Vol. 2, No. 4, 2009, (578-603)

ISSN 1307-5543 – www.ejpam.com

**Mixed Type Symmetric and Self-Duality for **

**Multiob-jective Variational Problems**

### I. Husain

1∗### and Rumana G. Mattoo

21 _{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.** In this paper, a new formulation of multiobjective symmetric dual pair, called

mixed type multiobjective symmetric dual pair, for multiobjective variational problems is

pre-sented. This mixed formulation unifies two existing Wolfe and Mond-Weir type symmetric

dual pairs of multiobjective variational problems. For this pair of mixed type

multiobjec-tive variational problems, various duality theorems are established under invexity-incavity

and pseudoinvexity-pseudoincavity of kernel functions appearing in the problems. Under

ad-ditional hypotheses, a self duality theorem is validated. It is also pointed that our duality

theorems can be viewed as dynamic generalization of the corresponding (static) symmetric

and self duality of multiobjective nonlinear programming already existing in the literature.

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

**Key Words and Phrases: Efficiency; Mixed type multiobjective symmetric dual variational**

problem; Mixed type symmetric duality; Mixed type self duality; Natural boundary values;

Multiobjective nonlinear programming.

∗_{Corresponding author.}

I. Husain and R. Mattoo Eur. J. Pure Appl. Math,**2**(2009), (578-603) 579

**1. Introduction**

Following Dorn[7], symmetric duality results in mathematical programming have been derived by a number of authors, notably, Dantzig et al[8], Mond[12], Bazaraa and Goode[1]. In these researches, the authors have studied symmetric duality under the hypothesis of convexity-concavity of the kernel function involved. Mond and Cottle[13]presented self duality for the problems of[8]by assuming skew symmetric of the kernel function. Later Mond-Weir[14]formulated a different pair of symmetric dual nonlinear program with a view to generalize convexity-concavity of the kernel function to pseudoconvexity-pseudoconcavity.

Symmetric duality for variational problems was first introduced by Mond and
Han-son[15]under the convexity-concavity conditions of a scalar functions like*ψ*(t,*x(t*),

˙

*x*(t), *y(t), ˙y*(t))with *x*(t)∈*Rn* _{and} _{y(t}_{)}_{∈}_{R}m_{. Bector, Chandra and Husain}_{[}_{3}_{]} _{}
pre-sented a different pair of symmetric dual variational problems in order to relax the
requirement of convexity-concavity to that of pseudoconvexity-pseudoconcavity while
in[6]Chandra and Husain gave a fractional analogue.

Bector and Husain [4] probably were the first to study duality for multiobjective variational problems under appropriate convexity assumptions. Subsequently, Gulati, Husain and Ahmed [9] presented two distinct pairs of symmetric dual multiobjec-tive variational problems and established various duality results under appropriate invexity requirements. In this reference, self duality theorem is also given under skew symmetric of the integrand of the objective functional. Husain and Jabeen[10] for-mulated a pair of mixed type symmetric dual variational problem in order to unify the Wolfe and Mond-Weir symmetric dual pairs of variational problems studied in[9].

of a pair of mixed type symmetric dual of Husain and Jabeen[10] and hence study symmetric and self duality for a pair of mixed multiobjective variational problem. This research is motivated by the work of Xu [18]. The problems, treated in this research are quite hard to solve. So to expect any immediate application of these problems would be far from reality. Unfortunately, there has not always been sufficient flow between the researchers in the multiple criteria decision making and the researchers applying it to their problems. Of course, one can find optimal control applications in galore which reflect the utility of our model. Special cases are deduced and it is also pointed out that our results can be considered as dynamic generalizations of corresponding (static) symmetric duality results of multiobjective nonlinear nonlinear treated by Bector et al.[3].

**2. Notations and Preliminaries**

The following notation will be used for vectors in*Rn*_{.}
*x* *<* *y* ⇔ *x _{i}*

*<*

*y*,

_{i}*i*=1, 2, . . . ,

*n.*

*x*≦

*y*⇔

*x*≦

_{i}*y*,

_{i}*i*=1, 2, . . . ,

*n.*

*x* ≤ *y* ⇔ *x _{i}* ≤

*y*,

_{i}*i*=1, 2, . . . ,

*n, but*

*x*6=

*y*

*x*6≤

*y, is the negation of*

*x*≤

*y*

I. Husain and R. Mattoo Eur. J. Pure Appl. Math,**2**(2009), (578-603) 581

respectively, by*φi _{t}*,

*φi*,

_{x}*φ*

_{x}_{˙}

*i*,

*φi*,

_{y}*φi*

_{˙}

*, that is,*

_{y}*φi*
*t*=

*∂ φi*
*∂t*

*φi _{x}* =

_{∂ φ}i

*∂x*_{1},

*∂ φi*

*∂x*_{2}, . . . ,

*∂ φi*

*∂x _{n}*

, *φ*_{˙}* _{x}i* =

_{∂ φ}i

*∂*˙*x*_{1},

*∂ φi*

*∂x*˙_{2}, . . . ,

*∂ φi*

*∂*˙*x _{n}*

*φi _{y}* =

_{∂ φ}i

*∂y*_{1},

*∂ φi*

*∂* *y*_{2}, . . . ,

*∂ φi*

*∂* *y _{n}*

, *φi*_{˙}* _{y}* =

_{∂ φ}i*i*

*∂*˙*y*_{1},

*∂ φi*

*∂* ˙*y*_{2}, . . . ,

*∂ φi*

*∂*˙*y _{n}*

.

The twice partial derivatives of*φi* _{with respect to}_{t}_{,}_{x}_{(t), ˙}_{x}_{(t),}_{y}_{(t}_{)}_{and ˙}_{y(t}_{), }
respec-tively are the matrices

*φi*
*x x*=

* _{∂}*2

_{φ}i*∂x _{k}x_{s}*

*n*×*n*
, *φi*

*xx*˙=

* _{∂}*2

_{φ}i*∂x _{k}x*˙

_{s}

*n*×*n*
, *φi*

*x y* =

* _{∂}*2

_{φ}i*∂x _{k}y_{s}*

*n*×*n*

*φi*
*x*˙*y* =

* _{∂}*2

_{φ}i*∂x _{k}*˙

*y*

_{s}

*n*×*n*
, *φi*

˙

*x y* =

* _{∂}*2

_{φ}i*∂x*˙_{k}y_{s}

*n*×*n*
, *φi*

*x*˙*y* =

* _{∂}*2

_{φ}i*∂x _{k}*˙

*y*

_{s}

*n*×*n*

*φi _{y y}* =

* _{∂}*2

_{φ}i*∂y _{k}y_{s}*

*n*×*n*

, *φi _{y}*

_{˙}

*=*

_{y} * _{∂}*2

_{φ}i*∂y _{k}*˙

*y*

_{s}

*n*×*n*

, *φi*_{˙}_{y}_{˙}* _{y}* =

* _{∂}*2

_{φ}i*∂*˙*y _{k}*˙

*y*

_{s}

*n*×*n*

for*i*=1, 2, . . . ,*p.*
Noting that

*d*
*d tφ*

*i*

˙*y* =*φ*
*i*

˙*y t*+*φ*
*i*

˙*y y*˙*y*+*φ*
*i*

˙*y*˙*y*¨*y*+*φ*
*i*

˙

*y xx*˙+*φ*
*i*

˙*y*˙*xx*¨
and hence

*∂*
*∂* *y*

*d*
*d tφ*

*i*

˙

*y* =
*d*
*d tφ*

*i*
*y*˙*y*,

*∂*
*∂*˙*y*

*d*
*d tφ*

*i*

˙

*y* =
*d*
*d tφ*

*i*

˙

*y*˙*y*+*φ*
*i*

˙

*y y*,

*∂*
*∂*¨*y*

*d*
*d tφ*

*i*

˙

*y* =*φ*
*i*
˙
*y y*
*∂*
*∂x*
*d*
*d tφ*

*i*

˙

*y* =
*d*
*d tφ*

*i*

˙

*y x*,

*∂*
*∂x*˙

*d*
*d tφ*

*i*

˙

*y* =
*d*
*d tφ*

*i*

˙

*yx*˙+*φ*

*i*

˙

*y x*,

*∂*
*∂*¨*x*

*d*
*d tφ*

*i*

˙*y* =*φ*
*i*

˙*y*˙*x*

In order to establish our main results, the following are needed.

**Definition 1** (Partially Invex). *If there exists a vector function* *η*(t,*x*(t),*y*(t),*u(t),*
*v(t*))∈**R***n*_{+} *withη*= 0*at x*(t) =*u(t)or y(t) =* *v(t), such that for the scalar function*
*h(t,x*(t), ˙*x(t*),*y(t), ˙y(t))the functional*

*H*(x, ˙*x*,*y, ˙y*) =

Z

*I*

*satisfies*

*H(x*, ˙*x*,*y, ˙y*)−*H(u, ˙u,v, ˙v)* ≥

Z

*I*

[*ηTh _{x}*(t,

*x*(t), ˙

*x(t)*,

*y(t)*, ˙

*y(t))*+ (

*Dη*)

*T*

_{h}˙

*x*(t,*x*(t), ˙*x*(t),*y*(t), ˙*y(t*))]d t

*then H(x*, ˙*x*,*y, ˙y*)*is said to be partially invex in x and* ˙*x on I with respect toη, for fixed*
*y. if H satisfies*

*H*(x, ˙*x*,*y, ˙y*)−*H*(x, ˙*x*,*v, ˙v)* ≥

Z

*I*

[*ηTh _{y}*(t,

*x*(t), ˙

*x(t*),

*v(t), ˙v(t*)) +(D

*η*)

*T*

_{h}˙

*y*(t,*x*(t), ˙*x(t)*,*v(t)*, ˙*v(t))]d t,*

*then H(x*, ˙*x*,*y, ˙y)is said to be partially invex in y and* ˙*y on I with respect toη, for fixed*
*x . If*−*H is partially invex in x and* ˙*x (or in y and* ˙*y) on I with respect toη, for fixed*
*y(or for fixed x ), then H is said to be partially incave in x and* ˙*x (or in y and* ˙*y) on I*
*with respect toη, for fixed y (or for fixed x ).*

**Definition 2**(Partially Pseudoinvex). *The functional H is said to be partially *
*pseudoin-vex in x and* ˙*x with respect toη, for fixed y if H satisfies*

Z

*I*

[*ηTh _{x}*(t,

*x*, ˙

*x,y, ˙y) + (Dη*)

*T*

_{h}˙

*x*(t,*x*, ˙*x,y, ˙y)]d t*≧0
*implies*

*H*(x, ˙*x*,*u, ˙u)*≧*H*(x, ˙*x*,*y, ˙y*)

*and H is said to be partially pseudoinvex in y and* ˙*y with respect toη, for fixed x If H*
*satisfies*

Z

*I*

[*ηTh _{y}*(t,

*x*, ˙

*x,y, ˙y) + (Dη*)

*Th*˙

*y*(t,

*x*, ˙

*x*,

*y, ˙y*)]d t≧0

*implies*

I. Husain and R. Mattoo Eur. J. Pure Appl. Math,**2**(2009), (578-603) 583

**Definition 3**(Partially Quasi-invex). *The functional H is said to be partially quasi-invex*
*in x and* ˙*x with respect toη, for fixed y if H satisfies*

*H*(x, ˙*x*,*u, ˙u)*≦*H*(x, ˙*x*,*y, ˙y*)
*implies*

Z

*I*

[*ηTh _{x}*(t,

*x*, ˙

*x,y, ˙y) + (Dη*)

*T*

_{h}˙

*x*(t,*x*, ˙*x,y, ˙y)]d t*≦0

*and H is said to be partially quasi-invex in y and* ˙*y with respect toη, for fixed x if H*
*satisfies*

*H(x*, ˙*x,v, ˙v)*≦*H*(x, ˙*x*,*y, ˙y*)

*implies*

Z

*I*

[*ηTh _{y}*(t,

*x*, ˙

*x,y, ˙y*) + (D

*η*)

*T*

_{h}˙

*y*(t,*x, ˙x,y, ˙y)]d t* ≦0.

If *h* is independent of *t*, then the above definitions become the usual definitions
of invexity and generalized invexity, discussed by several authors, notably Ben-Israel
and Mond[5], Martin[11], and Rueda and Hanson[16].

**Definition 4**(Skew Symmetry). *The function h*:*I* ×*Rn*×*Rn*×*Rn*×*Rn* →*R is said to*
*be skew symmetric if for all x and y in the domain of h if*

*h(t,x*(t), ˙*x(t)*,*y*(t), ˙*y*(t)) =−*h(t,y*(t), ˙*y(t)*,*x*(t), ˙*x(t*)), *t*∈*I*

*where x and y are piecewise smooth on I .*

Now consider the following multiobjective variational problem considered in[4]:

(VP0) Minimize

Z

*I*

*φ*1(t,*x*, ˙*x)d t*,

Z

*I*

*φ*2(t,*x*, ˙*x)d t*, . . . ,

Z

*I*

*φp*(t,*x*, ˙*x*)d t,

*x*(a) =*α*, *x*(b) =*β*

*h(t,x*, ˙*x)*≦0, *t*∈*I*,

where*φi* _{:} _{I}_{×}_{R}n_{×}_{R}n_{×}_{R}n_{→}_{R}_{(i} _{=}_{1, 2, . . . ,}_{p)}_{and} _{h}_{:} _{I}_{×}_{R}n_{×}_{R}n_{×}_{R}n_{→}_{R}m_{. Let}
the set of feasible solution of(V P_{0})be represented by*K.*

**Definition 5** (Efficiency). *A point* ¯*x* ∈ *K is an efficient (Pareto optimal) solution of*
*(VP*_{0}*) if for all x*∈*K,*

Z

*I*

*φi*(t,*x*, ˙*x)d t*6≤

Z

*I*

*φi*(t, ¯*x, ˙*¯*x*)d t, (i=1, 2, . . . ,*p)*

**3. Statement of the Problems**

For *N* ={1, 2, . . . ,*n*} and *M* = {1, 2, . . . ,*m*}, let *J*_{1} ⊂ *N*,*K*_{1} ⊂ *M*,*J*_{2} = *N*\*J*_{1} and
*K*_{2} = *M* \*K*_{1}. Let |*J*_{1}| denote the number of elements in the subset *J*_{1}. The other
symbol |*J*_{2}|,|*K*_{1}| and |*K*_{2}|are similarly defined. Let *x*1 : *I* → *R*|*J*1| and * _{x}*2 :

*→*

_{I}*|*

_{R}*J*2|,

then any *x* : *I* → *Rn* _{can be written as} _{x}_{= (x}1_{,}* _{x}*2

_{). Similarly for}

*1*

_{y}_{:}

_{I}_{→}

*|*

_{R}*K*1|

and *y*2 : *I* → *R*|*K*2| can be written as * _{y}* = (

*1,*

_{y}*2) where*

_{y}*:*

_{x}*→*

_{I}*,*

_{R}n*:*

_{y}*→*

_{I}*. Let*

_{R}m*f* : *I*×*R*|*J*1|×* _{R}*|

*K*1|→

_{R}p_{and}

_{g}_{:}

*×*

_{I}*|*

_{R}*J*2|×

*|*

_{R}*K*2|→

_{R}p_{be twice continuously differentiable}

functions.

We state the following pair of mixed type multiobjective symmetric dual
varia-tional problems involving vector functions *f* and *g.*

(Mix SP) Minimize *F(x*1,*x*2,*y*1,*y*2) =

Z

*I*

{*f*(t,*x*1, ˙*x*1,*y*1, ˙*y*1) +*g(t,x*2, ˙*x*2,*y*2, ˙*y*2)

−*y*1(t)*T*(*λTf _{y}*1(t,

*x*1, ˙

*x*1,

*y*1, ˙

*y*1) −

*DλT*

_{f}˙

*y*1(t,*x*1, ˙*x*1,*y*1, ˙*y*1))e}*d t*

Subject to

I. Husain and R. Mattoo Eur. J. Pure Appl. Math,**2**(2009), (578-603) 585

*λTf _{y}*1(t,

*x*1, ˙

*x*1,

*y*1, ˙

*y*1)−

*DλTf*

_{˙}

*1(t,*

_{y}*x*1, ˙

*x*1,

*y*1, ˙

*y*1)≦0,

*t*∈

*I*, (3)

*λTg _{y}*2(t,

*x*2, ˙

*x*2,

*y*2, ˙

*y*2)−

*DλTg*

_{˙}

*2(t,*

_{y}*x*2, ˙

*x*2,

*y*2, ˙

*y*2)≦0,

*t*∈

*I*, (4)

Z

*I*

*y*2(t)*T*_{(}_{λ}T_{g}

*y*2(t,*x*2, ˙*x*2,*y*2, ˙*y*2)

−*DλTg*_{˙}* _{y}*2(t,

*x*2, ˙

*x*2,

*y*2, ˙

*y*2))≧0, (5)

*λ*∈Λ+. (6)

(Mix SD) Maximize*G(u*1,*u*2,*v*1,*v*2) =

Z

*I*

{*f*(t,*u*1, ˙*u*1,*v*1, ˙*v*1) +*g(t,u*2, ˙*u*2,*v*2, ˙*v*2)

−*u*1(t)*T*(*λTf _{y}*1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1) −

*DλTf*

_{˙}

*1(t,*

_{y}*u*1, ˙

*u*1,

*v*1, ˙

*v*1))e}

*d t*

Subject to

*u*1(a) =0=*u*1(b), *v*1(a) =0=*v*1(b), (7)
*u*2(a) =0=*u*2(b), *v*2(a) =0=*v*2(b), (8)

*λTf _{u}*1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1)−

*DλTf*

_{u}_{˙}1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1)≧0,

*t*∈

*I*, (9)

*λTg _{u}*2(t,

*u*2, ˙

*u*2,

*v*2, ˙

*v*2)−

*DλTg*

_{u}_{˙}2(t,

*u*2, ˙

*u*2,

*v*2, ˙

*v*2)≧0,

*t*∈

*I*, (10)

Z

*I*

*u*2(t)*T*(*λTg _{u}*2(t,

*u*2, ˙

*u*2,

*v*2, ˙

*v*2)

−*DλTg*_{˙}* _{u}*2(t,

*u*2, ˙

*u*2,

*v*2, ˙

*v*2))≧0, (11)

*λ*∈Λ+. (12)

whereΛ+_{=}_{{}_{λ}_{∈}_{R}p_{|}_{λ >}_{0,}_{λ}T_{e}_{=}_{1,}_{e}_{= (}_{1, 1, . . . , 1}_{)}*T* _{∈}_{R}p_{}}_{.}

**4. Mixed Type Multiobjective Symmetric Duality**

In this section, we present various duality results and the appropriate invexity and generalized invexity assumptions.

**Theorem 1**(Weak Duality). *Let*(x1_{,}* _{x}*2

_{,}

*1*

_{y}_{,}

*2*

_{y}_{,}

_{λ}_{)}

_{be feasible for}_{(Mix SP)}

_{and}*Let*

H1

R

*I* *f*(t, ., .,*y*

1_{(t)}_{, ˙}* _{y}*1

_{(t))d t be partially invex in x}1

_{, ˙}

*1*

_{x}*1*

_{on I for fixed y}_{, ˙}

*1*

_{y}

_{with }*re-spect toη*1(t,*x*1,*u*1)∈*R*|*J*1|*.*

R

*I* *f*(t,*x*

1_{(t}_{)}_{, ˙}* _{x}*1

_{(t)}

_{, ., .}

_{)d t be partially incave in y}1

_{, ˙}

*1*

_{y}*1*

_{on I for fixed x}_{, ˙}

*1*

_{x}

_{with}*respect toη*2(t,*y*1,*v*1)∈*R*|*K*1|*.*

H_{2} R
*Iλ*

*T _{g}*

_{(t, ., .,}

*2*

_{y}_{(t), ˙}

*2*

_{y}_{(t))d t be partially pseudoinvex in x}2

_{, ˙}

*2*

_{x}*2*

_{on I for fixed y}_{, ˙}

*2*

_{y}*with respect* *η*3(t,*x*2,*u*2) ∈ *R*|*J*2| *and*

R
*Iλ*

*T _{g}*

_{(t}

_{,}

*2*

_{x}_{, ˙}

*2*

_{x}_{, . . .)d t be partially }

*pseudoin-cave in y*2, ˙*y*2 *on I for fixed x*2, ˙*x*2 *with respect toη*_{4}(t,*y*2,*v*2)∈*R*|*K*2|_{.}

H3

*η*1(t,*x*1,*u*1) +*u*1(t)≧0, *t*∈*I*, (13)

*η*2(t,*v*1,*y*1) +*y*1(t)≧0, *t*∈*I*, (14)

*η*3(t,*x*2,*u*2) +*u*2(t)≧0, *t*∈*I*, (15)

*η*4(t,*v*2,*y*2) +*y*2(t)≧0, *t*∈*I*, (16)

*then*

*F*(x1,*x*2,*y*1,*y*2)6≤*G*(u1,*u*2,*v*1,*v*2).

*Proof.* Because of the partial invexity-incavity of the function *f*, we have for each
*i*={1, 2, . . . ,*p*}.

Z

*I*

*fi*(t,*x*1, ˙*x*1,*v*1, ˙*v*1)d t−

Z

*I*

*fi*(t,*u*1, ˙*u*1,*v*1, ˙*v*1)d t

≧

Z

*I*

{*η*_{1}*Tf _{x}i*1(t,

*u*

1_{, ˙}* _{u}*1

_{,}

*1*

_{v}_{, ˙}

*1*

_{v}_{) + (D}

_{η}1)*Tf*_{˙}* _{x}i*1(t,

*u*

1_{, ˙}* _{u}*1

_{,}

*1*

_{v}_{, ˙}

*1*

_{v}_{)}

_{}}

_{d t}_{(17)}

Z

*I*

*fi*(t,*x*1, ˙*x*1,*v*1, ˙*v*1)d t−

Z

*I*

*fi*(t,*x*1, ˙*x*1,*y*1, ˙*y*1)d t

≦

Z

*I*

{*η*_{2}*Tf _{y}i*1(t,

*x*

1_{, ˙}* _{x}*1

_{,}

*1*

_{y}_{, ˙}

*1*

_{y}_{) + (D}

_{η}2)*Tf*_{˙}* _{y}i*1(t,

*x*

I. Husain and R. Mattoo Eur. J. Pure Appl. Math,**2**(2009), (578-603) 587

Multiplying (17) by*λi>*0 and summing over*i.*

Z

*I*

*λTf*(t,*x*1, ˙*x*1,*v*1, ˙*v*1)d t−

Z

*I*

*λTf*(t,*u*1, ˙*u*1,*v*1, ˙*v*1)d t

≧

Z

*I*

{*ηT*_{1}(*λTf _{x}*1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1) + (D

*η*

_{1})

*TλTf*

_{˙}

*1(t,*

_{x}*u*1, ˙

*u*1,

*v*1, ˙

*v*1))}

*d t*

Integrating by parts, the above inequality becomes

Z

*I*

*λTf*(t,*x*1, ˙*x*1,*v*1, ˙*v*1)−

Z

*I*

*λTf*(t,*u*1, ˙*u*1,*v*1, ˙*v*1)d t

≧

Z

*I*

*η*_{1}*TλTf _{x}*1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1)d t+

*η*

_{1}

*TλTf*

_{x}_{˙}1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1)|

*t*=*b*
*t*=*a*

−

Z

*I*

*ηT*_{1}*DλTf*_{˙}* _{x}*1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1)d t

Using the boundary conditions which at*t*=*a,t*= *b*gives*η*_{1}=0, we have

Z

*I*

*λTf*(t,*x*1, ˙*x*1,*v*1, ˙*v*1)−

Z

*I*

*λTf*(t,*u*1, ˙*u*1,*v*1, ˙*v*1)d t

≧

Z

*I*

*ηT*_{1}[*λTf _{x}*1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1)d t−

*D(λTf*

_{˙}

*1(t,*

_{x}*u*1, ˙

*u*1,

*v*1, ˙

*v*1))]d t (19)

Multiplying (18) by*λi*_{,}_{i}_{∈ {}_{1, 2, . . . ,}_{p}_{}}_{and summing over}* _{i, we get,}*
Z

*I*

*λT _{f}*

_{(t,}

*1*

_{x}_{, ˙}

*1*

_{x}_{,}

*1*

_{v}_{, ˙}

*1*

_{v}_{)}

_{−}

Z

*I*

*λT _{f}*

_{(t}

_{,}

*1*

_{x}_{, ˙}

*1*

_{x}_{,}

*1*

_{y}_{, ˙}

*1*

_{y}_{)d t}

≦

Z

*I*

{*ηT*_{2}(*λTf _{y}*1(t,

*x*1, ˙

*x*1,

*y*1, ˙

*y*1)) + (D

*η*

_{2})

*TλTf*

_{˙}

*1(t,*

_{y}*x*1, ˙

*x*1,

*y*1, ˙

*y*1)}

*d t*

On integrating by parts the R.H.S of the above inequality and using the boundary
conditions which at *t*=*a,t*=*b* gives*η*2 =0, we have

Z

*I*

*λTf*(t,*x*1, ˙*x*1,*v*1, ˙*v*1)−

Z

*I*

*λTf*(t,*x*1, ˙*x*1,*y*1, ˙*y*1)d t

≦

Z

*I*

*ηT*_{2}[(*λTf _{y}*1(t,

*x*1, ˙

*x*1,

*y*1, ˙

*y*1))−

*D(λTf*

_{˙}

*1(t,*

_{y}*x*1, ˙

*x*1,

*y*1, ˙

*y*1))]d t (20)

Multiplying (20) by (-1) and adding to (19), we have

Z

*I*

*λTf*(t,*x*1, ˙*x*1,*y*1, ˙*y*1)d t−

Z

*I*

≧

Z

*I*

*ηT*_{1}[(*λTf _{x}*1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1))−

*D(λTf*

_{˙}

*1(t,*

_{x}*u*1, ˙

*u*1,

*v*1, ˙

*v*1))]d t

−

Z

*I*

*η*_{2}*T*[(*λTf _{y}*1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1))−

*D(λTf*

_{˙}

*1(t,*

_{y}*u*1, ˙

*u*1,

*v*1, ˙

*v*1))]d t (21)

Now from the inequality (9) along with (13), it follows

Z

*I*

*ηT*_{1}(*λTf _{u}*1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1)−

*DλTf*

_{u}_{˙}1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1))d t

≧−

Z

*I*

*u*1(t)*T*[*λTf _{u}*1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1)−

*DλTf*

_{˙}

*1(t,*

_{u}*u*1, ˙

*u*1,

*v*1, ˙

*v*1)]d t (22)

Also from the inequality (3) together with (14) implies

−

Z

*I*

*ηT*_{2}(*λTf _{y}*1(t,

*x*1, ˙

*x*1,

*y*1, ˙

*y*1)−

*DλTf*

_{˙}

*1(t,*

_{y}*x*1, ˙

*x*1,

*y*1, ˙

*y*1))d t

≧

Z

*I*

*y*1(t)*T*_{[}_{λ}T_{f}

*y*1(t,*x*1, ˙*x*1,*y*1, ˙*y*1)−*DλTf*_{˙}* _{y}*1(t,

*x*1, ˙

*x*1,

*y*1, ˙

*y*1)]d t (23)

Using (22) and (23), in (21), we have

Z

*I*

*λTf*(t,*x*1, ˙*x*1,*y*1, ˙*y*1)d t− *y*1(t)*T*

Z

*I*

*λTf _{y}*1(t,

*x*1, ˙

*x*1,

*y*1, ˙

*y*1)d t

≧−

Z

*I*

*u*1(t)*T*[(*λTf _{u}*1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1))−

*D(λTf*

_{˙}

*1(t,*

_{u}*u*1, ˙

*u*1,

*v*1, ˙

*v*1))]d t

+

Z

*I*

*y*1(t)*T*_{[(}_{λ}T_{f}

*y*1(t,*x*1, ˙*x*1,*y*1, ˙*y*1))

−*D(λTf*_{˙}* _{y}*1(t,

*x*1, ˙

*x*1,

*y*1, ˙

*y*1))]d t, (24)

which implies

Z

*I*

{*λTf*(t,*x*1, ˙*x*1,*y*1, ˙*y*1)− *y*1(t)*T*_{(}_{λ}T_{f}

*y*1(t,*x*1, ˙*x*1,*y*1, ˙*y*1)
−*DλTf*_{˙}* _{y}*1(t,

*x*1, ˙

*x*1,

*y*1, ˙

*y*1))}

*d t*

≧

Z

*I*

{*λTf _{u}*1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1)−

*u*1(t)

*T*(

*λTf*1(t,

_{x}*u*1, ˙

*u*1,

*v*1, ˙

*v*1)

I. Husain and R. Mattoo Eur. J. Pure Appl. Math,**2**(2009), (578-603) 589

Now from the inequality (10) along with (15), we have

Z

*I*

(*η*_{3}*T*(t,*x*2, ˙*x*2,*u*2, ˙*u*2) +*u*2(t))(*λTg _{u}*2(t,

*u*2, ˙

*u*2,

*v*2, ˙

*v*2) −

*DλTg*

_{u}_{˙}2(t,

*u*2, ˙

*u*2,

*v*2, ˙

*v*2))≧0.

This implies

Z

*I*

*ηT*_{3}(*λTg _{u}*2(t,

*u*2, ˙

*u*2,

*v*2, ˙

*v*2)−

*D(λTg*

_{˙}

*2(t,*

_{u}*u*2, ˙

*u*2,

*v*2, ˙

*v*2)))d t

≧−

Z

*I*

*u*2(t)*T*[*λTg _{u}*2(t,

*u*2, ˙

*u*2,

*v*2, ˙

*v*2)−

*D(λTg*

_{u}_{˙}2(t,

*u*2, ˙

*u*2,

*v*2, ˙

*v*2))]d t

Integrating by parts and using the boundary conditions which at *t* = *a,t* = *b* gives

*η*3=0, we have

Z

*I*

{*η*_{3}*T*(*λTg _{u}*2(t,

*u*2, ˙

*u*2,

*v*2, ˙

*v*2) + (D

*η*

_{3})

*T*(

*λTg*

_{˙}

*2(t,*

_{u}*u*2, ˙

*u*2,

*v*2, ˙

*v*2)))}

*d t*≧0

Because of the partial pseudo-invexity ofR_{I}λT_{g}

*u*2*d t, this gives*

Z

*I*

*λTg(t,x*2, ˙*x*2,*y*2, ˙*y*2)d t ≧

Z

*I*

*λTg(t*,*u*2, ˙*u*2,*v*2, ˙*v*2)d t (26)
Also from (4) together with (16), we have

Z

*I*

(*η*_{4}*T*(t,*v*2, ˙*v*2,*y*2, ˙*y*2) +*y*2(t))(*λTg _{y}*2(t,

*x*2, ˙

*x*2,

*y*2, ˙

*y*2) −

*DλTg*

_{˙}

*2(t,*

_{y}*x*2, ˙

*x*2,

*y*2, ˙

*y*2))d t ≧0

This implies,

Z

*I*

*η*_{4}*T*(*λTg _{y}*2(t,

*x*2, ˙

*x*2,

*y*2, ˙

*y*2))−

*D(λTg*

_{˙}

*2(t,*

_{y}*x*2, ˙

*x*2,

*y*2, ˙

*y*2))d t

≦−

Z

*I*

*y*2(t)*T*[*λTg _{y}*2(t,

*x*2, ˙

*x*2,

*y*2, ˙

*y*2)−

*D(λTg*

_{˙}

*2(t,*

_{y}*x*2, ˙

*x*2,

*y*2, ˙

*y*2))]d t

This in view of (5) yields,

Z

*I*

On integrating by parts and using the boundary conditions which at*t*=*a,t*= *b*gives

*η*4=0, we have,

Z

*I*

*η*_{4}*T*{*λTg _{y}*2(t,

*x*2, ˙

*x*2,

*y*2, ˙

*y*2) + (D

*η*

_{4})

*T*(

*λTg*

_{˙}

*2(t,*

_{y}*x*2, ˙

*x*2,

*y*2, ˙

*y*2))}

*d t*≦0

Because of partial pseudo-incavity ofR* _{I}λTg_{y}*2

*d t, we have*

Z

*I*

(*λTg*(t,*x*2, ˙*x*2,*v*2, ˙*v*2))d t≦

Z

*I*

(*λTg(t,x*2, ˙*x*2,*y*2, ˙*y*2))d t (27)

From (26) and (27), we get,

Z

*I*

(*λT _{g}*

_{(t}

_{,}

*2*

_{x}_{, ˙}

*2*

_{x}_{,}

*2*

_{y}_{, ˙}

*2*

_{y}_{))d t}

_{≧}

Z

*I*

(*λT _{g(t,}_{u}*2

_{, ˙}

*2*

_{u}_{,}

*2*

_{v}_{, ˙}

*2*

_{v}_{))d t}

_{(28)}

Combining (25) and (28), we get

Z

*I*

{*λTf*(t,*x*1, ˙*x*1,*y*1, ˙*y*1)−*y*1(t)*T*(*λTf _{y}*1(t,

*x*1, ˙

*x*1,

*y*1, ˙

*y*1)) −

*DλTf*

_{˙}

*1(t,*

_{y}*x*1, ˙

*x*1,

*y*1, ˙

*y*1) +

*λTg*(t,

*x*2, ˙

*x*2,

*y*2, ˙

*y*2)}

*d t*

≧

Z

*I*

{*λT _{f}*

_{(t,}

*1*

_{u}_{, ˙}

*1*

_{u}_{,}

*1*

_{v}_{, ˙}

*1*

_{v}_{)}

_{−}

*1*

_{u}_{(t)}

*T*

_{(}

_{λ}T_{f}*x*1(t,*u*1, ˙*u*1,*v*1, ˙*v*1))
−*DλTf*_{˙}* _{x}*1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1) +

*λTg(t,x*2, ˙

*x*2,

*y*2, ˙

*y*2)}

*d t*

This implies,

*λT*

Z

*I*

{*f*(t,*x*1, ˙*x*1,*y*1, ˙*y*1) +*g*(t,*x*2, ˙*x*2,*y*2, ˙*y*2)

−*y*1(t)*T*(*λTf _{y}*1(t,

*x*1, ˙

*x*1,

*y*1, ˙

*y*1)−

*DλTf*

_{˙}

*1(t,*

_{y}*x*1, ˙

*x*1,

*y*1, ˙

*y*1))e}

*d t*

≧*λT*

Z

*I*

{*f*(t,*u*1, ˙*u*1,*v*1, ˙*v*1) +*g*(t,*u*2, ˙*u*2,*v*2, ˙*v*2)

−*u*1(t)*T*(*λTf _{x}*1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1)−

*DλTf*

_{˙}

*1(t,*

_{x}*u*1, ˙

*u*1,

*v*1, ˙

*v*1))e}

*d t*

This implies,

Z

*I*

I. Husain and R. Mattoo Eur. J. Pure Appl. Math,**2**(2009), (578-603) 591

−*y*1(t)*T*(*λTf _{y}*1(t,

*x*1, ˙

*x*1,

*y*1, ˙

*y*1)−

*DλTf*

_{˙}

*1(t,*

_{y}*x*1, ˙

*x*1,

*y*1, ˙

*y*1))e}

*d t*6≤

Z

*I*

{*f*(t,*u*1, ˙*u*1,*v*1, ˙*v*1) +*g*(t,*u*2, ˙*u*2,*v*2, ˙*v*2)

−*u*1(t)*T*(*λTf _{x}*1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1)−

*DλTf*

_{˙}

*1(t,*

_{x}*u*1, ˙

*u*1,

*v*1, ˙

*v*1))e}

*d t*

This was to be proved.

**Theorem 2**(Strong Duality). *Let*(¯*x*1, ¯*x*2, ¯*y*1, ¯*y*2, ¯*λ*)*be an efficient solution of* (Mix SP).
*Letλ*=*λ*¯ *be fixed in*(Mix SD)*and*

(C1)

Z

*I*

[{(*φ*1(t))*T*(*λTf _{y}*1

*1−*

_{y}*DλTf*1

_{y}_{˙}

*1)−*

_{y}*Dφ*1(t)

*T*(−

*DλTf*

_{˙}

*1*

_{y}_{˙}

*1)*

_{y}+*D*2*φ*1(t)*T*(−*λTf*˙*y*1_{˙}* _{y}*1)}

*φ*1(t)]d t

*>*0,

*and*

Z

*I*

[{(*φ*2(t))*T*_{(}_{λ}T_{g}

*y*2* _{y}*2−

*DλTg*2

_{y}_{˙}

*2)−*

_{y}*Dφ*2(t)

*T*(−

*DλTg*

_{˙}

*2*

_{y}_{˙}

*2)*

_{y}+D2*φ*2(t)*T*(−*λTg*_{˙}* _{y}*2

_{˙}

*2)}*

_{y}*φ*2(t)]d t

*>*0,

(C2)

Z

*I*

[{(*φ*1(t))*T*_{(}_{λ}T_{f}

*y*1* _{y}*1−

*DλTf*1

_{y}_{˙}

*1)−*

_{y}*Dφ*1(t)

*T*(−

*DλTf*

_{˙}

*1*

_{y}_{˙}

*1)*

_{y}+D2*φ*1(t)*T*(−*λTf*_{˙}* _{y}*1

_{˙}

*1)}*

_{y}*φ*1(t)]d t=0,

*t*∈

*I*⇒

*φ*1(t) =0,

*t*∈

*I*,

*and*

Z

*I*

[{(*φ*2(t))*T*(*λTg _{y}*2

*2−*

_{y}*DλTg*2

_{y}_{˙}

*2)−*

_{y}*Dφ*2(t)

*T*(−

*DλTg*

_{˙}

*2*

_{y}_{˙}

*2)*

_{y}+D2*φ*2(t)*T*(−*λTg*_{˙}* _{y}*2

_{˙}

*2)}*

_{y}*φ*2(t)]d t =0,

*t*∈

*I*⇒

*φ*1(t) =0,

*t*∈

*I*.

*and*

(C_{3}) *gi*

*y*2−*Dg*

*i*

˙

*y*2 =0,*i*=1, 2, . . . ,*p are linearly independent.*

*Let*R

*I* *f d t and*
R

*Iλ*

*T _{gd t satisfy the invexity and generalized invexity as stated in }*

*Proof.* Since(*x*¯1, ¯*x*2, ¯*y*1, ¯*y*2, ¯*λ*)is efficient, it is weak minimum. Hence there exists

*τ*∈*Rp*_{,}_{η}_{∈} _{R}p_{,}_{γ}_{∈}_{R}_{and piecewise smooth functions} * _{θ}*1

_{(t}

_{)}

_{:}

_{I}_{→}

*|*

_{R}*K*1|

_{,}

*2*

_{θ}_{(t)}

_{:}

*→*

_{I}*R*|*k*2| _{and} _{µ}_{:} * _{I}* →

_{R}m_{such that the following Fritz-John optimality conditions, in view}

of the analysis on[13, 9, 14], are satisfied

*H* = *τ*(*f* +*g*) + (*θ*1(t)−(*τTe)y*1(t)*T*_{)(}_{λ}T_{f}

*y*1−*DλTf*_{˙}* _{y}*1)

+(*θ*2(t)−*γy*2(t)*T*_{)(}_{λ}T_{g}

*y*2−*DλTg*_{˙}* _{y}*2) +

*ηTλ*

Satisfying

*H _{x}*1−

*DH*

_{˙}

*1+*

_{x}*D*2

*H*

_{¨}

*1 =0,*

_{x}*t*∈

*I*(29)

*H _{x}*2−

*DH*

_{˙}

*2+*

_{x}*D*2

*H*

_{¨}

*2 =0,*

_{x}*t*∈

*I*(30)

*H _{y}*1−

*DH*

_{˙}

*1+*

_{y}*D*2

*H*

_{¨}

*1 =0,*

_{y}*t*∈

*I*(31)

*H _{y}*2−

*DH*

_{˙}

*2+*

_{y}*D*2

*H*

_{¨}

*2 =0,*

_{y}*t*∈

*I*(32)

(*θ*1(t)−(*τTe)y*1(t))*T*(*f _{y}*1−

*D f*

_{˙}

*1) + (*

_{y}*θ*2(t)−

*γy*2(t))

*T*(g

*2−*

_{y}*Dg*

_{˙}

*2)−*

_{y}*η*=0,

*t*∈

*I*

(33)

*θ*1(t)(*λTf _{y}*1−

*DλTf*

_{˙}

*1) =0,*

_{y}*t*∈

*I*(34)

*θ*2(t)(*λTg _{y}*2−

*DλTg*

_{˙}

*2) =0,*

_{y}*t*∈

*I*(35)

*γ*

Z

*I*

*y*2(t)*T*_{(}_{λ}T_{g}

*y*2−*DλTg*_{˙}* _{y}*2) =0 (36)

*ηTλ*¯=0 (37)

(*τ*,*θ*1(t),*θ*2(t),*η*,*γ*)≧0, *t*∈*I* (38)
(*τ*,*θ*1(t),*θ*2(t),*η*,*γ*)6=0, *t*∈*I* (39)

hold throughout *I* (except at the corners of (¯*x*1(t), ¯*x*2(t), ¯*y*1(t), ¯*y*2(t)) where
(29)-(32) are valid for unique right and left hand limits). Here*θ*1 _{and} * _{θ}*2

_{are continuous}

I. Husain and R. Mattoo Eur. J. Pure Appl. Math,**2**(2009), (578-603) 593

The relations (29)-(32) are all deducible from the classical Euler-Lagrange and
Clebsch necessary optimality conditions. Particularly, the equations (29)-(32) are the
famous Euler-Lagrange differential equation when second order derivatives appear in
*H*. Using the analogies of the observation of *Dφ*¨*y* from the notational section, the
equations (29)-(32) become,

*τ*(*f _{x}*1−

*D f*

_{x}_{˙}1)−(

*θ*1(t)−(

*τTe)*¯

*y*1(t))

*T*(

*λTf*1

_{y}*1−*

_{x}*DλTf*

_{˙}

*1*

_{y}*1) −*

_{x}*D(θ*1(t)−(

*τTe)*¯

*y*1(t))

*T*

_{(}

_{λ}T_{f}*y*1_{˙}* _{x}*1−

*DλTf*

_{˙}

*1*

_{y}_{˙}

*1−*

_{x}*λTf*

_{˙}

*1*

_{y}*1)*

_{x}+*D*2((*θ*1(t)−(*τTe)*¯*y*1(t))*T*_{(}_{−}_{λ}T_{f}

˙*y*1_{˙}* _{x}*1)) =0 (40)

*τ*(g* _{x}*2−

*Dg*

_{˙}

*2) + (*

_{x}*θ*2(t)−

*γ*¯

*y*2(t))

*T*(

*λTg*2

_{y}*2−*

_{x}*DλTg*

_{˙}

*2*

_{y}*2) −*

_{x}*D(θ*2(t)−

*γ*¯

*y*2(t))

*T*

_{(}

_{λ}T_{g}*y*2_{˙}* _{x}*2−

*DλTg*

_{˙}

*2*

_{y}_{˙}

*2−*

_{x}*λTg*

_{˙}

*2*

_{y}*2)*

_{x}+D2_{((}* _{θ}*2

_{(t)}

_{−}

_{γ}_{y}_{¯}2

_{(t}

_{))}

*T*

_{(}

_{−}

_{λ}T_{g}˙

*y*2_{x}_{˙}2)) =0 (41)

(*τ*−(*τTe)λ*)*T*(*f _{y}*1−

*D f*

_{˙}

*1) + (*

_{y}*θ*1(t)−(

*τTe)*¯

*y*1(t))

*T*×(

*λTf*1

_{y}*1−*

_{y}*DλTf*

_{˙}

*1*

_{y}*1)*

_{y}−*D(θ*1(t)−(*τTe)*¯*y*1(t))*T*(−*DλTf*_{˙}* _{y}*1

_{˙}

*1)*

_{y}+D2((*θ*1(t)−(*τTe)*¯*y*1(t))*T*(−*λTf*_{˙}* _{y}*1

_{˙}

*1)) =0 (42)*

_{y}(*τ*−*γλ*)*T*(g* _{y}*2−

*Dg*

_{˙}

*2) + (*

_{y}*θ*2(t)−

*γ*¯

*y*2(t))

*T*(

*λTg*2

_{y}*2−*

_{y}*DλTg*

_{˙}

*2*

_{y}*2) −*

_{y}*D(θ*2(t)−

*γ*¯

*y*2(t))

*T*(−

*DλTg*

_{˙}

*2*

_{y}_{˙}

*2)*

_{y}+D2(*θ*2(t)−*γ*¯*y*2(t))*T*(−*λTg*_{˙}* _{y}*2

_{˙}

*2) =0 (43)*

_{y}Since*λ >*0, (37) implies*η*=0. Consequently, (33) reduces to

(*θ*1(t)−(*τTe)y*1(t))*T*(*f _{y}*1−

*D f*

_{˙}

*1) + (*

_{y}*θ*2(t)−

*γy*2(t))

*T*(g

*2−*

_{y}*Dg*

_{˙}

*2) =0,*

_{y}*t*∈

*I*(44)

Postmultiplying (42) by(*θ*1(t)−(*τTe)y*1(t)), (43) by(*θ*2(t)−*γy*2(t)) and then
adding, we have

−*D[(θ*1(t)−(*τTe)*¯*y*1(t))*T*(−*DλTf*_{˙}* _{y}*1

_{˙}

*1)]*

_{y}+*D*2[(*θ*1(t)−(*τTe)*¯*y*1(t))*T*(−*λTf*_{˙}* _{y}*1

_{˙}

*1)]}(*

_{y}*θ*1(t)−(

*τTe)¯y*1(t))

+{(*τ*−*γλ*)*T*(g* _{y}*2−

*Dg*

_{˙}

*2) + (*

_{y}*θ*2(t)−

*γy*¯2(t))

*T*(

*λTg*2

_{y}*2−*

_{y}*DλTg*

_{˙}

*2*

_{y}*2) −*

_{y}*D[(θ*2(t)−

*γ*¯

*y*2(t))

*T*(−

*DλTg*

_{˙}

*2*

_{y}_{˙}

*2)]*

_{y}+*D*2[(*θ*2(t)−*γ*¯*y*2(t))*T*(−*λTg*_{˙}* _{y}*2

_{˙}

*2)]}(*

_{y}*θ*2(t)−

*γ*¯

*y*2(t)) =0 (45)

Now multiplying (44) by ¯*λ*and then using (35) and (36) we have

Z

*I*

(*θ*1(t)−(*τTe)*¯*y*1(t))*T*_{(}_{λ}T_{f}

*y*1−*DλTf*_{˙}* _{y}*1)d t =0

that is

Z

*I*

(*θ*1(t)−(*τT _{e)}*

_{¯}

*1*

_{y}_{(t))}

*T*

_{(}

_{λ}T_{f}*y*1−*DλTf _{y}*

_{˙}1)(

*τTe)d t*=0 (46)

Multiplying (44) by*τ*, we have

Z

*I*

[(*θ*1(t)−(*τTe)y*1(t))*T*(*τf _{y}*1−

*Dτf*

_{˙}

*1)*

_{y}+(*θ*2(t)−*γy*2(t))*T*_{(}_{τ}_{g}

*y*2−*Dτg*_{˙}* _{y}*2)]d t=0 (47)

Subtracting (46) and (47) and using (35) and (36), we have

Z

*I*

[(*θ*1(t)−(*τTe)y*1(t))*T*(*f _{y}*1−

*D f*

_{˙}

*1)(*

_{y}*τ*−(

*τTe)λ*¯)

+(*θ*2(t)−*γy*2(t))*T*(*τg _{y}*2−

*Dτg*

_{˙}

*2)(*

_{y}*τ*−

*γλ*¯)]d t=0 (48)

From (45) and (48), we obtain

Z

*I*

[{(*θ*1(t)−(*τTe)*¯*y*1(t))*T*(*λTf _{y}*1

*1−*

_{y}*DλTf*1

_{y}_{˙}

*1) −*

_{y}*D[(θ*1(t)−(

*τTe)*¯

*y*1(t))

*T*(−

*DλTf*˙

*y*1

_{˙}

*1)]*

_{y}+*D*2[(*θ*1(t)−(*τTe)*¯*y*1(t))*T*(−*λTf*_{˙}* _{y}*1

_{˙}

*1)]}.(*

_{y}*θ*1(t)−(

*τTe)*¯

*y*1(t))]d t

+

Z

*I*

I. Husain and R. Mattoo Eur. J. Pure Appl. Math,**2**(2009), (578-603) 595

−*D[(θ*2(t)−*γ*¯*y*2(t))*T*(−*DλTg*_{˙}* _{y}*2

_{˙}

*2)]*

_{y}+*D*2[(*θ*2(t)−*γ*¯*y*2(t))*T*(−*λTg*_{˙}* _{y}*2

_{˙}

*2)]}(*

_{y}*θ*2(t)−

*γ*¯

*y*2(t))]d t =0

In view of the hypothesis(C_{1}), we have

Z

*I*

[{(*θ*1(t)−(*τTe)*¯*y*1(t))*T*(*λTf _{y}*1

*1−*

_{y}*DλTf*1

_{y}_{˙}

*1) −*

_{y}*D[(θ*1(t)−(

*τTe)*¯

*y*1(t))

*T*(−

*DλTf*

_{˙}

*1*

_{y}_{˙}

*1)]*

_{y}+*D*2[(*θ*1(t)−(*τTe)*¯*y*1(t))*T*(−*λTf*_{˙}* _{y}*1

_{˙}

*1)]}.(*

_{y}*θ*1(t)−(

*τTe)*¯

*y*1(t))]d t=0

and

Z

*I*

[{(*θ*2(t)−*γ*¯*y*2(t))*T*(*λTg _{y}*2

*2−*

_{y}*DλTg*2

_{y}_{˙}

*2) −*

_{y}*D[(θ*2(t)−

*γ*¯

*y*2(t))

*T*(−

*DλTg*

_{˙}

*2*

_{y}_{˙}

*2)]*

_{y}+D2[(*θ*2(t)−*γ*¯*y*2(t))*T*(−*λTg _{y}*

_{˙}2

_{˙}

*2)]}(*

_{y}*θ*2(t)−

*γ*¯

*y*2(t))]d t=0

This in view of the hypothesis(C_{2})yields,

*φ*1(t) =*θ*1(t)−(*τTe)¯y*1(t) =0, *t*∈*I* (49)

*φ*2(t) =*θ*2(t)−*γ*¯*y*2(t) =0, *t*∈*I* (50)
From (50) and (43), we have

(*τ*−*γλ*)*T*(g* _{y}*2−

*Dg*

_{˙}

*2) =0*

_{y}that is

*p*
X

*i*=1

(*τi*−*γλi*)*T*_{(}_{g}

*y*2−*Dg*_{˙}* _{y}*2) =0

This in view of the(C_{3})yields

Let if possible, *γ*= 0. Then from (51), we have*τ*=0 and therefore, from (49) and
(50), we have

*φ*1(t) =0,*θ*2(t) =0,*t*∈*I*

Hence (*τ*,*θ*1(t),*θ*2(t),*η*,*γ*) = 0, contradicting Fritz-John conditions (39). Hence

*γ >*0 and consequently*τ >*0.

From (40) and (4.29) along with (51), we obtain
(*λ*¯*T _{f}*

*x*1−*Dλ*¯*Tf*_{˙}* _{x}*1) =0,

*t*∈

*I*(52)

(*λ*¯*Tg _{x}*2−

*Dλ*¯

*Tg*

_{˙}

*2) =0,*

_{x}*t*∈

*I*(53)

which implies

Z

*I*

*x*2(t)*T*(*λ*¯*Tg _{x}*2−

*Dλ*¯

*Tg*

_{˙}

*2)d t=0 (54)*

_{x}From (52)-(54) together with (49), we have

*y*1(t)*T*(*λ*¯*Tf _{y}*1−

*Dλ*¯

*Tf*

_{˙}

*1) =0,*

_{y}*t*∈

*I*(55)

From the primal objective with (55)

Z

*I*

{*f*(t,*x*1, ˙*x*1,*y*1, ˙*y*1) +*g(t,x*2, ˙*x*2,*y*2, ˙*y*2)

− *y*1(t)*T*(*λTf _{y}*1−

*DλTf*

_{˙}

*1)}*

_{y}*d t*

=

Z

*I*

{*f*(t,*x*1, ˙*x*1,*y*1, ˙*y*1) +*g*(t,*x*2, ˙*x*2,*y*2, ˙*y*2)}*d t* (56)

From the dual objective in view of (52), we have

Z

*I*

{*f*(t,*x*1, ˙*x*1,*y*1, ˙*y*1) +*g(t,x*2, ˙*x*2,*y*2, ˙*y*2)

− *x*1(t)*T*(*λTf _{x}*1−

*DλTf*

_{˙}

*1)}*

_{x}*d t*

=

Z

*I*

I. Husain and R. Mattoo Eur. J. Pure Appl. Math,**2**(2009), (578-603) 597

From (55) and (57), the equality of objective values is evident. Consequently, in view
of the hypothesis of Theorem 1, the efficiency of(*x*¯1, ¯*x*2, ¯*y*1, ¯*y*2, ¯*λ*)follows.

We now state converse duality whose proof follows by symmetry.

**Theorem 3**(Converse Duality). *Let*(*x*¯1, ¯*x*2, ¯*y*1, ¯*y*2, ¯*λ*)*be an efficient solution of*(Mix SP).
*Letλ*=*λ*¯ *be fixed in*(Mix SD)*and*

(A1)

Z

*I*

[{*ψ*1(t)*T*(*λTf _{x}*1

*1−*

_{x}*DλTf*1

_{x}

_{x}_{˙}1)−

*Dψ*1(t)

*T*(−

*DλTf*

_{x}_{˙}1

_{x}_{˙}1)

+D2*ψ*1(t)*T*(−*λTf*_{˙}* _{x}*1

_{˙}

*1)}*

_{x}*ψ*1(t)]d t

*>*0,

*and*

Z

*I*

[{*ψ*2(t)*T*(*λTg _{x}*2

*2−*

_{x}*DλTg*2

_{x}

_{x}_{˙}2)−

*Dψ*2(t)

*T*(−

*DλTg*

_{˙}

*2*

_{x}_{˙}

*2)*

_{x}+D2*ψ*2(t)*T*_{(}_{−}_{λ}T_{g}

˙

*x*2_{˙}* _{x}*2)}

*ψ*2(t)]d t

*>*0,

(A_{2})

Z

*I*

[{*ψ*1(t)*T*(*λTf _{x}*1

*1−*

_{x}*DλTf*1

_{x}

_{x}_{˙}1)−

*Dψ*1(t)

*T*(−

*DλTf*

_{˙}

*1*

_{x}_{˙}

*1)*

_{x}+D2* _{ψ}*1

_{(t)}

*T*

_{(}

_{−}

_{λ}T_{f}˙

*x*1_{˙}* _{x}*1)}

*ψ*1(t)]d t=0,

*t*∈

*I*⇒

*ψ*1(t) =0,

*t*∈

*I*,

*and*

Z

*I*

[{*ψ*2(t)*T*_{(}_{λ}T_{g}

*x*2* _{x}*2−

*DλTg*2

_{x}_{˙}

*2)−*

_{x}*Dψ*2(t)

*T*(−

*DλTg*

_{˙}

*2*

_{x}

_{x}_{˙}2)

+*D*2*ψ*2(t)*T*(−*λTg*_{˙}* _{x}*2

_{˙}

*2)}*

_{x}*ψ*2(t)]d t=0,

*t*∈

*I*⇒

*ψ*2(t) =0,

*t*∈

*I*

*and*

(A3) *g _{x}i*2−

*Dg*

*i*

˙

*x*2 =0,*i*=1, 2, . . . ,*p are linearly independent.*

**5. Self Duality**

A problem is said to be self-dual if it is formally identical with its dual, in general,
the problems (Mix SP) and (Mix SD) are not formally in the absence of an additional
restrictions of the function *f* and *g. Hence skew symmetric of* *f* and *g* is assumed in
order to validate the following self-duality theorem.

**Theorem 4** (Self Duality). *Let fi* *and gi*,*i* =1, 2, . . . ,*p, be skew symmetric. Then the*
*problem*(Mix SP)*is self dual. If the problems*(Mix SP)*and*(Mix SD)*are dual problems*
*and*(¯*x*1(t), ¯*x*2(t), ¯*y*(t), ¯*y*2(t), ¯*λ*)*is a joint optimal solution of* (Mix SP)*and*(Mix SD),
*then so is*(¯*y(t*), ¯*y*2_{(t), ¯}* _{x}*1

_{(t}

_{), ¯}

*2*

_{x}_{(t), ¯}

_{λ}_{), and the common functional value is zero, i.e.}

Minimum(Mix SP) =

Z

*I*

{*f*(*x*1, ˙*x*1,*y*1, ˙*y*1) +*g*(x2_{, ˙}* _{x}*2

_{,}

*2*

_{y}_{, ˙}

*2*

_{y}_{)}

_{}}

_{d t}_{=}

_{0}

*Proof.* By skew symmetric of *fi* _{and} _{g}i_{, we have}

*f _{x}i*1(t,

*x*

1_{(t}_{)}_{, ˙}* _{x}*1

_{(t}

_{)}

_{,}

*1*

_{y}_{(t}

_{)}

_{, ˙}

*1*

_{y}_{(t}

_{)) =}

_{−}

_{f}i*y*1(t,

*y*

1_{(t)}_{, ˙}* _{y}*1

_{(t)}

_{,}

*1*

_{x}_{(t)}

_{, ˙}

*1*

_{x}_{(t))}

*gi _{x}*2(t,

*x*

2_{(t)}_{, ˙}* _{x}*2

_{(t)}

_{,}

_{y}_{(t)}

_{, ˙}

*2*

_{y}_{(t)) =}

_{−}

_{g}i˙

*y*2(t,*y*

2_{(t)}_{, ˙}* _{y}*2

_{(t)}

_{,}

*2*

_{x}_{(t)}

_{, ˙}

*2*

_{x}_{(t))}

*f _{y}i*1(t,

*x*

1_{(t)}_{, ˙}* _{x}*1

_{(t)}

_{,}

*1*

_{y}_{(t)}

_{, ˙}

*1*

_{y}_{(t)) =}

_{−}

_{f}i*x*1(t,

*y*

1_{(t)}_{, ˙}* _{y}*1

_{(t)}

_{,}

*1*

_{x}_{(t)}

_{, ˙}

*1*

_{x}_{(t))}

*gi _{y}*2(t,

*x*2

_{(t)}

, ˙*x*2(t),*y*(t), ˙*y*2(t)) =−*gi*_{˙}* _{x}*2(t,

*y*2

_{(t)}

, ˙*y*2(t),*x*2(t), ˙*x*2(t))
*f _{x}i*1(t,

*x*

1_{(t}_{)}_{, ˙}* _{x}*1

_{(t}

_{)}

_{,}

*1*

_{y}_{(t}

_{)}

_{, ˙}

*1*

_{y}_{(t}

_{)) =}

_{−}

_{f}i˙

*y*1(t,*y*

1_{(t)}_{, ˙}* _{y}*1

_{(t)}

_{,}

*1*

_{x}_{(t)}

_{, ˙}

*1*

_{x}_{(t))}

*gi*_{˙}* _{x}*2(t,

*x*2

_{(t)}

, ˙*x*2(t),*y*(t), ˙*y*2(t)) =−*gi*_{˙}* _{y}*2(t,

*y*2

_{(t)}

, ˙*y*2(t),*x*2(t), ˙*x*2(t))
*f _{y}i*1(t,

*x*

1_{(t), ˙}* _{x}*1

_{(t),}

*1*

_{y}_{(t), ˙}

*1*

_{y}_{(t)) =}

_{−}

_{f}i˙

*x*1(t,*y*

1_{(t), ˙}* _{y}*1

_{(t),}

*1*

_{x}_{(t), ˙}

*1*

_{x}_{(t))}

*gi*_{˙}* _{y}*2(t,

*x*2

_{(t)}

, ˙*x*2(t),*y*(t), ˙*y*2(t)) =−*gi*_{˙}* _{x}*2(t,

*y*2

_{(t)}

, ˙*y*2(t),*x*2(t), ˙*x*2(t))

Recasting the dual problem (Mix SD) as a minimization problem and using the above relations, we have

(Mix SD1) Minimize−

Z

*I*

I. Husain and R. Mattoo Eur. J. Pure Appl. Math,**2**(2009), (578-603) 599

*x*1(t)*T*(*λTf _{x}*1(t,

*y*1, ˙

*y*1,

*x*1, ˙

*x*1) −

*DλTf*

_{˙}

*1(t,*

_{x}*y*1, ˙

*y*1,

*x*1, ˙

*x*1))e}

*d t*

Subject to

*x*1(a) =0=*x*1(b),*y*1(a) =0= *y*1(b)
*x*2(a) =0=*x*2(b),*y*2(a) =0= *y*2(b)

*λTf _{x}*1(t,

*y*1, ˙

*y*1,

*x*1, ˙

*x*1)−

*DλTf*

_{˙}

*1(t,*

_{x}*y*1, ˙

*y*1,

*x*1, ˙

*x*1)≦0,

*t*∈

*I*

*λTg _{x}*2(t,

*y*2, ˙

*y*2,

*x*2, ˙

*x*2)−

*DλTg*

_{x}_{˙}2(t,

*y*2, ˙

*y*2,

*x*2, ˙

*x*2)≦0,

*t*∈

*I*

Z

*I*

*x*2(t)*T*_{(}_{λ}T_{g}

*x*2(t,*y*2, ˙*y*2,*x*2, ˙*x*2)
−*DλTg*_{˙}* _{x}*1(t,

*y*2, ˙

*y*2,

*x*2, ˙

*x*2))d t≧0

*λ*∈Λ+

This shows that the problem (Mix SD_{1}) is just the primal problem (Mix SP).
There-fore, (¯*x*1(t), ¯*x*2(t), ¯*y*1(t), ¯*y*2(t), ¯*λ*) is an optimal solution of (Mix SD) implies that
(¯*y*1(t), ¯*y*2(t), ¯*x*1(t), ¯*x*2(t), ¯*λ*) is an optimal solution for (Mix SP), and by symmetric
duality also for (Mix SD).

Now from (55)

Minimum (Mix SP)=

Z

*I*

{*f*(t,*x*1, ˙*x*1,*y*1, ˙*y*1) +*g(t,x*2, ˙*x*2,*y*2, ˙*y*2)}*d t*

Correspondingly with the solution(¯*y*(t), ¯*y*2_{(t), ¯}* _{x}*1

_{(t), ¯}

*2*

_{x}_{(t), ¯}

_{λ}_{), we have}

Minimum (Mix SP)=

Z

*I*

{*f*(t,*y*1, ˙*y*1,*x*1, ˙*x*1) +*g(t,y*2, ˙*y*2,*x*2, ˙*x*2)}*d t*

By the skew symmetric of *fi* and *gi*, we have
Minimum (Mix SP) =

Z

*I*

{*f*(t,*x*1, ˙*x*1,*y*1, ˙*y*1) +*g*(t,*x*2, ˙*x*2,*y*2, ˙*y*2)}*d t*

=

Z

*I*

{*f*(t,*y*1, ˙*y*1,*x*1, ˙*x*1) +*g*(t,*y*2, ˙*y*2,*x*2, ˙*x*2)}*d t*

= −

Z

*I*

this yields,

Minimum (Mix SP)=

Z

*I*

{*f*(t,*x*1, ˙*x*1,*y*1, ˙*y*1) +*g(t*,*x*2, ˙*x*2,*y*2, ˙*y*2)}*d t* =0

This accomplishes the proof of the theorem.

**6. Natural Boundary Conditions**

The pair of mixed symmetric multiobjective variational problem with natural bound-ary values rather than fixed points may be formulated as,

Primal problem (Mix SP0)

Minimize

Z

*I*

{*f*(t,*x*1, ˙*x*1,*y*1, ˙*y*1) +*g*(t,*x*2, ˙*x*2,*y*2, ˙*y*2).

−*y*1(t)*T*(*λTf _{y}*1(x1, ˙

*x*1,

*y*1, ˙

*y*1) −

*DλTf*

_{˙}

*1(x1, ˙*

_{y}*x*1,

*y*1, ˙

*y*1))e}

*d t*

Subject to

*λTf _{y}*1(t,

*x*1, ˙

*x*1,

*y*1, ˙

*y*1)−

*DλTf*

_{˙}

*1(t,*

_{y}*x*1, ˙

*x*1,

*y*1, ˙

*y*1)≦0

*λTg _{y}*2(t,

*x*2, ˙

*x*2,

*y*2, ˙

*y*2)−

*DλTg*

_{˙}

*2(t,*

_{y}*x*2, ˙

*x*2,

*y*2, ˙

*y*2)≦0

Z

*I*

*y*2(t)*T*(*λTg _{y}*2(t,

*x*2, ˙

*x*2,

*y*2, ˙

*y*2) −

*DλTg*

_{˙}

*2(t,*

_{y}*x*2, ˙

*x*2,

*y*2, ˙

*y*2))≧0

*λTf _{y}*1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1)|

_{t}_{=}

*=0,*

_{a}*λTf*

_{˙}

*1(t,*

_{y}*u*1, ˙

*u*1,

*v*1, ˙

*v*1)|

_{t}_{=}

*=0*

_{b}*λTg _{y}*2(t,

*x*2, ˙

*x*2,

*y*2, ˙

*y*2)|

_{t}_{=}

*=0,*

_{a}*λTg*

_{˙}

*2(t,*

_{y}*x*2, ˙

*x*2,

*y*2, ˙

*y*2)|

_{t}_{=}

*=0*

_{b}*λ*∈Λ+
Dual problem (Mix SD0)

Maximize

Z

*I*

{*f*(u1, ˙*u*1,*v*1, ˙*v*1) +*g*(u2, ˙*u*2,*v*2, ˙*v*2)

I. Husain and R. Mattoo Eur. J. Pure Appl. Math,**2**(2009), (578-603) 601

−*DλTf*_{˙}* _{u}*1(u1, ˙

*u*1,

*v*1, ˙

*v*1))e}

*d t*

Subject to

*λTf _{u}*1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1)−

*DλTf*

_{u}_{˙}1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1)≧0

*λTg _{u}*2(t,

*u*2, ˙

*u*2,

*v*2, ˙

*v*2)−

*DλTg*

_{˙}

*2(t,*

_{u}*u*2, ˙

*u*2,

*v*2, ˙

*v*2)≧0

Z

*I*

*u*2(t)*T*(*λTg _{u}*2(t,

*u*2, ˙

*u*2,

*v*2, ˙

*v*2) −

*DλTg*

_{u}_{˙}2(t,

*u*2, ˙

*u*2,

*v*2, ˙

*v*2))d t ≦0,

*λTf _{x}*

_{˙}1(t,

*u*1, ˙

*u*1,

*v*1, ˙

*v*1)|

_{t}_{=}

*=0,*

_{a}*λTf*

_{˙}

*1(t,*

_{x}*u*1, ˙

*u*1,

*v*1, ˙

*v*1)|

_{t}_{=}

*=0*

_{b}*λTg*_{˙}* _{x}*2(t,

*x*2, ˙

*x*2,

*y*2, ˙

*y*2)|

_{t}_{=}

*=0,*

_{a}*λTg*

_{˙}

*2(t,*

_{x}*x*2, ˙

*x*2,

*y*2, ˙

*y*2)|

_{t}_{=}

*=0*

_{b}*λ*∈Λ+

For these problems, Theorem 1-3 will remain true except that some slight modifica-tions in the arguments for these theorems are to be indicated.

**7. Mathematical Programming**

If the time dependency of (Mix SP) and (Mix SD) is removed and *b*−*a*= 1, we
obtain following pair of static mixed type multiobjective dual problems studied by
Bector, Chandra and Abha[2].

Primal (Mix SP_{1}) Minimize *f*(x1,*y*1) +*g(x*2,*y*2)−(*y*1)*T*(*λTf _{y}*1(x1,

*y*1)

Subject to

*λTf _{y}*1(x1,

*y*1)≦0,

*λTg _{y}*2(

*x*2,

*y*2)≦0,

*y*2(t)*T*(*λTg _{y}*2(x2,

*y*2))≧0,

*λ*∈Λ+

Subject to

*λTf _{u}*1(u1,

*v*1)≧0,

*λTg _{u}*2(u2,

*v*2)≧0,

(u2)*T*(*λTg _{u}*2(u2,

*v*2).≦0,

*λ*∈Λ+.

**ACKNOWLEDGEMENTS** The authors are grateful to the anonymous referee for
his/her valuable comments that have substantially improved the presentation of this
research.

**References**

[1] M.S. Bazaraa and J.J. Goode, On Symmetric Duality in Nonlinear Programming,
*Oper-ations Research***21(1) (1973), 1–9.**

[2] C.R. Bector, Chandra and Abha, On Mixed Type Symmetric Duality in Multiobjective

Programming,*Opsearch***36(4) (1999), 399–407.**

[3] C.R. Bector, S. Chandra and I. Husain, Generalized Concavity and Duality in Continuous

Programming,*Utilitas, Mathematica***25(1984), 171–190.**

[4] C.R. Bector and I. Husain, Duality for Multiobjective Variational Problems, *Journal of*
*Math. Anal and Appl.***166(1) (1992), 214–224.**

[5] A. Ben-Israel and B. Mond, What is Invexity? *J. Austral. Math. Soc. Ser. B***28(1986),**
1–9.

[6] S. Chandra and I. Husain, Symmetric Dual Continuous Fractional Programming,*J. Inf.*
*Opt. Sc.***10(1989), 241–255.**

[7] W.S. Dorn, Asymmetric Dual Theorem for Quadratic Programs, *Journal of Operations*
*Research Society of Japan***2(1960) 93–97.**

[9] Gulati, I. Husain and A. Ahmed, Multiobjective Symmetric Duality with Invexity,*Bulletin*
*of the Australian Mathematical Society***56(1997) 25–36.**

[10] I. Husain and Z. Jabeen, Mixed Type Symmetric and Self Duality for Variational

Prob-lems,*Congressus Numerantum***171(2004), 77–103.**

[11] D.H. Martin, The Essence of Invexity, *Journal of Optimization Theory and Applicatins*
**47(1) (1985), 65–76.**

[12] B. Mond, A Symmetric Dual Theorem for Nonlinear Programs, *Quaterly Jounal of *
*ap-plied Mathematics***23(1965) 265–269.**

[13] B. Mond and R.W. Cottle, Self Duality in Mathematical Programming, *SIAM J. Appl.*
*Math.***14(1966), 420–423.**

[14] B. Mond and T. Weir, Generalized Concavity and Duality, in : S.Sciable, W.T.Ziemba

(Eds.),*Generalized Concavity in Optimization and Economics*, Academic Press, New York,
(1981).

[15] B. Mond and M.A. Hanson, Symmetric Duality for Variational problems,*J. Math. Anal.*
*Appl.***18(1967) 161-172.**

[16] N.G. Rueda and M.A. Hanson, Optimality Criteria in Mathematical Programming

In-volving Generalized Invexity.*Journal of Mathematical Analysis and Applications***130(2),**
375–385.

[17] F.A. Valentine, The Problem of Lagrange with Differential Inequalities as Added Side

Conditions, *Contributions to calculus of variations*, 1933-37, Univ. Of Chicago Press,
(1937), 407–448.