Efficiency conditions for multiobjective fractional
variational problems
Ştefan Mititelu
Abstract. In this paper we define the notion of normal efficiency solution.
Necessary conditions for normal efficient solutions of a class of multiobjective fractional variational problem (MFP) with nonlinear equality and
inequality constraints are are established using a parametric approach to
relate efficient solutions of fractional problems and a non-fractional problems. Also, the sufficiency of these conditions for efficiency solutions in
problem (MFP) is established using the (ρ, b)-quasiinvexity notion. Particularly, for the multiobjective variational problem (MP), similar results
are derived.
M.S.C. 2000: 65K10, 90C29, 26B25.
Key words: multiobjective fractional variational problem, efficient solution, quasiinvexity.
1
Introduction
The first result on the necessity of the optimal solutions of the scalar variational
problems was established by Valentine [15] in 1937. The papers of Mond and Hanson [9], Bector [1], Mond, Chandra and Husain [11], Mond and Husain [10], Preda
[13] developed the duality of the scalar variational problems involving convex and
generalized convex functions. Mukherjee and Purnachandra [12], Preda and Gramatovici [14], Mititelu [8] established weak efficiency conditions and developed different
types of dualities for multiobjective variational problems generated by various types of
generalized convex functions. Kim and Kim [6] used the efficiency property of the nondifferentiable multiobjective variational problems in duality theory. In this paper we
introduce the notion of normal efficient solution and establish necessary conditions
of normal efficiency for the multiobjective fractional and non-fractional variational
problems. For these problems we also establish sufficient efficiency conditions using
(ρ, b)-quasiinvexity hypotheses.
Applied Sciences, Vol.10, 2008, pp.
162-175.
c Balkan Society of Geometers, Geometry Balkan Press 2008.
°
Efficiency conditions
2
163
Notions and statement of the problem
Let I = [a, b] be a real interval and f = (f1 , . . . , fp ) : I × Rn × Rn → Rp , k =
(k1 , . . . , kp ) : I × Rn × Rn → Rp , g = (g1 , . . . , gm ) : I × Rn × Rn → Rm , h =
(h1 , . . . , hq ) : I × Rn × Rn → Rq be twice differentiable functions.
Consider the vector-valued function f (t, x, ẋ), where t ∈ I, x : I → Rn with
derivative ẋ with respect to t. fx and fẋ denote the p × n matrices of first-order
partial derivatives of the components with respect to x and ẋ, i.e.
µ
fix =
∂fi
∂fi
,...,
∂x1
∂xn
¶
µ
and
fiẋ =
∂fi
∂fi
,...,
∂ ẋ1
∂ ẋn
¶
,
i = 1, 2, . . . , p.
Similarly, kx , gx , hx and kẋ , gẋ , hẋ denotes the p × n, m × n, q × n matrices of the first
order of k, g and h respectively, with respect to x and ẋ.
For any vectors v = (v1 , . . . , vn ) and w = (w1 , . . . , wn ) the relations 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 ⇔ u = w and u 6= v.
<
Denote be X the space of piecewise smooth functions x with the norm ||x|| :=
||x||∞ + ||Dx||∞ , where the differential operator D is given by
Z
t
u = Dx ⇔ x(t) = x(a) +
u(s)ds.
a
d
, except at discontinuities.
dt
The mathematical model of study is the following multiobjective variational problem

Z b

Z b



fp (t, x, ẋ)dt 
f1 (t, x, ẋ)dt




 a

a

,
.
.
.
,
Minimize
(Pareto)


Z
Z b



 b

k1 (t, x, ẋ)dt
kp (t, x, ẋ)dt
(MFP)
a
a





subject to x(a) = a0 , x(b) = b0 ,





<
g(t, x, ẋ) = 0, h(t, x, ẋ) = 0, ∀t ∈ I
Therefore, D =
Z
Assume that
b
ki (t, x, ẋ)dt > 0 for all i = 1, 2, . . . , p. Let
a
<
D = {x ∈ X|x(a) = a0 , x(b) = b0 , f (t, x, ẋ) = 0, h(t, x, ẋ) = 0, ∀t ∈ I}
be the set of all feasible solutions (domain) of (MFP).
164
3
Ştefan Mititelu
Preliminaries: optimality and efficiency for variational problems, (ρ, b)-quasiinvexity
Let s : I × Rn × Rm → R be a scalar twice differentiable function and consider now
the following scalar variational problem:

Z b


Minimize
s(t, x, ẋ)dt



a
(SP)
subject to x(a) = a0 , x(b) = b0 ,





<
g(t, x, ẋ) = 0, h(t, x, ẋ) = 0, ∀t ∈ I
The first result in literature for the necessity of the optimality of x0 to (SP) was
established by Valentine in the following form:
Theorem 3.1 (Necessary Valentine’s conditions [15]). Let x0 be an optimal solution to (SP) and let s, g and h be twice differentiable functions. Then there exists
scalar λ and the piecewise smooth functions µ0 (t) and ν 0 (t) satisfying the following
conditions:

λsx (t, x0 , ẋ0 ) + µ0 (t)0 gx (t, x0 , ẋ0 ) + ν 0 (t)0 hx (t, x0 , ẋ0 ) =




d
= [λsẋ (t, x0 , ẋ0 ) + µ0 (t)0 gẋ (t, x0 , ẋ0 ) + ν 0 (t)0 hẋ (t, x0 , ẋ0 )]
(VC)

dt


 0 0
>
>
µ (t) g(t, x0 , ẋ0 ) = 0, µ0 (t) = 0, ∀t ∈ I, [λ = 0].
For the relation (VC) there are excluded the values of t corresponding to the
corners of x0 (t) where µ0 (t) and ν 0 (t) cannot vanish.
Definition 3.1. The optimal solution x0 of (SP) is called normal if λ 6= 0.
According to this definition in what follows without loss of generality, we can take
λ = 1.
Consider now the following multiple variational problem
ÃZ
!

Z b
Z b
b



min
f (t, x, ẋ)dt = min
f1 (t, x, ẋ)dt, . . . ,
fp (t, x, ẋ)dt



a
a
a
(MP)


 subject to x(a) = a0 , x(b) = b0 ,



<
g(t, x, ẋ) = 0, h(t, x, ẋ) = 0, ∀t ∈ I
The domain of (MP) is also D.
Definition 3.2 [3]. A feasible solution x0 ∈ D is said to be an efficient solution
(or a Pareto minimum) to the problem (MP) if there exists no other feasible solution
x ∈ D such that
Z b
Z b
f (t, x, ẋ)dt ≤
f (t, x0 , ẋ0 )dt.
a
a
Z
Let ρ ∈ R be and the functions b : X × X → [0, ∞) and H(x) =
b
h(t, x, ẋ)dt.
a
Efficiency conditions
165
Definition 3.3 [13]. The function H is said to be [strictly] (ρ, b)-quasiinvex at
x0 if there exist vector functions η : I × X × X → Rn with η(t, x(t), ẋ(t)) = 0 for
<
x(t) = x0 (t) and θ : X × X → Rn such that for any x[x 6= x0 ], H(x) = H(x0 )
Z
b
0
⇒ b(x, x )
<
[η 0 hx (t, x0 , ẋ0 , u0 ) + (Dη)0 hẋ (t, x0 , ẋ0 , u0 )]dt[<] =
a
−ρb(x, x0 )||θ(x, x0 )||2 .
4
Necessary conditions of normal efficiency for the
problems (MFP) and (MP)
In this section we establish necessary conditions for the normal efficiency of a point
x0 ∈ D in the problem (MFP). Consider the following scalar problem
(FP)i (x0 )

Z b



fi (t, x, ẋ)dt



a

min

Z b

x



ki (t, x, ẋ)dt



a




 subject to x(a) = a0 , x(b) = b0 ,
<


g(t, x, ẋ) = 0, h(t, x, ẋ) = 0, ∀t ∈ I




Z b
Z b




fj (t, x0 , ẋ0 )dt
f
(t,
x,
ẋ)dt

j


<
a


= Zab
, j = 1, p, j 6= i.
Z b





kj (t, x, ẋ)dt
kj (t, x0 , ẋ0 )dt
a
a
Lemma 4.1 (Chankong, Haimes [2]). x0 ∈ D is an efficient solution to problem
(MFP) if and only if x0 is an optimal solution to problems (FP)i , i = 1, p.
We denote
Z
b
Z
0
0
b
fi (t, x, ẋ)dt
fi (t, x , ẋ )dt
Ri0 = Zab
0
0
ki (t, x , ẋ )dt
a
= min Zab
x,u
,
i = 1, p,
ki (t, x, ẋ)dt
a
and then, (Remark 4.1) the problem (FP)i (x0 ) can be written under the next equiv-
166
Ştefan Mititelu
alently form:

Z b


fi (t, x, ẋ)dt



a


min Z b


x



ki (t, x, ẋ)dt



a
(FPR)i
[= Ri0 ]
subject to x(a) = a0 , x(b) = b0 ,




<


g(t, x, ẋ) = 0, h(t, x, ẋ) = 0, ∀t ∈ I




Z b



<


[fj (t, x, ẋ) − Rj0 kj (t, x, ẋ)]dt = 0, j = 1, p, j 6= i.
a
Consider now the problem:

Z b


[fi (t, x, ẋ) − Ri0 ki (t, x, ẋ)]dt
min


x

a




 subject to x(a) = a0 , x(b) = b0 ,
(SPR)i
<


g(t, x, ẋ) = 0, h(t, x, ẋ) = 0, ∀t ∈ I




Z b


<


[fj (t, x, ẋ) − Rj0 kj (t, x, ẋ)]dt = 0, j = 1, p, j 6= i.
a
Lemma 4.2 (Jagannathan [5]). x0 ∈ D is optimal to (FPR)i if and only if x0 is
optimal to (SPR)i .
Lemma 4.3. If x0 is a [normal] optimal solution for the scalar problem (SRP)i ,
then there exist real scalars λji ≥ 0 [λii = 1] and the piecewise smooth functions µi
and νi such that
 p
X



λji [fjx (t, x0 , ẋ0 ) − Rj0 kjx (t, x0 , ẋ0 )]+




j=1






+µi (t)0 gx (t, x0 , ẋ0 ) + νi (t)0 hx (t, x0 , ẋ0 ) =




p


d X
{
λji [fj ẋ (t, x0 , ẋ0 ) − Rj0 kj ẋ (t, x0 , ẋ0 )]+
=
(V F )i
dt

j=1






+µi (t)0 gx (t, x0 , ẋ0 ) + νi (t)0 hx (t, x0 , ẋ0 )}





>


µi (t)0 g(t, x0 , ẋ0 ) = 0, µi (t) = 0, ∀t ∈ I





>
λji = 0, j = 1, p [λii = 1].
Proof. For j = 1, p, j 6= i we define a twice differentiable function ωj :
I × Rn × Rn → R by
Z b
Fj (x) =
[fj (t, x, ẋ) − Rj0 kj (t, x, ẋ) + ωj (t, x, ẋ)]dt = 0,
a
Efficiency conditions
167
where ωj (t, x, ẋ) ≥ 0, j = 1, p, j 6= i. Then x0 is a [normal] optimal solution to (SPR)i
if and only if x0 is a [normal] optimal solution to the problem
P̄i (x0 )

Z b



Maximize
[fi (t, x, ẋ) − Rj0 ki (t, x, ẋ)]dt


a
subject to x ∈ D, Fj (x) = 0,





ωj (t, x, ẋ) ≥ 0, j = 1, p, j 6= i.
According to an Euler’s theorem, there exist scalars λji ≥ 0, j = 1, p such that x0
is [normal] optimal to problem

Z b


Maximize
{fi (t, x, ẋ) − Rj0 ki (t, x, ẋ)+



a



=
p
X
0
P i (x )
+
λji [fj (t, x, ẋ) − Rj0 kj (t, x, ẋ) + ωj (t, x, ẋ)]}dt




j=1,j6=i




subject to x ∈ D, ωj (t, x, ẋ) ≥ 0, j = 1, p, j 6= i.
or equivalently, to

Z b


Maximize
{fi (t, x, ẋ) − Ri0 ki (t, x, ẋ)+



a




p

X

0
λji [fj (t, x, ẋ) − Rj0 kj (t, x, ẋ) + ωj (t, x, ẋ)]}dt
+
P̃i (x )

j=1,j6=i






subject to x(a) = a0 , x(b) = b0 , g(t, x, ẋ) ≤ 0, h(t, x, ẋ) = 0,




−ωj (t, x, ẋ) ≥ 0, j = 1, p, j 6= i.
We associate to P̃i (x0 ) the function
Vi (t, x, ẋ) = γi {fi (t, x, ẋ) − Ri0 ki (t, x, ẋ)+
+
p
X
λji [fj (t, x, ẋ) − Rj0 kj (t, x, ẋ) + ωj (t, x, ẋ)]}dt+
j=1,
j6=i
+µ(t)0 g(t, x, ẋ) + ν(t)0 h(t, x, ẋ) −
p
X
αj (t)ωj (t, x, ẋ),
j=1,
j6=i
where γi ∈ R and µ : I → Rm , ν : I → Rq and αj : I → R, j = 1, p, j 6= i are
piecewise smooth functions.
Because x0 is a [normal] optimal to P̃i (x0 ) then there are scalars γi ≥ 0 [= 1]
and the functions µi , νj and αj ≥ 0, j = 1, p, j 6= i such that the next Valentine’s
168
Ştefan Mititelu
conditions are true

d

Vix (t, x0 , ẋ0 ) = Viẋ (t, x0 , ẋ0 )



dt



>

µj (t)0 g(t, x0 , ẋ0 ) = 0, µi (t) = 0, ∀t ∈ I
(4.1)

>


αj (t)0 ωj (t, x0 , ẋ0 ) = 0, αj (t) = 0, j = 1, p, j 6= i, ∀t ∈ I





>
γi ≥ 0[= 1], λji = 0, j = 1, p, j 6= i.
Developed, the first condition of (4.1) can be written so:
0
0
γi [fix (t, x , ẋ ) −
Ri0 kix (t, x0 , ẋ0 )]
+
p
X
γi λji [fjx (t, x0 , ẋ0 ) − Rj0 kjx (t, x0 , ẋ0 )]+
j=1
j6=i
(E)
+
p
X
[γi λji − αj (t)]ωjx (t, x0 , ẋ0 ) = µi (t)0 gx (t, x0 , ẋ0 ) + νi (t)0 hx (t, x0 , ẋ0 ) =
j=1
j6=i
p
X
d
hγi [fiẋ (t, x0 , ẋ0 ) − Ri0 kiẋ (t, x0 , ẋ0 )] +
γi λji [fj ẋ (t, x0 , ẋ0 ) − Rj0 kj ẋ (t, x0 , ẋ0 )]+
dt
j=1
j6=i
+
p
X
[γi λji − αj (t)]ωj ẋ (t, x0 , ẋ0 ) = µi (t)0 gẋ (t, x0 , ẋ0 ) + νi (t)0 hẋ (t, x0 , ẋ0 )i.
j=1
j6=i
(E) is a linear system by n equations and 2(p − 1) + m + q unknowns (coefficients).
Consider n < 2(p − 1) + m + q and we choose a set of values for these coefficients
putting conditions γi λji − αj = 0, j = 1, p, j 6= 1. Then we define λii = γi and
>
λji ≡ γi λji = 0, j = 1, p, j 6= 1 and (E) becomes
λii fix (t, x0 , ẋ0 ) − Rj0 kix (t, x0 , ẋ0 )]+
+
p
X
λji [fjx (t, x0 , ẋ0 ) − Rj0 kjx (t, x0 , ẋ0 )] + µi (t)0 gx (t, x0 , ẋ0 )+
j=1
j6=i
νl (t)0 hx (t, x0 , ẋ0 ) =
p
X
d
hλii [fix (t, x0 , ẋ0 ) − Ri0 kiẋ (t, x0 , ẋ0 )]+
dt
λji [fj ẋ (t, x0 , ẋ0 ) − Rj0 kj ẋ (t, x0 , ẋ0 ) + µi (t)0 gẋ (t, x0 , ẋ0 ) + νi (t)0 hẋ (t, x0 , ẋ0 )i.
j=1
j6=i
Conditions γi λji −αj = 0, j = 1, p, j 6= 1 corresponding modify the second relation
of (4.1). So, from (4.1) there result conditions (VF)i .
Theorem 4.4. x0 ∈ D is an efficient solution to (MFP) if and only if it is an
optimal solution for each of problems (SPR)i , i = 1, p.
Efficiency conditions
169
Proof. x0 is efficient in (MFP) if and only if it is optimal in the problems (FPR)i ,
i = 1, p (according to Lemma 4.1 and Remark 4.1 ) and also, for each i, x0 is optimal
to (FPR)i if and only if it is optimal to (SPR)i (according to Lemma 4.2).
Definition 4.1. x0 ∈ D is a normal efficient solution to (MFP) if it is a normal
optimal solution at least one of the scalar problems (FP)i , i = 1, p.
Now it follows the main result of this section.
Theorem 4.5 (Necessary efficiency conditions for (MFP)). Let x0 ∈ D be a normal efficient solution to problem (MFP). Then there exist λ0 ∈ Rp and the piecewise
smooth functions µ0 : I → Rm and ν 0 : I → Rq that satisfy the following conditions:
(MFV)
 p
X



λ0i [fix (t, x0 , ẋ0 ) − Ri0 kix (t, x0 , ẋ0 )] + µ(t)0 gx (t, x0 , ẋ0 )+




i=1



p



d X

0
0
0
 +ν(t) hx (t, x , ẋ ) = {
λi [fiẋ (t, x0 , ẋ0 ) − Ri0 hiẋ (t, x0 , ẋ0 )]+
dt i=1



0
0
0
0
0
0


 +µ(t) gẋ (t, x , ẋ ) + ν(t) hẋ (t, x , ẋ )},



>
 µ0 (t)g(t, x0 , ẋ0 ) = 0, µ0 (t) =

0, ∀t ∈ I



 0
0 0
0
λ ≥ 0, e λ = 1, e = (1, . . . , 1) ∈ Rp .
Proof. According to Lemma 4.1 x0 is an optimal solution for each problem (SPR)i
where i = 1, p. If x0 is a [normal] optimal solution to (SPR)i , then relations (VF)i of
Lemma 4.3 are true.
Summing over i = 1, p all the relations of (VF)i and setting
λj =
p
X
λji , i = 1, p,
µ(t) =
p
X
i=1
µi (t),
ν(t) =
p
X
νi (t),
i=1
i=1
the next relations are obtained
 p
X


λj [fjx (t, x0 , ẋ0 ) − Rj0 kjx (t, x0 , ẋ0 )] + µ(t)0 gx (t, x0 , ẋ0 )+




j=1




p


d X
0
0
0
λj [fj ẋ (t, x0 , ẋ0 ) − Rj0 kj ẋ (t, x0 , ẋ0 )]+
+ν(t) hx (t, x , ẋ ) = {
(FV)
dt j=1







+µ(t)0 gẋ (t, x0 , ẋ0 ) + ν(t)0 hẋ (t, x0 , ẋ0 )},





>
>
µ(t)0 g(t, x0 , ẋ0 ) = 0, µ(t) = 0 ∀t ∈ I, λj ≥ 0, i = 1, p, [λj = 1].
By dividing all relations of (FV) by S =
p
X
>
λj = 1 and setting
j=1
λ0j = λj /S,
µ0 (t) = µ(t)/S,
ν 0 (t) = ν(t)/S
170
Ştefan Mititelu
then, from (FV) its result the relations (MFV) (j was replaced by i).
Denote
Z b
Z b
Fi (x0 ) =
fi (t, x0 , ẋ0 )dt, Ki (x0 ) =
ki (t, x0 , ẋ0 )dt
a
a
and we obtain
(4.2)
Ri0 =
Fj (x0 )
,
Ki (x0 )
i = 1, p.
By replacing the numbers Ri0 by (4.2) in the relations (MFV)i and redefining µ0
and ν 0 then the following result is obtained:
Theorem 4.6 (Necessary efficiency conditions for (MFP)). Let x0 be a normal
efficient solution to the problem (MFP). Then there exist λ0 ∈ Rn and the piecewise
smooth functions µ0 : I → Rm and ν 0 : I → Rq that satisfy
 p
X



λ0i [Ki (x0 )fix (t, x0 , ẋ0 ) − Fi (x0 )kix (t, x0 , ẋ0 )] + µ0 (t)0 gx (t, x0 , u0 )+




i=1



p


d X 0
0
0
0
0
λi [Ki (x0 )fiẋ (t, x0 , ẋ0 )−
{
+ν
(t)
h
(t,
x
,
u
)
=
x
(MFV)0
dt


i=1




0
0
0
0

−Fi (x )kiẋ (t, x , ẋ )] + µ (t)0 gẋ (t, x0 , u0 ) + ν 0 (t)0 hẋ (t, x0 , u0 )}




 0 0
>
µ (t) g(t, x0 , ẋ0 ) = 0, µ0 (t) = 0, ∀t ∈ I, λ0 ≥ 0, e0 λ0 = 1.
In Theorem 4.5 or Theorem 4.6 we put Ki (x) ≡ 1, i = 1, p and then there are obtained the following conditions for the of normal efficiency of x0 in the multiobjective
variational problem (MP) :
Theorem 4.7 (Necessary efficiency conditions for (MP)). Let x0 ∈ D be a normal
efficient solution to (MP). Then there exist a vector λ0 ∈ Rp and the piecewise smooth
functions µ0 : I → Rm and ν 0 : I → Rq that satisfy the following conditions:
 0

λ0 fx (t, x0 , ẋ0 ) + µ0 (t)0 gx (t, x0 , ẋ0 ) + ν 0 (t)hx (t, x0 , ẋ0 ) =






 = d [λ00 fẋ (t, x0 , ẋ0 ) + µ0 (t)gẋ (t, x0 , ẋ0 ) + ν 0 (t)hx (t, x0 , ẋ0 )]
dt
(MV)

>


 µ0 (t)0 g(t, x0 , ẋ0 ) = 0, µi (t) = 0, ∀t ∈ I,



 0
λ ≥ 0, e0 λ0 = 1.
5
Sufficient conditions of efficiency for the
problems (MP) and (MFP)
First, we establish sufficient efficiency conditions for(MP).
Theorem 5.1 (Sufficient efficiency for (MP)). Let x0 , λ0 , µ0 and ν 0 be satisfying
the relations (MV) from Theorem 4.6. We suppose that exist vector functions η and
θ satisfying Definition 3.3 and that the following conditions are fulfilled:
Efficiency conditions
171
Z
b
a) For each i = 1, p,
a
fi (t, x, ẋ)dt is (ρ1i , b)-quasiinvex at x0 with respect to η
and θ. Z
b
b)
µ0 (t)0 g(t, x, ẋ)dt is (ρ2 , b)-quasiinvex at x0 with respect to η and θ.
a
Z b
c)
ν 0 (t)0 h(t, x, ẋ)dt is (ρ3 , b)-quasiinvex at x0 with respect to η and θ.
a
d) One of the integral of a)-c) is strictly (ρ, b)-quasiinvex at x0
p
X
>
e)
λ0i ρ1i + ρ2 + ρ3 = 0 (ρ1i , ρ2 , ρ3 ∈ R).
i=1
Then x0 is an efficient solution to (MP).
Proof. Let us suppose toward a contradiction, that x0 is not an efficient solution
for (MP). For each i = 1, p there exists x, a feasible solution to (MP), such that:
Z b
Z b
fi (t, x, ẋ)dt ≤
fi (t, x0 , ẋ0 )dt.
a
a
0
The case x = x is excluded. According to a) it follows that:
Z b
<
0
b(x, x )
[η(t, x, x0 )fix (t, x0 , ẋ0 ) + Dη(t, x, x0 )fiẋ (t, x0 , ẋ0 )]dt =
a
−ρ1i b(x, x0 )||θ(x, x0 )||2 .
Multiplying this inequality by λ0i ≥ 0 and summing over i = 1, p we obtain
Z b
0
0
<
b(x, x0 )
[η(t, x, x0 )λ0 fx (t, x0 , ẋ0 ) + Dη(t, x, x0 )λ0 fẋ (t, x0 , ẋ0 )]dt =
a
(5.1)
Ã
−
p
X
!
λ0i ρ1i
b(x, x0 )||θ(x, x0 )||2 .
i=1
We have
Z
a
b
<
Z
b
µ0 (t)0 g(t, x, ẋ)dt =
µ0 (t)0 g(t, x0 , ẋ0 )dt
a
and taking into account the condition b) it results
Z b
<
b(x, x0 )
[η(t, x, x0 )µ0 (t)0 gx (t, x0 , ẋ0 ) + Dη(t, x, x0 )µ0 (t)0 gẋ (t, x0 , ẋ0 )]dt =
(5.2)
a
−ρ2 b(x, x0 )||θ(x, x0 )||2 .
Taking into account the condition c), from
Z b
Z b
ν 0 (t)0 h(t, x, ẋ)dt =
ν 0 (t)0 h(t, x0 , ẋ0 )dt
a
it results
(5.3)
Z
b
b(x, x0 )
a
<
[η(t, x, x0 )ν 0 (t)0 hx (t, x, ẋ) + Dη(t, x, x0 )ν 0 (t)0 hẋ (t, x0 , ẋ0 )]dt =
a
−ρ3 b(x, x0 )||θ(x, x0 )||2 .
172
Ştefan Mititelu
Summing side by side relations (5.1), (5.2) and (5.3) and taking into account d) we
obtain
(5.4)
Z b
0
b(x, x0 )
[η(t, x, x0 )[ν 0 fx (t, x0 , ẋ0 ) + µ0 (t)0 gx (t, x0 , ẋ0 ) + ν 0 (t)0 hx (t, x0 , ẋ0 )]dt+
Z
a
b
+b(x, x0 )
0
Dη(t, x, x0 )[λ0 fẋ (t, x0 , ẋ0 ) + µ0 (t)0 gẋ (t, x0 , ẋ0 ) + ν 0 (t)0 hẋ (t, x0 , ẋ0 )]dt <
a
<−
à p
X
!
λ0i ρ1i
+ ρ2 + ρ3
b(x, x0 )||θ(x, x0 )||2 .
i=1
¿From (6.4) it results b(x, x0 ) > 0 and then (5.4) becomes
Z b
0
η(t, x, x0 )[λ0 fx (t, x0 , ẋ0 ) + µ0 (t)0 gx (t, x0 , ẋ0 ) + ν 0 (t)0 hx (t, x0 , ẋ0 )]dt+
a
Z
(5.5) +
a
<−
b
0
Dη(t, x, x0 )[λ0 fẋ (t, x0 , ẋ0 ) + µ0 (t)0 gẋ (t, x0 , ẋ0 ) + ν 0 (t)0 hẋ (t, x0 , ẋ0 )] <
Ã
p
X
!
λ0i ρ0i
+ ρ2 + ρ3
||θ(x, x0 )||2 .
i=1
Using an integration by parts, (5.5) becomes
Z b
0
η(t, x, x0 )[λ0 fx (t, x0 , ẋ0 ) + µ0 (t)0 gx (t, x0 , ẋ0 ) + ν 0 (t)0 hx (t, x0 , ẋ0 )]dt+
a
0
+η(t, x, x0 )[λ0 fẋ (t, x, ẋ0 ) + µ0 (t)0 gẋ (t, x0 , ẋ0 ) + ν 0 (t)0 hẋ (t, x0 , ẋ0 )]|ba −
Z b
(5.6)
0
d
−
η(t, x, x0 ) [λ0 fẋ (t, x0 , ẋ0 ) + µ0 (t)0 gẋ (t, x0 , ẋ0 ) + ν 0 (t)0 hẋ (t, x0 , ẋ0 )]dt <
dt
a
à p
!
X
0 1
λi ρi + ρ2 + ρ3 ||θ(x, x0 )||2 .
<−
i=1
Using now the boundary conditions η(a, x(a), ẋ0 (a)) = 0 (where x(a) = x0 (a) = a0
and η(b, x(b), ẋ0 (b) = 0) (where x(b) = x0 (b) = b0 ) from (5.6) we obtain
Z b
0
η(t, x, x0 ){λ0 fx (t, x0 , ẋ0 ) + µ0 (t)0 gx (t, x0 , ẋ0 ) + ν 0 (t)0 hx (t, x0 , ẋ0 )]−
a
(5.7)
−
d 00
[λ fẋ (t, x0 , ẋ0 ) + µ0 (t)0 gẋ (t, x0 , ẋ0 ) + ν 0 (t)0 hẋ (t, x0 , ẋ0 )]}dt <
dt
à p
!
X
0 1
<−
λi ρi + ρ2 + ρ3 ||θ(x, x0 )||2 .
i=1
Taking into account the first relation of (MV), (5.7) becomes
à p
!
X
0
0<−
λ0i λ0 ρ1 + ρ2 + ρ3 b(x, x0 )||θ(x, x0 )||2 .
i=1
Efficiency conditions
173
≥
Having ||θ(x, x0 )|| = 0 and the hypothesis e), the obtained inequality 0 < 0 is
false. Therefore x0 is an efficient solution to (MP).
In what follows we establish efficiency sufficient conditions for the problem (MFP).
Theorem 5.2 (Sufficient efficiency for (MFP)). Let x0 , λ0 , µ0 and ν 0 be satisfying the relations (MFV) from Theorem 4.4. Also, we suppose that exist the vector
functions η and θ as in Definition 3.3 and the following conditions are fulfilled:
Z b
[fi (t, x, ẋ)−Rj0 ki (t, x, ẋ)]dt is (ρ1i , b)-quasiinvex at x0 with
a0 ) For each i = 1, p,
a
respect to η and θ.
Z b
b)
µ0 (t)0 g(t, x, ẋ)dt is (ρ2 , b)-quasiinvex at x0 with respect to η and θ.
a
Z b
c)
ν 0 (t)0 h(t, x, ẋ)dt is (ρ3 , b)-quasiinvex at x0 with respect to η and θ.
a
d0 ) One of the integral by a0 ), b) and c) is strictly (ρ, b)-quasiinvex at x0 with
respect to η and ν (ρ = ρ1i , ρ2 or ρ3 , respectively).
p
X
>
e)
λ0i ρ1i + ρ3 + ρ4 = 0.
i=1
Then x0 is an efficient solution to (MFP).
Proof. It is similarly to those of Theorem 5.1 where, for each i = 1, p,
Z b
is replaced by
[fi (t, x, ẋ) − Ri0 ki (t, x, ẋ)]dt.
Z
b
fi (t, x, ẋ)dt,
a
a
Theorem 5.3 (Sufficient efficiency for (MFP). Let x0 , λ0 , µ0 and ν 0 be satisfying
the relations (MFV) from Theorem 4.5. Also, assume that exist the vector functions
η and θ as in Definition 3.3 such that the following conditions are satisfied:
Z b
00
a ) For each i = 1, p,
[Ki (x0 )fi (t, x, ẋ)−Fi (x0 )ki (t, x, ẋ)]dt is (ρ1i , b)-quasiinvex
a
at x0 with respect to η and θ.
b), c) and e) of Theorem 5.2.
d00 ) One of the integrals from a00 ), b) and c) is strictly (ρ, b)-quasiinvex at x0 with
respect to η and θ (ρ = ρ1i , ρ2 or ρ3 respectively).
Then x0 is an efficient solution to (MFP).
Proof. It is similarly to those of Theorem 5.1, where the hypothesis a) is replaced
by hypothesis a00 ) of this theorem.
If in Theorems 5.1-5.3 the integrals from the hypotheses b) and c) are replaced
Z b
[µ0 (t)0 g(t, x, ẋ) + ν 0 (t)0 h(t, x, ẋ)]dt, then there are the following
by the integral
results:
a
Corollary 5.10 (Sufficient efficiency conditions (MP)). Let x0 , λ0 , µ0 and ν 0 be
satisfying the relations (MV) from Theorem 4.6. We suppose that exist the vector
functions η and θ as in Definition 3.3 and the following conditions are satisfied:
Z b
a) For each i = 1, p,
fi (t, x, ẋ)dt is (ρ1i , b)-quasiinvex at x0 with respect to η
and θ.
a
174
Ştefan Mititelu
Z
b
0
b)
[µ0 (t)0 g(t, x, ẋ) + ν 0 (t)0 h(t, x, ẋ)]dt is (ρ2 , b)-quasiinvex at x0 with respect to
a
η and θ.
d0 ) One of the integrals from a) and b0 ) is strictly (ρ, b)-quasiinvex at x0 with
respect to η and θ or (ρ = ρ1i or ρ2 , respectively).
p
X
>
e0 )
λ0i ρ1i + ρ2 = 0.
i=1
Then x0 is an efficient solution to (MP).
Corollary 5.20 (Sufficient efficiency conditions for (MFP). Let x0 , λ0 , µ0 and ν 0
be satisfying the relations (MFV) from Theorem 4.4. Also, we suppose that exist the
vector functions η and θ as in Definition 3.3 such that the following conditions are
satisfied:
Z b
[fi (t, x, ẋ)−Ri0 ki (t, x, ẋ)]dt is (ρ1i , b)-quasiinvex at x0 with
a0 ) For each i = 1, p,
a
respect to η and θ.
b0 ) and e0 ) from Corollary 5.10 .
d00 ) One of the integrals from a0 ) and b0 ) is strictly (ρ, b)-quasiinvex at x0 with
respect to η and θ.
Then x0 is an efficient solution to (MFP).
Corollary 5.30 (Sufficient efficiency conditions for (MFP)). Let x0 , λ0 , µ0 and ν 0
be satisfying the relations (MFV) from Theorem 4.5. Also, suppose that exist the
vector functions η and θ as in Definition 3.3 such that the following conditions are
satisfied:
Z b
a00 ) For each i = 1, p,
[Ki (x0 )fi (t, x, ẋ)−Fi (x0 )ki (t, x, ẋ)]dt is (ρ1i , b)-quasiinvex
a
at x0 with respect to η and θ.
b0 ) and e0 ) from Corollary 5.10 .
d00 ) One of the integrals from a00 ) and b0 ) is strictly (ρ, b)-quasiinvex at x0 with
respect to η and θ.
Then x0 is an efficient solution to (MFP).
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Author’s address:
Ştefan Mititelu
Technical University of Civil Engineering
Department of Mathematics and Informatics
124 Lacul Tei , 020396 Bucharest 2, Romania.
E-mail: st mititelu@yahoo.com