(t){_{0}}I^{1-\alpha }_{t}[y^{(1)}](t). \end{aligned}$$

We observe that $\gamma _{j} \in C([0,\pi ];\mathbb {R}), \;j=1,2,3$ and ${_{0}}D^{1-\alpha }_{t}[\gamma _{3}]\in L^{2}(0,\pi ;\mathbb {R})$ because

$$ {_0}D^{1-\alpha }_{t}[\gamma _{3}]={_{0}}D^{1-\alpha }_{t}\left[ p \cdot {_{0}}I^{1-\alpha }_{t}[y^{(1)}]\right] = {_{0}}I^{\alpha }_{t} \left[ \frac{d}{\mathrm{d}t}\left( p \cdot {_{0}}I^{1-\alpha }_{t}[y^{(1)}]\right) \right] $$

$$ = {_{0}}I^{\alpha }_{t} \left[ p' \cdot {_{0}}I^{1-\alpha }_{t}[y^{(1)}] + p\cdot \; {^{C}_{0}}D^{\alpha }_{t}[y^{(1)}]\right] . $$

Both parts of the above function belong to the $L^{2}(0,\pi ;\mathbb {R})$ space.

Assuming that function $h$ in (6.27) is an arbitrary function fulfilling assumptions of Lemma 6.3 and applying Lemma 6.3 part (a), we conclude that $\gamma _3 =-p\cdot {_{0}}I^{1-\alpha }_{t}[y^{(1)}]\in C^{1}([0,\pi ];\mathbb {R})$. From this fact it follows that $p\cdot \frac{d}{\mathrm{d}t} {_{0}}I^{\alpha }_{t}[y^{(1)}]\in C([0,\pi ];\mathbb {R})$ and integral (6.27) can be rewritten as

$$\begin{aligned} \int \limits _{0}^{\pi }\left[ p(t){^{C}_{0}} {D}_t^{\alpha }[y^{(1)}](t){^{C}_{0}} {D}_t^{\alpha }[h](t) +(q(t)-\lambda ^{(1)}w_{\alpha }(t))y^{(1)}(t)h(t)\right] \,\mathrm{d}t = 0. \end{aligned}$$

Now, we apply Lemma 6.3 part (b) defining

$$\begin{aligned}&\bar{\gamma }_{1} (t) : = \gamma _1 (t)=(q(t)-\lambda ^{(1)}w_{\alpha })y^{(1)}(t),\\&\bar{\gamma }_{2} (t):= p(t)\frac{d}{\mathrm{d}t}{_{0}}I^{1-\alpha }_{t}[y^{(1)}](t). \end{aligned}$$

This time $\bar{\gamma }_{1},\bar{\gamma }_2\in C([0,\pi ];\mathbb {R})$ and from Lemma 6.3 part (b) it follows that

$$\begin{aligned} {^{C}_{t}} {D}_{\pi }^{\alpha }\left[ p(\tau ){^{c}_{0}} {D}_{\tau }^{\alpha }[y^{(1)}](\tau )\right] (t) +q(t)y^{(1)}(t)= \lambda ^{(1)}w_{\alpha }(t)y^{(1)}(t). \end{aligned}$$

By construction, this solution fulfills the Dirichlet boundary conditions

$$\begin{aligned} y^{(1)}(0)=y^{(1)}(\pi )=0 \end{aligned}$$

and is nontrivial because

$$\begin{aligned} \mathcal {J}(y^{(1)})=\int \limits _{0}^{\pi }w_{\alpha }(t)\left( y^{(1)}(t)\right) ^{2}\,\mathrm{d}t=1. \end{aligned}$$

In addition, we also have for the solution

$$ {_{0}}D^{\alpha }_{t}[y^{(1)}] ={^{C}_{0}}D^{\alpha }_{t}[y^{(1)}]\in C([0,\pi ];\mathbb {R}). $$

Let us observe that from the Dirichlet boundary conditions it follows that $y^{(1)}$ also solves the FSLP (6.16) and (6.17) in $[0,\pi ]$.

Step 5. Now, let us restore the superscript on $y_{m}^{(1)}$ and show that $\left( y_{m}^{(1)}\right) _{m\in \mathbb {N}}$ itself converges to $y^{(1)}$. First, let us point out that for given $\lambda$ the solution of

$$\begin{aligned} {^{C}_{t}} {D}_{\pi }^{\alpha }\left[ p(\tau ){^{c}_{0}} {D}_{\tau }^{\alpha }[y](\tau )\right] (t)+q(t)y(t) = \lambda ^{(1)}w_{\alpha }(t)y(t), \end{aligned}$$

(6.31)

subject to the boundary conditions

$$\begin{aligned} y(a)=y(b)=0 \end{aligned}$$

(6.32)

and the normalization condition

$$\begin{aligned} \int \limits _0^{\pi } w_{\alpha }(t)y^2(t)\,\mathrm{d}t=1, \end{aligned}$$

(6.33)

is unique except for a sign. Next, let us assume that $y^{(1)}$ solves the Sturm–Liouville equation (6.31) and that the corresponding eigenvalue is $\lambda =\lambda ^{(1)}$. In addition, suppose that $y^{(1)}$ is nontrivial, i.e., we can find $t_0\in [0,\pi ]$ such that $y^{(1)}(t_0)\ne 0$ and choose the sign, so that $y^{(1)}(t_0)>0$. Similarly, for all $m\in \mathbb {N}$, let $y_m^{(1)}$ solve (6.31) with corresponding eigenvalue $\lambda =\lambda _m^{(1)}$ and let us choose the signs, so that $y_m^{(1)}(t_0)\ge 0$. Now suppose that $\left( y_{m}^{(1)}\right) _{m\in \mathbb {N}}$ does not converge to $y^{(1)}$. It means that we can find another subsequence of $\left( y_{m}^{(1)}\right) _{m\in \mathbb {N}}$ such that it converges to another solution $\bar{y}^{(1)}$ of (6.31) with $\lambda =\lambda ^{(1)}$. We know that for $\lambda =\lambda ^{(1)}$ solution of (6.31) subject to (6.32) and (6.33) must be unique except for a sign, thence

$$\begin{aligned} \bar{y}^{(1)}=-y^{(1)} \end{aligned}$$

and we must have $\bar{y}^{(1)}(t_0)<0$. However, it is impossible because for all $m\in \mathbb {N}$ the value of $y_m^{(1)}$ in $t_0$ is greater or equal than zero. It means that we have contradiction and hence, choosing each $y_m^{(1)}$ with adequate sign, we obtain $y_m^{(1)}\rightarrow y^{(1)}$.

Step 6. In order to find eigenfunction $y^{(2)}$ and the corresponding eigenvalue $\lambda ^{(2)}$, we again minimize functional (6.18) subject to (6.19) and (6.17), but now with an extra orthogonality condition

$$\begin{aligned} \int \limits _0^{\pi } w_{\alpha }(t)y(t) y^{(1)}(t)\,\mathrm{d}t=0. \end{aligned}$$

(6.34)

If we approximate solution by

$$\begin{aligned} y_{m}(t)= \frac{1}{\sqrt{w_{\alpha }}}\sum _{k=1}^{m} \beta _{k}\sin (kt), \quad y_{m}(0)=y_{m}(\pi )=0, \end{aligned}$$

then we again receive quadratic form (6.21). However, in this case admissible solutions are points satisfying (6.22) together with

$$\begin{aligned} \frac{\pi }{2}\sum \limits _{k=1}^{m}\beta _k\beta _k^{(1)}=0, \end{aligned}$$

(6.35)

i.e., they lie in the $(m-1)$-dimensional sphere. As before, we find that function $I([\beta ])$ has a minimum $\lambda _m^{(2)}$ and there exists $\lambda ^{(2)}$ such that

$$\begin{aligned} \lambda ^{(2)}=\lim \limits _{m\rightarrow \infty }\lambda _m^{(2)}, \end{aligned}$$

because $J(y)$ is bounded from below. Moreover, it is clear that the following relation:

$$\begin{aligned} \lambda ^{(1)}\le \lambda ^{(2)} \end{aligned}$$

(6.36)

holds. Now, let us denote by

$$\begin{aligned} y_{m}^{(2)}(t)= \frac{1}{\sqrt{w_{\alpha }}}\sum _{k=1}^{m} \beta _{k}^{(2)}\sin (kt) \end{aligned}$$

the linear combination achieving the minimum $\lambda _m^{(2)}$, where

$$ \beta ^{(2)}=(\beta _1^{(2)},\ldots ,\beta _m^{(2)}) $$

is the point satisfying (6.22) and (6.35). By the same argument as before, we can prove that the sequence $(y_m^{(2)})_{m\in \mathbb {N}}$ converges uniformly to a limit function $y^{(2)}$, which satisfies the Sturm-Liouville equation (6.16) with $\lambda ^{(2)}$, the boundary conditions (6.17), normalization condition (6.19), and the orthogonality condition (6.34). Therefore, solution $y^{(2)}$ of the FSLP corresponding to the eigenvalue $\lambda ^{(2)}$ exists. Furthermore, because orthogonal functions cannot be linearly dependent, and since only one eigenfunction corresponds to each eigenvalue (except for a constant factor), we have the strict inequality

$$\begin{aligned} \lambda ^{(1)}<\lambda ^{(2)} \end{aligned}$$

instead of (6.36). Finally, if we repeat the above procedure, with similar modifications, we can obtain eigenvalues $\lambda ^{(3)},\lambda ^{(4)},\ldots$ and corresponding eigenfunctions $y^{(3)},y^{(4)},\ldots$.

6.2.2 The First Eigenvalue

In this section we prove two theorems showing that the first eigenvalue of problem (6.16) and (6.17) is a minimum value of certain functionals. As in the proof of Theorem 6.5 in the sequel, for simplicity, we assume that $a=0$ and $b=\pi$ in the problem (6.16) and (6.17).

Theorem 6.6

Let $y^{(1)}$ denote the eigenfunction, normalized to satisfy the isoperimetric constraint

$$\begin{aligned} \mathcal {J}(y)=\int \limits _0^{\pi }w_{\alpha }(t)y^2(t)\,\mathrm{d}t=1, \end{aligned}$$

(6.37)

associated to the first eigenvalue $\lambda ^{(1)}$ of problem (6.16) and (6.17) and assume that function ${_{t}}D_{\pi }^{\alpha }\left[ p\cdot {^{C}_0}D_{t}^{\alpha }[y]\right]$ is continuous. Then, $y^{(1)}$ is a minimizer of the following variational functional

$$\begin{aligned} \mathcal {I}(y)=\int \limits _0^{\pi }\left[ p(t) ({^{C}_{0}} {D}_{t}^{\alpha }[y](t))^{2}+q(t)y^{2}(t)\right] \,\mathrm{d}t, \end{aligned}$$

(6.38)

in the class $C([0,\pi ];\mathbb {R})$ with ${^{C}_{0}} {D}_{t}^{\alpha }[y]\in C([0,\pi ];\mathbb {R})$ subject to the boundary conditions

$$\begin{aligned} y(0)=y(\pi )=0 \end{aligned}$$

(6.39)

and an isoperimetric constraint (6.37). Moreover, $\mathcal {I}(y^{(1)})=\lambda ^{(1)}$.

Proof

Suppose that $y\in C([0,\pi ];\mathbb {R})$ is a minimizer of $\mathcal {I}$ and ${^{C}_{0}} {D}_{t}^{\alpha }[y]\in C([0,\pi ];\mathbb {R})$. Then, by Theorem 6.4, there is number $\lambda$ such that $y$ satisfies equation

$$\begin{aligned} {_{t}} {D}_{\pi }^{\alpha }\left[ p(\tau ){^{c}_{0}} {D}_{\tau }^{\alpha }[y](\tau )\right] (t) +q(t)y(t)= \lambda w_{\alpha }(t)y(t), \end{aligned}$$

(6.40)

and conditions (6.37) and (6.39). Since ${_{t}}D_{\pi }^{\alpha }\left[ p\cdot {^{C}_0}D_{t}^{\alpha }[y]\right]$ and ${^{C}_{t}}D_{\pi }^{\alpha }\left[ p\cdot {^{C}_0}D_{t}^{\alpha }[y]\right]$ are continuous, it follows that $\left. p(t)\cdot {^{C}_0}D_{t}^{\alpha }[y](t)\right| _{t=\pi }=0$. Therefore, Eq. (6.40) is equivalent to

$$\begin{aligned} {^{C}_{t}} {D}_{\pi }^{\alpha }\left[ p(\tau ){^{c}_{0}} {D}_{\tau }^{\alpha }[y](\tau )\right] (t) +q(t)y(t)= \lambda w_{\alpha }(t)y(t). \end{aligned}$$

(6.41)

Let us multiply (6.40) by $y$ and integrate it on the interval $[0,\pi ]$. Then

$$\begin{aligned} \int \limits _0^{\pi }\left( y(t)\cdot {_{t}} {D}_{\pi }^{\alpha }\left[ p(\tau ){^{c}_{0}} {D}_{\tau }^{\alpha }[y](\tau )\right] (t)+q(t)y^2(t)\right) \,\mathrm{d}t =\lambda \int \limits _0^{\pi }w_{\alpha }(t)y^2(t)\,\mathrm{d}t. \end{aligned}$$

Applying the integration by parts formula for fractional derivatives (cf. (2.9)) and having in mind that conditions (6.39), (6.37) and

$$ \left. p(t)\cdot {^{C}_0}D_{t}^{\alpha }[y](t)\right| _{t=\pi }=0 $$

hold, one has

$$\begin{aligned} \int \limits _0^{\pi }\left( \left( {^{c}_{0}} {D}_{t}^{\alpha } [y](t)\right) ^2 p(t) +q(t)y^2(t)\right) \,\mathrm{d}t=\lambda . \end{aligned}$$

Hence, $\mathcal {I}(y)=\lambda$. Any solution to problem (6.37)–(6.39) that satisfies Eq. (6.41) must be nontrivial since (6.37) holds, so $\lambda$ must be an eigenvalue. Moreover, according to Theorem 6.5 there is the least element in the spectrum being eigenvalue $\lambda ^{(1)}$ and the corresponding eigenfunction $y^{(1)}$ normalized to meet the isoperimetric condition. Therefore $J(y^{(1)})=\lambda ^{(1)}$.

Definition 6.7

We call to functional $\mathcal {R}$ defined by

$$\begin{aligned} \mathcal {R}(y)=\frac{\mathcal {I}(y)}{\mathcal {J}(y)}, \end{aligned}$$

where $\mathcal {I}(y)$ is given by (6.38) and $\mathcal {J}(y)$ by (6.37), the Rayleigh quotient for the fractional Sturm–Liouville problem (6.16) and (6.17).

Theorem 6.8

Let us assume that function $y\in C([0,\pi ];\mathbb {R})$ with ${^{C}_{0}} {D}_{t}^{\alpha }[y]\in C([0,\pi ];\mathbb {R})$, satisfying boundary conditions $y(0)=y(\pi )=0$ and being nontrivial, is a minimizer of the Rayleigh quotient $\mathcal {R}$ for the Sturm–Liouville problem (6.16) and (6.17). Moreover, assume that function ${_{t}}D_{\pi }^{\alpha }\left[ p\cdot {^{C}_0}D_{t}^{\alpha }[y]\right]$ is continuous. Then, the value of $\mathcal {R}$ in $y$ is equal to the first eigenvalue $\lambda ^{(1)}$, i.e., $\mathcal {R}(y)=\lambda ^{(1)}$.

Proof

Suppose that function $y\in C([0,\pi ];\mathbb {R})$ with ${^{C}_{0}} {D}_{t}^{\alpha }[y]\in C([0,\pi ];\mathbb {R})$, satisfying $y(0)=y(\pi )=0$ and nontrivial, is a minimizer of the Rayleigh quotient $\mathcal {R}$ and that the value of $\mathcal {R}$ in $y$ is equal to $\lambda$, i.e.,

$$ \mathcal {R}(y)=\frac{\mathcal {I}(y)}{\mathcal {J}(y)}=\lambda . $$

Consider a one-parameter family of curves $\hat{y}=y+h\eta$, $\left| h\right| \le \varepsilon$, where $\eta \in C([0,\pi ];\mathbb {R})$ with ${^{C}_{0}} {D}_{t}^{\alpha }[\eta ]\in C([0,\pi ];\mathbb {R})$ is such that $\eta (0)=\eta (\pi )=0$, $\eta \ne 0$, and define the following functions:

$$\begin{aligned} \begin{array}{rcl}\phi :[-\varepsilon ,\varepsilon ] &{}\longrightarrow &{}\mathbb {R}\\ h&{} \longmapsto &{}\mathcal {J}(y+h\eta ) =\displaystyle \int \limits _0^{\pi }w_{\alpha }(t)(y(t)+h\eta (t))^2\,\mathrm{d}t, \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{rcl}\psi :[-\varepsilon ,\varepsilon ] &{}\longrightarrow &{}\mathbb {R}\\ h&{} \longmapsto &{}\mathcal {I}(y+h\eta ) =\displaystyle \int \limits _0^{\pi }\left[ p(t) ({^{C}_{0}} {D}_{t}^{\alpha }[y+h\eta ](t))^{2}\right. \end{array}\qquad \\ \left. +q(t)(y(t)+h\eta (t))^{2}\right] \,\mathrm{d}t \end{aligned}$$

and

$$\begin{aligned} \begin{array}{rcl}\zeta :[-\varepsilon ,\varepsilon ] &{}\longrightarrow &{}\mathbb {R}\\ h&{} \longmapsto &{}\mathcal {R}(y+h\eta ) =\frac{\mathcal {I}(y+h\eta )}{\mathcal {J}(y+h\eta )}. \end{array} \end{aligned}$$

Since $\zeta$ is of class $C^1$ on $[-\varepsilon ,\varepsilon ]$ and

$$ \zeta (0)\le \zeta (h),\quad \left| h\right| \le \varepsilon , $$

we deduce that

$$\begin{aligned} \zeta '(0)=\left. \frac{d}{\mathrm{d}h}\mathcal {R}(y+h\eta )\right| _{h=0}=0. \end{aligned}$$

Moreover, notice that

$$\begin{aligned} \zeta '(h)=\frac{1}{\phi (h)}\left( \psi '(h)-\frac{\psi (h)}{\phi (h)}\phi '(h)\right) \end{aligned}$$

and that

$$\begin{aligned} \begin{aligned} \psi '(0)&=\left. \frac{d}{\mathrm{d}h}\mathcal {I}(y+h\eta )\right| _{h=0}\\&=2\int \limits _0^{\pi }\left[ p(t)\cdot {^{C}_{0}} {D}_{t}^{\alpha }[y](t) \cdot {^{C}_{0}} {D}_{t}^{\alpha }[\eta ](t)+q(t)y(t)\eta (t)\right] \,\mathrm{d}t,\\ \phi '(0)&=\left. \frac{d}{\mathrm{d}h}\mathcal {J}(y+h\eta )\right| _{h=0} =2\int \limits _0^{\pi }\left[ w_{\alpha }(t)y(t)\eta (t)\right] \,\mathrm{d}t. \end{aligned} \end{aligned}$$

Therefore

$$\begin{aligned} \begin{aligned} \zeta '(0)&=\left. \frac{d}{\mathrm{d}h}\mathcal {R}(y+h\eta )\right| _{h=0}\\&=\frac{2}{\mathcal {J}(y)}\left( \int \limits _0^{\pi }p(t) \cdot {^{C}_{0}} {D}_{t}^{\alpha }[y](t)\cdot {^{C}_{0}} {D}_{t}^{\alpha }[\eta ](t) +q(t)y(t)\eta (t)\,\mathrm{d}t\right. \\&\quad \left. -\frac{\mathcal {I}(y)}{\mathcal {J}(y)}\int \limits _0^{\pi } w_{\alpha }(t)y(t)\eta (t)\,\mathrm{d}t\right) =0. \end{aligned} \end{aligned}$$

Having in mind that $\frac{\mathcal {I}(y)}{\mathcal {J}(y)}=\lambda$ and $\eta (0)=\eta (\pi )=0$, using the integration by parts formula (2.9) we obtain

$$\begin{aligned} \int \limits _0^{\pi }\left( {_{t}} {D}_{\pi }^{\alpha } \left[ p{^{c}_{0}} {D}_{\tau }^{\alpha }[y]\right] (t) +q(t)y(t)-\lambda w_{\alpha }(t)y(t)\right) \eta (t)\,\mathrm{d}t=0. \end{aligned}$$

Now, applying the fundamental lemma of the calculus of variations, we arrive to

$$\begin{aligned} {_{t}} {D}_{\pi }^{\alpha }\left[ p(\tau ) \cdot {^{C}_{0}} {D}_{\tau }^{\alpha }[y](\tau )\right] (t) +q(t)y(t)=\lambda w_{\alpha }(t)y(t). \end{aligned}$$

(6.42)

Under our assumptions, $\left. p(t)\cdot {^{C}_0}D_{t}^{\alpha }[y](t)\right| _{t=\pi }=0$ and therefore Eq. (6.42) is equivalent to

$$\begin{aligned} {^{C}_{t}} {D}_{\pi }^{\alpha }\left[ p(\tau ) \cdot {^{C}_{0}} {D}_{\tau }^{\alpha }[y](\tau )\right] (t)+q(t)y(t) =\lambda w_{\alpha }(t)y(t). \end{aligned}$$

(6.43)

Since $y\ne 0$ we deduce that number $\lambda$ is an eigenvalue of (6.43). On the other hand, let $\lambda ^{(m)}$ be an eigenvalue and $y^{(m)}$ the corresponding eigenfunction. Then

$$\begin{aligned} {^{C}_{t}} {D}_{\pi }^{\alpha }\left[ p(\tau ){^{C}_{0}} {D}_{\tau }^{\alpha }[y^{(m)}](\tau )\right] (t) +q(t)y^{(m)}(t)=\lambda ^{(m)} w_{\alpha }(t)y^{(m)}(t). \end{aligned}$$

(6.44)

Similarly to the proof of Theorem 6.6, we can obtain

$$\begin{aligned} \frac{\int \limits _0^{\pi }\left( \left( {^{C}_{0}} {D}_{t}^{\alpha } [y^{(m)}](t)\right) ^2 p(t) +q(t)(y^{(m)}(t))^2\right) \,\mathrm{d}t}{\int \limits _0^{\pi }\lambda ^{(m)} w_{\alpha }(t)(y^{(m)}(t))^2\,\mathrm{d}t} =\lambda ^{(m)}, \end{aligned}$$

for any $m\in \mathbb {N}$. That is $\mathcal {R}(y^{(m)})=\frac{\mathcal {I}(y^{(m)})}{\mathcal {J}(y^{(m)})} =\lambda ^{(m)}$. Finally, since the minimum value of $\mathcal {R}$ at $y$ is equal to $\lambda$, i.e.,

$$\begin{aligned} \lambda \le \mathcal {R}(y^{(m)})=\lambda ^{(m)}\quad \forall m\in \mathbb {N}, \end{aligned}$$

we have $\lambda =\lambda ^{(1)}$.

6.2.3 An Illustrative Example

Let us consider the following fractional oscillator equation:

$$\begin{aligned} {_{t}} {D}_{b}^{\alpha }\left[ {^{c}_{a}} {D}_{\tau }^{\alpha }[y](\tau )\right] (t)-\lambda y(t)=0, \end{aligned}$$

(6.45)

where $y(a)=y(b)=0$. One can easily check that problem of finding nontrivial solutions to Eq. (6.45) and corresponding values of parameter $\lambda$ is a particular case of problem (6.16) and (6.17) with $p(t)\equiv 1$, $q(t)\equiv 0$ and $w_{\alpha }(t)\equiv 1$. The corresponding variational functional is

$$\begin{aligned} \mathcal {I}_{\alpha }(y) = \int \limits _{a}^{b}p(t) \cdot ({^{C}_{a}} {D}_t^\alpha [y](t))^{2}\,\mathrm{d}t = ||\sqrt{p}\;\;{^{C}_{a}} {D}_t^\alpha [y]||^{2}_{L^{2}} \end{aligned}$$

with the isoperimetric condition

$$\begin{aligned} \int \limits _{a}^{b}y^{2}(t)\,\mathrm{d}t = 1. \end{aligned}$$

Let us fix the value of parameter $p$ and assume that orders $\alpha _{1}, \alpha _{2}$ fulfill the condition $\frac{1}{2}<\alpha _{1}<\alpha _{2}<1$. Then, we obtain for functionals $\mathcal {I}_{\alpha _{1}}, \mathcal {I}_{\alpha _{2}}$ the following relation

$$\begin{aligned} \begin{aligned} \mathcal {I}_{\alpha _{1}}(y)&= ||\sqrt{p}{^{C}_{a}} {D}_t^{\alpha _1}[y]||^{2}_{L^{2}} = ||\sqrt{p}{_{a}} {I}_t^{1-\alpha _1}\left[ \frac{d}{\mathrm{d}t}y\right] ||^{2}_{L^{2}}\\&=||\sqrt{p}{_{a}}I^{\alpha _2-\alpha _1}_{t}{_{a}}I^{1-\alpha _2}_{t}\left[ \frac{d}{\mathrm{d}t}y\right] ||^{2}_{L^{2}}\\&\le K_{\alpha _{2}-\alpha _{1}}^{2}\cdot ||\sqrt{p}{^{C}_{a}} {D}_t^{\alpha _2}[y]||^{2}_{L^{2}}\\&=K_{\alpha _{2}-\alpha _{1}}^{2}\mathcal {I}_{\alpha _{2}}(y), \end{aligned} \end{aligned}$$

where we denoted

$$\begin{aligned} K_{\alpha _{2}-\alpha _{1}}:= \frac{(b-a)^{\alpha _2-\alpha _1}}{\varGamma (\alpha _2-\alpha _1+1)}. \end{aligned}$$

We observe that in the above estimation two cases occur:

$$\begin{aligned}&\text {if }K_{\alpha _{2}-\alpha _{1}}\le 1, \quad \text {then } \; \mathcal {I}_{\alpha _{1}}(y) \le \mathcal {I}_{\alpha _{2}}(y);\\&\text {if }K_{\alpha _{2}-\alpha _{1}}>1,\quad \text {then } \; \mathcal {I}_{\alpha _{1}}(y) \le K^{2}_{\alpha _{2}-\alpha _{1}} \cdot \mathcal {I}_{\alpha _{2}}(y). \end{aligned}$$

The relations between functionals for different values of fractional order lead to the set of inequalities for eigenvalues $\lambda _{j}$ valid for any $j\in \mathbb {N}$:

$$\begin{aligned}&\text {if }K_{\alpha _{2}-\alpha _{1}}\le 1,\quad \text {then }\; \lambda _{j}(\alpha _1)\le \lambda _{j}(\alpha _2);\\&\text {if } K_{\alpha _{2}-\alpha _{1}}>1,\quad \text {then }\; \lambda _{j}(\alpha _1)\le K_{\alpha _{2}-\alpha _{1}}^{2} \cdot \lambda _{j}(\alpha _2). \end{aligned}$$

In particular when order $\alpha _{2}=1$ we get

$$\begin{aligned} \begin{aligned} \mathcal {I}_{\alpha _{1}}(y)&=||\sqrt{p}{^{C}_{a}} {D}_t^{\alpha _1}[y]||^{2}_{L^{2}} = ||\sqrt{p}{_{a}} {I}_t^{1-\alpha _1}\left[ Dy\right] ||^{2}_{L^{2}} \\&\le K^{2}_{1-\alpha _{1}}\cdot ||\sqrt{p}Dy||^{2}_{L^{2}} =K^{2}_{1-\alpha _{1}}\mathcal {I}_{1}(y) \end{aligned} \end{aligned}$$

and the following relations dependent on the value of constant $K_{1-\alpha _{1}}$:

$$\begin{aligned}&\text {if }K_{1-\alpha _{1}}\le 1,\quad \text {then }\; \mathcal {I}_{\alpha _{1}}(y) \le \mathcal {I}_{1}(y);\\&\text {if } K_{1-\alpha _{1}}>1,\quad \text {then }\; \mathcal {I}_{\alpha _{1}}(y) \le K^{2}_{1-\alpha _{1}}\cdot \mathcal {I}_{1}(y). \end{aligned}$$

Thus, comparing the eigenvalues for the fractional and the classical harmonic oscillator equation for boundary conditions $y(a)=y(b)=0$, we conclude that the respective classical eigenvalues are higher than the ones resulting from the fractional problem for any $j\in \mathbb {N}$. Namely,

$$\begin{aligned}&\text {if } K_{1-\alpha _{1}}\le 1,~\text {then } \lambda _{j}(\alpha _1)\le \lambda _{j}(1) =p\left( \frac{j\pi }{b-a}\right) ^{2};\\&\text {if } K_{1-\alpha _{1}}>1,~\text {then } \lambda _{j}(\alpha _1)\le K^{2}_{1-\alpha _{1}} \cdot \lambda _{j}(1)=p\left( \frac{j\pi }{(b-a)^{\alpha _{1}}\varGamma (2-\alpha _{1})}\right) ^{2}. \end{aligned}$$