Hahn's Symmetric Quantum Variational Calculus

We introduce and develop the Hahn symmetric quantum calculus with applications to the calculus of variations. Namely, we obtain a necessary optimality condition of Euler-Lagrange type and a sufficient optimality condition for variational problems within the context of Hahn's symmetric calculus. Moreover, we show the effectiveness of Leitmann's direct method when applied to Hahn's symmetric variational calculus. Illustrative examples are provided.


1.
Introduction. Due to its many applications, quantum operators are recently subject to an increase number of investigations [24][25][26]. The use of quantum differential operators, instead of classical derivatives, is useful because they allow to deal with sets of nondifferentiable functions [4,10]. Applications include several fields of physics, such as cosmic strings and black holes [27], quantum mechanics [12,29], nuclear and high energy physics [18], just to mention a few. In particular, the q-symmetric quantum calculus has applications in quantum mechanics [17].
In 1949, Hahn introduced his quantum difference operator [13], which is a generalization of the quantum q-difference operator defined by Jackson [14]. However, only in 2009, Aldwoah [1] defined the inverse of Hahn's difference operator, and short after, Malinowska and Torres [24] introduced and investigated the Hahn quantum variational calculus. For a deep understanding of quantum calculus, we refer the reader to [2,5,6,11,15,16] and references therein.
For a fixed q ∈ ]0, 1[ and an ω ≥ 0, we introduce here the Hahn symmetric difference operator of function f at point t = ω 1 − q bỹ Our main aim is to establish a necessary optimality condition and a sufficient optimality condition for the Hahn symmetric variational problem where α and β are fixed real numbers, and extremize means maximize or minimize. Problem (P) will be clear and precise after definitions of Section 2. We assume that the Lagrangian L satisfies the following hypotheses: (H1) (u, v) → L (t, u, v) is a C 1 R 2 , R function for any t ∈ I; (H2) t → L t, y σ (t) ,D q,ω [y] (t) is continuous at ω 0 for any admissible function y; (H3) functions t → ∂ i+2 L t, y σ (t) ,D q,ω [y] (t) belong to Y 1 [a, b] q,ω , R for all admissible y, i = 0, 1; where I is an interval of R containing ω 0 := ω 1 − q , a, b ∈ I, a < b, and ∂ j L denotes the partial derivative of L with respect to its jth argument. In Section 2 we introduce the necessary definitions and prove some basic results for the Hahn symmetric calculus. In Section 3 we formulate and prove our main results for the Hahn symmetric variational calculus. New results include a necessary optimality condition (Theorem 3.8) and a sufficient optimality condition (Theorem 3.10) to problem (P). In Section 3.3 we show that Leitmann's direct method can also be applied to variational problems within Hahn's symmetric variational calculus. Leitmann introduced his direct method in the sixties of the 20th century [19], and the approach has recently proven to be universal: see, e.g., [3,8,9,[20][21][22][23]28].

2.
Hahn's symmetric calculus. Let q ∈ ]0, 1[ and ω ≥ 0 be real fixed numbers. Throughout the text, we make the assumption that I is an interval (bounded or unbounded) of R containing ω 0 := ω 1 − q . We denote by I q,ω the set I q,ω := qI+ω := {qt + ω : t ∈ I}. Note that I q,ω ⊆ I and, for all t ∈ I q,ω , one has q −1 (t − ω) ∈ I . For k ∈ N 0 , Definition 2.1. Let f be a real function defined on I. The Hahn symmetric difference operator of f at a point t ∈ I q,ω \ {ω 0 } is defined bỹ , provided f is differentiable at ω 0 (in the classical sense). We call toD q,ω [f ] the Hahn symmetric derivative of f .

Remark 2.
If ω > 0 and we let q → 1 in Definition 2.1, then we obtain the well known symmetric difference operatorD ω : Remark 3. If f is differentiable at t ∈ I q,ω in the classical sense, then In what follows we make use of the operator σ defined by σ (t) := qt + ω, t ∈ I. Note that the inverse operator of σ, σ −1 , is defined by σ −1 (t) := q −1 (t − ω). Moreover, Aldwoah [1, Lemma 6.1.1] proved the following useful result.

Remark 4.
With above notations, if t ∈ I q,ω \ {ω 0 }, then the Hahn symmetric difference operator of f at point t can be written as Lemma 2.3. Let n ∈ N 0 and t ∈ I. Then, where σ 0 ≡ id is the identity function.
Proof. The equality follows by direct calculations: The Hahn symmetric difference operator has the following properties.
Proof. For each t ∈ I\{ω 0 } we havẽ We now present two technical results that will be useful to prove the fundamental theorem of Hahn's symmetric integral calculus (Theorem 2.8).
The next result tell us that if a function f is continuous at ω 0 , then f is Hahn's symmetric integrable.
Corollary 1 (cf. [1]). Let a, b ∈ I, a < b, and f : I → R be continuous at ω 0 . Then, for s ∈ [a, b], the series +∞ n=0 q 2n+1 f σ 2n+1 (s) is uniformly convergent on I. Theorem 2.8 (Fundamental theorem of the Hahn symmetric integral calculus). Assume that f : I → R is continuous at ω 0 and, for each x ∈ I, define for all a, b ∈ I.
Proof. We note that function F is continuous at ω 0 by Corollary 1. Let us begin by considering x ∈ I\{ω 0 }. Then, where in the third equality we use Lemma 2.3, then The Hahn symmetric integral has the following properties.
Theorem 2.9. Let f, g : I → R be Hahn's symmetric integrable on I, a, b, c ∈ I, and α, β ∈ R. Then, Proof. Properties 1 to 4 are trivial. Property 5 follows from Theorem 2.4 and Theorem 2.8: sincẽ Proposition 1. Let c ∈ I, f and g be Hahn's symmetric integrable on I.
providing the desired equality.
As an immediate consequence, we have the following result.
Corollary 2. Let c ∈ I and f be Hahn's symmetric integrable on I. Suppose that As an example, consider the function f defined in [−5, 5] by For q = 1 2 and ω = 1, this function is Hahn's symmetric integrable because is continuous at ω 0 = 2. However, This example also proves that, in general, it is not true that b a f (t)d q,ω t ≤ b a |f (t)|d q,ω t for any a, b ∈ I.

3.
Hahn's symmetric variational calculus. We begin this section with some useful definitions and notations. For s ∈ I we set Let a, b ∈ I with a < b. We define the Hahn symmetric interval from a to b by Definition 3.1. We say that y is an admissible function to problem (P) if y ∈ Y 1 [a, b] q,ω , R and y satisfies the boundary conditions y (a) = α and y (b) = β.
Definition 3.2. We say that y * is a local minimizer (resp. local maximizer) to problem (P) if y * is an admissible function and there exists δ > 0 such that for all admissible y with y * − y 1 < δ. Before proving our main results, we begin with three basic lemmas.
3.1. Basic Lemmas. The following results are useful to prove Theorem 3.8.
(c) The case a = ω 0 and b = ω 0 is similar to (b). 2. If p = ω 0 , we assume, without loss of generality, that f (p) > 0. Since Therefore, there exists an order n 0 ∈ N such for all n > n 0 the inequalities (c) If b = ω 0 , the proof is similar to the previous case.

Optimality Conditions.
In this section we present a necessary optimality condition (the Hanh symmetric Euler-Lagrange equation) and a sufficient optimality condition to problem (P).
Definition 3.9. Given a Lagrangian L, we say that L (t, u, v) is jointly convex (resp. concave) in (u, v) if, and only if, ∂ i L, i = 2, 3, exist and are continuous and verify the following condition: Theorem 3.10. Suppose that a < b and a, b ∈ [c] q,ω for some c ∈ I. Also, assume that L is a jointly convex (resp. concave) function in (u, v). If y * satisfies the Hahn symmetric Euler-Lagrange equation (4), then y * is a global minimizer (resp. maximizer) to problem (P).
Proof. Let L be a jointly convex function in (u, v) (the concave case is similar). Then, for any admissible variation η, we have Using the integration by parts formula (2) and Lemma 2.5, we get Since y * satisfies (4) and η is an admissible variation, we obtain proving that y * is a minimizer to problem (P).
Example 1. Let q ∈ ]0, 1[ and ω ≥ 0 be fixed real numbers. Also, let I ⊆ R be an interval such that a := ω 0 , b ∈ I and a < b. Consider the problem If y * is a local minimizer to the problem, then y * satisfies the Hahn symmetric Euler-Lagrange equatioñ It is simple to check that function y * (t) = t is a solution to (6) satisfying the given boundary conditions. Since the Lagrangian is jointly convex in (u, v), then we conclude from Theorem 3.10 that function y * (t) = t is indeed a minimizer to problem (5).
The desired result follows immediately because the right-hand side of the above equality is a constant, depending only on the fixed-endpoint conditions y (a) = α and y (b) = β.