7_457_husain.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 1, 2010, 81-97

ISSN 1307-5543 – www.ejpam.com

On Mixed Type Duality for Nondifferentiable Multiobjective

Variational Problems

I. Husain1∗ and Rumana G. Mattoo2

1 Department of Mathematics, Jaypee Institute of Engineering and Technology, Guna, MP, India. (A

constituent centre of Jaypee University of Information Technology, Waknaghat, Solan, HP, India)
2 Department of Statistics, University of Kashmir, Srinagar, Kashmir, India.

Abstract. A mixed type dual to a nondifferentiable variational problem involving higher order deriva-

tive is formulated and duality results are proved under generalized invexity conditions. Special cases

are generated from our results.

2000 Mathematics Subject Classifications: Primary 90C30, Secondary 90C11, 90C20, 90C26.

Key Words and Phrases: Mixed type dual; Nondifferentiable Variational problem; Duality; Invexity;

Generalized Invexity.

1. Introduction

In [9], Husain et al considered the following nondifferentiable multiobjective variational

problem containing square root terms:

(VP) Minimize

�
∫

I

( f 1(t, x , ẋ , ẍ) + (x(t)T B1(t)x(t))
1

2 )d t,

. . . ,

∫

I

( f p(t, x , ẋ , ẍ) + (x(t)T Bp(t)x(t))
1

2 )d t

�

subject to

x(a) = 0= x(b)

ẋ(a) = 0= ẋ(b)

g(t, x , ẋ , ẋ) ≦ 0, t ∈ I ,

where

∗Corresponding author.

Email address: iqbal.husain�jiet.a
.in (I. Husain)

http://www.ejpam.com 81 c© 2009 EJPAM All rights reserved.



I. Husain, R. Mattoo / Eur. J. Pure Appl. Math, 3 (2010), 81-97 82

1 f i : I × Rn × Rn × Rn → R, (i = 1,2, . . . , p), g : I × Rn × Rn × Rn → Rm, j = 1, . . . , m

are assumed to be continuously differentiable functions, and for each t ∈ I , i ∈ P =

{1,2, . . . , p}, Bi(t) is an n×n positive semidefinite (symmetric) matrix, with Bi(·)continuous

on I .

The following Wolfe type dual to the problem (VP) is presented in [9]:

(MWD) Maximize

�
∫

I

( f 1(t,u, u̇, ü) + (u(t)T B1(t)z1(t)) + y(t)T g(t,u, u̇, ü))d t

, . . . ,

∫

I

( f p(t,u, u̇, ü) + (u(t)T Bp(t)zp(t)) + y(t)T g(t,u, u̇, ü))d t

�

Subject to

u(a) = 0= u(b)

u̇(a) = 0= u̇(b)
p
∑

i=1

λi
�

f i
u (t,u, u̇, ü) d t + Bi (t) z i (t) + y (t)T gu (t,u, u̇, ü)

�

−D
�

λT fu̇ (t,u, u̇, ü) + y (t)T gu̇ (t,u, u̇, ü)
�

+D2
�

λT fü (t,u, u̇, ü) + y (t)T gü (t,u, u̇, ü)
�

= 0 , t ∈ I ,

y(t) ≧ 0, t ∈ I ,

λ > 0, λT e = 1.

z̄ i(t)T Bi(t)z̄ i(t)≦ 1, t ∈ I , i ∈ P,

The problem (MWD) is a dual to (VP) assuming that

p
∑

i=1

λi

∫

I

( f i(t, ., ., .) + (·)T Bi(t)z i(t) + y(t)T g(t, ., ., .))d t

is pseudoinvex with respect to η. The authors in [9] further weakened the invexity required

in Wolfe type duality by constructing the following Mond- Weir type vector dual.

The following is the Mond-Weir vector type dual to (VP):

(M-WVD) Maximize

�
∫

I

( f 1(t,u, u̇, ü) + (u(t)T B1(t)z1(t)))d t

, . . . ,

∫

I

( f p(t,u, u̇, ü) + (u(t)T Bp(t)zp(t)))d t

�

Subject to

u(a) = 0= u(b)

u̇(a) = 0= u̇(b)
p
∑

i=1

λi
�

f i
u (t,u, u̇, ü) d t + Bi(t)z i(t) + y(t)T gu(t,u, u̇, ü)

�



I. Husain, R. Mattoo / Eur. J. Pure Appl. Math, 3 (2010), 81-97 83

−D
�

λT fu̇(t,u, u̇, ü) + y(t)T gu̇(t,u, u̇, ü)
�

+D2
�

λT fü(t,u, u̇, ü) + y(t)T gü(t,u, u̇, ü)
�

= 0 , t ∈ I ,

z̄ i(t)T Bi(t)z̄ i(t) ≦ 1, t ∈ I , i ∈ P

∫

I

y(t)T g(t,u, u̇, ü)d t ≧ 0,

y(t) ≧ 0, t ∈ I ,

λ > 0.

Mond-Weir established usual duality theorems under the hypotheses that

p
∑

i=1

λi

∫

I

( f i(t, ., ., .) + (·)T Bi(t)z i(t))d t

is pseudoinvex and
∫

I
y(t)T g(t, ., ., .)d t is quasi-invex with respect to the same η.

In this paper, we propose, in the spirit of Husain and Jabeen [7] and Xu [18], a mixed type

dual to (VP) and established various duality results under generalized invexity requirements.

Special cases are generalized and it is also pointed out that our duality results can be viewed

as dynamic generalizations of those of static case, already existing in the literature.

2. Pre-requisites

Consider the real interval I = [a, b], and the continuously differentiable function φ :

I×Rn×Rn×Rn→ R. In order to consider φ(t, x , ẋ , ẍ), where x : I → Rn is twice differentiable

with its first and second order derivatives ẋ and ẍ respectively, we denote the partial derivative

φ by φt .

φx =

�

∂ φ

∂ x1
, . . . ,

∂ φ

∂ xn

�T

, φ ẋ =

�

∂ φ

∂ ẋ1
, . . . ,

∂ φ

∂ ẋn

�T

, φ ẍ =

�

∂ φ

∂ ẍ1
, . . . ,

∂ φ

∂ ẍn

�T

.

The partial derivative of other function will be written similarly. Let K designates the space

of piecewise functions x : I → Rn possessing derivatives ẋ and ẍ with the norm ‖x‖ =
‖x‖∞‖x‖∞ + ‖D

2 x‖∞, where the differentiation operator D is given by

u = Dx⇔ x(t) = α+

∫ t

a

u(s)ds,

Where α is given boundary value; thus D ≡ d

d t
except at discontinuities.

In the results to follow, we use C(I ,Rk) to denote the space of continuous functions with

the uniform norm; superscript T denotes matrix transpose.



I. Husain, R. Mattoo / Eur. J. Pure Appl. Math, 3 (2010), 81-97 84

Before stating our mixed type multiobjective variational problem, we mention the follow-

ing conventions for vectors x and y in n-dimensional Euclidian space Rn to be used through-

out the analysis of this research together with some definitions of invexity and generalized

invexity for easy reference.

x < y, ⇔ x i < yi, i = 1,2, . . . , n.

x ≦ y, ⇔ x i ≤ yi, i = 1,2, . . . , n.

x ≤ y, ⇔ x i ≦ yi , i = 1,2, . . . , n. but x 6= y

x � y, is the negation of x ≤ y

For x , y ∈ R, x ≤ y and x < y have the usual meaning.

Definition 1 (Invexity). If there exists vector function η(t, x ,u) ∈ Rn with η = 0 and x(t) =

u(t), t ∈ I and Dη = 0 for ẋ(t) = u̇(t), t ∈ I such that for a scalar function φ(t, x , ẋ , ẍ), the

functional Φ(x , ẋ , ẍ) =
∫

I
φ(t, x , ẋ , ẍ)d t satisfies

Φ(x , u̇, ü)−Φ(x , ẋ, ẍ)

≧

∫

I

{ηTφx(t, x , ẋ , ẍ) + (Dη)Tφ ẋ (t, x , ẋ , ẍ) + (D2η)Tφ ẍ (t, x , ẋ , ẍ)}d t,

Φ is said to be invex in x , ẋ and ẍ on I with respect to η.

Definition 2 (Pseudoinvexity). Φ is said to be pseudoinvex in x , ẋ and ẍ with respect to η if

∫

I

{ηTφx (t, x , ẋ , ẍ) + (Dη)Tφ ẋ(t, x , ẋ , ẍ) + (D2η)Tφ ẍ (t, x , ẋ , ẍ)}d t ≧ 0

implies

Φ(x , u̇, ü)≧ Φ(x , ẋ , ẍ).

Definition 3 (Quasi-invex). The functional Φ is said to quasi-invex in x , ẋ and ẍ with respect

to η if

Φ(x , u̇, ü)≦ Φ(x , ẋ , ẍ)

implies

∫

I

{ηTφx (t, x , ẋ , ẍ) + (Dη)Tφ ẋ(t, x , ẋ , ẍ) + (D2η)Tφ ẍ(t, x , ẋ , ẍ)}d t ≦ 0.

We require the following definition of efficient solution for our further analysis. The set of

feasible solutions for (VP) is designated by X .



I. Husain, R. Mattoo / Eur. J. Pure Appl. Math, 3 (2010), 81-97 85

Definition 4 (Efficiency). A point x̄ ∈ X is said to be efficient solution of (VP) if for all feasible

x̄ ∈ X
∫

I

( f i(t, x(t), ẋ(t), ẍ(t))d t + (x(t)T Bi(t)x(t))
1

2 )d t

�

∫

I

( f i(t, x̄(t), ˙̄x(t), ¨̄x(t))d t + ( x̄(t)T Bi(t) x̄(t))
1

2 )d t

for all i ∈ P.

In order to prove the strong duality theorem we will invoke the following lemma due to

Changkong and Haimes [5].

Lemma 1. A point x̄ ∈ X be an efficient solution of (VP) if and only if x̄ ∈ X is an optimal

solution of the following problem (Pk( x̄)) for all k

(Pk( x̄)) Minimize

�
∫

I

( f k(t, x , ẋ , ẍ)d t + (x(t)T Bk(t)x(t))
1

2

�

d t

Subject to

x(a) = 0= x(b),

ẋ(a) = 0= ẋ(b),

g(t, x , ẋ , ẋ)≦ 0, t ∈ I ,
∫

I

( f i(t, x(t), ẋ(t), ẍ(t))d t + (x(t)T Bi(t)x(t))
1

2 )d t

≤

∫

I

( f i(t, x̄(t), ˙̄x(t), ¨̄x(t))d t + ( x̄(t)T Bi(t) x̄(t))
1

2 )d t, i 6= k

3. Mixed Type Duality

We formulate the following type dual (Mix D) to (VP):

(Mix D) Maximize

�
∫

I

�

f 1(t,u, u̇, ü) + u(t)T B1(t)z1(t) +
∑

j∈J◦

y j(t)g j(t,u, u̇, ü)

�

d t

, . . . ,

∫

I

�

f p(t,u, u̇, ü) + (u(t)T Bp(t)zp(t)) +
∑

j∈J◦

y j(t)g j(t,u, u̇, ü)

�

d t

�

Subject to

u(a) = 0= u(b), (1)

u̇(a) = 0= u̇(b), (2)

p
∑

i=1

λi( f i
u (t,u, u̇, ü)d t + Bi(t)z i(t) + y(t)T gu(t,u, u̇, ü))



I. Husain, R. Mattoo / Eur. J. Pure Appl. Math, 3 (2010), 81-97 86

−D(λT fu̇ + y(t)T gu̇) + D2(λT fü + y(t)T gü) = 0, t ∈ I (3)
∑

j∈Jα

∫

I

y j(t)g j(t,u, u̇, ü)d t ≧ 0, α = 1,2, . . . , r (4)

z̄ i(t)T Bi(t)z̄ i(t) ≦ 1, t ∈ I , i ∈ P (5)

y(t) ≧ 0, t ∈ I , (6)

λ ∈ Λ+ (7)

where

(i) Λ+ = {λ ∈ Rp|λ > 0,λT e = 1, e = (1,1, . . . , 1)T ∈ Rp}

(ii) Jα ⊆ M = {1,2, . . . , m}, α= 0,1,2, . . . , r with
r
⋃

α=0

Jα = M and Jα
⋂

Jβ = φ, if α 6= β .

If J◦ = M , then (Mix D) becomes Wolfe type dual considered in [9], if J◦ = φ and Jα = M for

some α ∈ {1,2, . . . , r}, then (Mix D) becomes Mond-Weir type dual considered in [9].

Theorem 1 (Weak Duality). Let x̄ be feasible for (VP) and (u, y, z1, . . . , zp,λ) be feasible for

(Mix D). If for feasible (x ,u, z1, . . . , zp,λ) ,

p
∑

i=1

λi

∫

I

�

f i(t, ., ., .) +
∑

j∈J◦

y j(t)g j(t, ., ., .) + (·)T Bi(t)z i(t)

�

d t

is pseudoinvex and

∑

j∈Jα

∫

I

y j(t)g j(t, ., ., .)d t, α = 1,2, . . . , r

is quasi-invex with respect to same η, then the following cannot hold:

∫

I

( f i(t, x , ẋ , ẍ)d t + (x(t)T Bi(t)x(t))
1

2 )d t

≦

∫

I

�

f i(t,u, u̇, ü)d t + (u(t)T Bi(t)z i(t)) +
∑

j∈J◦

y j(t)g j(t,u, u̇, ü)

�

d t (8)

For all i ∈ P, and

∫

I

( f k(t, x , ẋ , ẍ)d t + (x(t)T Bk(t)x(t))
1

2 )d t

≦

∫

I

�

f k(t,u, u̇, ü)d t + (u(t)T Bk(t)zk(t)) +
∑

j∈J◦

y j(t)g j(t,u, u̇, ü)

�

d t (9)

for some k.



I. Husain, R. Mattoo / Eur. J. Pure Appl. Math, 3 (2010), 81-97 87

Proof. Suppose contrary to the result, that (8) and (9) hold. In view of y(t) ≧ 0, t ∈ I ,

z̄ i(t)T Bi(t)z̄ i(t) ≦ 1, t ∈ I , i ∈ P and g(t, x , ẋ , ẋ)≦ 0, t ∈ I , these inequalities yield,

∫

I

�

f i(t, x , ẋ , ẍ)d t + (x(t)T Bi(t)z i(t)) +
∑

j∈J◦

y j(t)g j(t,u, u̇, ü)

�

d t

≦

∫

I

�

f i(t,u, u̇, ü)d t + (u(t)T Bi(t)z i(t)) +
∑

j∈J◦

y j(t)g j(t,u, u̇, ü)

�

d t

for all i ∈ P, and
∫

I

�

f k(t, x , ẋ , ẍ)d t + (x(t)T Bk(t)zk(t)) +
∑

j∈J◦

y j(t)g j(t,u, u̇, ü)

�

d t

≦

∫

I

�

f k(t,u, u̇, ü)d t + (u(t)T Bk(t)zk(t)) +
∑

j∈J◦

y j(t)g j(t,u, u̇, ü)

�

d t

for some k.

Now using λ > 0 and λT e = 1, these inequalities yield

p
∑

i=1

λi

∫

I

�

f i(t, x , ẋ , ẍ)d t + (x(t)T Bi(t)z i(t)) +
∑

j∈J◦

y j(t)g j(t, x , ẋ , ẍ)

�

d t

<

p
∑

i=1

λi

∫

I

�

f i(t,u, u̇, ü)d t + (u(t)T Bi(t)z i(t)) +
∑

j∈J◦

y j(t)g j(t, x , ẋ , ẍ)

�

d t

By pseudo invexity of

p
∑

i=1

λi

∫

I

�

f i(t, ., ., .) +
∑

j∈J◦

y j(t)g j(t, ., ., .) + (·)T Bi(t)z i(t)

�

d t

with respect to η, we have

0>

p
∑

i=1

λi

∫

I

ηT

��

f i
x + Bi(t)z i(t) +

∑

j∈J◦

y j(t)g j
x

�

+(Dη)T
�

f i
ẋ +
∑

j∈J◦

y j(t)g
j

ẋ

�

+ (D2η)T
�

f i
ẍ +
∑

j∈J◦

y j(t)g
j

ẍ

��

d t

This, by integration by parts, gives,

=

p
∑

i=1

λi

∫

I

ηT

��

( f i
x + Bi(t)z i(t) +

∑

j∈J◦

y j(t)g j
x )− D

�

f i
ẋ +
∑

j∈J◦

y j(t)g
j

ẋ

��

d t

+ηT

�

f i
ẋ +
∑

j∈J◦

y j(t)g
j

ẋ

�
�

�

�

�

t=b

t=a

+ (Dη)T
�

f i
ẍ +
∑

j∈J◦

y j(t)g
j

ẍ

�
�

�

�

�

t=b

t=a



I. Husain, R. Mattoo / Eur. J. Pure Appl. Math, 3 (2010), 81-97 88

+

∫

I

(Dη)T D

�

f i
ẍ +
∑

j∈J◦

y j(t)g
j

ẍ

�

d t

�

Using the boundary conditions which at t = a, t = b gives Dη = 0= η

=

p
∑

i=1

λi

∫

I

�

ηT

��

f i
x + Bi(t)z i(t) +

∑

j∈J◦

y j(t)g j
x

�

− D

�

f i
ẋ +
∑

j∈J◦

y j(t)g
j

ẋ

��

d t

+

∫

I

(Dη)T D

�

f i
ẍ +
∑

j∈J◦

y j(t)g
j

ẍ

��

d t

=

p
∑

i=1

λi

∫

I

�

ηT

��

f i
x + Bi(t)z i(t) +

∑

j∈J◦

y j(t)g j
x

�

− D

�

f i
ẋ +
∑

j∈J◦

y j(t)g
j

ẋ

�

+D2

�

f i
ẍ +
∑

j∈J◦

y j(t)g
j

ẍ

���

d t −ηT

�

f i
ẍ +
∑

j∈J◦

y j(t)g
j

ẍ

�
�

�

�

�

t=b

t=a

0 >

p
∑

i=1

λi

∫

I

�

ηT

��

f i
x + Bi(t)z i(t) +

∑

j∈J◦

y j(t)g j
x

�

− D

�

f i
ẋ +
∑

j∈J◦

y j(t)g
j

ẋ

�

+D2

�

f i
ẍ +
∑

j∈J◦

y j(t)g
j

ẍ

���

d t (10)

Now, from the feasibility of (VP) and (Mix-D), we have

∑

j∈Jα

∫

I

y j(t)g j(t, x , ẋ , ẍ)d t ≦
∑

j∈Jα

∫

I

y j(t)g j(t,u, u̇, ü)d t

This, because of quasi-invexity of

∑

j∈Jα

∫

I

y j(t)g j(t, ., ., .)d t

with η implies

∑

j∈Jα

∫

I

{ηT (y j(t)g j
x) + (Dη)

T y j(t)g
j

ẋ
+ (D2η)T y j(t)g

j

ẍ}d t ≦ 0

As earlier, integrating by parts and using the boundary conditions, we have

∑

j∈Jα

∫

I

ηT {(y j(t)g j
x ) + DT y j(t)g

j

ẋ
+ D2 y j(t)g

j

ẍ
}d t ≦ 0 (11)

Combining (10) and (11), we have

∫

I

ηT

� p
∑

i=1

λi( f i
x + Bi(t)z i(t) + y(t)T gx

�



I. Husain, R. Mattoo / Eur. J. Pure Appl. Math, 3 (2010), 81-97 89

−D(λT f ẋ + y(t)T g ẋ) + D2(λT f ẍ + y(t)T g ẍ))d t < 0

From the equality constraint of the dual, we have

∫

I

ηT (

p
∑

i=1

λi( f i
x + Bi(t)z i(t) + y(t)T gx)

−D(λT f ẋ + y(t)T g ẋ) + D2(λT f ẍ + y(t)T g ẍ))d t = 0

which is a contradiction. Hence the conclusion of the theorem is true.

Theorem 2 (Strong Duality). Let x̄ ∈ X be an efficient solution of (VP) and for at least one

i ∈ P, x̄ satisfies the regularity condition [3] for the problem (Pk( x̄)). Then there exist multipliers

λ̄ ∈ Rp, piecewise smooth ȳ ∈ Rm and z i(t) ∈ Rn, i = {1,2, . . . , p} such that ( x̄ , ȳ , z̄1, . . . , z̄p,λ)

is feasible for (Mix D) and the objectives of (VP) and (Mix D) are equal.

Further, if the hypotheses of Theorem 1 are met, then ( x̄ , ȳ , z̄1, . . . , z̄p,λ) is an efficient solu-

tion of (Mix D).

Proof. x̄ ∈ X is an optimal solution of (Pk( x̄)). This implies that there exist ξ̄ ∈ Rp with

ξ̄1, . . . , ξ̄p, z i(t) ∈ Rn, i = {1,2, . . . , p} and piecewise smooth ῡ ∈ Rmsuch that, the following

optimality conditions [3] hold:

ξ̄k( f k
x + Bk(t)zk(t)− D f k

ẋ + D2 f k
ẍ )

+

p
∑

i=1
i 6=k

ξ̄i( f i
x + Bi(t)z i(t)− D f i

ẋ + D2 f i
ẍ )

+ῡ(t)T gx − D(ῡ(t)T g ẋ) + D2(ῡ(t)T g ẍ) = 0 (12)

ῡ(t)T g(t, x̄ , ˙̄x , ¨̄x)d t = 0 (13)

( x̄(t)T Bi(t) x̄(t))
1

2 = ( x̄(t)T Bi(t)z̄ i(t)), i = 1, . . . , p (14)

(z̄(t)T Bi(t)z̄ i(t)) ≦ 1, t ∈ I , i = 1,2, . . . , p (15)

ξ̄ > 0, ῡ(t) ≧ 0, t ∈ I (16)

Dividing (12), (13) and (16) by
p
∑

i=1

ξ̄i, and setting λ̄i =
ξ̄

p
∑

i=1

ξ̄i

, i = 1, . . . , p and

ȳ(t) =
ῡ(t)
p
∑

i=1

ξ̄i

, we have

p
∑

i=1

λ̄i( f i
x + Bi(t)z̄ i(t)− D f i

ẋ + D2 f i
ẍ )

+ ȳ(t)T gx − D( ȳ(t)T g ẋ) + D2( ȳ(t)T g ẍ) = 0 (17)

ȳ(t)T g(t, x̄ , ˙̄x , ¨̄x)d t = 0 (18)

λ̄ > 0 , ȳ(t) ≧ 0, t ∈ I (19)



I. Husain, R. Mattoo / Eur. J. Pure Appl. Math, 3 (2010), 81-97 90

The equation (18) implies

∑

j∈J◦

y j(t)g j(t, x , ẋ , ẍ) = 0 and
∑

j∈Jα

y j(t)g j(t, x , ẋ , ẍ) = 0, α = 1, . . . , r (20)

This implies,

∑

j∈Jα

∫

I

y j(t)g j(t, x , ẋ , ẍ) = 0, α= 1, . . . , r (21)

Consequently (15), (17), (19) and (21) implies that ( x̄ , ȳ , z̄1, . . . , z̄p, λ̄) is feasible for (Mix D).

∫

I

�

f i(t, x̄ , ˙̄x , ¨̄x) + ( x̄(t)T Bi(t) x̄(t))
1

2 +
∑

j∈J◦

y j(t)g j(t, x̄ , ˙̄x , ¨̄x)

�

d t

=

∫

I

( f i(t, x̄ , ˙̄x , ¨̄x)+ ( x̄(t)T Bi(t)z̄ i(t)))d t, i = 1,2, . . . , p

This in view of Theorem 1, the efficiency of ( x̄ , ȳ , z̄1, . . . , z̄p, λ̄) follows.

As in [15], by employing chain rule in calculus, it can be easily seen that the expression

p
∑

i=1

λi( f i
x (t,u, u̇, ü)d t + Bi(t)z i(t) + y(t)T gx(t,u, u̇, ü))

−D(λT f ẋ + y(t)T g ẋ) + D2(λT f ẍ + y(t)T g ẍ) = 0, , t ∈ I

may be regarded as a function θ of variables t, x , ẋ , ẍ ,
...
x , y, ẏ , ÿ and λ, where

...
x = D3 x and

ÿ = D2 y. That is, we can write

θ(t, x , ẋ , ẍ ,
...
x , y, ẏ , ÿ ,λ) =

p
∑

i=1

λi( f i
x (t,u, u̇, ü)d t + Bi(t)z i(t) + y(t)T gx(t,u, u̇, ü))

−D(λT f ẋ + y(t)T g ẋ) + D2(λT f ẍ + y(t)T g ẍ) = 0

In order to prove converse duality between (VP) and (Mix D), the space X is now replaced

by a smaller space X2 of piecewise smooth thrice differentiable function x : I → Rn with the

norm ‖x‖∞+ ‖Dx‖∞+ ‖D
2 x‖∞+ ‖D

3 x‖∞. The problem (Mix D) may now be briefly written

as,

Minimize

�

−

∫

I

( f 1(t,u, u̇, ü) + u(t)T B1(t)z1(t) +
∑

j∈J◦

y j(t)g j(t,u, u̇, ü)

�

d t

, . . . ,

∫

I

−( f p(t,u, u̇, ü) + (u(t)T Bp(t)zp(t)) +
∑

j∈J◦

y j(t)g j(t,u, u̇, ü)

�

d t

�

Subject to



I. Husain, R. Mattoo / Eur. J. Pure Appl. Math, 3 (2010), 81-97 91

u(a) = 0= u(b)

u̇(a) = 0= u̇(b)

θ(t, x , ẋ , ẍ ,
...
x , y, ẏ , ÿ ,λ) = 0, t ∈ I

∑

j∈Jα

∫

I

y j(t)g j(t,u, u̇, ü)d t ≧ 0, α= 1,2, . . . , r

z̄ i(t)T Bi(t)z̄ i(t) ≦ 1, t ∈ I , i ∈ P

y(t) ≧ 0, t ∈ I

λ > 0, λT e = 1

where

θ(t, x , ẋ , ẍ ,
...
x , y, ẏ , ÿ ,λ) =

p
∑

i=1

λi( f i
x (t,u, u̇, ü) + Bi(t)z i(t) + y(t)T gx(t,u, u̇, ü))

−D(λT f ẋ + y(t)T g ẋ) + D2(λT f ẍ + y(t)T g ẍ) = 0

Consider θ(t, x(·), ẋ(·), ẍ(·),
...
x (·), y(·), ẏ(·), ÿ(·),λ) = 0 as defining a mapping ψ : X × Y ×

Rp → B where Y is a space of piecewise twice differentiable function and B is the Banach

Space. In order to apply Theorem 1 [10] to the problem (Mix D), the infinite dimensional

inequality must be restricted. In the following theorem, we use ψ′ to represent the Frèchèt

derivative
�

ψx(x , y,λ),ψy(x , y,λ),ψλ(x , y,λ)
�

.

Theorem 3 (Converse Duality). Let ( x̄ , ȳ , z̄1, . . . , z̄p, λ̄) be an efficient solution for (Mix D).

Assume that

A1 The Frèchèt derivative ψ′ has a (weak*) closed range,

A2 f and g are twice continuously differentiable,

A3

�

f i
x + Bi(t)z i(t) +

∑

j∈J◦

y j(t)g i
x

�

− D

�

f i
ẋ
+
∑

j∈J◦

y j(t)g i
ẋ

�

+D2

�

f i
ẋ
+
∑

j∈J◦

y j(t)g i
ẍ

�

,

i ∈ {1,2, . . . , p} are linearly independent, and

A4 (β(t)
Tθx − Dβ(t)Tθ ẋ + D2β(t)Tθ ẍ − D3β(t)Tθ...

x )β(t) = 0,⇒ β(t) = 0, t ∈ I Further,

if the hypotheses of Theorem 1 are met then x̄ is an efficient solution of (VP).

Proof. Since ( x̄ , ȳ , z̄1, . . . , z̄p, λ̄) with ψ′having a (weak∗) closed range, is an efficient so-

lution of (Mix D), then by Theorem 1 [9], there exist multipliers τ ∈ Rp, piecewise smooth

β(t) : I → Rn, z i(t) ∈ Rn, i = 1, . . . , p, γ ∈ R for each of rconstraints, η ∈ Rp, δ ∈ R and

piecewise smooth µ(t) : I → Rm satisfying the following Fritz-John optimality conditions [9].

−
p
∑

i=1

τi

��

f i
x + Bi(t)z i(t) +

∑

j∈J◦

y j(t)g j
x

�

− D

�

f i
ẋ +
∑

j∈J◦

y j(t)g
j

ẋ

�



I. Husain, R. Mattoo / Eur. J. Pure Appl. Math, 3 (2010), 81-97 92

+D2

�

f i
ẍ +
∑

j∈J◦

y j(t)g
j

ẍ

��

− γ
r
∑

α=1

∑

j∈Jα

(y j(t)g j
x − D y j(t)g

j

ẋ
+ D2 y j(t)g

j

ẍ
)

+β(t)Tθx − Dβ(t)Tθ ẋ + D2β(t)Tθ ẍ − D3β(t)Tθ...
x = 0, t ∈ I (22)

−(τT e)g j + β(t)Tθy j − Dβ(t)Tθ ẏ j + D2β(t)Tθ ÿ j −µ j(t) = 0, j ∈ J◦ (23)

−γg j + β(t)Tθy j − Dβ(t)Tθ ẏ j+D2β(t)Tθ ÿ j −µ j(t)=0, j∈Jα,α=1, . . . , r (24)

−τi x(t)T Bi(t) +λiβ(t)T Bi(t) + 2δiBi(t)z i(t) = 0, i = 1, . . . , p (25)
�

f i
x + Bi(t)z i(t) +

∑

j∈J◦

y j(t)g j
x − D

�

f i
ẋ +
∑

j∈J◦

y j(t)g
j

ẋ

�

+D2

�

f i
ẍ +
∑

j∈J◦

y j(t)g
j

ẍ

��

−ηi = 0 (26)

µT (t) ȳ(t) = 0, t ∈ I (27)

ηTλ= 0 (28)

γ
∑

j∈Jα

∫

I

y(t)T g(t, x̄ , ˙̄x , ¨̄x)d t = 0, α= 1,2, . . . , p (29)

δi(z i(t)T Bi(t)z i(t)− 1) = 0, t ∈ I (30)

(τ,γ,µ(t),δ,η) ≧ 0 (31)

(τ,β(t),γ,µ(t),δ,η) 6= 0 (32)

Since λ > 0, (28) implies η = 0. Consequently (26) implies

�

f i
x + Bi(t)z i(t) +

∑

j∈J◦

y j(t)g j
x − D( f i

ẋ +
∑

j∈J◦

y j(t)g
j

ẋ
) + D2( f i

ẍ +
∑

j∈J◦

y j(t)g
j

ẍ)

�

= 0 (33)

Using the duality constraint of (Mix D) equation (22) reduces to

−
p
∑

i=1

(τi − γλi)

��

f i
x + Bi(t)z i(t) +

∑

j∈J◦

y j(t)g j
x )− D( f i

ẋ +
∑

j∈J◦

y j(t)g
j

ẋ

�

+D2

�

f i
ẍ +
∑

j∈J◦

y j(t)g
j

ẍ

��

+ β(t)Tθx − Dβ(t)Tθ ẋ

+D2β(t)Tθ ẍ − D3β(t)Tθ...
x = 0, t ∈ I (34)

Post multiplying (34) by β(t) and then using (33), we obtain,

(β(t)Tθx − Dβ(t)Tθ ẋ + D2β(t)Tθ ẍ − D3β(t)Tθ...
x )β(t) = 0, t ∈ I



I. Husain, R. Mattoo / Eur. J. Pure Appl. Math, 3 (2010), 81-97 93

This because of the hypothesis (A4) yields,

β(t) = 0, t ∈ I (35)

Using (35) in (34) in (22), we have

−
p
∑

i=1

(τi − γλi){( f i
x + Bi(t)z i(t) +

∑

j∈J◦

y j(t)g j
x )− D( f i

ẋ +
∑

j∈J◦

y j(t)g
j

ẋ
)

+D2( f i
ẍ +
∑

j∈J◦

y j(t)g
j

ẍ
)}= 0

This, due to the hypothesis (A3) gives,

τi − γλi = 0, i = 1,2, . . . , p (36)

Suppose γ= 0, then from (36) we have τ = 0. Consequently from (23) and (24) implies

µ(t) = 0, t ∈ I .

Also from (25) we have δiBi(t)z i(t) = 0, which together with (30) implies δi = 0, i.e.

δ = 0. Thus, (τ,β(t),µ(t),η,γ,δ, ) = 0, which is a contradiction to (32). Hence γ > 0. From

(34) it implies that τ > 0.

By the generalized Schwartz inequality [17],

( x̄(t)T Bi(t)z̄ i(t))¶ ( x̄(t)T Bi(t) x̄(t))
1

2 (z̄ i(t)Bi(t)z̄ i(t))
1

2 , i = 1,2, . . . , p (37)

Now let δ
i

τi = α
i . Then αi ≧ 0 and from (25), we have

Bi(t) x̄(t) = 2αiBi(t)z̄ i(t), i = 1,2, . . . , p

which, is the condition for the equality in (37). Therefore, we have

( x̄(t)T Bi(t)z̄ i(t)) = ( x̄(t)T Bi(t) x̄(t))
1

2 (z i(t)Bi(t)z i(t))
1

2

From (30), either δi = 0 or z̄ i(t)T Bi(t)z̄ i(t) = 1 or δi = 0, i = 1, . . . , p and hence

Bi(t) x̄(t) = 0. Therefore, in either case

(x(t)T Bi(t)z i(t)) = (x(t)T Bi(t)x(t))
1

2 , i = 1,2, . . . , p (38)

From (23) and (24) we readily obtain,

g(t, x̄ , ˙̄x , ¨̄x)≦ 0, t ∈ I

which gives the feasibility of x̄ for (P). Using (27) in (23) and (24) we have

ȳ(t)T g = 0, i.e., ȳ j(t)g j = 0, j = 1,2, . . . , m, t ∈ I



I. Husain, R. Mattoo / Eur. J. Pure Appl. Math, 3 (2010), 81-97 94

This obviously gives

∑

j∈J◦

ȳ j(t)g j = 0, j = 1,2, . . . , m (39)

Hence
∫

I

( f i(t, x̄ , ˙̄x , ¨̄x)d t + x̄(t)T Bi(t)z̄ i(t) +
∑

j∈J◦

ȳ j(t)g j)d t

=

∫

I

( f i(t, x̄ , ˙̄x , ¨̄x)d t + ( x̄(t)T Bi(t) x̄(t))
1

2 )d t, i = 1,2, . . . , p

(by using (38) and (39))

The efficiency of x̄ for (VP) follows by an application of Theorem 1.

4. Variational Problems with Natural Boundary Conditions

It is possible to extend the duality theorems established in the previous sections to the

corresponding variational problems with natural boundary values rather that fixed points.

(VP)0 Minimize

�
∫

I

�

f 1(t, x , ẋ , ẍ)d t + (x(t)T B1(t)x(t))
1

2

�

d t

, . . . ,

∫

I

�

f p(t, x , ẋ , ẍ)d t + (x(t)T Bp(t)x(t))
1

2

�

d t

�

Subject to

g j(t, x , ẋ , ẋ) ≦ 0, t ∈ I , j = 1, . . . , m

(M IX D)0 Maximize

�
∫

I

�

f 1(t,u, u̇, ü) + u(t)T B1(t)z1(t) +
∑

j∈J◦

y j(t)g j(t,u, u̇, ü)

�

d t

, . . . ,

∫

I

�

f p(t,u, u̇, ü) + (u(t)T Bp(t)zp(t)) +
∑

j∈J◦

y j(t)g j(t,u, u̇, ü)

�

d t

�

Subject to
p
∑

i=1

λi( f i
u (t,u, u̇, ü) + Bi(t)z i(t) + y(t)T gu(t,u, u̇, ü))

−D(λT fu̇(t,u, u̇, ü) + y(t)T gu̇(t,u, u̇, ü))

+D2(λT fü(t,u, u̇, ü) + y(t)T gü(t,u, u̇, ü)) = 0, t ∈ I ,
�

f i
ẋ +
∑

j∈J◦

y j(t)g
j

ẋ

�
�

�

�

�

t=a

= 0

�

f i
ẋ +
∑

j∈J◦

y j(t)g
j

ẍ

�
�

�

�

�

t=b

= 0



REFERENCES 95

�

f i
ẍ +
∑

j∈J◦

y j(t)g
j

ẋ

�
�

�

�

�

t=a

= 0

�

f i
ẍ

∑

j∈J◦

y j(t)g
j

ẍ

�
�

�

�

�

t=b

= 0

∀ i ∈ {1,2, . . . , p}

z̄ i(t)T Bi(t)z̄ i(t)≦ 1, t ∈ I , i ∈ P
∑

j∈Jα

∫

I

y j(t)g j(t,u, u̇, ü)d t ≧ 0, α = 1,2, . . . , r,

y(t) ≧ 0, t ∈ I .

5. Nonlinear Programming

If all the functions are independent of t then the problems (VP) 0 and (MIX D) 0 become

the following problems.

(VP)1 Minimize( f 1(x)+ (x T B1 x)
1

2 , . . . , f p(x)+ (x T Bp x)
1

2 )

Subject to

g(x)≦ 0

(MixD)1 Maximize

�

f 1(u) + uT B1z1 +
∑

j∈J◦

y j g j(u), . . . , f P(u) + uT BPzP +
∑

j∈J◦

y j g j(u)

�

Subject to
p
∑

i=1

λi( f i
u (u)d t + Biz i + yT gu(u)) = 0

z̄ iT Bi z̄ i ≦ 1, i ∈ P
∑

j∈Jα

y j g j(u)≧ 0, α = 1,2, . . . , r

y(t) ≧ 0, t ∈ I

λ ∈ Λ+

where Λ+ = {λ ∈ Rp|λ > 0,λT e = 1, e = (1,1, . . . , 1)T ∈ Rp}
The duality results for this pair of problems are not explicitly reported in the literature but

can be derived easily on the lines of the analysis of this research.

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