Directory UMM :Journals:Journal_of_mathematics:BJGA:
Efficiency and duality for multitime vector fractional
variational problems on manifolds
S¸tefan Mititelu and Mihai Postolache
Abstract. We consider a multitime scalar variational problem (SVP), a
multitime multiobjective variational problem (VVP) and a multitime vector fractional variational problem (VFP). For (SVP) we establish necessary optimality conditions. For the two vector variational problems (VVP)
and (VFP), we define the notions of efficient solution and of normal efficient solution and using these notions we establish necessary efficiency
conditions. Using the notion of (ρ, b)-quasiinvexity adapted for variational
problems, we introduce a duality of Mond-Weir-Zalmai type for the fractional problem (VFP) through weak, direct and converse duality theorems.
M.S.C. 2010: 53C65, 65K10, 90C29, 26B25.
Key words: Riemannian manifold, multitime vector fractional variational problem,
efficient solution, normal efficient solution, (ρ, b)-quasiinvexity.
1
Introduction
In [9], Mititelu and Stancu-Minasian considered the following multiobjective fractional variational problem:
Z b
Z b
fp (t, x, x)dt
˙
f1 (t, x, x)dt
˙
, . . . , Zab
Maximize Zab
(MSP)
k1 (t, x, x)dt
˙
kp (t, x, x)dt
˙
a
a
subject to x(a) = a0 , x(b) = b0 ,
g(t, x, x)
˙ ≦ 0, h(t, x, x)
˙ = 0, ∀t ∈ I,
where I = [a, b] is interval, x = (x1 , . . . , xn ) : I → Rn is piecewise smooth function on
dx
I with x˙ =
its derivative, f1 , k1 , . . . , fp , kp : I ×Rn ×Rn → R, g : I ×Rn ×Rn → Rm
dt
and h : I × Rn × Rn → Rq are functions of C 2 -class.
∗
Balkan
Journal of Geometry and Its Applications, Vol.16, No.2, 2011, pp. 90-101.
c Balkan Society of Geometers, Geometry Balkan Press 2011.
°
Multitime vector fractional variational problems on manifolds
91
For this problem, they established necessary efficiency (Pareto minimum) conditions, and using generalized quasiinvex functions, developed a duality theory through
weak, direct and converse duality theorems, see also [6].
In [14], Udri¸ste studied a control variational problem into multidimensional (multitime) framework, establishing some optimality conditions, that is he gave a multitime
maximum principle, see also [15]. Pitea, Udri¸ste and Mititelu [11], [12] considered
the multitime vector version of problem (MSP) within a geometrical framework, using curvilinear integrals, and they established necessary optimality conditions and
developed a duality theory for this problem.
In this work, we study properties of a multitime multiobjective fractional variational problem, within a geometrical framework [11], [12] using multiple integrals.
Let (T, h) and (M, g) be two Riemannian manifolds of dimensions m and n, respectively. Denote by t = (t1 , . . . , tm ) = (tα ) the elements of a measurable set Ω in T
and by x = (x1 , . . . , xn ) = (xk ) ∈ Rn the elements of M . Consider J 1 (T, M ) the first
order jet bundle associated to T and M and the functions
x : Ω → M, X : J 1 (T, M ) → R, f = (f1 , . . . , fp ) : J 1 (T, M ) → Rp ,
k = (k1 , . . . , kp ) : J 1 (T, M ) → Rp , Xαi : J 1 (T, M ) → Rm , Yβ : J 1 (T, M ) → Rq ,
where m, q ∈ N∗ , i = 1, n, α = 1, m and β = 1, q.
The arguments of X, f , g, Xαi , Yβ are (t, x, xγ ) = (t, x(t), xγ (t)), where
x = x(t) = (x1 (t), . . . , xn (t)) = (xk (t)), t ∈ Ω,
∂x
xγ (t) = γ (t), γ = 1, m.
∂t
We suppose that X, f , g, Xαi , Yβ belong to C 2 (Ω). We define the set of functions
Φ = {x : Ω → M | x is piecewise smooth on Ω},
where Ω is a normed space with kxk = kxk∞ +
n
X
kxkγ k∞ .
k=1
Throughout in the paper, for two vectors v = (v1 , . . . , vn ) and w = (w1 , . . . , wn )
the relations of the form v = w, v < w, v ≦ w, v ≤ w are defined as follows:
v = w ⇔ vi = wi , i = 1, n;
v ≦ w ⇔ vi ≦ wi , i = 1, n;
v < w ⇔ vi < wi , i = 1, n;
v ≤ w ⇔ v ≦ w and v 6= w.
The aim of present work is to establish necessary efficiency conditions and develop
a duality theory for the following multitime fractional variational vector problem:
Z
Z
f
(t,
x(t),
x
(t))dv
f
(t,
x(t),
x
(t))dv
p
γ
γ
Ω 1
Z
ZΩ
,
.
.
.
,
Maximize Pareto
(VFP)
k1 (t, x(t), xγ (t))dv
kp (t, x(t), xγ (t))dv
Ω
Ω
subject to Xαi (t,
x(t), xγ (t)) = 0, Yβ (t, x(t), xγ (t)) ≦ 0,
¯
x(t)¯ = u(t) (given), ∀t ∈ Ω,
∂Ω
where dv = dt1 dt2 · · · dtn .
First, there are established necessary optimality conditions for a multitime scalar
variational problem (SVP) and after that necessary efficiency conditions for a multitime vector variational problem (VVP), both having the same constraints as (VFP).
92
2
S¸tefan Mititelu and Mihai Postolache
Necessary optimality conditions for (SVP)
The first model studied in this paper is the scalar multitime variational problem
Z
X(t, x(t), xγ (t))dv
Minimize
I[x]
=
Ω
(SVP)
subject to Xαi (t,
¯ x(t), xγ (t)) = 0, Yβ (t, x(t), xγ (t)) ≦ 0, t ∈ Ω,
x(t)¯ = u(t).
∂Ω
The set of feasible solutions for this problem (the domain of (SVP)) is given by
¯
D = {x ∈ Φ | Xαi (t, x(t), xγ (t)) = 0, Yβ (t, x(t), xγ (t)) ≦ 0, x(t)¯ = u(t), ∀t ∈ Ω}.
∂Ω
Theorem 2.1 (Necessary optimality for (SVP)). If x = x(t) ∈ D is an optimal
solution for problem (SVP), then there exist a real scalar τ and the piecewise smooth
nm
functions Λ(t) = (Λα
and M (t) = (M β (t)) ∈ Rq which satisfy the following
i (t)) ∈ R
conditions:
∂Xαi
∂Yβ
∂X
α
+
Λ
(t)
+ M β (t) k −
τ
i
k
k
∂x
∂x µ
∂x
¶
∂Xαi
∂Yβ
∂X
α
β
= 0,
+
Λ
(t)
+
M
(t)
−D
τ
(SFJ)
γ
i
∂xkγ
∂xkγ
∂xkγ
M β (t)Yβ (t, x(t), xγ (t)) = 0, β = 1, q,
τ ≧ 0, (M β (t)) ≧ 0, t ∈ Ω,
where Dγ (·) =
d
∂X
∂X
∂X
∂X
(·),
:=
(t, x(t), xγ (t)),
:=
(t, x(t), xγ (t)) etc..
k
k
k
dt
∂x
∂x
∂xγ
∂xkγ
Proof. Consider the following auxiliary multitime variational problem
Z
X(t, x(t), xγ (t))dv
Minimize
I[x]
=
Z
Z
Ω
α
i
(AVP)
µβ (t)Yβ (t, x(t), xγ (t))dv ≦ 0,
λi (t)Xα (t, x(t), xγ (t))dv = 0,
s. t.
Ω
Ω
¯
(µβ (t)) ≧ 0, x(t)¯∂Ω = u(t), t ∈ Ω,
β
where λα
i and µ are piecewise smooth real functions on Ω.
Problems (SVP) and (AVP) have the same domain D and the same optimal solution x(t).
Let ε > 0 be a scalar and let p : Ω → Rn be a function of C 1 (Ω)-class. Consider
the next neighborhood of the optimal solution x(t):
¯
Vε = {¯
x ∈ X |x
¯ = x(t) + εp(t), p¯ = 0}.
∂D
Then x = x(t) is an optimal solution of problem (SVP) if ε = 0 is a minimal
solution to the next nonlinear problem:
Z
X(t, x(t) + εp(t), xγ (t) + εp(t))dv
˙
Minimize
F
(ε)
=
Ω
subject toZ
i
˙
= 0, i = 1, n, α = 1, m,
Giα (ε) = λα
(2.1)
i (t)Xα (t, x(t)+εp(t), xγ (t)+εp(t))dv
Ω
Z
µβ (t)Yβ (t, x(t) + εp(t), xγ (t) + εp(t))dv
˙
≦ 0, β = 1, q,
Hβ (ε) =
¯
¯
Ω
(µβ (t)) ≧ 0,
u¯∂Ω = u, p¯∂Ω = 0, t ∈ Ω.
Multitime vector fractional variational problems on manifolds
93
The Fritz John conditions for (2.1) at ε = 0 are the following: there exists a scalar
τ ∈ R, a matrix B = (Biα ) ∈ Rnm and a vector C = (C β ) ∈ Rq , C ≧ 0, such that:
τ ∇F (0) + Biα ∇Giα (0) + C β ∇Hβ (0) = 0
C β Hβ (0) = 0
(FJ)
τ ≧ 0, µβ ≧ 0,
µ
¶
¶
Z
∂Xαi k ∂Xαi k
∂X k ∂X k
α
i
λ
(t)
dv;
∇G
(0)
=
dv;
p
+
p
p
+
p
i
α
∂xk
∂xkγ γ
∂xk
∂xkγ γ
Ω
µ
¶
ZΩ
∂Yβ k
∂h k
µβ (t)
and ∇Hβ (0) =
p +
p dv.
k
∂x
∂xkγ γ
Ω
Consequently, the first condition of (FJ) becomes
µ
¶
¶
Z
Z µ
∂Xαi k ∂Xαi k
∂X k ∂X k
α
α
λi (t)
p +
p dv + Bi
p +
p dv +
τ
∂xk
∂xkγ γ
∂xk
∂xkγ γ
Ω
Ω
µ
¶
Z
∂Yβ k ∂Yβ k
µβ (t)
+C β
dv = 0,
p
+
p
∂xk
∂xkγ γ
Ω
with ∇F (0) =
(2.2)
Z µ
Z ·
Ω
¸
∂Xαi
∂Yβ k
∂X
α
β
τ k + λi (t) k + µ (t) k p dv+
∂x
∂x
¸
Z ·∂x
∂X
∂Yβ k
∂Xαi
α
β
τ k + λi (t) k + µ (t) k pλ dv = 0,
+
∂xγ
∂xγ
∂xγ
Ω
α α
β
β β
where we denoted Λα
i (t) = Bi λi (t) and M (t) = C µ (t).
α
i
We denote E(t, x, xγ ) = τ X(t, x, xγ ) + Λi (t)Xα (t, x, xγ ) + M β (t)Yβ (t, x, xγ ) and
(2.2) can be written
Z
Z
∂E k
∂E k
(2.3)
p
dv
+
p dv = 0.
k
k γ
∂x
∂x
Ω
Ω
γ
For an useful version of the second integral in (2.3) we have
¶
¶
µ
µ
∂E k
∂E
∂E k
pk
=
p
p + Dγ
Dγ
∂xkγ
∂xkγ γ
∂xkγ
and integrating, we obtain
µ
¶
¶
µ
Z
Z
Z
∂E k
∂E
∂E k
pk dv.
Dγ
Dγ
p dv =
p dv −
k γ
∂xkγ
∂xkγ
Ω ∂xγ
Ω
Ω
But using the divergence formula, we have
µ
¶
¸
Z
Z ·
∂E k
∂E k
Dγ
dv
=
~ndσ = 0,
p
p
k
∂xkγ
Ω
∂Ω ∂xγ
where ~¯n = (n1 , . . . , nm ) is the unit normal vector to ∂Ω at the current point of ∂Ω
and pk ¯∂Ω = 0, k = 1, n.
94
S¸tefan Mititelu and Mihai Postolache
Then we obtain
Z
Ω
∂E k
p dv = −
∂xkγ γ
Z
Dγ
Ω
µ
∂E
∂xkγ
¶
pk dv,
therefore(2.3) becomes
(2.4)
Z ·
Ω
∂E
− Dγ
∂xk
µ
∂E
∂xkγ
¶¸
pk dv = 0.
By the fundamental lemmaµof variational
calculus, from (2.4) it follows the Euler¶
∂E
∂E
Lagrange equation
= 0, that is the first relation of (SFJ).
− Dγ
∂xk
∂xkγ
The second condition of (FJ) becomes
M β (t)Yβ (t, x(t), xγ (t)) = 0.
Definition 2.1 ([4]). A point x∗ ∈ D is called normal optimal solution of problem
(SVP) if τ 6= 0.
3
Necessary efficiency conditions for (VVP)
•Efficiency for multitime vector variational problems. In the framework
of problem (SVP), we consider the vector functional
I[x] =
Z
f (t, x(t), xγ (t))dv,
Ω
which can be written on components as
¶
µZ
Z
fp (t, x(t), xγ (t))dv .
f1 (t, x(t), xγ (t))dv, . . . ,
I[x] = (I1 [x], . . . , Ip [x]) =
Ω
Ω
Consider now the multitime variational vector problem
Z
f (t, x(t), xγ (t))dv
Minimize Pareto I[x] =
Ω
(VVP)
i
subject to X¯α (t, x(t), xγ (t)) = 0, Yβ (t, x(t), xγ (t)) ≦ 0,
x¯ = u(t), ∀t ∈ Ω.
∂Ω
The domain of (VVP) is also D.
Definition 3.1 ([2]). A point x∗ ∈ D is an efficient solution (Pareto minimum) of
problem (VVP) if there exist no x ∈ D such that I[x] ≤ I[x∗ ].
Theorem 3.1 (Necessary efficiency of (VVP)). Consider the vector multitime
variational problem (VVP) and let x = x(t) ∈ D be an efficient solution of (VVP).
95
Multitime vector fractional variational problems on manifolds
Then there are a vector τ ∈ Rp , and the piecewise smooth functions λ(t) in Rnm and
µ(t) in Rq , defined on Ω, which satisfy the conditions
∂Xαi
∂Yβ
α
r ∂fr
+
λ
(t)
+ µβ (t) k −
τ
i
k
k
∂x
∂x µ
∂x
¶
∂Xαi
∂Yβ
α
β
r ∂fr
τ
−D
= 0,
+
λ
(t)
+
µ
(t)
(VFJ)
γ
i
∂xkγ
∂xkγ
∂xkγ
µβ (t)Yβ (t, x(t), xγ (t)) = 0, β = 1, q,
τ ≧ 0, (µβ (t)) ≧ 0, t ∈ Ω.
Proof. If x = x(t) ∈ D is an efficient solution of (VVP), the inequality I[¯
x] ≤ I[x],
∀¯
x ∈ D, is false. Then there exists r ∈ {1, . . . , p} and a neighborhood Nr in D of
the point x such that Ir [¯
x] ≧ Ir [x], ∀¯
x ∈ Nr . Therefore, x is an optimal solution to
following scalar variational problem
(SVP)r
Z
fr (t, x(t), xγ (t))dv
Minimize Ir [x] =
Ω
subject to Xαi (t, x(t),
¯ xγ (t)) = 0, Yβ (t, x(t), xγ (t)) ≦ 0,
x ∈ Φ, x¯∂Ω = u(t), ∀t ∈ Ω.
Then, according to Theorem 2.1, there are scalars νr ∈ R and the piecewise smooth
β
real functions λα
i,r and µr , such that the next conditions are true:
(3.1)
∂Xαi
∂Yβ
∂fr
(t)T
+ µβr (t)T k −
νr k + λα
i,r
k
∂x
∂x
∂x
¶
µ
i
∂fr
α
T ∂Xα
β
T ∂Yβ
= 0,
+ µr (t)
−Dγ νr k + λi,r (t)
∂xγ
∂xkγ
∂xkγ
µβ (t)T Yβ (t, x(t), xγ (t)) = 0,
r
νr ≧ 0, µβr (t) ≧ 0, t ∈ Ω,
∂fr
∂fr
:=
(t, x(t), xγ (t)), while i, α, β are variables.
∂x
∂x
(ν
p
r
X
, when Ir [¯
x] ≧ Ir [x]
r
We denote by S =
νr and τ =
S
0, when Ir [¯
x] < Ir [x].
r=1
Also we denote by
where
(3.2)
τ = (τ 1 , . . . , τ p ),
λα
i (t) =
λα
i,r (t)
,
S
µβ (t) =
µβr (t)
.
S
Taking into account relations (3.2) in (3.1), we find (VFJ).
Definition 3.2 ([4]). The point x0 ∈ D is a normal efficient solution of (VVP) if
within the conditions (VFJ) there exist τ ≥ 0 with e′ τ = 1.
•Efficiency for multitime fractional vector variational problems.
First of all, we recall some definitions and auxiliary results which will be needed
later in our discussion about efficiency conditions for problem (VFP), defined by the
96
S¸tefan Mititelu and Mihai Postolache
following multitime fractional variational vector problem
Z
Z
fp (t, x(t), xγ (t))dv
Ω f1 (t, x(t), xγ (t))dv
Ω
Z
Z
,
.
.
.
,
Minimize J[x]=
(VFP)
k1 (t, x(t), xγ (t))dv
kp (t, x(t), xγ (t))dv
Ω
Ω
subject to Xαi (t,
x(t), xγ (t)) = 0, Yβ (t, x(t), xγ (t)) ≦ 0,
¯
x(t)¯ = u(t), ∀t ∈ Ω.
∂Ω
From now on, we assume that
Z
kr (t, x(t), xγ (t))dv > 0 for all r = 1, p. The
Ω
domain of (VFP) is the same set D.
Definition 3.3 ([2]). A point x0 ∈ D is said to be an efficient solution of (VFP) if
there is no x ∈ D, x 6= x0 , such that J[x] ≤ J[x0 ].
We present now the necessary efficiency conditions for (VFP). Let x0 (t) be an
efficient solution of (FVP). Consider the problem
Z
fr (t, x(t), xγ (t))dv
Ω
Z
Minimize
x
kr (t, x(t), xγ (t))dv
Ω ¯
subject to x(t)
¯ = u(t),
∂Ω
(FP)r (x0 )
i
Xα (t, x(t), xγ (t)) = 0, Yβ (t, x(t), xγ (t)) ≦ 0,
Z
Z
f
(t,
x(t),
x
(t))dv
fj (t, x0 (t), x0γ (t))dv
j
γ
Ω
Ω
Z
≦Z
, j = 1, p, j 6= r.
0
0
kj (t, x(t), xγ (t))dv
kj (t, x (t), xγ (t))dv
Ω
Ω
Definition 3.4. The efficient solution x0 ∈ D is a normal efficient solution of (VFP)
if x0 is optimal point to at least one of scalar problems (FP)r , r = 1, p.
Theorem 3.2 (Necessary efficiency in (FVP)). Let x = x(t) ∈ D be a normal
efficient solution of problem (FVP). Then there exist a vector τ = (τ r ) ∈ Rp and
nm
piecewise smooth functions λ = (λα
and µ = (µβ (t)) ∈ Rq , defined on Ω,
i (t)) ∈ R
which satisfy the conditions
·
¸
∂X i
∂Yβ
0 ∂kr
r ∂fr
(t) kα + µβ (t) k −
+ λα
−
R
τ
r
i
k
k
∂x
∂x
∂x
∂x
¸
¶
µ ·
∂Xαi
∂Yβ
0 ∂kr
β
α
r ∂fr
+
λ
(t)
= 0,
−
R
+
µ
(t)
−D
τ
γ
(MFJ)
r
i
∂xkγ
∂xkγ
∂xkγ
∂xkγ
µβ (t)Y (t, x(t), x (t)) = 0, β = 1, q,
β
γ
τ ≥ 0, er τ r = 1, (µβ (t)) ≧ 0, t ∈ Ω.
Proof. It is similar
to that of Theorem 4.1 Zfrom [9].
Z
Fr (x)
,
kr (t, x, xγ )dv. Then, Rr0 =
fr (t, x, xγ )dv, Kr (x) =
Put Fr (x) =
Kr (x)
Ω
Ω
r = 1, p, and Theorem 3.2 becomes
97
Multitime vector fractional variational problems on manifolds
Theorem 3.3 (Necessary efficiency in (VFP)). Let x = x(t) ∈ D be a normal
efficient solution of problem (VFP). Then there exist vector τ = (τ r ) ∈ Rp and
pm
and µ = (µβ (t)) ∈ Rq , defined on Ω,
piecewise smooth functions λ = (λα
i (t)) ∈ R
which satisfy the conditions
·
¸
∂X i
∂kr
∂Yβ
∂fr
r
+ λα
(t) kα + µβ (t) k −
−
F
(x)
τ
K
(x)
r
r
i
k
k
∂x
∂x
∂x
∂x
· µ
¶
¸
i
∂f
∂X
∂k
∂Yβ
r
r
α
β
r
α
K
(x)
τ
−D
+
λ
(t)
= 0,
−
F
(x)
+
µ
(t)
r
γ
r
(MFJ)0
i
∂xkγ
∂xkγ
∂xkγ
∂xkγ
µβ (t)Y (t, x(t), x (t)) = 0, β = 1, q,
β
γ
τ ≥ 0, er τ r = 1, (µβ (t)) ≧ 0, t ∈ Ω.
α
β
β
In relations (MFJ)0 , we put λα
i (t) : = Kr (x)λi (t), µ (t) : = Kr (x)µ (t).
Definition 3.5. If conditions (MFJ) or (MFJ)0 hold with τ ≥ 0, e′ τ = 1, then the
point x0 ∈ D is called normal efficient solution of (VFP).
Z
Let ρ ∈ R and b : Φ×Φ → [0, ∞) a function. Consider H(x) = h(t, x(t), xγ (t))dv.
Ω
0
Definition 3.6 ([14]). The function H is called (strictly) (ρ, b)-quasiinvex at
¯ x if
there exist vector functions η(t) = (η1 (t), . . . , ηn (t)) ∈ Rn of C 1 -class with η(t)¯∂Ω = 0
and θ(x, x0 ) ∈ Rn such that for any x (x 6= x0 ),
0
½ ≦ H(x ) ⇒
¾
Z H(x)
∂h
∂h
0
0
0
0
0
ηi i (t, x (t), xγ (t)) + (Dγ ηi ) i (t, x (t), xγ (t) dv( 0, which can be written
¸
Z ·
∂V
∂V
ηs s + (Dγ ηs ) s dt < −kθ(x, y)k2 {τ r ρr + ρ},
(4.7)
∂y
∂yγ
Ω
i
β
α
where V = τ r [Kr (y)fr (t, y,µ
v) − Fr (y)k
¶ r (t, y, v)]
µ + λ¶i (t)Xα (t, y, v) + µ (t)Yβ (t, y, v).
∂V
∂V
∂V
, it follows
Since (Dγ ηs ) s = Dγ ηs s − ηs Dγ
∂yγ
∂yγ
∂yγs
µ
µ
¶
¶
Z
Z
Z
∂V
∂V
∂V
(Dγ ηs ) s dv =
ηs Dγ
Dγ ηs s dv −
dv.
∂yγ
∂yγ
∂yγs
Ω
Ω
Ω
Using the divergence formula, we have
µ
¶
¶
Z µ
Z
∂V
∂V
ηs s ~η (t)dσ = 0,
Dγ ηs s dv =
∂yγ
∂yγ
∂Ω
Ω
¯
where ~η (t) is unit vector to surface ∂Ω at the current point, and ηs (t)¯∂Ω = 0.
Then relation (4.7) becomes
¶¸
µ
Z b ·
∂V
∂V
dv < −kθ(x, y)k2 {τ r ρr + ρ}.
− Dγ
(4.8)
ηs
∂y s
∂yγs
a
Taking into account the first constraint of problem (WFD), we have
¶
µ
∂V
∂V
= 0,
−
D
γ
∂y s
∂yγs
and relation (4.8) becomes 0 < −kθ(x, y)k2 {τ r ρr + ρ}.
100
S¸tefan Mititelu and Mihai Postolache
According to the hypothesis d) of the theorem, this inequality becomes 0 < 0,
which is false. Then, from (4.6)
Z
i
β
τ r [Fr (x)Kr (y) − Kr (x)Fr (y)]+ [λα
i (t)Xα (t, x, xγ ) + µ (t)Yβ (t, x, xγ )]dv−
Z
Ω
(4.9)
i
β
− [λα
i (t)Xα (t, y, yγ ) + µ (t)Yβ (t, y, yγ )]dv > 0.
Ω
Taking into account relation (4.4), from (4.9) it results τ r [Fr (x)Kr (y)−Kr (x)Fr (y)]> 0,
which contradict relation (4.3). Therefore π(x) ≤ δ(y, λ, µ, ν) is false.
Theorem 4.2 (Direct duality). Let x0 be a normal efficient solution of the primal (VFP) and suppose satisfied the hypotheses of Theorem 4.1. Then there are
0
nm
vector τ 0 ∈ Rp and the piecewise smooth functions λ0 = (λα
and
i ) : Ω → R
0
0
q
0
0
0
0
µ = (µ ) : Ω → R such that (x , λ , µ , ν ) is an efficient solution of the dual
(MWFD) and π(x0 ) = δ(x0 , λ0 , µ0 , ν 0 ).
Proof. Because x0 is a normal efficient solution to (VFP), according to Theorem 3.3 there are vector τ 0 = (τ r )0 ∈ Rp and the piecewise smooth functions
0
nm
and µ0 = (µβ )0 : Ω → Rq which satisfy relations (MFJ)0 . It
λ0 = (λα
i ) : Ω→R
0 0
0
follows that (x , τ , λ , µ0 ) ∈ ∆ and π(x0 ) = δ(x0 , τ 0 , λ0 , µ0 ).
Theorem 4.3 (Converse duality). Let (x0 , τ 0 , λ0 , µ0 ) ∈ ∆ be an efficient solution
of dual and (MWFD) and assume satisfied the next conditions
i) x
¯ is a normal efficient solution of primal (VFP);
ii) the hypotheses of Theorem 4.1 are satisfied with (y, τ, λ, µ) = (x0 , τ 0 , λ0 , µ0 ).
Then x
¯ = x0 and π(x0 ) = δ(x0 , τ 0 , λ0 , µ0 ).
Proof. On the contrary, suppose that x
¯ 6= x0 and we will obtain a contradiction.
According to Theorem 3.3, because x
¯ is normal efficient solution of (VFP), then there
¯ = (λ
¯ α ) : Ω → Rnm and µ
are vector τ¯ ∈ Rp and vector functions λ
¯ = (¯
µβ ) : Ω → Rq
i
α
i
β
¯ (t)X (t, x
that satisfy conditions (MFJ)0 . It results λ
¯, x
¯γ ) + µ
¯ (t)Yβ (t, x
¯, x
¯γ ) = 0,
α
i
¯ µ
¯ µ
therefore (¯
x, τ¯, λ,
¯) ∈ ∆. Moreover, π(¯
x) = δ(¯
x, τ¯, λ,
¯). According to Theorem 4.1
¯ µ
relation π(¯
x)≤ δ(x0 , τ 0 , λ0 , µ0 ) is false. Then the relation δ(¯
x, τ¯, λ,
¯)≤ δ(x0 , τ 0 , λ0 , µ0 )
0 0
0
0
is false. Therefore, the maximal efficiency of (x , τ , λ , µ ) is contradicted. Then, it
results that the assumption x
¯ 6= x0 , above made, is false. It follows x
¯ = x0 and also
0 ¯
0
0
0
0
0
0
τ¯ = τ , λ = λ , µ
¯ = µ . Finally, we have π(¯
x) = δ(x , λ , µ , ν ).
For other advances on this field, we address the reader to [1], [3], [7], [8], [16].
Acknowledgements. The authors are grateful to Professor Constantin Udri¸ste
for his remarks and encouragements during the preparation of various versions of this
work. This paper could not have been refined without his help.
References
[1] V. Chankong and Y. Y. Haimes, Multiobjective Decisions Making: Theory and
Methodology, North Holland, New York, 1983.
[2] M. A. Geoffrion, Proper efficiency and the theory of vector minimization, J. Math.
Anal. Appl., 149(1968), No. 6, 618-630.
Multitime vector fractional variational problems on manifolds
101
[3] M. A. Hanson, Duality for variational problems, J. Math. Anal. Appl., 18(1967),
355-364.
[4] R. Jagannathan, Duality for nonlinear fractional programming, Z. Oper. Res.,
17(1973), 1-3.
[5] S¸t. Mititelu, Invex sets and preinvex functions, J. Adv. Math. Stud. 2, 2 (2009),
41-52.
[6] S¸t. Mititelu, Efficiency conditions for multiobjective fractional variational problems, Appl. Sci., 10 (2008), 162-175.
[7] S¸t. Mititelu, Optimality and duality for invex multi-time control problems with
mixed constraints, J. Adv. Math. Stud., 2, 1 (2009), 25-34.
[8] S¸t. Mititelu and M. Postolache, Mond-Weir dualities with Lagrangians for multiobjective fractional and nonfractional variational problems, J. Adv. Math. Stud.,
3, 1 (2010), 41-58.
[9] S¸t. Mititelu and I. M. Stancu-Minasian, Efficiency and duality for multiobjective
fractional variational problems with (ρ, b)-quasiinvexity, Yugoslav J. Oper. Res.,
19, 1 (2009), 85-99.
[10] B. Mond and T. Weir, Generalized concavity and duality, in Generalized Concavity in Optimization and Economics (S. Schaible and W. I. Ziemba (Eds.)),
Academic Press, 1981, 263-279.
[11] Ariana Pitea, C. Udri¸ste and S¸t. Mititelu, PDE-constrained optimization problems with curvilinear functional quotients as objective vectors, Balkan J. Geom.
Appl., 14, 2 (2009), 75-88.
[12] Ariana Pitea, C. Udri¸ste and S¸t. Mititelu, New type dualities in PDI and PDE
constrained optimization problems, J. Adv. Math. Stud., 2, 1 (2009), 81-90.
[13] V. Preda, On Mond-Weir duality for variational problems, Rev. Roum. Math.
Pures Appl., 28, 2 (1993), 155-164.
[14] C. Udri¸ste, Simplified multiple maximum principle, Balkan J. Geom. Appl., 14,
1 (2009), 102-119.
[15] C. Udri¸ste, Equivalence of multitime optimal control problems, Balkan J. Geom.
Appl., 15, 1 (2010), 155-162.
[16] F. A. Valentine, The problem of Lagrange with differentiable inequality as added
side conditions, in Contributions to the Calculus of Variations”, 1933-37, Univ.
”
of Chicago Press, 1937, 407-448.
[17] G. J. Zalmai, Generalized (F, b, ϕ, ρ, θ)-univex n-set functions and semiparametric duality models in multiobjective fractional subset programming, Int. J. Math.
Sci. 7 (2006), 1109-1133.
Authors’ addresses:
S
¸ tefan Mititelu
Technical University of Civil Engineering,
124 Lacul Tei, 020396 Bucharest (RO)
E-mail: st [email protected]
Mihai Postolache
University Politehnica of Bucharest,
313 Splaiul Independent¸ei, 060042 Bucharest, Roamnia.
E-mail: [email protected]
variational problems on manifolds
S¸tefan Mititelu and Mihai Postolache
Abstract. We consider a multitime scalar variational problem (SVP), a
multitime multiobjective variational problem (VVP) and a multitime vector fractional variational problem (VFP). For (SVP) we establish necessary optimality conditions. For the two vector variational problems (VVP)
and (VFP), we define the notions of efficient solution and of normal efficient solution and using these notions we establish necessary efficiency
conditions. Using the notion of (ρ, b)-quasiinvexity adapted for variational
problems, we introduce a duality of Mond-Weir-Zalmai type for the fractional problem (VFP) through weak, direct and converse duality theorems.
M.S.C. 2010: 53C65, 65K10, 90C29, 26B25.
Key words: Riemannian manifold, multitime vector fractional variational problem,
efficient solution, normal efficient solution, (ρ, b)-quasiinvexity.
1
Introduction
In [9], Mititelu and Stancu-Minasian considered the following multiobjective fractional variational problem:
Z b
Z b
fp (t, x, x)dt
˙
f1 (t, x, x)dt
˙
, . . . , Zab
Maximize Zab
(MSP)
k1 (t, x, x)dt
˙
kp (t, x, x)dt
˙
a
a
subject to x(a) = a0 , x(b) = b0 ,
g(t, x, x)
˙ ≦ 0, h(t, x, x)
˙ = 0, ∀t ∈ I,
where I = [a, b] is interval, x = (x1 , . . . , xn ) : I → Rn is piecewise smooth function on
dx
I with x˙ =
its derivative, f1 , k1 , . . . , fp , kp : I ×Rn ×Rn → R, g : I ×Rn ×Rn → Rm
dt
and h : I × Rn × Rn → Rq are functions of C 2 -class.
∗
Balkan
Journal of Geometry and Its Applications, Vol.16, No.2, 2011, pp. 90-101.
c Balkan Society of Geometers, Geometry Balkan Press 2011.
°
Multitime vector fractional variational problems on manifolds
91
For this problem, they established necessary efficiency (Pareto minimum) conditions, and using generalized quasiinvex functions, developed a duality theory through
weak, direct and converse duality theorems, see also [6].
In [14], Udri¸ste studied a control variational problem into multidimensional (multitime) framework, establishing some optimality conditions, that is he gave a multitime
maximum principle, see also [15]. Pitea, Udri¸ste and Mititelu [11], [12] considered
the multitime vector version of problem (MSP) within a geometrical framework, using curvilinear integrals, and they established necessary optimality conditions and
developed a duality theory for this problem.
In this work, we study properties of a multitime multiobjective fractional variational problem, within a geometrical framework [11], [12] using multiple integrals.
Let (T, h) and (M, g) be two Riemannian manifolds of dimensions m and n, respectively. Denote by t = (t1 , . . . , tm ) = (tα ) the elements of a measurable set Ω in T
and by x = (x1 , . . . , xn ) = (xk ) ∈ Rn the elements of M . Consider J 1 (T, M ) the first
order jet bundle associated to T and M and the functions
x : Ω → M, X : J 1 (T, M ) → R, f = (f1 , . . . , fp ) : J 1 (T, M ) → Rp ,
k = (k1 , . . . , kp ) : J 1 (T, M ) → Rp , Xαi : J 1 (T, M ) → Rm , Yβ : J 1 (T, M ) → Rq ,
where m, q ∈ N∗ , i = 1, n, α = 1, m and β = 1, q.
The arguments of X, f , g, Xαi , Yβ are (t, x, xγ ) = (t, x(t), xγ (t)), where
x = x(t) = (x1 (t), . . . , xn (t)) = (xk (t)), t ∈ Ω,
∂x
xγ (t) = γ (t), γ = 1, m.
∂t
We suppose that X, f , g, Xαi , Yβ belong to C 2 (Ω). We define the set of functions
Φ = {x : Ω → M | x is piecewise smooth on Ω},
where Ω is a normed space with kxk = kxk∞ +
n
X
kxkγ k∞ .
k=1
Throughout in the paper, for two vectors v = (v1 , . . . , vn ) and w = (w1 , . . . , wn )
the relations of the form v = w, v < w, v ≦ w, v ≤ w are defined as follows:
v = w ⇔ vi = wi , i = 1, n;
v ≦ w ⇔ vi ≦ wi , i = 1, n;
v < w ⇔ vi < wi , i = 1, n;
v ≤ w ⇔ v ≦ w and v 6= w.
The aim of present work is to establish necessary efficiency conditions and develop
a duality theory for the following multitime fractional variational vector problem:
Z
Z
f
(t,
x(t),
x
(t))dv
f
(t,
x(t),
x
(t))dv
p
γ
γ
Ω 1
Z
ZΩ
,
.
.
.
,
Maximize Pareto
(VFP)
k1 (t, x(t), xγ (t))dv
kp (t, x(t), xγ (t))dv
Ω
Ω
subject to Xαi (t,
x(t), xγ (t)) = 0, Yβ (t, x(t), xγ (t)) ≦ 0,
¯
x(t)¯ = u(t) (given), ∀t ∈ Ω,
∂Ω
where dv = dt1 dt2 · · · dtn .
First, there are established necessary optimality conditions for a multitime scalar
variational problem (SVP) and after that necessary efficiency conditions for a multitime vector variational problem (VVP), both having the same constraints as (VFP).
92
2
S¸tefan Mititelu and Mihai Postolache
Necessary optimality conditions for (SVP)
The first model studied in this paper is the scalar multitime variational problem
Z
X(t, x(t), xγ (t))dv
Minimize
I[x]
=
Ω
(SVP)
subject to Xαi (t,
¯ x(t), xγ (t)) = 0, Yβ (t, x(t), xγ (t)) ≦ 0, t ∈ Ω,
x(t)¯ = u(t).
∂Ω
The set of feasible solutions for this problem (the domain of (SVP)) is given by
¯
D = {x ∈ Φ | Xαi (t, x(t), xγ (t)) = 0, Yβ (t, x(t), xγ (t)) ≦ 0, x(t)¯ = u(t), ∀t ∈ Ω}.
∂Ω
Theorem 2.1 (Necessary optimality for (SVP)). If x = x(t) ∈ D is an optimal
solution for problem (SVP), then there exist a real scalar τ and the piecewise smooth
nm
functions Λ(t) = (Λα
and M (t) = (M β (t)) ∈ Rq which satisfy the following
i (t)) ∈ R
conditions:
∂Xαi
∂Yβ
∂X
α
+
Λ
(t)
+ M β (t) k −
τ
i
k
k
∂x
∂x µ
∂x
¶
∂Xαi
∂Yβ
∂X
α
β
= 0,
+
Λ
(t)
+
M
(t)
−D
τ
(SFJ)
γ
i
∂xkγ
∂xkγ
∂xkγ
M β (t)Yβ (t, x(t), xγ (t)) = 0, β = 1, q,
τ ≧ 0, (M β (t)) ≧ 0, t ∈ Ω,
where Dγ (·) =
d
∂X
∂X
∂X
∂X
(·),
:=
(t, x(t), xγ (t)),
:=
(t, x(t), xγ (t)) etc..
k
k
k
dt
∂x
∂x
∂xγ
∂xkγ
Proof. Consider the following auxiliary multitime variational problem
Z
X(t, x(t), xγ (t))dv
Minimize
I[x]
=
Z
Z
Ω
α
i
(AVP)
µβ (t)Yβ (t, x(t), xγ (t))dv ≦ 0,
λi (t)Xα (t, x(t), xγ (t))dv = 0,
s. t.
Ω
Ω
¯
(µβ (t)) ≧ 0, x(t)¯∂Ω = u(t), t ∈ Ω,
β
where λα
i and µ are piecewise smooth real functions on Ω.
Problems (SVP) and (AVP) have the same domain D and the same optimal solution x(t).
Let ε > 0 be a scalar and let p : Ω → Rn be a function of C 1 (Ω)-class. Consider
the next neighborhood of the optimal solution x(t):
¯
Vε = {¯
x ∈ X |x
¯ = x(t) + εp(t), p¯ = 0}.
∂D
Then x = x(t) is an optimal solution of problem (SVP) if ε = 0 is a minimal
solution to the next nonlinear problem:
Z
X(t, x(t) + εp(t), xγ (t) + εp(t))dv
˙
Minimize
F
(ε)
=
Ω
subject toZ
i
˙
= 0, i = 1, n, α = 1, m,
Giα (ε) = λα
(2.1)
i (t)Xα (t, x(t)+εp(t), xγ (t)+εp(t))dv
Ω
Z
µβ (t)Yβ (t, x(t) + εp(t), xγ (t) + εp(t))dv
˙
≦ 0, β = 1, q,
Hβ (ε) =
¯
¯
Ω
(µβ (t)) ≧ 0,
u¯∂Ω = u, p¯∂Ω = 0, t ∈ Ω.
Multitime vector fractional variational problems on manifolds
93
The Fritz John conditions for (2.1) at ε = 0 are the following: there exists a scalar
τ ∈ R, a matrix B = (Biα ) ∈ Rnm and a vector C = (C β ) ∈ Rq , C ≧ 0, such that:
τ ∇F (0) + Biα ∇Giα (0) + C β ∇Hβ (0) = 0
C β Hβ (0) = 0
(FJ)
τ ≧ 0, µβ ≧ 0,
µ
¶
¶
Z
∂Xαi k ∂Xαi k
∂X k ∂X k
α
i
λ
(t)
dv;
∇G
(0)
=
dv;
p
+
p
p
+
p
i
α
∂xk
∂xkγ γ
∂xk
∂xkγ γ
Ω
µ
¶
ZΩ
∂Yβ k
∂h k
µβ (t)
and ∇Hβ (0) =
p +
p dv.
k
∂x
∂xkγ γ
Ω
Consequently, the first condition of (FJ) becomes
µ
¶
¶
Z
Z µ
∂Xαi k ∂Xαi k
∂X k ∂X k
α
α
λi (t)
p +
p dv + Bi
p +
p dv +
τ
∂xk
∂xkγ γ
∂xk
∂xkγ γ
Ω
Ω
µ
¶
Z
∂Yβ k ∂Yβ k
µβ (t)
+C β
dv = 0,
p
+
p
∂xk
∂xkγ γ
Ω
with ∇F (0) =
(2.2)
Z µ
Z ·
Ω
¸
∂Xαi
∂Yβ k
∂X
α
β
τ k + λi (t) k + µ (t) k p dv+
∂x
∂x
¸
Z ·∂x
∂X
∂Yβ k
∂Xαi
α
β
τ k + λi (t) k + µ (t) k pλ dv = 0,
+
∂xγ
∂xγ
∂xγ
Ω
α α
β
β β
where we denoted Λα
i (t) = Bi λi (t) and M (t) = C µ (t).
α
i
We denote E(t, x, xγ ) = τ X(t, x, xγ ) + Λi (t)Xα (t, x, xγ ) + M β (t)Yβ (t, x, xγ ) and
(2.2) can be written
Z
Z
∂E k
∂E k
(2.3)
p
dv
+
p dv = 0.
k
k γ
∂x
∂x
Ω
Ω
γ
For an useful version of the second integral in (2.3) we have
¶
¶
µ
µ
∂E k
∂E
∂E k
pk
=
p
p + Dγ
Dγ
∂xkγ
∂xkγ γ
∂xkγ
and integrating, we obtain
µ
¶
¶
µ
Z
Z
Z
∂E k
∂E
∂E k
pk dv.
Dγ
Dγ
p dv =
p dv −
k γ
∂xkγ
∂xkγ
Ω ∂xγ
Ω
Ω
But using the divergence formula, we have
µ
¶
¸
Z
Z ·
∂E k
∂E k
Dγ
dv
=
~ndσ = 0,
p
p
k
∂xkγ
Ω
∂Ω ∂xγ
where ~¯n = (n1 , . . . , nm ) is the unit normal vector to ∂Ω at the current point of ∂Ω
and pk ¯∂Ω = 0, k = 1, n.
94
S¸tefan Mititelu and Mihai Postolache
Then we obtain
Z
Ω
∂E k
p dv = −
∂xkγ γ
Z
Dγ
Ω
µ
∂E
∂xkγ
¶
pk dv,
therefore(2.3) becomes
(2.4)
Z ·
Ω
∂E
− Dγ
∂xk
µ
∂E
∂xkγ
¶¸
pk dv = 0.
By the fundamental lemmaµof variational
calculus, from (2.4) it follows the Euler¶
∂E
∂E
Lagrange equation
= 0, that is the first relation of (SFJ).
− Dγ
∂xk
∂xkγ
The second condition of (FJ) becomes
M β (t)Yβ (t, x(t), xγ (t)) = 0.
Definition 2.1 ([4]). A point x∗ ∈ D is called normal optimal solution of problem
(SVP) if τ 6= 0.
3
Necessary efficiency conditions for (VVP)
•Efficiency for multitime vector variational problems. In the framework
of problem (SVP), we consider the vector functional
I[x] =
Z
f (t, x(t), xγ (t))dv,
Ω
which can be written on components as
¶
µZ
Z
fp (t, x(t), xγ (t))dv .
f1 (t, x(t), xγ (t))dv, . . . ,
I[x] = (I1 [x], . . . , Ip [x]) =
Ω
Ω
Consider now the multitime variational vector problem
Z
f (t, x(t), xγ (t))dv
Minimize Pareto I[x] =
Ω
(VVP)
i
subject to X¯α (t, x(t), xγ (t)) = 0, Yβ (t, x(t), xγ (t)) ≦ 0,
x¯ = u(t), ∀t ∈ Ω.
∂Ω
The domain of (VVP) is also D.
Definition 3.1 ([2]). A point x∗ ∈ D is an efficient solution (Pareto minimum) of
problem (VVP) if there exist no x ∈ D such that I[x] ≤ I[x∗ ].
Theorem 3.1 (Necessary efficiency of (VVP)). Consider the vector multitime
variational problem (VVP) and let x = x(t) ∈ D be an efficient solution of (VVP).
95
Multitime vector fractional variational problems on manifolds
Then there are a vector τ ∈ Rp , and the piecewise smooth functions λ(t) in Rnm and
µ(t) in Rq , defined on Ω, which satisfy the conditions
∂Xαi
∂Yβ
α
r ∂fr
+
λ
(t)
+ µβ (t) k −
τ
i
k
k
∂x
∂x µ
∂x
¶
∂Xαi
∂Yβ
α
β
r ∂fr
τ
−D
= 0,
+
λ
(t)
+
µ
(t)
(VFJ)
γ
i
∂xkγ
∂xkγ
∂xkγ
µβ (t)Yβ (t, x(t), xγ (t)) = 0, β = 1, q,
τ ≧ 0, (µβ (t)) ≧ 0, t ∈ Ω.
Proof. If x = x(t) ∈ D is an efficient solution of (VVP), the inequality I[¯
x] ≤ I[x],
∀¯
x ∈ D, is false. Then there exists r ∈ {1, . . . , p} and a neighborhood Nr in D of
the point x such that Ir [¯
x] ≧ Ir [x], ∀¯
x ∈ Nr . Therefore, x is an optimal solution to
following scalar variational problem
(SVP)r
Z
fr (t, x(t), xγ (t))dv
Minimize Ir [x] =
Ω
subject to Xαi (t, x(t),
¯ xγ (t)) = 0, Yβ (t, x(t), xγ (t)) ≦ 0,
x ∈ Φ, x¯∂Ω = u(t), ∀t ∈ Ω.
Then, according to Theorem 2.1, there are scalars νr ∈ R and the piecewise smooth
β
real functions λα
i,r and µr , such that the next conditions are true:
(3.1)
∂Xαi
∂Yβ
∂fr
(t)T
+ µβr (t)T k −
νr k + λα
i,r
k
∂x
∂x
∂x
¶
µ
i
∂fr
α
T ∂Xα
β
T ∂Yβ
= 0,
+ µr (t)
−Dγ νr k + λi,r (t)
∂xγ
∂xkγ
∂xkγ
µβ (t)T Yβ (t, x(t), xγ (t)) = 0,
r
νr ≧ 0, µβr (t) ≧ 0, t ∈ Ω,
∂fr
∂fr
:=
(t, x(t), xγ (t)), while i, α, β are variables.
∂x
∂x
(ν
p
r
X
, when Ir [¯
x] ≧ Ir [x]
r
We denote by S =
νr and τ =
S
0, when Ir [¯
x] < Ir [x].
r=1
Also we denote by
where
(3.2)
τ = (τ 1 , . . . , τ p ),
λα
i (t) =
λα
i,r (t)
,
S
µβ (t) =
µβr (t)
.
S
Taking into account relations (3.2) in (3.1), we find (VFJ).
Definition 3.2 ([4]). The point x0 ∈ D is a normal efficient solution of (VVP) if
within the conditions (VFJ) there exist τ ≥ 0 with e′ τ = 1.
•Efficiency for multitime fractional vector variational problems.
First of all, we recall some definitions and auxiliary results which will be needed
later in our discussion about efficiency conditions for problem (VFP), defined by the
96
S¸tefan Mititelu and Mihai Postolache
following multitime fractional variational vector problem
Z
Z
fp (t, x(t), xγ (t))dv
Ω f1 (t, x(t), xγ (t))dv
Ω
Z
Z
,
.
.
.
,
Minimize J[x]=
(VFP)
k1 (t, x(t), xγ (t))dv
kp (t, x(t), xγ (t))dv
Ω
Ω
subject to Xαi (t,
x(t), xγ (t)) = 0, Yβ (t, x(t), xγ (t)) ≦ 0,
¯
x(t)¯ = u(t), ∀t ∈ Ω.
∂Ω
From now on, we assume that
Z
kr (t, x(t), xγ (t))dv > 0 for all r = 1, p. The
Ω
domain of (VFP) is the same set D.
Definition 3.3 ([2]). A point x0 ∈ D is said to be an efficient solution of (VFP) if
there is no x ∈ D, x 6= x0 , such that J[x] ≤ J[x0 ].
We present now the necessary efficiency conditions for (VFP). Let x0 (t) be an
efficient solution of (FVP). Consider the problem
Z
fr (t, x(t), xγ (t))dv
Ω
Z
Minimize
x
kr (t, x(t), xγ (t))dv
Ω ¯
subject to x(t)
¯ = u(t),
∂Ω
(FP)r (x0 )
i
Xα (t, x(t), xγ (t)) = 0, Yβ (t, x(t), xγ (t)) ≦ 0,
Z
Z
f
(t,
x(t),
x
(t))dv
fj (t, x0 (t), x0γ (t))dv
j
γ
Ω
Ω
Z
≦Z
, j = 1, p, j 6= r.
0
0
kj (t, x(t), xγ (t))dv
kj (t, x (t), xγ (t))dv
Ω
Ω
Definition 3.4. The efficient solution x0 ∈ D is a normal efficient solution of (VFP)
if x0 is optimal point to at least one of scalar problems (FP)r , r = 1, p.
Theorem 3.2 (Necessary efficiency in (FVP)). Let x = x(t) ∈ D be a normal
efficient solution of problem (FVP). Then there exist a vector τ = (τ r ) ∈ Rp and
nm
piecewise smooth functions λ = (λα
and µ = (µβ (t)) ∈ Rq , defined on Ω,
i (t)) ∈ R
which satisfy the conditions
·
¸
∂X i
∂Yβ
0 ∂kr
r ∂fr
(t) kα + µβ (t) k −
+ λα
−
R
τ
r
i
k
k
∂x
∂x
∂x
∂x
¸
¶
µ ·
∂Xαi
∂Yβ
0 ∂kr
β
α
r ∂fr
+
λ
(t)
= 0,
−
R
+
µ
(t)
−D
τ
γ
(MFJ)
r
i
∂xkγ
∂xkγ
∂xkγ
∂xkγ
µβ (t)Y (t, x(t), x (t)) = 0, β = 1, q,
β
γ
τ ≥ 0, er τ r = 1, (µβ (t)) ≧ 0, t ∈ Ω.
Proof. It is similar
to that of Theorem 4.1 Zfrom [9].
Z
Fr (x)
,
kr (t, x, xγ )dv. Then, Rr0 =
fr (t, x, xγ )dv, Kr (x) =
Put Fr (x) =
Kr (x)
Ω
Ω
r = 1, p, and Theorem 3.2 becomes
97
Multitime vector fractional variational problems on manifolds
Theorem 3.3 (Necessary efficiency in (VFP)). Let x = x(t) ∈ D be a normal
efficient solution of problem (VFP). Then there exist vector τ = (τ r ) ∈ Rp and
pm
and µ = (µβ (t)) ∈ Rq , defined on Ω,
piecewise smooth functions λ = (λα
i (t)) ∈ R
which satisfy the conditions
·
¸
∂X i
∂kr
∂Yβ
∂fr
r
+ λα
(t) kα + µβ (t) k −
−
F
(x)
τ
K
(x)
r
r
i
k
k
∂x
∂x
∂x
∂x
· µ
¶
¸
i
∂f
∂X
∂k
∂Yβ
r
r
α
β
r
α
K
(x)
τ
−D
+
λ
(t)
= 0,
−
F
(x)
+
µ
(t)
r
γ
r
(MFJ)0
i
∂xkγ
∂xkγ
∂xkγ
∂xkγ
µβ (t)Y (t, x(t), x (t)) = 0, β = 1, q,
β
γ
τ ≥ 0, er τ r = 1, (µβ (t)) ≧ 0, t ∈ Ω.
α
β
β
In relations (MFJ)0 , we put λα
i (t) : = Kr (x)λi (t), µ (t) : = Kr (x)µ (t).
Definition 3.5. If conditions (MFJ) or (MFJ)0 hold with τ ≥ 0, e′ τ = 1, then the
point x0 ∈ D is called normal efficient solution of (VFP).
Z
Let ρ ∈ R and b : Φ×Φ → [0, ∞) a function. Consider H(x) = h(t, x(t), xγ (t))dv.
Ω
0
Definition 3.6 ([14]). The function H is called (strictly) (ρ, b)-quasiinvex at
¯ x if
there exist vector functions η(t) = (η1 (t), . . . , ηn (t)) ∈ Rn of C 1 -class with η(t)¯∂Ω = 0
and θ(x, x0 ) ∈ Rn such that for any x (x 6= x0 ),
0
½ ≦ H(x ) ⇒
¾
Z H(x)
∂h
∂h
0
0
0
0
0
ηi i (t, x (t), xγ (t)) + (Dγ ηi ) i (t, x (t), xγ (t) dv( 0, which can be written
¸
Z ·
∂V
∂V
ηs s + (Dγ ηs ) s dt < −kθ(x, y)k2 {τ r ρr + ρ},
(4.7)
∂y
∂yγ
Ω
i
β
α
where V = τ r [Kr (y)fr (t, y,µ
v) − Fr (y)k
¶ r (t, y, v)]
µ + λ¶i (t)Xα (t, y, v) + µ (t)Yβ (t, y, v).
∂V
∂V
∂V
, it follows
Since (Dγ ηs ) s = Dγ ηs s − ηs Dγ
∂yγ
∂yγ
∂yγs
µ
µ
¶
¶
Z
Z
Z
∂V
∂V
∂V
(Dγ ηs ) s dv =
ηs Dγ
Dγ ηs s dv −
dv.
∂yγ
∂yγ
∂yγs
Ω
Ω
Ω
Using the divergence formula, we have
µ
¶
¶
Z µ
Z
∂V
∂V
ηs s ~η (t)dσ = 0,
Dγ ηs s dv =
∂yγ
∂yγ
∂Ω
Ω
¯
where ~η (t) is unit vector to surface ∂Ω at the current point, and ηs (t)¯∂Ω = 0.
Then relation (4.7) becomes
¶¸
µ
Z b ·
∂V
∂V
dv < −kθ(x, y)k2 {τ r ρr + ρ}.
− Dγ
(4.8)
ηs
∂y s
∂yγs
a
Taking into account the first constraint of problem (WFD), we have
¶
µ
∂V
∂V
= 0,
−
D
γ
∂y s
∂yγs
and relation (4.8) becomes 0 < −kθ(x, y)k2 {τ r ρr + ρ}.
100
S¸tefan Mititelu and Mihai Postolache
According to the hypothesis d) of the theorem, this inequality becomes 0 < 0,
which is false. Then, from (4.6)
Z
i
β
τ r [Fr (x)Kr (y) − Kr (x)Fr (y)]+ [λα
i (t)Xα (t, x, xγ ) + µ (t)Yβ (t, x, xγ )]dv−
Z
Ω
(4.9)
i
β
− [λα
i (t)Xα (t, y, yγ ) + µ (t)Yβ (t, y, yγ )]dv > 0.
Ω
Taking into account relation (4.4), from (4.9) it results τ r [Fr (x)Kr (y)−Kr (x)Fr (y)]> 0,
which contradict relation (4.3). Therefore π(x) ≤ δ(y, λ, µ, ν) is false.
Theorem 4.2 (Direct duality). Let x0 be a normal efficient solution of the primal (VFP) and suppose satisfied the hypotheses of Theorem 4.1. Then there are
0
nm
vector τ 0 ∈ Rp and the piecewise smooth functions λ0 = (λα
and
i ) : Ω → R
0
0
q
0
0
0
0
µ = (µ ) : Ω → R such that (x , λ , µ , ν ) is an efficient solution of the dual
(MWFD) and π(x0 ) = δ(x0 , λ0 , µ0 , ν 0 ).
Proof. Because x0 is a normal efficient solution to (VFP), according to Theorem 3.3 there are vector τ 0 = (τ r )0 ∈ Rp and the piecewise smooth functions
0
nm
and µ0 = (µβ )0 : Ω → Rq which satisfy relations (MFJ)0 . It
λ0 = (λα
i ) : Ω→R
0 0
0
follows that (x , τ , λ , µ0 ) ∈ ∆ and π(x0 ) = δ(x0 , τ 0 , λ0 , µ0 ).
Theorem 4.3 (Converse duality). Let (x0 , τ 0 , λ0 , µ0 ) ∈ ∆ be an efficient solution
of dual and (MWFD) and assume satisfied the next conditions
i) x
¯ is a normal efficient solution of primal (VFP);
ii) the hypotheses of Theorem 4.1 are satisfied with (y, τ, λ, µ) = (x0 , τ 0 , λ0 , µ0 ).
Then x
¯ = x0 and π(x0 ) = δ(x0 , τ 0 , λ0 , µ0 ).
Proof. On the contrary, suppose that x
¯ 6= x0 and we will obtain a contradiction.
According to Theorem 3.3, because x
¯ is normal efficient solution of (VFP), then there
¯ = (λ
¯ α ) : Ω → Rnm and µ
are vector τ¯ ∈ Rp and vector functions λ
¯ = (¯
µβ ) : Ω → Rq
i
α
i
β
¯ (t)X (t, x
that satisfy conditions (MFJ)0 . It results λ
¯, x
¯γ ) + µ
¯ (t)Yβ (t, x
¯, x
¯γ ) = 0,
α
i
¯ µ
¯ µ
therefore (¯
x, τ¯, λ,
¯) ∈ ∆. Moreover, π(¯
x) = δ(¯
x, τ¯, λ,
¯). According to Theorem 4.1
¯ µ
relation π(¯
x)≤ δ(x0 , τ 0 , λ0 , µ0 ) is false. Then the relation δ(¯
x, τ¯, λ,
¯)≤ δ(x0 , τ 0 , λ0 , µ0 )
0 0
0
0
is false. Therefore, the maximal efficiency of (x , τ , λ , µ ) is contradicted. Then, it
results that the assumption x
¯ 6= x0 , above made, is false. It follows x
¯ = x0 and also
0 ¯
0
0
0
0
0
0
τ¯ = τ , λ = λ , µ
¯ = µ . Finally, we have π(¯
x) = δ(x , λ , µ , ν ).
For other advances on this field, we address the reader to [1], [3], [7], [8], [16].
Acknowledgements. The authors are grateful to Professor Constantin Udri¸ste
for his remarks and encouragements during the preparation of various versions of this
work. This paper could not have been refined without his help.
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Authors’ addresses:
S
¸ tefan Mititelu
Technical University of Civil Engineering,
124 Lacul Tei, 020396 Bucharest (RO)
E-mail: st [email protected]
Mihai Postolache
University Politehnica of Bucharest,
313 Splaiul Independent¸ei, 060042 Bucharest, Roamnia.
E-mail: [email protected]