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Generalizing the variational theory on time scales to include the delta indefinite integral

We prove necessary optimality conditions of Euler-Lagrange type for generalized problems of the calculus of variations on time scales with a Lagrangian depending not only on the independent variable, an unknown function and its delta derivative, but also on a delta indefinite integral that depends on the unknown function. Such kind of variational problems were considered by Euler himself and have been recently investigated in [Methods Appl. Anal. 15 (2008), no. 4, 427-435]. Our results not only provide a generalization to previous results, but also give some other interesting optimality conditions as special cases.


Introduction
In what follows, T denotes a time scale with operators σ, ρ, µ, ν, ∆, and ∇ [1,2]. We also assume that there exist at least three points on the time scale: a, b, s ∈ T with a < b < s, and that the operator σ is delta differentiable. The main purpose of this paper is to generalize the Calculus of Variations on time scales (see [3][4][5][6][7][8] and references therein) by considering the variational problem where "extr" denotes "extremize" (i.e., minimize or maximize) and the variable z in the integrand is itself expressed in terms of an indefinite integral z(t) = t a g τ, y σ (τ ), y ∆ (τ ) ∆τ.
In Subsection 3.1 we obtain the Euler-Lagrange equation for problem (1) in the class of functions y ∈ C 1 rd (T, R) satisfying the boundary conditions y(a) = α and y(b) = β for some fixed α, β ∈ R (cf. Theorem 4). Accordingly to Fraser [9], the ideia of generalizing the basic problem of the calculus of variations by considering a variational integral depending also on an indefinite integral (in the classical setting, that is, when T = R) was first considered by Euler in 1741. Our Euler-Lagrange equation is a generalization of the Euler-Lagrange equations obtained by Euler [9,Eq. (8)], Bohner [3], and Gregory [10]. The transversality conditions for problem (1) are obtained in Subsection 3.2. In Subsection 3.3 we prove a necessary optimality condition for the isoperimetric problem: problem (1)- (2) subject to the delta integral constraint J (y) = b a F t, y σ (t), y ∆ (t), z(t) ∆t = γ for some given γ ∈ R. In Subsection 3.4 we explain how it is possible to prove backward versions of our results by means of Caputo's duality [11] (see also [12]). Finally, in Section 4 we provide some applications of our main results.

Preliminaries
For definitions, notations and results concerning the theory of time scales we refer the readers to the comprehensive books [1,2]. All the intervals in this paper are time scale intervals. Throughout the text we denote by ∂ i f the partial derivative of a function f with respect to its ith argument. We assume that 1. the admissible functions y belong to the class C 1 rd (T, R); 2. (t, y, v, z) → L(t, y, v, z) and (t, y, v, z) → F (t, y, v, z) have continuous partial derivatives with respect to y, v, z for all t ∈ [a, b]; 3. (t, y, v) → g(t, y, v) has continuous partial derivatives with respect to y, v for all t ∈ [a, b]; 4. t → L(t, y σ (t), y ∆ (t), z(t)) and t → F (t, y σ (t), y ∆ (t), z(t)) belong to the class C rd (T, R) for any admissible function y; , z(t)) and t → ∂ 3 g(t, y σ (t), y ∆ (t)) belong to the class C 1 rd (T, R) for any admissible function y. Definition 1. An admissible function y * ∈ C 1 rd (T, R) is said to be a local minimizer (resp. local maximizer) to problem (1)-(2) if there exists δ > 0 such that L(y * ) ≤ L(y) (resp. L(y * ) ≥ L(y)) for all admissible y satisfying the boundary conditions (2) and y − y * < δ, where Definition 2. We say that η ∈ C 1 rd (T, R) is an admissible variation to problem (1)-(2) provided η (a) = η (b) = 0.
The following result, known as the fundamental lemma of the calculus of variations on time scales, is an important tool in the proofs of our main results. The proof of Lemma 3 follows immediately from [13,Theorem 15] and the duality arguments of Caputo [11].

Euler-Lagrange equation
Theorem 4 (Necessary optimality condition to (1)- (2)). Suppose that y * is a local minimizer or local maximizer to problem (1)- (2). Then y * satisfies the Euler-Lagrange equation Proof. Suppose that y * is a local minimizer (resp. maximizer) to problem (1)- (2). Let η be an admissible variation and define the function φ : R → R by φ (ǫ) := L(y * + ǫη). It is clear that a necessary condition for y * to be an extremizer is given by φ ′ (0) = 0. Note that Using the integration by parts formula, we obtain Using again integration by parts in the last integral we obtain b a ∂ 3 g{y * }(t) · Since η(a) = η(b) = 0, then From the optimality condition φ ′ (0) = 0 we conclude, by the fundamental lemma of the calculus of variations on time scales (Lemma 3), that for all t ∈ [a, b] κ , proving the desired result. 3. The Euler-Lagrange equation for the basic problem of the Calculus of Variations on time scales (see, e.g., [3]) is easily obtained from Theorem 4: in this case, ∂ 4 L = 0 and therefore we get the equation Remark 6. Theorem 4 gives the Euler-Lagrange equation in the delta-differential form. As in the classical case, one can obtain the Euler-Lagrange equation in the integral form. More precisely, the Euler-Lagrange equation in the delta-integral form to problem (1)- (2) is

Natural boundary conditions
We now consider the case when the values y(a) and y(b) are not necessarily specified.

if y(a) is free, then the natural boundary condition
holds; 2. if y(b) is free, then the natural boundary condition holds.
Proof. Suppose that y * is a local minimizer (resp. maximizer) to problem (1). Let η ∈ C 1 rd (T, R) and define the function φ : R → R by φ (ǫ) := L(y * + ǫη). It is clear that a necessary condition for y * to be an extremizer is given by φ ′ (0) = 0. From the arbitrariness of η, and using similar arguments as the ones used in the proof of Theorem 4, we conclude that y * satisfies the Euler-Lagrange equation (3).
is free, then we restrict ourselves to those η for which η(b) = 0. Therefore, Using the Euler-Lagrange equation (3) into (6) we obtain From the arbitrariness of η it follows that 2. Suppose now that y(b) is free. If y(a) = α, then η(a) = 0; if y(a) is free, then we restrict ourselves to those η for which η(a) = 0. Thus, Using the Euler-Lagrange equation (3) into (7), and from the arbitrariness of η, it follows that Remark 8. In the classical setting, T = R and L does not depend on z. Then, equations (4) and (5) reduce to the well-known natural boundary conditions ∂ 3 L(a, y * (a), y ′ * (a)) = 0 and ∂ 3 L(b, y * (b), y ′ * (b)) = 0, respectively.

Isoperimetric problem
We now study the isoperimetric problem on time scales with a delta integral constraint, both for normal and abnormal extremizers. The problem consists of minimizing or maximizing the functional where the variable z in the integrand is itself expressed in terms of an indefinite delta integral in the class of functions y ∈ C 1 rd (T, R), satisfying the boundary conditions y(a) = α and y(b) = β (9) and the delta integral constraint for some given α, β, γ ∈ R.
Definition 10. We say that y ∈ C 1 rd (T, R) is an extremal to J if y satisfies the Euler-Lagrange equation (3) relatively to J . An extremizer (i.e., a local minimizer or a local maximizer) to problem (8)-(10) that is not an extremal to J is said to be a normal extremizer; otherwise (i.e., if it is an extremal to J ), the extremizer is said to be abnormal.
Theorem 11 (Necessary optimality condition for normal extremizers of (8)- (10)). Suppose that y * ∈ C 1 rd (T, R) gives a local minimum or a local maximum to the functional L subject to the boundary conditions (9) and the integral constraint (10). If y * is not an extremal to J , then there exists a real λ such that y * satisfies the equation Proof. Suppose that y * ∈ C 1 rd (T, R) is a normal extremizer to problem (8)- (10). Define the real functions φ, ψ : where η 2 is a fixed variation (that we will choose later) and η 1 is an arbitrary variation. Note that Integration by parts and η 2 (a) = η 2 (b) = 0 gives Since, by hypothesis, y * is not an extremal to J , then we can choose η 2 such that ∂ψ ∂ǫ 2 (0, 0) = 0.
Theorem 12 (Necessary optimality condition for normal and abnormal extremizers of (8)- (10)). Suppose that y * ∈ C 1 rd (T, R) gives a local minimum or a local maximum to the functional L subject to the boundary conditions (9) and the integral constraint (10). Then there exist two constants λ 0 and λ, not both zero, such that y * satisfies the equation Proof. Following the proof of Theorem 11, since (0, 0) is an extremizer of φ subject to the constraint ψ = 0, the abnormal Lagrange multiplier rule (cf., e.g., [14]) guarantees the existence of two reals λ 0 and λ, not both zero, such that λ 0 ∇φ = λ∇ψ. Therefore, From the arbitrariness of η 1 and Lemma 3 it is clear that equation (13) holds for all t ∈ [a, b] κ , where H = λ 0 L − λF .

Corollary 14.
Let h > 0 and suppose that y * is a solution to the discrete-time problem in the class of functions y satisfying the boundary conditions y(a) = α and y(b) = β and the constraint for some given α, β, γ ∈ R. Then there exist two constants λ 0 and λ, not both zero, such that Proof. Choose T = hZ, where a, b ∈ T. The result follows from Theorem 12. F (t, y(qt), D q y(t), z(t)) = γ for some given α, β, γ ∈ R. Then there exist two constants λ 0 and λ, not both zero, such that The result follows from Theorem 12.

Duality
In the paper [11] (see also [19,20]) Caputo states that the delta calculus and the nabla calculus on time scales are the "dual" of each other. A Duality Principle is presented, that basically asserts that it is possible to obtain results for the nabla calculus directly from results on the delta calculus and vice versa. Using the duality arguments of Caputo it is possible to prove easily the nabla versions of Theorem 4, Theorem 7, Theorem 11 and Theorem 12.
In what follows we assume that there exist at least three points on the time scale: r, a, b ∈ T with r < a < b, and that the operator ρ is nabla differentiable. The following theorem is the nabla version of Theorem 12, where the variational problem consists of minimizing or maximizing the functional the variable z in the integrand being itself expressed in terms of a nabla indefinite integral in the class of functions y ∈ C 1 ld (T, R) satisfying the boundary conditions y(a) = α and y(b) = β (15) and the nabla integral constraint for some given α, β, γ ∈ R. We assume that 1. the admissible functions y belong to the class C 1 ld (T, R); 2. (t, y, v, z) → L(t, y, v, z) and (t, y, v, z) → F (t, y, v, z) have continuous partial derivatives with respect to y, v, z for all t ∈ [a, b]; 3. (t, y, v) → g(t, y, v) has continuous partial derivatives with respect to y, v for all t ∈ [a, b]; 4. t → L(t, y ρ (t), y ∇ (t), z(t)) and t → F (t, y ρ (t), y ∇ (t), z(t)) belong to the class C ld (T, R) for any admissible function y; , z(t)) and t → ∂ 3 g(t, y ρ (t), y ∇ (t)) belong to the class C 1 ld (T, R) for any admissible function y.
Theorem 16 (Necessary optimality condition for normal and abnormal extremizers of (14)- (16)). Suppose that y * ∈ C 1 ld (T, R) gives a local minimum or a local maximum to the functional L subject to the boundary conditions (15) and the integral constraint (16). Then there exist two constants λ 0 and λ, not both zero, such that y * satisfies the equation Remark 17. Theorem 2 of [21] is a corollary of our Theorem 16: in that case ∂ 4 H = 0, and one obtains From Theorem 4, via duality, one can easily obtain the Euler-Lagrange equation for the nabla problem (14)-(15) (or from Theorem 16 noting that, since there is no nabla integral constraint, F = 0 and γ = 0).
Theorem 18 (Necessary optimality condition to (14)- (15)). Suppose that y * is a local minimizer or local maximizer to problem (14)- (15). Then y * satisfies the Euler-Lagrange equation Remark 19. As a corollary of Theorem 18 we obtain the Euler-Lagrange equation for the basic problem of the calculus of variations on nabla calculus [13] (see also [22]). In that case ∂L 4 = 0 and one obtains that Applying the duality arguments of Caputo to Theorem 7 the following result is obtained.
Theorem 21 (Natural boundary conditions to (14)). Suppose that y * is a local minimizer (resp. local maximizer) to problem (14). Then y * satisfies the Euler-Lagrange equation (17). Moreover, 1. if y(a) is free, then the natural boundary condition is free, then the natural boundary condition holds.

Applications
From now on we assume that T satisfies the following condition (H): (H) for each t ∈ T, ρ(t) = a 1 t + a 0 for some a 1 ∈ R + and a 0 ∈ R.
Proposition 23. Suppose that T satisfies condition (H), ξ is a real parameter, and k ∈ R is a given constant. Suppose that f : R 2 → R is a C 2 function that satisfies the conditions: , for all ξ and for all admissible function y; (A2) ∂ 2 1,1 f (y ρ (t), ξ) = 0 for all t in some non-degenerate interval I ⊆ [a, b], for all ξ and for all admissible function y.
Observe that the solution to the class of problems considered in Proposition 23 is a constant function that depends only on the boundary conditions (and the isoperimetric constraint) but not explicitly on the integrand function and its parameters.
Remark 24. By the isoperimetric constraint (16), a necessary condition for the problem of Proposition 23 to have a solution is that α = γ b − a .
Remark 25. Let b be a left dense point. Then, by the boundary conditions (15), a necessary condition for the problem of Proposition 23 to have solution is that α = β.
We end the paper with an example of application of the nabla version of Theorem 11.
Note that any extremal to J does not satisfy the isoperimetric constraint (20). Hence, this problem has no abnormal extremizers and, by the nabla version of Theorem 11, there exists λ ∈ R such that y * satisfies the equation L(t, y, v, z) = v 2 − q(t)y 2 + 2z, g(t, y, v) = v, and F (t, y, v, z) = y 2 .
Note that in the basic problem of calculus of variations on time scales, ∂ 4 H = 0, and we obtain the nabla version of the well known Sturm-Liouville eigenvalue equation: y ∇ 2 (t) + q(t)y ρ (t) + λy ρ (t) = 0, t ∈ [a, b] κ 2 (see [15,24]). The study of solutions to equation (22) in the case ∂ 4 H = 0 is an interesting open problem.