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Yugoslav Journal of Operations Research 23 (2013) Number 3, 387-417 DOI: 10.2298/YJOR110517009H MIXED TYPE SYMMETRIC AND SELF DUALITY FOR MULTIOBJECTIVE VARIATIONAL PROBLEMS WITH SUPPORT FUNCTIONS I.HUSAIN Department of Mathematics, Jaypee University of Engineering and Technology, Guna, MP, India. Email: ihusain11@yahoo.com Rumana, G.MATTOO Department of Statistics, University of Kashmir, Srinagar, Kashmir, India. Email: rumana_research@yahoo.co.in Received: May 2011 / Accepted: March 2013 Abstact: In this paper, a pair of mixed type symmetric dual multiobjective variational problems containing support functions is formulated. This mixed formulation unifies two existing pairs Wolfe and Mond-Weir type symmetric dual multiobjective variational problems containing support functions. For this pair of mixed type nondifferentiable multiobjective variational problems, various duality theorems are established under convexity-concavity and pseudoconvexity-pseudoconcavity of certain combination of functionals appearing in the formulation. A self duality theorem under additional assumptions on the kernel functions that occur in the problems is validated. A pair of mixed type nondifferentiable multiobjective variational problem with natural boundary values is also formulated to investigate various duality theorems. It is also pointed that our duality theorems can be viewed as dynamic generalizations of the corresponding (static) symmetric and self duality of multiobjective nonlinear programming with support functions. Kеywords: Efficiency, mixed type symmetric duality, mixed type self duality, natural boundary values, multiobjective nonlinear programming, convexity-convexity, pseudoconvexitypseudoconcavity, support functions. MSC: 90C30, 90C11, 90C20, 90C26. 388 I.Husain, R.G.Mattoo / Mixed Type Symmetric 1. INTRODUCTION Following Dorn [6], symmetric duality results in mathematical programming have been derived by a number of authors, notably, Dantzig et al [7], Mond [13], 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 [14] presented self duality for the problems of [7] by assuming skew symmetric of the kernel function. Later Mond-Weir [16] 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 Hanson [17] under the convexity-concavity conditions of scalar functions like ψ (t , x(t ), x& (t ), y (t ), y& (t )) with x(t ) ∈ R n and y (t ) ∈ R m . Bector, Chandra and Husain [3] presented a different pair of symmetric dual variational problems in order to relax the requirement of convexity-concavity to that of pseudoconvexity-pseudoconcavity while in [5] Chandra and Husain gave a fractional analogue. Bector and Husain [4] were probably the first to study duality for multiobjective variational problems under appropriate convexity assumptions. Subsequently, Gulati, Husain and Ahmed [8] presented two distinct pairs of symmetric dual multiobjective 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 [12] formulated a pair of mixed type symmetric dual variational problem in order to unify the Wolfe and MondWeir symmetric dual pairs of variational problems studied in [8]. The purpose of this research is to unify the formulations of the pairs of Wolfe and Mond-Weir type symmetric dual multiobjective variational problems involving support functions recently treated by Husain and Rumana [11] and study symmetric and self duality for these pairs of nondifferentiable variational problem under appropriate assumptions. Our duality results reported in this research extend the results of Husain and Rumana [11] to nondifferentiable setting by introducing support functions. The support functions which appear in the problems of facility location and related problems of decision theory are quite significant functions amongst well known nondifferentiable convex functions. The dual problems presented in this research are pretty hard to solve. So, expecting any immediate application of these problems would be premature. 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 varieties of contexts, which reflects the utility of our models. It is also indicated that our results can be viewed as dynamic generalizations of corresponding (static) symmetric duality results of multiobjective nonlinear programming with support functions. I.Husain, R.G.Mattoo / Mixed Type Symmetric 389 2. NOTATIONS AND PRELIMINARIES n The following notation will be used for vectors in R . x < y, ⇔ xi < yi , i = 1, 2,K , n. x < y, ⇔ xi < yi , i = 1, 2,K , n. x ≤ y, ⇔ xi ≤ yi , i = 1, 2,K , n, but x ≠ y x ≤ y, is the negation of x ≤ y Let I = [ a, b ] be the real interval, and φ i (t , x(t ), x& (t ), y(t ), y& (t )) , i = 1, 2,K , p be a scalar function and twice differentiable function where x : I → R n and y : I → R n with derivatives partial x& and y& . In order to consider each φ i (t , x(t ), x& (t ), y(t ), y& (t )) denote the first derivatives of φi with respect to t , x(t ), x& (t ), y (t ), y& (t ) respectively, by φ , φ , φ , φ , φ , that is, i t i x i x& i y φti = i y& ∂φ i ∂t ⎡ ∂φ i ∂φ i ∂φ i ⎤ , ,K , ⎥, ∂xn ⎦ ⎣ ∂x1 ∂x2 ⎡ ∂φ i ∂φ i ∂φ i ⎤ , ,K , φ x&i = ⎢ ⎥ ∂x&n ⎦ ⎣ ∂x&1 ∂x&2 ⎡ ∂φ i ∂φ i ∂φ i ⎤ , ,K , ⎥, ∂yn ⎦ ⎣ ∂y1 ∂y2 φ y&i = ⎢ φxi = ⎢ ⎡ ∂φ i i ∂φ i ∂φ i ⎤ , ,K , ⎥. ∂y& n ⎦ ⎣ ∂y&1 ∂y& 2 φ yi = ⎢ The twice partial derivatives of φ , i = 1, 2,K , p with respect to t , x ( t ) , i x& ( t ) , y ( t ) and y& ( t ) , respectively are the matrices ⎛ ∂ 2φ i ⎞ ⎟ , ⎝ ∂xk xs ⎠n× n φxxi & = ⎜ ⎛ ∂ 2φ i ⎞ ⎟ , ⎝ ∂xk y& s ⎠ n× n φxy&i = ⎜ ⎛ ∂ 2φ i ⎞ ⎟ , ⎝ ∂yk ys ⎠ n× n φ yyi & = ⎜ φxxi = ⎜ φxyi & = ⎜ φ yyi = ⎜ ⎛ ∂ 2φ i ⎞ ⎟ , ⎝ ∂xk x&s ⎠ n×n φxyi = ⎜ ⎛ ∂ 2φ i ⎞ ⎟ , ⎝ ∂x&k ys ⎠n× n φ xyi & = ⎜ ⎛ ∂ 2φ i ⎞ ⎟ , ⎝ ∂yk y& s ⎠n× n φ yy&i & = ⎜ Noting that d i φ y& = φ yt&i + φ yy&i y& + φ yy&i & &&y + φ yx&i x& + φ yx&i & &&x dt ⎛ ∂ 2φ i ⎞ ⎟ ⎝ ∂xk ys ⎠n× n ⎛ ∂ 2φ i ⎞ ⎟ ⎝ ∂xk y& s ⎠ n× n ⎛ ∂ 2φ i ⎞ ⎟ ⎝ ∂y& k y& s ⎠ n× n 390 I.Husain, R.G.Mattoo / Mixed Type Symmetric and hence ∂ d d ∂ d d d d i φ y& = φ yy& , φ y& = φ yy& & + φ yy& , φ y& = φ yy&i ∂y dt dt dt dy&& dt ∂y& dt ∂ d d ∂ d d ∂ d φ y& = φ yx& , φ y& = φ yx& & + φ yx& , φ y& = φ yx& & ∂x dt dt dt ∂&& x dt ∂x& dt In order to establish our main results, the following concepts are needed. Definition 1. (Support function): Let K be a compact set in R n , then the support function of K is defined by { s ( x ( t ) K ) = max x ( t ) v ( t ) : v ( t ) ∈ K , t ∈ I T } A support function, being convex everywhere finite, has a subdifferential in the sense of convex analysis i.e., there exists z ( t ) ∈ R n , t ∈ I , such that s ( y (t ) C ) − s ( x(t ) C ) ≥ ( y (t ) − x(t ) ) z ( t ) T From [15], subdifferential of s ( x ( t ) K ) is given by { } ∂s ( x ( t ) K ) = z ( t ) ∈ K , t ∈ I x ( t ) z ( t ) = s ( x ( t ) K ) . T For any set Γ ⊂ R n , the normal cone to Γ at a point x ( t ) ∈ Γ is defined by { } N Γ ( x (t ) ) = y(t ) ∈ R n y (t ) ( z ( t ) − x ( t ) ) ≤ 0, ∀z ( t ) ∈ Γ It can be verified that for a compact convex set K, y (t ) ∈ N K ( x(t ) ) if and only if s ( y (t ) K ) = x ( t ) y (t ) , t ∈ I T Definition 2. (Skew Symmetric function): The function h : I × R n × R n × R n × R n → 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 ( piecewise smooth are on I ) are of the same dimension. Now consider the following multiobjective variational problem (VPo): (VP0) Minimize ∫ F ( t , x, x& ) dt I Subject to 391 I.Husain, R.G.Mattoo / Mixed Type Symmetric x ( a) = α , x (b) = β g ( t , x, x& ) < 0 , t ∈ I , where F : I × R n × R n → R p and g : I × R n × R n → R m . Definition 3. (Efficient Solution): A feasible solution x is efficient for (VPo) if there exists no other feasible x for (VP) such that for some i ∈ P = {1, 2,..., p} , ∫ F ( t , x, x& ) dt < ∫ F ( t, x , x& ) dt i for all i ∈ P. i I I and ∫ F ( t, x, x& ) dt < ∫ F ( t, x , x& ) dt j j I for all j ∈ P , j ≠ i . I 3. STATEMENT OF THE PROBLEMS N = {1, 2,K , n} For M = {1, 2, K , m} , and J 1 ⊂ N , L1 ⊂ M , J 2 = N \ J 1 and L2 = M \ L1 . Let let J1 denote the number of elements in the subset J1 . The other symbols J 2 , L1 and L2 are similarly defined. Let x1 : I → R J and 1 x2 : I → R J2 , then any y1 : I → R L1 and be written as x = ( x1 , x 2 ) . Similarly for n x : I → R can y 2 : I → R L2 can be written as y = ( y1 , y 2 ) . Let f : I × R × R → R and g : I × R × R → R be twice continuously differentiable functions. We state the following pair of mixed type multiobjective symmetric dual variational problems with support functions involving vector functions f and g . J1 L1 J2 p (Mix SP): Minimize: L2 p ∫ ( H (t, x , x , y , y 1 1 2 1 2 , x&1 , x& 2 , y& 1 , y& 2 , z1 , z 2 , λ ) ,K , I ) H p ( t , x1 , x 2 , y1 , y 2 , x&1 , x& 2 , y& 1 , y& 2 , z1 , z 2 , λ ) dt Subject to: x1 ( a ) = 0 = x1 ( b ) , y1 ( a ) = 0 = y1 ( b ) , (1) x2 ( a ) = 0 = x2 ( b ) , y2 ( a ) = 0 = y2 (b) . (2) p ∑λ i =1 i ⎡ f i1 ( t , x1 , x&1 , y1 , y& 1 ) − z1i ( t ) − Df i1 ( t , x1 , x&1 , y1 , y& 1 ) ⎤ < 0 , t ∈ I , y& ⎣ y ⎦ (3) 392 I.Husain, R.G.Mattoo / Mixed Type Symmetric ⎡ g i t , x 2 , x& 2 , y 2 , y& 2 ) − zi2 ( t ) − Dg i 2 ( t , x 2 , x& 2 , y 2 , y& 2 ) ⎤ < 0 , t ∈ I , y& ⎣ y2 ( ⎦ p ∑λ i i =1 ⎡ p ∫ y ( t ) ⎢⎣∑ λ T 2 i i =1 I ( g ( t , x , x& , y , y& ) − z i y2 2 ( x (t ) , x (t )) > 0 , t ∈ I 1 2 zi1 ( t ) ∈ Ki1 and 2 2 2 2 i ( t ) − Dg iy& ( t , x , x& , y , y& ) )⎥ > 0 , 2 2 (4) 2 2 2 ⎤ ⎦ , (5) (6) zi2 ( t ) ∈ Ki2 , (7) λ > 0, λ T e = 1 , eT = (1,.....,1) . (8) where ( ) ( H i = f i ( t , x1 , x&1 , y1 , y& 1 ) + g i ( t , x 2 , x& 2 , y 2 , y& 2 ) + s x1 ( t ) Ci1 + s x 2 ( t ) Ci2 ) − y1 ( t ) ∑ λ i ⎡ f yi1 ( t , x1 , x&1 , y1 , y& 1 ) − zi1 ( t ) − Df y&i1 ( t , x1 , x&1 , y1 , y& 1 ) ⎤ ⎣ ⎦ i =1 p − zi1 ( t ) y1 ( t ) − zi2 ( t ) y 2 ( t ) (Mix SD): Maximize ∫ ( G ( t, u , u 1 1 2 , v1 , v 2 , u&1 , u& 2 , v&1 , v& 2 , w1 , w2 , λ ) ,K , I ) G p ( t , u1 , u 2 , v1 , v 2 , u&1 , u& 2 , v&1 , v& 2 , w1 , w2 , λ ) dt Subject to: u1 ( a ) = 0 = u1 ( b ) , v1 ( a ) = 0 = v1 ( b ) , (9) u 2 ( a ) = 0 = u 2 (b ) , v2 ( a ) = 0 = v2 ( b ) . (10) p ∑λ i =1 i ⎡ f i1 ( t , u1 , u&1 , v1 , v&1 ) + ωi1 ( t ) − Df i1 ( t , u1 , u&1 , v1 , v&1 ) ⎤ > 0 , t ∈ I , u& ⎣ u ⎦ , ∑ λ i ⎡⎣ gui 2 ( t , u 2 , u& 2 , v 2 , v& 2 ) + ωi2 ( t ) − Dgui& 2 ( t , u 2 , u& 2 , v 2 , v& 2 ) ⎤⎦ > 0 , t ∈ I , (11) p (12) i =1 ∫ u (t ) 2 I T ⎡ g i 2 ( t , u 2 , u& 2 , v 2 , v& 2 ) + ωi2 ( t ) − Dg i& 2 ( t , u 2 , u& 2 , v 2 , v& 2 ) ⎤ < 0 , t ∈ I , u ⎣ u ⎦ (13) ( v (t ) , v (t )) > 0 , t ∈ I , (14) ωi1 ( t ) ∈ Ci1 and ωi2 ( t ) ∈ Ci2 , (15) λ > 0, λ T e = 1 , eT = (1,.....,1) , (16) 1 2 393 I.Husain, R.G.Mattoo / Mixed Type Symmetric where, G i = f i ( t , x1 , x&1 , y1 , y& 1 ) + g i ( t , x 2 , x& 2 , y 2 , y& 2 ) ( ) ( ) + s v1 ( t ) Ki1 + s v 2 ( t ) Ki2 + u1 ( t ) ωi1 ( t ) + ui2 ( t ) ω 2 ( t ) − u1 ( t ) ∑ λ i ⎡⎣ f ui1 ( t , x1 , x&1 , y1 , y& 1 ) + ωi1 ( t ) − Dfu&i1 ( t , x1 , x&1 , y1 , y& 1 ) ⎤⎦ p i =1 4. MIXED TYPE MULTIOBJECTIVE SYMMETRIC DUALITY In this section, we present various duality results for a pair of mixed type multiobjective symmetric problems, (Mix SP) and (Mix SD) under pseudo-concavitypseudo-concavity assumptions. Theorem 1. (Weak Duality): Let ( x1 ( t ) , x 2 ( t ) , y1 ( t ) , y 2 ( t ) , z1 ( t ) , z 2 ( t ) , λ ) be feasible ( ) for (Mix SP) and u1 ( t ) , u 2 ( t ) , v1 ( t ) , v 2 ( t ) , ω ( t ) , ω 2 ( t ) , λ be feasible for (Mix SD). 1 Assume that ∫ { f ( t,.,., y ( t ) , y& ( t ) ) dt (H1): for each i 1 i 1 be convex in x1 , x&1 for fixed y1 , y& 1 and I ∫ { f ( t , x ( t ) , x& ( t ) ,.,.) dt be concave in 1 i 1 y1 , y& 1 on I for fixed x1 , x&1 . I ( ( p ) (H2): ∑ λi ∫ gui 2 t ,.,., y 2 ( t ) , y& 2 ( t ) + ( i =1 T ) ωi2 ( t ) ) dt pseudo-convex in x 2 , x& 2 for fixed I y , y& 2 2 and ∑ λ ∫ ( g ( t, x ( t ) , x& ( t ) ,.,.) − ( ) p i =1 i y2 i 2 2 T ) zi2 ( t ) dt pseudo-concave in y 2 , y& 2 for fixed I 2 x 2 , x& . Then, ∫ Hdt ≤ ∫ Gdt . I I where H = ( H 1 , H 2 ,..., H i ,..., H p ) and G = ( G1 , G 2 ,..., G i ,..., G p ) . Proof: Using the convexity of each ∫ f i ( t , . , . , y, y& ) dt in ( x, x& ) for fixed ( y, y& ) , we have I 394 I.Husain, R.G.Mattoo / Mixed Type Symmetric ∫ f ( t , x , x& , v , v& ) dt − ∫ f ( t , u , u& , v , v& ) dt i 1 1 1 1 i I 1 1 1 1 I T T ≥ ∫ ⎡( x1 ( t ) − u1 ( t ) ) fui1 ( t , u1 , u&1 , v1 , v&1 ) + ( x&1 ( t ) − u&1 ( t ) ) fu&i1 ( t , u1 , u&1 , v1 , v&1 ) ⎤ dt ⎢⎣ ⎥⎦ I = ∫ ⎡( x1 ( t ) − u1 ( t ) ) ⎢⎣ I T { f (t , u , u& , v , v& ) − Df (t , u , u& , v , v& )}⎤⎥⎦ dt 1 1 1 + ( x1 ( t ) − u1 ( t ) ) f u&i1 ( t , u1 , u&1 , v1 , v&1 ) t =a i u1 1 T 1 i u&1 1 1 1 t =b Using (1) and (9), this yields ∫ f ( t, x , x& , v , v& ) dt − ∫ f ( t, u , u& , v , v& ) dt i 1 1 1 1 i I 1 1 1 1 I { } T ≥ ∫ ⎡⎢( x1 ( t ) − u1 ( t ) ) fui1 ( t , u1 , u&1 , v1 , v&1 ) − Dfu&i1 ( t , u1 , u&1 , v1 , v&1 ) ⎣ I { ( ⎤ dt ⎥⎦ (17) ) Also by concavity of ∫ f i t , x1 ( t ) , x&1 ( t ) , . , . dt , we have I − ∫ f i ( t , x1 , x&1 , v1 , v&1 ) dt − ∫ f i ( t , x1 , x&1 , y1 , y& 1 ) dt I I T T > − ∫ ⎡( v1 ( t ) − y1 ( t ) ) f yi1 ( t , x1 , x&1 , y1 , y& 1 ) + ( v&1 ( t ) − y& 1 ( t ) ) f y&i1 ( t , x1 , x&1 , y1 , y& 1 ) ⎤ dt ⎢⎣ ⎥⎦ I = − ∫ ⎡( v1 ( t ) − y1 ( t ) ) ⎣⎢ T I { f ( t, x , x& , y , y& ) − Df ( t, x , x& , y , y& )}⎤⎦⎥ dt i y1 1 1 1 + ( v1 ( t ) − y1 ( t ) ) f y&i1 ( t , x1 , x&1 , y1 , y& 1 ) T 1 i y&1 1 1 1 1 t =a t =b , which by using (2) and (10), gives − ∫ f i ( t , x1 , x&1 , v1 , v&1 ) dt − ∫ f i ( t , x1 , x&1 , y1 , y& 1 ) dt I I { } T > − ∫ ⎡⎢( v1 ( t ) − y1 ( t ) ) f yi1 ( t , x1 , x&1 , y1 , y& 1 ) − Df y&i1 ( t , x1 , x&1 , y1 , y& 1 ) ⎤⎥ dt ⎣ ⎦ I The addition of (17) and (18) implies ∫ f ( t , x , x& , y , y& ) dt − ∫ f ( t , u , u& , v , v& ) dt i 1 1 1 1 i I 1 1 1 1 I { } T ≥ ∫ ⎡( x1 ( t ) − u1 ( t ) ) f ui1 ( t , u1 , u&1 , v1 , v&1 ) − Df u&i1 ( t , u1 , u&1 , v1 , v&1 ) ⎢⎣ I − ( v1 ( t ) − y1 ( t ) ) T { f ( t, x , x& , y , y& ) − Df ( t, x , x& , y , y& )}⎥⎦⎤ dt i y1 1 1 1 1 i y&1 1 1 1 1 (18) 395 I.Husain, R.G.Mattoo / Mixed Type Symmetric = ∫ ⎡( x1 ) ⎢⎣ I T − ( u1 ( t ) ) T { f ( t, u , u& , v , v& ) − Df (t , u , u& , v , v& )} i u1 1 1 1 i u&1 1 1 1 1 1 { f ( t, u , u& , v , v& ) − Df (t , u , u& , v , v& )} i u1 1 1 1 i u&1 1 1 1 1 1 { f ( t , x , x& , y , y& ) − Df ( t , x , x& , y , y& )} + ( y ( t ) ) { f ( t , x , x& , y , y& ) − Df ( t , x , x& , y , y& )}⎥⎤ dt ⎦ − ( v1 ( t ) ) T i y1 1 1 1 1 i y&1 1 1 1 1 T i y1 1 1 1 1 i y&1 1 1 1 1 1 Multiplying this by λ i and summing over i , i = 1, 2,.., p , we get ∑ λ ∫ f ( t , x , x& , y , y& ) dt − ∑ λ ∫ f ( t , u , u& , v , v& ) dt p p i 1 1 1 i 1 i i =1 T ⎡ > ∫ ⎢( x1 ( t ) ) I ⎣ 1 1 1 1 i i =1 I I ∑ λ { f ( t , u , u& , v , v& ) − Df ( t , u , u& , v , v& )} p i =1 i u1 i 1 1 1 i u&1 1 1 1 1 1 − ( u1 ( t ) ) ∑ λ { f ( t , u , u& , v , v& ) − Df ( t , u , u& , v , v& )} − ( v1 ( t ) ) ∑ λ { f ( t , x , x& , y , y& ) − Df ( t , x , x& , y , y& )} T p i =1 T T T 1 1 1 i u&1 1 1 1 1 1 i i y1 1 1 1 i y&1 1 1 1 1 1 ∑ λ { f ( t , x , x& , y , y& ) − Df ( t , x , x& , y , y& )}⎥ dt ⎤ p i =1 T ⎡ = ∫ ⎢( x1 ( t ) ) I ⎣ − ( u1 ( t ) ) i u1 p i =1 + ( y1 ( t ) ) i i y1 i 1 1 1 i y&1 1 1 1 1 1 ⎦ ∑ λ { f ( t , u , u& , v , v& ) + ω ( t ) − Df ( t , u , u& , v , v& )} p i =1 i u1 i 1 1 1 1 i u&1 1 i 1 1 1 1 ∑ λ { f ( t , u , u& , v , v& ) + ω ( t ) − Df ( t , u , u& , v , v& )} p i =1 i i u1 p 1 1 1 1 1 i i u&1 1 1 1 1 p −∑ λi x1 ( t ) ωi1 ( t ) + ∑ λi u ( t ) ωi1 ( t ) i =1 i =1 − ( v1 ( t ) ) T ∑ λ { f ( t , x , x& , y , y& ) − z ( t ) − Df ( t , x , x& , y , y& )} p i =1 + ( y1 ( t ) ) T i i y1 1 1 1 1 1 i i y&1 1 1 1 1 ∑ λ { f ( t , x , x& , y , y& ) − z ( t ) − Df ( t , x , x& , y , y& )} p i =1 i i y1 1 1 1 1 p p ⎤ − ∑ λi v1 ( t ) zi1 ( t ) + ∑ λi y1 ( t ) z1i ( t ) ⎥ dt i =1 i =1 ⎦ Using (3), (6), (11) and (14), we get 1 i i y&1 1 1 1 1 396 I.Husain, R.G.Mattoo / Mixed Type Symmetric ∑ λ ∫ ⎡⎣ f (t , x , x& , y , y& ) − ( y ) ∑ λ { f ( t , x , x& , y , y& ) − z ( t ) − Df ( t , x , x& , y , y& )} p i 1 1 1 p 1 T 1 i i =1 i =1 I p p i =1 i =1 i y1 i 1 1 1 1 i y&1 1 i 1 1 1 1 + ∑ λi ( x1 ( t ) ωi1 ( t ) ) − ∑ λi ( y1 ( t ) zi1 ( t ) ) > ∑ λ ∫ ⎡⎣ f (t , u , u& , v , v& ) − ( u ( t ) ) ∑ λ { f ( t , u , u& , v , v& ) + ω ( t ) − Df ( t , u , u& , v , v& )} p i 1 1 1 1 T 1 i i =1 p i =1 I i u1 i 1 1 1 1 i u&1 1 i 1 1 1 1 +u ( t ) ω ( t ) − y ( t ) z ( t ) ⎤⎦ dt 1 1 i 1 1 i In view of ( ) ( ) s x1 ( t ) Ci1 > ( x1 ( t ) ) ωi1 , i = 1,..., p T and s v1 ( t ) K i1 > ( v1 ( t ) ) zi1 , i = 1,..., p , this yields T ∑ λ ∫ ⎡⎣ f ( t , x , x& , y , y& ) + s ( x (t ) C ) p 1 i T T 1 1 1 i ∑ λ { f ( t , x , x& , y , y& ) − z ( t ) − Df ( t, x , x& , y , y& )} − ( y ( t ) ) p i =1 i y1 i 1 1 1 1 i y&1 1 i 1 1 1 1 1 T zi1 ( t ) ⎤⎥ dt ⎦ ∑ λ ∫ ⎡⎣ f ( t , u , u& , v , v& ) − s ( v ( t ) K ) p i 1 1 1 1 1 1 i i i =1 − ( u1 ) 1 I − ( y1 ( t ) ) > 1 i i =1 I ∑ λ { f ( t, u , u& , v , v& ) + ω ( t ) − Df ( t, u , u& , v , v& )} +u (t ) ω ( t )⎤⎦ dt p i =1 i u1 i 1 1 1 1 1 i 1 i u&1 1 1 1 1 1 i (19) From (12) together with (6) and (13), we have T ⎡ 2 2 ∫I ⎢⎣( x ( t ) − u ( t ) ) ∑ λ ( g (t, u i u2 i i =1 ) 2 , u& 2 , v 2 , v& 2 ) − ωi2 ( t ) − Dg ui& 2 ( t , u 2 , u& 2 , v 2 , v& 2 ) dt > 0 2 , u& 2 , v 2 , v& 2 ) − ωi2 ( t ) p Which integrated by parts implies ⎡ ∫ ⎢⎣( x ( t ) − u ( t ) ) ∑ λ ( g ( t , u 2 p T 2 i =1 I i u2 i + ( x& 2 ( t ) − u& 2 ( t ) ) ∑ λ g (t , u 2 − ( x (t ) − u (t )) ∑ λ g (t , u 2 T 2 2 T p i =1 i i u&1 ⎤ , u& 2 , v 2 , v& 2 ) ⎥ dt ⎦ p i =1 i i u& 2 , u& , v , v& 2 2 2 ) t =b >0 t =a )