Variational methods for fractional q-Sturm-Liouville problems

In the joint paper of Sturm and Liouville [3], they studied the problem with certain boundary conditions at a and b. Here, the functions p, w are positive on \([a,b]\) and r is a real valued function on \([a,b]\). They proved the existence of non-zero solutions (eigenfunctions) only for special values of the parameter λ which are called eigenvalues. For a comprehensive study of the contribution of Sturm and Liouville to the theory, see [4]. Recently, many mathematicians have become interested in a fractional version of (1.1), i.e., when the derivative is replaced by a fractional derivative like Riemann-Liouville derivative or Caputo derivative; see [5–10]. Iterative methods, variational method, and the fixed point theory are three different approaches used in proving the existence and uniqueness of solutions of Sturm-Liouville problems, cf. [4, 11, 12]. The calculus of variations has recently been developed to calculate the extremum of a functional that contains fractional derivatives, which is called the fractional calculus of variations; see for example [13–20]. In [5], Klimek et al. applied the methods of fractional variational calculus to prove the existence of a countable set of orthogonal solutions and corresponding eigenvalues. In [1, 2], Annaby and Mansour introduced a q-version of (1.1), i.e., when the derivative is replaced by Jackson q-derivative. Their results are applied and developed in different respects; for example, see [21–26]. Throughout this paper q is a positive number less than 1. The set of non-negative integers is denoted by \(\mathbb{N}_{0}\), and the set of positive integers is denoted by \(\mathbb{N}\). For \(t>0\), and When \(t=1\), we simply use \(A_{q}\), \(A_{q}^{*}\), and \(\mathcal{A}_{q}\) to denote \(A_{q,1}\), \(A_{q,1}^{*}\), and \(\mathcal{A}_{q,1}\), respectively. In the following, we state the basic q-notations and notions we use in this article, cf. [27, 28].

For \(n \in\mathbb{N}_{0}\), the q-shifted factorial \((a;q)_{n}\) of \(a\in\mathbb{C}\) is defined by The multiple q-shifted factorial for complex numbers \(a_{1},\ldots,a_{k}\) is defined by The limit \(\lim_{n\to\infty}(a;q)_{n}\) exists and is denoted by \((a;q)_{\infty}\). For \(\alpha\in\mathbb{R}\), The q-gamma function, [28, 29], is defined for \(z\in\mathbb{C}\), \(z\neq-n\), \(n\in\mathbb{N}\) by Here we take the principal values of \(q^{z}\) and \((1-q)^{1-z}\). Then \(\Gamma_{q}(z)\) is a meromorphic function with poles at \(z=-n\), \(n\in\mathbb{N}\).

Let \(\mu\in\mathbb{R}\) be fixed. A set \(A\subseteq\mathbb{R}\) is called a μ-geometric set if for \(x\in A\), \(\mu x\in A\). If f is a function defined on a q-geometric set \(A\subseteq\mathbb{R}\), the q-difference operator, \(D_{q}\), is defined by If \(0\in A\), we say that f has q-derivative at zero if exists and does not depend on x. In this case, we shall denote this limit by \(D_{q}f(0)\). In some literature the q-derivative at zero is defined to be \(f'(0)\) if it exists, cf. [30, 31], but the above definition is more suitable for our approach. The non-symmetric Leibniz rule holds. Equation (1.7) can be symmetrized using the relation \(f(qx)=f(x)-x(1-q)D_{q}f(x)\), giving the additional term \(-x(1-q)D_{q}f(x)D_{q}g(x)\). The q-integration of Jackson [32] is defined for a function f defined on a q-geometric set A to be where provided that the series converges. A function f defined on X is called q-regular at zero if Let \(C(X)\) denote the space of all q-regular at zero functions defined on X with values in \(\mathbb{R}\). \(C(X)\) associated with the norm function is a normed space. The q-integration by parts rule [33] is where f, g are q-regular at zero functions.

For \(p>0\), and Y is \(A_{q,t}\) or \(A_{q,t}^{*}\), the space \(L_{q}^{p}(Y)\) is the normed space of all functions defined on Y such that If \(p=2\), then \(L_{q}^{2}(Y)\) associated with the inner product is a Hilbert space. A weighted \(L_{q}^{2}(Y, w)\) space is the space of all functions f defined on Y, such that where w is a positive function defined on Y. \(L_{q}^{2}(Y, w)\) associated with the inner product is a Hilbert space. The space of all q-absolutely functions on \(A_{q,t}^{*}\) is denoted by \(\mathcal{A}C_{q}(A_{q,t}^{*})\) and defined as the space of all q-regular at zero functions f satisfying and K is a constant depending on the function f, cf. [33], Definition 4.3.1. That is, The space \(\mathcal{A}C_{q}^{(n)}(A_{q,t}^{*})\) (\(n\in\mathbb{N}\)) is the space of all functions defined on X such that \(f, D_{q}f, \ldots, D_{q}^{n-1}f\) are q-regular at zero and \(D_{q}^{n-1}f\in\mathcal {A}C_{q}(A_{q,t}^{*})\), cf. [33], Definition 4.3.2. Also it has been proved in [33], Theorem 4.6, that a function \(f\in\mathcal{A}C_{q}^{(n)}(A_{q,t}^{*})\) if and only if there exists a function \(\phi\in L_{q}^{1}(A_{q,t}^{*})\) such that In particular, \(f\in\mathcal{A}C(A_{q,t}^{*})\) if and only if f is q-regular at zero such that \(D_{q}f\in L_{q}^{1}(A_{q,t}^{*})\). It is worth noting that in [33], all the definitions and results we have just mentioned are defined and proved for functions defined on the interval \([0,a]\) instead of \(A_{q,t}^{*}\). In [34], Mansour studied the problem where \(p(x)\neq0\) and \(w_{\alpha}>0\) for all \(x\in A_{q,a}^{*}\), p, r, \(w_{\alpha}\) are real valued functions defined in \(A_{q,a}^{*}\) and the associated boundary conditions are with \(c_{1}^{2}+c_{2}^{2}\neq0\) and \(d_{1}^{2}+d_{2}^{2}\neq0\). it is proved that the eigenvalues are real and the eigenfunctions associated to different eigenvalues are orthogonal in the Hilbert space \(L_{q}^{2}(A_{q,a}^{*}, w_{\alpha})\). A sufficient condition on the parameter λ to guarantee the existence and uniqueness of the solution is introduced by using the fixed point theorem, also a condition is imposed on the domain of the problem in order to prove the existence and uniqueness of solution for any λ. This paper is organized as follows. Section 2 is on the q-fractional operators and their properties which we need in the sequel. Cardoso [35] introduced basic Fourier series for functions defined on a q-linear grid of the form \(\{\pm q^{n}: n\in \mathbb{N}_{0} \}\cup \{0 \}\). In Section 3, we reformulate Cardoso’s results for functions defined on a q-linear grid of the form \(\{\pm aq^{n}: n\in\mathbb {N}_{0} \}\cup \{0 \}\). In Section 4, we introduce a fractional q-analog for Euler-Lagrange equations for functionals defined in terms of Jackson q-integration and the integrand contains the left-sided Caputo fractional q-derivative. We also introduce a fractional q-isoperimetric problem. In Section 5, we use the variational q-calculus developed in Section 4 to prove the existence of a countable number of eigenvalues and orthogonal eigenfunctions for the fractional q-Sturm-Liouville problem with the boundary condition \(y(0)=y(a)=0\). We also define the Rayleigh quotient and prove a criterion for the smallest eigenvalue.

The left-sided Riemann-Liouville q-fractional operator is defined by This definition was introduced by Agarwal in [36] when \(a=0\) and by Rajković et al. [37] for \(a\neq0\). The right-sided Riemann-Liouville q-fractional operator is defined by see [34]. The left-sided Riemann-Liouville q-fractional operator satisfies the semigroup property The case \(a=0\) is proved in [36], while the case \(a>0\) is proved in [37].

The right-sided Riemann-Liouville q-fractional operator satisfies the semigroup property [34] for any function defined on \(A_{q,b}\) and for any values of α and β.

For \(\alpha>0\) and \(\ulcorner\alpha\urcorner=m\), the left- and right-sided Riemann-Liouville fractional q-derivatives of order α are defined by the left- and right-sided Caputo fractional q-derivatives of order α are defined by see [34]. From now on, we shall consider left-sided Riemann-Liouville and Caputo fractional q-derivatives when the lower point \(a=0\) and right-sided Riemann-Liouville and Caputo fractional q-derivatives when \(b=a\). According to [33],pp.124, 148, \(D_{q,0^{+}}^{\alpha}f(x)\) exists if and \({}^{\mathrm{c}}D_{q,a^{+}}^{\alpha}f\) exists if

Set \(X=A_{q,a}\) or \(A_{q,a}^{*}\). Then Moreover, if \(f\in C(X)\) then

We also have the following inequalities:

1. 1.
   If \(f\in C(A_{q,a}^{*})\) then \(I_{q,0^{+}}^{\alpha}f\in C(A_{q,a}^{*})\) and

   $$ \bigl\Vert I_{q,0^{+}}^{\alpha}f \bigr\Vert \leq \frac{a^{\alpha }}{\Gamma_{q}(\alpha+1)}\Vert f\Vert . $$

   (2.12)
    
2. 2.
   If \(f\in L_{q}^{1}(X)\) then \(I_{q,0^{+}}^{\alpha}f\in L_{q}^{1}(X)\) and

   $$ \bigl\Vert I_{q,0^{+}}^{\alpha}f \bigr\Vert _{1}\leq M_{\alpha,1}\Vert f\Vert _{1},\quad M_{\alpha ,1}:=\frac{a^{\alpha}(1-q)^{\alpha}}{(1-q^{\alpha })(q;q)_{\infty}}. $$

   (2.13)
    
3. 3.
   If \(f\in L_{q}^{2}(X)\) then \(I_{q,0^{+}}^{\alpha}f\in L_{q}^{2}(X)\) and

   $$ \bigl\Vert I_{q,0^{+}}^{\alpha}f\bigr\Vert _{2}\leq M_{\alpha,2} \Vert f\Vert _{2}, $$

   (2.14)

   where

   $$M_{\alpha,2}:=\frac{a^{\alpha}}{\Gamma_{q}(\alpha)}\sqrt{\frac {(1-q)}{(1-q^{2\alpha})}} \biggl( \int_{0}^{1}(q\xi;q)^{2}_{\alpha-1} \,d_{q}\xi \biggr)^{1/2}. $$
    
4. 4.
   If \(\alpha>\frac{1}{2}\) and \(f\in L_{q}^{2}(X)\) then \(I_{q,0^{+}}^{\alpha}f\in C(X)\) and

   $$ \bigl\Vert I_{q,0^{+}}^{\alpha}f\bigr\Vert \leq \widetilde {M}_{\alpha} \Vert f\Vert ,\quad \widetilde{M}_{\alpha}:= \frac{a^{\alpha-\frac{1}{2}}}{\Gamma _{q}(\alpha)} \biggl( \int_{0}^{1}(q\xi;q)^{2}_{\alpha-1} \,d_{q}\xi \biggr)^{1/2}. $$

   (2.15)
    
5. 5.
   Since \(\Vert f\Vert _{2}\leq\sqrt{a} \Vert f\Vert \), we conclude that if \(f\in C(X)\) then \(I_{q,0^{+}}^{\alpha}f\in L_{q}^{2}(X)\) and

   $$ \bigl\Vert I_{q,0^{+}}^{\alpha}f\bigr\Vert _{2}\leq K_{\alpha} \Vert f\Vert , \quad K_{\alpha }:= \sqrt{a}M_{\alpha,2}. $$

   (2.16)
    
6. 6.
   If \(f\in C(A_{q,a}^{*})\) then \(I_{q,a^{-}}^{\alpha}f\in C(A_{q,a}^{*})\) and

   $$\bigl\Vert I_{q,a^{-}}^{\alpha}f\bigr\Vert \leq c_{\alpha,0} \Vert f\Vert ,\quad c_{\alpha ,0}:=\frac{a^{\alpha}(1-q)^{\alpha}}{(1-q^{\alpha})(q;q)_{\infty}}. $$
    
7. 7.
   If \(f\in L_{q}^{1}(X)\) then \(I_{q,a^{-}}^{\alpha}f\in L_{q}^{1}(X)\) and

   $$\bigl\Vert I_{q,a^{-}}^{\alpha}f\bigr\Vert _{1}\leq \left \{ \textstyle\begin{array}{l@{\quad}l} \frac{(1-q)^{\alpha}a^{\alpha}}{(1-q^{\alpha })(q;q)_{\infty}} \Vert f\Vert _{1},& \mbox{if } \alpha< 1, \\ \frac{(1-q)^{\alpha-1}a^{\alpha-1}}{(q;q)_{\infty}} \Vert f\Vert _{1},& \mbox{if } \alpha\geq1. \end{array}\displaystyle \right . $$
    
8. 8.
   If \(\alpha\neq\frac{1}{2}\) and \(f\in L_{q}^{2}(X)\) then \(I_{q,a^{-}}^{\alpha}f\in L_{q}^{1}(X)\) and

   $$\bigl\Vert I_{q,a^{-}}^{\alpha}f\bigr\Vert _{2}\leq \left \{ \textstyle\begin{array}{l@{\quad}l} \frac{(1-q)^{\alpha-\frac{1}{2}}a^{\alpha}}{\sqrt {1-q^{2\alpha-1}}(q;q)_{\infty}} \Vert f\Vert _{2},& \mbox{if } \alpha < \frac{1}{2}, \\ \frac{(1-q)^{\alpha}a^{\alpha}}{(q;q)_{\infty}\sqrt{(1-q^{2\alpha -1})(1-q^{2\alpha})}}\Vert f\Vert _{2},& \mbox{if } \alpha > \frac{1}{2}. \end{array}\displaystyle \right . $$
    

The following lemmas are needed in the remaining sections.

One can verify that where \(z\in\mathbb{C}\) and \(w\in\mathbb{C}\) is a fixed parameter. A modification of the orthogonality relation given in [38], Theorem 4.1, is the following.

Cardoso introduced a sufficient condition for the uniform convergence of the basic Fourier series where \(a_{0}=\int_{-1}^{1}f(t) \,d_{q}t\) and for \(k=1,2,\ldots\) , on the q-linear grid \(\mathcal{A}_{q}\), where \(\{w_{k} : k\in \mathbb{N} \}\) is the set of positive zeros of \(S_{q}(z)\). Cardoso proved that \(\mu_{k}=O(q^{-2k^{2}})\) as \(k\to\infty\) for any \(q\in(0,1)\). In the following we give a modified version of Cardoso’s result for any function defined on the q-linear grid \(\mathcal{A}_{q,a}\), \(a>0\).

The calculus of variations is as old as the calculus itself, and has many applications in physics and mechanics. As the calculus has two forms, the continuous calculus with the power concept of limits, and the discrete calculus which also is called the calculus of finite differences, the calculus of variations has also both the discrete and the continuous forms. For a brief history of the continuous calculus of variations, see [39]. The discrete calculus of variations started in 1948 by Fort in his book [40] where he devoted a chapter to the finite analog of the calculus of variations, and he introduced a necessary condition analog to the Euler equation and also a sufficient condition. The paper of Cadzow [41], 1969, was the first paper published in this field, then Logan developed the theory in his PhD thesis [42], 1970, and in a series of papers [43–46]. See also the PhD thesis of Harmsen [47] for a brief history for the discrete variational calculus; and for the developments in the theory, see [48–54]. In 2004, a q-version of the discrete variational calculus is introduced by Bangerezako in [55] for functions defined in the form where \(q^{\alpha}\) and \(q^{\beta}\) are in the uniform lattice \(A_{q,a}^{*}\) for some \(a>0\) such that \(\alpha>\beta\), provided that the boundary conditions He introduced a q-analog of the Euler-Lagrange equation which he applied to solve certain isoperimetric problem. Then, in 2005, Bangerezako [56] introduced certain q-variational problems on a nonuniform lattice. In [57, 58], Malinowska, and Torres introduced the Hahn quantum variational calculus. They derived the Euler-Lagrange equation associated with the variational problem under the boundary condition \(y(a)=\alpha\), \(y(b)=\beta\) where α and β are constants and \(D_{q,w}\) is the Hahn difference operator defined by Problems of the classical calculus of variations with integrand depending on fractional derivatives instead of ordinary derivatives are first introduced by Agrawal [17] in 2002. Then he extended his result for variational problems including Riesz fractional derivatives in [18]. Numerous works have been dedicated to the subject since Agrawal’s work. See for example [5, 13–16, 59–61].

In this section, we shall derive Euler-Lagrange equation for a q-variational problem when the integrand includes a left-sided q-Caputo fractional derivative and we also solve a related isoperimetric problem. From now on, we fix \(\alpha \in(0,1)\), and define a subspace of \(C(A_{q,a}^{*})\) by and the space of variations \({}^{\mathrm{c}}\operatorname{Var}(0,a)\) for the Caputo q-derivative by For a function \(f(x_{1},x_{2},\ldots,x_{n})\) (\(n\in\mathbb{N}\)) by \(\partial_{i} f\) we mean the partial derivative of f with respect to the ith variable, \(i=1,2,\ldots,n\). In the sequel, we shall need the following definition from [62].

We now present first order necessary conditions of optimality for functionals, defined on \({}_{0}E_{a}^{\alpha}\), of the type where \(F:A_{q,a}^{*}\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}\) is a given function. We assume that:

1. 1.

   The functions \((u,v)\to F(t,u,v)\) and \((u,v)\to\partial_{i} F(t,u,v)\) (\(i=2,3\)) are continuous functions uniformly on \(A_{q,a}\).

    
2. 2.

   \(F (\cdot,y(\cdot),{}^{\mathrm{c}}D_{q,0^{+}}^{\alpha }(\cdot) )\), \(\delta_{i} F (\cdot,y(\cdot),{}^{\mathrm{c}}D_{q,0^{+}}^{\alpha }(\cdot) )\) (\(i=2,3\)) are q-regular at zero.

    
3. 3.

   \(\delta_{3} F\) has a right Riemann-Liouville fractional q-derivative of order α which is q-regular at zero.

    

In this section, we use the q-calculus of variations we developed in Section 4 to investigate the existence of solutions of the qFSLP under the boundary condition The proof of the main result of this section depends on the Arzelà-Ascoli theorem [64], p.156. The setting of this theorem is a compact metric space X. Let \(C(X)\) denote the space of all continuous functions on X with values in \(\mathbb{C}\) or \(\mathbb{R}\). \(C(X)\) is associated with the metric function

In our q-setting, we take \(X=A_{q,a}^{*}\). Hence \(f\in C(A_{q,a}^{*}) \) if and only if f is q-regular at zero, i.e.,

This paper is the first paper that deals with variational problems of functionals defined in terms of Jackson q-integral on a finite domain and where the left-sided Caputo q-derivative appears in the integrand. We give a fractional q-analog of the Euler-Lagrange equation and a q-isoperimetric problem is defined and solved. We use these results in recasting the qFSLP under consideration as a q-isoperimetric problem, and then we solve it by a technique similar to the one used in solving regular Sturm-Liouville problems in [11] and fractional Sturm-Liouville problems in [5]. This completes the work started by the author in [34], and it generalizes the study of integer Sturm-Liouville problem introduced by Annaby and Mansour in [1, 2]. A similar study for the fractional Sturm-Liouville problem is in progress.