Variational methods for fractional $q$-Sturm--Liouville Problems

In this paper, we formulate a regular $q$-fractional Sturm--Liouville problem (qFSLP) which includes the left-sided Riemann--Liouville and the right-sided Caputo $q$-fractional derivatives of the same order $\alpha$, $\alpha\in (0,1)$. We introduce the essential $q$-fractional variational analysis needed in proving the existence of a countable set of real eigenvalues and associated orthogonal eigenfunctions for the regular qFSLP when $\alpha>1/2$ associated with the boundary condition $y(0)=y(a)=0$. A criteria for the first eigenvalue is proved. Examples are included. These results are a generalization of the integer regular $q$-Sturm--Liouville problem introduced by Annaby and Mansour in [1].


Introduction
In the joint paper of Sturm and Liouville [2], they studied the problem . They proved the existence of non-zero solutions (eigenfunctions) only for special values of the parameter λ which is called eigenvalues. For a comprehensive study for the contribution of Sturm and Liouville to the theory, see [3]. Recently, many mathematicians were 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 [4][5][6][7][8][9]. 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, c.f. [3,10,11]. The calculus of variations has recently developed to calculate extremum of functional contains fractional derivatives, which is called fractional calculus of variations, see for example [12][13][14][15][16][17][18][19]. In [4], 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] 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 aspects, for example, see [20][21][22][23][24][25]. Throughout this paper q is a positive number less than 1. The set of non negative integers is denoted by N 0 , and the set of positive integers is denoted by N. For t > 0, A q,t := {tq n : n ∈ N 0 } , A * q,t := A q,t ∪ {0} , and A q,t := {±tq n : n ∈ N 0 } .
When t = 1, we simply use A q , A * q , and A q to denote A q,1 , A * q,1 , and A q,1 , respectively. We follow [26] for the definitions and notations of the q-shifted factorial, the q-gamma and q-beta functions, the basic hypergeometric series, and Jackson q-difference operator and integrals. A set A is called a q-geometric set if qx ∈ A whenever x ∈ A. Let X be a q-geometric set containing zero. A function f defined on X is called q-regular at zero if lim n→∞ f (xq n ) = f (0) for all x ∈ X.
Let C(X) denote the space of all q-regular at zero functions defined on X with values in R. C(X) associated with the norm function f = sup {|f (xq n )| : x ∈ X, n ∈ N 0 } , is a normed space. The q-integration by parts rule [27] is and f, g are q-regular at zero functions. For p > 0, and Y is A q,t or A * q,t , the space L p q (Y ) is the normed space of all functions defined on Y such that If p = 2, then L 2 q (Y ) associated with the inner product is a Hilbert space. By a weighted L 2 q (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 2 q (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 AC 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 , c.f. [27,Definition 4.3.1]. I.e.
The space AC (n) q (A * q,t ) (n ∈ N) is the space of all functions defined on X such that f, D q f, . . . , D n−1 q f are q-regular at zero and D n−1 q f ∈ AC q (A * q,t ), c.f. [27,Definition 4.3.2]. Also it is proved in [27,Theorem 4.6] that a function f ∈ AC D k q f (0) Γ q (k + 1) In particular, f ∈ AC(A * q,t ) if and only if f is q-regular at zero such that D q f ∈ L 1 q (A * q,t ). It is worth noting that in [27], 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 [28], Mansour studied the problem where p(x) = 0 and w α > 0 for all x ∈ A * q,a , p, r, w α are real valued functions defined in A * q,a and the associated boundary conditions are with c 2 1 +c 2 2 = 0 and d 2 1 +d 2 2 = 0. it is proved that the eigenvalues are real and the eigenfunctions associated to different eigenvalues are orthogonal in the Hilbert space L 2 q (A * q,a , w α ). 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 [29] introduced basic Fourier series for functions defined on a q-linear grid of the form {±q n : n ∈ N 0 } ∪ {0}.
In Section 3, we reformulate Cardoso's results for functions defined on a qlinear grid of the form {±aq n : n ∈ N 0 } ∪ {0}. In Section 4, we introduce a fractional q-analogue 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 criteria for the smallest eigenvalue.

Fractional q-Calculus
This section includes the definitions and properties of the left sided and right sided Riemann-Liouville q-fractional operators which we need in our investigations.
The left sided Riemann-Liouville q-fractional operator is defined by This definition is introduced by Agarwal in [30] when a = 0 and by Rajković et.al [31] for a = 0. The right sided Riemann-Liouville q-fractional operator by see [28]. The left sided Riemann-Liouville q-fractional operator satisfies the semigroup property I α q,a + I β q,a + f (x) = I α+β q,a + f (x). The case a = 0 is proved in [30] and the case a > 0 is proved in [31].
The right sided Riemann-Liouville q-fractional operator satisfies the semigroup property [28] for any function defined on A q,b and for any values of α and β. For α > 0 and α = m, the left and right side 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 [28]. 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 [27, pp. 124, 148], The following proposition is proved in [28] Proposition 2.1. Let α ∈ (0, 1).
(2.11) Set X = A q,a or A * q,a . Then Moreover, if f ∈ C(X) then We have also the following inequalities: 1. If f ∈ C(A * q,a ) then I α q,0 + f ∈ C(A * q,a ) and 2. If f ∈ L 1 q (X) then I α q,0 + f ∈ L 1 q (X) and 3. If f ∈ L 2 q (X) then I α q,0 + f ∈ L 2 q (X) and where 4. If α > 1 2 and f ∈ L 2 q (X) then I α q,0 + f ∈ C(X) and 5. Since f 2 ≤ √ a f , we conclude that if f ∈ C(X) then I α q,0 + f ∈ L 2 q (X) and (2.16) 6. If f ∈ C(A * q,a ) then I α q,a − f ∈ C(A * q,a ) and 7. If f ∈ L 1 q (X) then I α q,a − f ∈ L 1 q (X) and 8. If α = 1 2 and f ∈ L 2 q (X) then I α q,a − f ∈ L 1 q (X) and The following lemmas are introduced and proved in [28] Lemma 2.2. Let α > 0. If (a) f ∈ L 1 q (X) and g is a bounded function on A q,a , or (b) α = 1 2 and f, g are L 2 q (X) functions then , and g is a bounded function on A * q,a such that D α q,a − g ∈ L 1 q (A * q,a ) then

Basic Fourier series on q-Linear grid and some properties
The purpose of this section is to reformulate Cardoso's results of Fourier series expansions for functions defined on the q-linear grid A q := {q n , n ∈ N 0 } to functions defined on q-linear grids A q,a :=:= {±aq n , n ∈ N 0 }, a > 0. Cardoso in [29] defined the space of all q-linear Hölder functions on the q-linear grid A q . We generalize his definition for functions defined on a q-linear grid of the form A q,a , a > 0.
Definition 3.2. The q-trigonometric functions S q (z) and C q (z) are defined for z ∈ C by, see [29,32] One can verify that where z ∈ C and w ∈ C is a fixed parameter. A modification of the orthogonality relation given in [32,Theorem 4.1] is Cardoso introduced a sufficient condition for the uniform convergence of the basic Fourier series as k → ∞ for any q ∈ (0, 1). In the following we give a modified version of Cardoso's result for any function defined on the q-linear grid A q,a , a > 0.
is a q-linear Hölder function of order λ > 1 2 , then the q-Fourier series converges uniformly to the function f on the q-linear grid A q,a .
Proof. The proof is a modification of the proof of [29, Theorem 4.1] and is omitted.
in [29,Theorem 4.1] by the weakest condition that f is q-regular at zero. Because he needs this condition only to guarantee that lim n→∞ f (q n−1/2 ) = lim n→∞ f (−q n−1/2 ) and this holds if f is q-regular at zero. See [27, (1.22)] for a function which is q-regular at zero but not continuous at zero.

A modified version of [29, Theorem 3.5] is
then the q-Fourier series (3.1) converges uniformly on A q,a .

A modified version of [29, Corollary 4.3] is
Corollary 3.7. If f is continuous and piecewise smooth on a neighborhood of the origin, then the corresponding q-Fourier series S q (f ) converges uniformly to f on the set of points A q,a . where converges uniformly to the function f on the q-linear grid A q,a .
Proof. The proof follows from (3.4) by considering the function g( making the substitution u = qt and using that g is an odd function, we obtain the required result. Definition 3. 9. Let (f n ) n be a sequence of functions in C(A * q,a ). We say that is an odd function satisfying D k q g (k = 0, 1, 2) is continuous and piecewise smooth function in a neighborhood of zero, and satisfying the boundary condition then g can be approximated in the q−mean by a linear combination where at the same time D k q g n (k=1,2) converges in q-mean to the D k q g. Moreover, the coefficients c (n) r need not depend on n and can be written simply as c r .
Proof. We consider the q-sine Fourier transform of D 2 q g. Hence Consequently, Applying the q-integration by parts rule (1.2) gives Note that a 0 (D q g) = 0 because g(0) = g(a) = 0. Again by q-integrating the two sides of (3.4), we obtain One can verify that Hence the right hand sides of (3.4) and (3.5) are the q-Fourier series of D q g and g, respectively. Hence the convergence is uniform in C(A * q,a ) and L q 2 (A * q,a ) norms.

q-Fractional Variational Problems
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 called the calculus of finite difference, the calculus of variations has also both of the discrete and continuous forms. For a brief history of the continuous calculus of variations, see [33]. The discrete calculus of variations starts in 1948 by Fort in his book [34] where he devoted a chapter to the finite analogue of the calculus of variations, and he introduced a necessary condition analogue to the Euler equation and also a sufficient condition. The paper of Cadzow [35,1969] was the first paper published in this field, then Logan developed the theory in his Ph.D. thesis [36,1970] and in a series of papers [37][38][39][40]. See also the Ph.D. thesis of Harmsen [41] for a brief history for the discrete variational calculus, and for the developments in the theory, see [42][43][44][45][46][47][48]. In 2004, a q-version of the discrete variational calculus is firstly introduced by Bangerezako in [49] for a functions defined in the form where q α and q β are in the uniform lattice A * q,a for some a > 0 such that α > β, provided that the boundary conditions He introduced a q-analogue of the Euler-Lagrange equation and applied it to solve certain isoperimetric problem.
under the boundary condition y(a) = α, y(b) = β 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 [16] in 2002. Then, he extends his result for variational problems include Riez fractional derivatives in [17]. Numerous works have been dedicated to the subject since Agarwal work. See for example [4,[12][13][14][15][51][52][53].
In this Section, we shall derive Euler-Lagrange equation for a q-variational problem when the integrand include a left-sided q-Caputo fractional derivative and we also solve a related isoperimetric problem. From now on, we fix α ∈ (0, 1), and define a subspace of C(A * q,a ) by  N) by ∂ i f we mean the partial derivative of f with respect to the ith variable, i = 1, 2, . . . , n. In the sequel, we shall need the following definition from [54].
Definition 4.1. Let A ⊆ R and g : A×] − θ, θ[→ R. We say that g(t, ·) is continuous at θ 0 uniformly in t, if and only if ∀ǫ > 0 ∃δ > 0 such that Furthermore, we say that g(t, ·) is differentiable at θ 0 uniformly in t if and only if ∀ǫ > 0 ∃δ > 0 such that We now present first order necessary conditions of optimality for functionals, defined on 0 E α a , of the type where F : A * q,a × R × R → R is a given function. We assume that 2. F ·, y(·), c D α q,0 + (·) , δ i F ·, y(·), c D α q,0 + (·) (i = 2, 3) are q-regular at zero. 3. δ 3 F has a right Riemann-Liouville fractional q-derivative of order α which is q-regular at zero.
Then J has a local maximum at y 0 if ∃δ > 0 such that J(y) ≤ J(y 0 ) for all y ∈ 0 E α a with y − y 0 < δ, and J has a local minimum at y 0 if ∃δ > 0 such that J(y) ≥ J(y 0 ) for all y ∈ S with y − y 0 < δ.
J is said the have a local extremum at y 0 if it has either a local maximum or local minimum. Moreover, in the two cases, if γ is q-regular at zero, then γ(0) = 0.
Proof. To prove (i), we fix k ∈ N 0 and set h k (x) = 1, x = aq k 0, otherwise . Then h k ∈ L 2 q (0, a). Substituting in (4.2) yields Thus, γ(aq k ) = 0 for all k ∈ N 0 . Clearly if γ is q-regular at zero, then The proof of (ii) is similar and is omitted. Proof. Let c be the constant defined by the relation c = 1 So, h and D q h are q-regular at zero functions such that Therefore, α(x) = c for all x ∈ A q,a . But α is q-regular at zero, hence α(0) = 0. This yields the required result.
Theorem 4.5. Let y ∈ c Var(0, a) be a local extremum of J. Then, y satisfies the Euler-Lagrange equation Proof. Let y be a local extremum of J and let η be arbitrary but fixed variation function of y. Define Φ(ǫ) = J(y + ǫη).

A q-Fractional Isoperimetric Problem
In the following, we shall solve the q-fractional isoperimetric problem: Given a functional J as in (4.1), which function minimize (or maximize) J, when subject to the boundary conditions y(0) = y 0 , y(a) = y a (4.6) and the q-integral constraint where l is a fixed real number. Here, similarly as before, 2. G ·, y(·), c D α q,0 + (·) , δ i G ·, y(·), c D α q,0 + (·) (i = 2, 3) are q-regular at zero. 3. δ 3 G has a right Riemann-Liouville fractional q-derivative of order α which is q-regular at zero.
A function y ∈ E that satisfies (4.6) and (4.7) is called admissible.
Definition 4.6. An admissible function y is an extremal for I in (4.7) if it satisfies the equation Theorem 4.7. Let y be a local extremum for J given by (4.1), subject to the conditions (4.6) and (4.7). If y is not an extremal of the function I, then there exits a constant λ such that y satisfies where H := F − λ G.
Proof. Let η 1 , η 2 ∈ c Var(0, a) be two functions, and let ǫ 1 and ǫ 2 be two real numbers, and consider the new function of two parameters.
The reason why we consider two parameters is because we can choose one of them as a function of the other in order toy satisfies the q-integral constraint (4.7). LetȊ It follows by the q-integration by parts rule (1.2) that Since y is not an extremal of I, then there exists a function η 2 satisfying the condition ∂Ȋ ∂ǫ2 | (0,0) = 0. Hence, from the fact thatȊ(0, 0) = 0 and the Implicit Function Theorem, there exists a C 1 function ǫ 2 (·), defined in some neighborhood of zero, such thatȊ (ǫ 1 , ǫ 2 (ǫ 1 )) = 0.

Existence of Discrete Spectrum for a fractional q-Sturm-Liouville problem
In this section, we use the q-calculus of variations we developed in Sections 4 to investigate the existence of solutions of the qFSLP Theorem 5.1 (Arzela-Ascoli Theorem). If a sequence {f n } n in C(X) is bounded and equicontinuous then it has a uniformly convergent subsequence.
In our q-setting, we take X = A * q,a . Hence f ∈ C(A * q,a ) if and only if f is q-regular at zero, i.e. f (0) := lim n→∞ f (aq n ).
Remark 5.2. A question may be raised why in (5.1) we have only x ∈ A * q,qa instead of A * q,a . The reason for that is the qFSLP (5.1)-(5.2) will be solved by using the q-fractional isoperimetric problem developed in Theorem 4.7, and its q-Euler-Lagrange equation (4.9) holds only for x ∈ A * q,qa . Also, in order that (5.1) holds at x = a, we should have D α q,a − (p(·) c D α q,0 + y(·))(a) = 0 and this holds only if p(a) c D α q,0 + y(a) = 0 which may not hold.
Theorem 5.3. Let 1 2 < α < 1. Assume that the functions p, r, w α are defined on A * q,a and satisfying the conditions (i) w α is a positive continuous function on [0, a] such that D k q 1 wα (k = 0, 1, 2) are bounded functions on A q,a , (ii) r is a bounded function on A q,a , The q-fractional Sturm-Liouville problem (5.1)-(5.2) has an infinite number of eigenvalues λ (1) , λ (2) , . . . , and to each eigenvalue λ (n) there is a corresponding eigenfunction y (n) which is unique up to a constant factor. Furthermore, eigenfunctions y (n) form an orthogonal set of solutions in the Hilbert space L 2 q (A * q,a , w α ).
Proof. The qFSLP (5.1)-(5.2) can be recast as a q-fractional variational problem. Let and consider the problem of finding the extremals of J subject to the boundary condition y(0) = y(a) = 0, (5.4) and the isoperimetric constraint The q-fractional Euler-Lagrange equation for the functional I is which is satisfied only for the trivial solution y = 0, because w α is positive on A q,a . So, no extremals for I can satisfy the q-isoperimetric condition. If y is an extremal for the q-fractional isoperimetric problem, then from Theorem 4.7, there exists a constant λ such that y satisfies the q-fractional Euler-Lagrange equation (4.9) in A * q,qa but this is equivalent to the qFSLP (5.1) In the following, we shall derive a method for approximating the eigenvalues and the eigenfunctions at the same time similar to the technique in [4,10]. The proof follows in 6 steps.
Step 1. First let us point out that functional (5.3) is bounded from below. Indeed, since p, w α are positive on A q,a , then According to Ritz method [10, P. 201], we approximate a solution of (5.3)-(5.4) using the following q-trigonometric functions with the coefficients depending on w α : Observe that y m (0) = y m (a) = 0. By substituting (5.6) into (5.3) and (5.5) we obtain If this procedure is carried out for m = 1, 2, . . ., we obtain a sequence of numbers λ 2 , . . . , and a corresponding sequence of functions 2 (x), y 3 (x), . . . .
Noting that σ m is the subset of σ m+1 obtained by setting β m+1 = 0, while Since increasing the domain of definition of a function can only decrease its minimum. It follows from (5.9) and the fact that J(y) is bounded from below that its limit Step 2. We shall prove that the sequence (y (1) m ) m∈N contains a uniformly convergent subsequence. From now on, for simplicity, we shall write y m instead of y (1) m . Recall that is convergent, so it must be bounded, i.e., there exists a constant M 0 > 0 such that Therefore, for all m ∈ N it holds the inequality Since y m (0) = 0, then from (2.15) and (5.10) for α > 1/2. Hence, (y m ) m is uniformly bounded on A * q,a . Now we prove that the sequence (y m ) m is equicontinuous. Let x 1 , x 2 ∈ A q,a . Assume that x 1 < x 2 . Applying the Schwarz's inequality and (2.9) . Since x 1 < x 2 , then we have Using the inequality Hence {y m } is equicontinuous. Therefore, from Arzelà-Ascoli Theorem for metric spaces, a uniformly convergent subsequence (y mn ) n∈N exists. It means that we can find y (1) ∈ C(A * q,a ) such that Step 3 From the Lagrange multiplier at [β] = (β , j = 1, 2, . . . , m.
By multiplying each equation by an arbitrary constant c j and summing from 1 to m, we obtain According to Proposition 3.10, we can choose the coefficients c j such that there exists a function g satisfying lim m→∞ D k q g m = D k q g (k = 0, 1, 2) and the convergence is in L 2 q (A * q,a ) norm. Hence lim m→∞ D k q h m − D k q h 2 = 0 (k = 0, 1, 2). (5.13) We can write (5.12) in the form Since y m (0) = 0, then from (2.11) c D α q,0 + y m = D α q,0 + y m = D q I 1−α q,0 + y m . Then replacing c D α q,0 + y m by D α q,0 + y m in (5.14) and applying the q-integration by parts rule (1.2), we obtain .
In the following we shall prove that (5.16) For the first q-integral in (5.16), by adding and subtracting the term to the integrand, we obtain and the first q-integral vanishes as m → ∞. As, for the second q-integral, we add and subtract the term p(qx)D q c D α q,0 + h(x) I 1−α q,0 + y m (qx). This gives if D q f (0) = 0, and since lim m→∞ D 2 q h m − D 2 q h 2 = 0 then from (2.14), the second q-integral tends to zero as m tends to ∞. For the next two terms, we have for x = 0, a resulting from the convergence of the sequence y m − y → 0, and at the points Therefore, Similarly, the last term in the estimation (5.16) vanishes as m → ∞.
Step 4 Since Hence, from Lemma 4.4 there is a constant c such that Acting on the two sides of (5.19) by − 1 q D q −1 , we obtain Hence, y is a solution of the qFSLP.
Step 5 In the following, we show that (y (1) m ) m∈N itself converges to y (1) . First, from Theorem [28, Theorem 3.12], for a given λ the solution of (5.20) subject to the boundary conditions y(0) = y(a) = 0 (5.21) and the normalization condition is unique except for a sign. Let us assume that y (1) solves (5.20) and the corresponding eigenvalue is λ = λ (1) . Suppose that y (1) is nontrivial, i.e., there exists x 0 ∈ A * q,qa such that y(x 0 ) = 0 and choose the sign so that y (1) (x 0 ) > 0. Similarly, for all m ∈ N, let y  m ) does not converges to y (1) . It means that we can find another subsequence of y (1) m such that it converges to another solution y (1) . But for λ = λ (1) , the solution of (5.20)-(5.22) is unique except for a sign, hence y (1) = −y (1) and we must have y (1) (x 0 ) < 0. However, this is impossible because for all m ∈ N, y (1) m (x 0 ) ≥ 0. A contradiction, hence the solution is unique.
Step 6 In order to find eigenfunction y (2) and the corresponding eigenvalue λ (2) , we minimize the functional (5.3) subject to (5.4) and (5.5) but now with an extra orthogonality condition a 0 y(x)y (1) If we approximate the solution by i.e. they lay in the (m − 1)-dimensional sphere. As before, we find that the function J([β]) has a minimum λ (2) m and there exists λ (2) such that because J(y) is bounded from below. Moreover, it is clear that The function y (2) m defined by achieves its minimum λ 1 , . . . , β (2) m is the point satisfying (5.8) and (5.23). By the same argument as before, we can prove that the sequence (y (2) m ) converges uniformly to a limit function y (2) , which satisfies the qFSLP (5.1) with λ (2) , boundary conditions (5.4) and orthogonality condition (5.5). Therefore, solution y (2) of the qFSLP corresponding to the eigenvalue λ (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 λ (1) < λ (2) instead of (5.24). Finally, if we repeat the above procedure with similar modifications, we can obtain eigenvalues λ (3) , λ (4) , . . . and corresponding eigenvectors y (3) , y (4) , . . .. where J(y) and I(y) are given by (5.3) and (5.5), respectively.

The first eigenvalue
Theorem 5.5. Let y be a non zero function satisfying y and c D α q,0 + y are in C(A q,a * ) and y(0) = y(a) = 0. Then, y is a minimizer of R(y) and R(y) = λ if and only if y is an eigenfunction of problem (5.1)-(5.2) associated with λ. That is, the minimum value of R at y is the first eigenvalue λ (1) .
This proves the necessity. Now we prove the sufficiency. Assume that y is an eigenfunction of (5.1)-(5.2) associated with an eigenvalue λ. Then D α q,a − P (x) c D α q,0 + y(x) + r(x)y(x) = λw α (x)y(x), x ∈ A * q,qa . (5.25) Multiply (5.25) by y and calculate the q-integration from 0 to a, we obtain a 0 y(x)D α q,a − P (x) c D α q,0 + y(x) + r(x)y 2 (x) d q x = λ a 0 w α (x)y 2 (x) d q x.
I.e. R(y) = λ. Therefore, any minimum value of J is an eigenvalue and it is attained at the associated eigenfunction. Therefore the minimum value of J is the smallest eigenvalue.

Conclusion and future work
This paper is the first paper deals with variational problems of functionals defined in terms of Jackson q-integral on finite domain and the left sided Caputo q-derivative appears in the integrand. We give a fractional q-analogue 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 [10] and fractional Sturm-Liouville problems in [4]. This complete the work started by the author in [28], and generalizes the study of integer Sturm-Liouville problem introduced by Annaby and Mansour in [1]. A similar study for the fractional Sturm-Liouville problem c D q,a − p(x)D α q,0 + y(x) + r(x)y(x) = λw α (x)y(x), is in progress.