Directory UMM :Journals:Journal_of_mathematics:BJGA:

variational problems on manifolds

S

¸tefan Mititelu and Mihai Postolache

Abstract. We consider a multitime scalar variational problem (SVP), a multitime multiobjective variational problem (VVP) and a multitime vec-tor fractional variational problem (VFP). For (SVP) we establish neces-sary optimality conditions. For the two vector variational problems (VVP) and (VFP), we define the notions of efficient solution and of normal ef-ficient solution and using these notions we establish necessary efficiency conditions. Using the notion of (ρ, b)-quasiinvexity adapted for variational problems, we introduce a duality of Mond-Weir-Zalmai type for the frac-tional problem (VFP) through weak, direct and converse duality theorems.

M.S.C. 2010: 53C65, 65K10, 90C29, 26B25.

Key words: Riemannian manifold, multitime vector fractional variational problem, efficient solution, normal efficient solution, (ρ, b)-quasiinvexity.

1

Introduction

In [9], Mititelu and Stancu-Minasian considered the following multiobjective frac-tional variafrac-tional problem:

(MSP)

        

       

Maximize



   

Z b a

f1(t, x,x˙)dt

Z b a

k1(t, x,x˙)dt , . . . ,

Z b a

fp(t, x,x˙)dt Z b

a

kp(t, x,x˙)dt 

   

subject to x(a) =a0, x(b) =b0,

g(t, x,x˙)≦0, h(t, x,x˙) = 0, ∀t∈I,

whereI= [a, b] is interval,x= (x1, . . . , xn) :I→Rn is piecewise smooth function on

Iwith ˙x=dx

dt its derivative,f1, k1, . . . , fp, kp:I×R

n_×Rn_→R,g:I×Rn_×Rn_→Rm

andh:I×Rn_×Rn_→Rq _{are functions of}C2 -class.

∗

Balkan Journal of Geometry and Its Applications, Vol.16, No.2, 2011, pp. 90-101. c

For this problem, they established necessary efficiency (Pareto minimum) condi-tions, and using generalized quasiinvex funccondi-tions, developed a duality theory through weak, direct and converse duality theorems, see also [6].

In [14], Udri¸ste studied a control variational problem into multidimensional (multi-time) framework, establishing some optimality conditions, that is he gave a multitime maximum principle, see also [15]. Pitea, Udri¸ste and Mititelu [11], [12] considered the multitime vector version of problem (MSP) within a geometrical framework, us-ing curvilinear integrals, and they established necessary optimality conditions and developed a duality theory for this problem.

In this work, we study properties of a multitime multiobjective fractional varia-tional problem, within a geometrical framework [11], [12] using multiple integrals.

Let (T, h) and (M, g) be two Riemannian manifolds of dimensions m and n, re-spectively. Denote byt= (t1

, . . . , tm_{) = (t}α_{) the elements of a measurable set Ω inT}

and byx= (x1

, . . . , xn_{) = (x}k_)∈Rn_{the elements ofM. ConsiderJ}1₍

T, M) the first order jet bundle associated toT andM and the functions

x: Ω→M, X:J1

(T, M)→R, f = (f1, . . . , fp) :J1(T, M)→Rp,

k= (k1, . . . , kp) :J1(T, M)→Rp, X_αi:J1(T, M)→Rm, Yβ:J1(T, M)→Rq, wherem, q∈N∗

,i= 1, n,α= 1, mandβ= 1, q. The arguments ofX,f,g,Xi

α,Yβ are (t, x, xγ) = (t, x(t), xγ(t)), where x=x(t) = (x1₍

t), . . . , xn_{(t)) = (x}k_{(t)), t∈Ω,} xγ(t) = ∂x

∂tγ(t), γ = 1, m.

We suppose thatX, f, g,X_αi,Yβ belong toC2_{(Ω). We define the set of functions} Φ ={x: Ω→M|xis piecewise smooth on Ω},

where Ω is a normed space withkxk=kxk∞+

n X

k=1

kxk_γk∞.

Throughout in the paper, for two vectors v = (v1, . . . , vn) andw = (w1, . . . , wn)

the relations of the formv=w,v < w,v≦w,v≤ware defined as follows:

v=w⇔vi=wi, i= 1, n; v < w⇔vi< wi, i= 1, n;

v≦w⇔vi≦wi, i= 1, n; v≤w⇔v≦wand v6=w.

The aim of present work is to establish necessary efficiency conditions and develop a duality theory for the following multitime fractional variational vector problem:

(VFP)

       

      

Maximize Pareto



   Z

Ω

f1(t, x(t), xγ(t))dv Z

Ω

k1(t, x(t), xγ(t))dv , . . . ,

Z

Ω

fp(t, x(t), xγ(t))dv Z

Ω

kp(t, x(t), xγ(t))dv 

  

subject to Xi

α(t, x(t), xγ(t)) = 0, Yβ(t, x(t), xγ(t))≦0, x(t)¯

¯

∂Ω=u(t) (given),∀t∈Ω, wheredv=dt1dt2· · ·dtn.

2

Necessary optimality conditions for (SVP)

The first model studied in this paper is the scalar multitime variational problem

(SVP)

The set of feasible solutions for this problem (the domain of (SVP)) is given by

D={x∈Φ|X_αi(t, x(t), xγ(t)) = 0, Yβ(t, x(t), xγ(t))≦0, x(t)¯ ¯

∂Ω=u(t), ∀t∈Ω}.

Theorem 2.1 (Necessary optimality for (SVP)). Ifx=x(t)∈ Dis an optimal solution for problem(SVP), then there exist a real scalarτ and the piecewise smooth functionsΛ(t) = (Λα

Proof. Consider the following auxiliary multitime variational problem

(AVP) whereλα_i andµβ are piecewise smooth real functions on Ω.

Problems (SVP) and (AVP) have the same domainDand the same optimal solu-tionx(t).

Letε > 0 be a scalar and letp: Ω→Rn _{be a function of}C1

(Ω)-class. Consider the next neighborhood of the optimal solutionx(t):

Vε={x¯∈X|x¯=x(t) +εp(t), p¯¯

∂D= 0}.

Then x = x(t) is an optimal solution of problem (SVP) if ε = 0 is a minimal solution to the next nonlinear problem:

The Fritz John conditions for (2.1) atε= 0 are the following: there exists a scalar

Consequently, the first condition of (FJ) becomes

τ

(2.2) can be written

(2.3)

For an useful version of the second integral in (2.3) we have

Dγ

and integrating, we obtain

Z

But using the divergence formula, we have

Then we obtain

Z

Ω

∂E ∂xk γ

pk_γdv=−

Z

Ω

Dγ µ

∂E ∂xk γ

¶ pkdv,

therefore(2.3) becomes

(2.4)

Z

Ω

· ∂E ∂xk −Dγ

µ ∂E ∂xk γ

¶¸

pkdv= 0.

By the fundamental lemma of variational calculus, from (2.4) it follows the

Euler-Lagrange equation ∂E

∂xk −Dγ µ

∂E ∂xk γ

¶

= 0, that is the first relation of (SFJ).

The second condition of (FJ) becomes

Mβ(t)Yβ(t, x(t), xγ(t)) = 0. Definition 2.1 ([4]). A pointx∗

∈ D is callednormal optimal solution of problem (SVP) ifτ6= 0.

3

Necessary efficiency conditions for (VVP)

•Efficiency for multitime vector variational problems_{. In the framework} of problem (SVP), we consider the vector functional

I[x] =

Z

Ω

f(t, x(t), xγ(t))dv,

which can be written on components as

I[x] = (I1[x], . . . , Ip[x]) =

µZ

Ω

f1(t, x(t), xγ(t))dv, . . . , Z

Ω

fp(t, x(t), xγ(t))dv ¶

.

Consider now the multitime variational vector problem

(VVP)

   

  

Minimize Pareto I[x] =

Z

Ω

f(t, x(t), xγ(t))dv

subject to Xi

α(t, x(t), xγ(t)) = 0, Yβ(t, x(t), xγ(t))≦0, x¯

¯

∂Ω=u(t), ∀t∈Ω. The domain of (VVP) is alsoD.

Definition 3.1 ([2]). A pointx∗

∈ D is anefficient solution (Pareto minimum) of problem (VVP) if there exist nox∈ Dsuch thatI[x]≤I[x∗

].

Then there are a vectorτ∈Rp_{, and the piecewise smooth functionsλ(t)inR}nm _and µ(t)inRq_{, defined onΩ, which satisfy the conditions}

(VFJ)

       

      

τr∂fr ∂xk +λ

α i(t)

∂Xαi ∂xk +µ

β_(t)∂Yβ ∂xk−

−Dγ µ

τr∂fr ∂xk γ

+λα_i(t)∂X

i α ∂xk

γ

+µβ(t)∂Yβ

∂xk γ

¶

= 0,

µβ_{(t)Yβ(t, x(t), xγ(t)) = 0, β = 1, q,} τ≧0, (µβ_{(t))≧0, t∈Ω.}

Proof. Ifx=x(t)∈ Dis an efficient solution of (VVP), the inequalityI[¯x]≤I[x], ∀x¯ ∈ D, is false. Then there exists r ∈ {1, . . . , p} and a neighborhood Nr in D of the pointxsuch thatIr[¯x]≧Ir[x], ∀x¯ ∈Nr. Therefore,xis an optimal solution to following scalar variational problem

(SVP)_r

   

  

MinimizeIr[x] = Z

Ω

fr(t, x(t), xγ(t))dv

subject to Xi

α(t, x(t), xγ(t)) = 0, Yβ(t, x(t), xγ(t))≦0, x∈Φ, x¯¯

∂Ω=u(t), ∀t∈Ω.

Then, according to Theorem 2.1, there are scalarsνr∈R_{and the piecewise smooth}

real functionsλα_i,r andµβ_r, such that the next conditions are true:

(3.1)

       

      

νr∂fr ∂xk +λ

α i,r(t)T

∂Xi α ∂xk +µ

β r(t)T

∂Yβ ∂xk−

−Dγ µ

νr∂fr ∂xk γ

+λαi,r(t)T ∂Xi

α ∂xk

γ

+µβr(t)T ∂Yβ ∂xk γ

¶

= 0,

µβ

r(t)TYβ(t, x(t), xγ(t)) = 0, νr≧0, µβr(t)≧0, t∈Ω,

where ∂fr

∂x := ∂fr

∂x(t, x(t), xγ(t)), whilei,α,β are variables.

We denote byS=

p X

r=1

νr andτr= (νr

S, whenIr[¯x]≧Ir[x]

0, whenIr[¯x]< Ir[x].

Also we denote by

(3.2) τ= (τ1

, . . . , τp), λα_i(t) = λ

α i,r(t)

S , µ

β_{(t) =} µβr(t) S .

Taking into account relations (3.2) in (3.1), we find (VFJ).

Definition 3.2 ([4]). The pointx0 _{∈ D is a normal efficient solution of (VVP) if} within the conditions (VFJ) there existτ≥0 withe′

τ= 1.

following multitime fractional variational vector problem From now on, we assume that

Z

Ω

kr(t, x(t), xγ(t))dv > 0 for all r = 1, p. The

domain of (VFP) is the same setD.

Definition 3.3 ([2]). A pointx0

∈ D is said to be anefficient solutionof (VFP) if there is nox∈ D,x6=x0

, such thatJ[x]≤J[x0 ].

We present now the necessary efficiency conditions for (VFP). Let x0

(t) be an efficient solution of (FVP). Consider the problem

(FP)_r(x0

Definition 3.4. The efficient solutionx0_{∈ Dis anormal efficient solutionof (VFP)} ifx0_{is optimal point to at least one of scalar problems (FP)}

r,r= 1, p.

Theorem 3.2 (Necessary efficiency in (FVP)). Let x=x(t)∈ D be a normal efficient solution of problem (FVP). Then there exist a vector τ = (τr_{) ∈ R}p _and

piecewise smooth functions λ= (λα

i(t))∈Rnm and µ= (µβ(t))∈Rq, defined on Ω,

which satisfy the conditions

(MFJ)

Proof. It is similar to that of Theorem 4.1 from [9].

Theorem 3.3 (Necessary efficiency in (VFP)). Let x=x(t)∈ D be a normal efficient solution of problem (VFP). Then there exist vector τ = (τr) ∈ Rp _and

piecewise smooth functions λ= (λα_i(t))∈Rpm _{andµ= (µ}β_(t))∈Rq_{, defined on Ω,}

which satisfy the conditions

(MFJ)0

Definition 3.5. If conditions (MFJ) or (MFJ)0 hold withτ ≥0,e′

τ = 1, then the pointx0_{∈ Dis callednormal efficient solutionof (VFP).}

Letρ∈R_andb: Φ×Φ→[0,∞) a function. ConsiderH(x) =

Z

Ω

h(t, x(t), xγ(t))dv. Definition 3.6 ([14]). The function H is called (strictly) (ρ, b)-quasiinvex at x0 _if there exist vector functionsη(t) = (η1(t), . . . , ηn(t))∈Rn_ofC1_{-class with}

For a specialized study of invexity, we address the reader to [5].

4

Mond-Weir-Zalmai type duality for (VFP)

Consider a function y ∈ Φ and we associate to (VFP) the following multitime fractional variational vector dual problem

Theorem 4.1 (Weak duality). Let x∈ Dand(y, λ, µ, ν)∈∆ be feasible points of problems(VFP)and(WFD). Assume satisfied the conditions:

a)for eachr= 1, p, the integral

Z

Ω

[Kr(y)fr(t, x(t), xγ(t))−Fr(y)kr(t, x(t), xγ(t))]dv

is(ρ′

r, b)-quasiinvex aty with respect toη andθ;

b)

Z

Ω

[λi_α(t)X_iα(t, x, xγ) +µβ(t)Yβ(t, x, xγ)]dvis(ρ, b)-quasiinvex aty with respect toη andθ;

c) one of the functions of a)-b) is strictly (ρ, b)-quasiinvex aty with respect toη

andθ; d)τrρ′

r+ρ≧0.

Thenπ(x)≤δ(y, λ, µ, ν)is false.

Proof. By reductio ad absurdum, supposeπ(x)≤δ(y, λ, µ, ν), or componentwise

Z

Ω

fr(t, x, xγ)dv

Z

Ω

kr(t, x, xγ)dv

≦

Z

Ω

fr(t, y, yγ)dv

Z

Ω

kr(t, y, yγ)dv

, r= 1, p.

This relation can be written

(4.1)

Z

Ω

[Kr(y)fr(t, x, xγ)−Fr(y)kr(t, x, xγ)]dv≦0 ·

=

Z

Ω

[Kr(y)fr(t, y, yγ)−Fr(y)kr(t, y, yγ)]dv ¸

.

According to hypothesis a), (4.1) implies

(4.2)

b(x, y)

Z

Ω

½ ηs

·

Kr(y)∂fr

∂ys −Fr(y) ∂kr ∂ys ¸

+

+(Dγηs)′

·

Kr(y)∂fr

∂ys γ

−Fr(y)∂kr

∂ys γ

¸¾

dv≦−ρ′

rb(x, y)kθ(x, y)k

2

.

Multiplying (4.1) and (4.2) byτr≧0, and summing overr= 1, p, it results (4.3) τr[Fr(x)Kr(y)−Kr(x)Fr(y)]≦0 ⇒

b(x, y)

Z

Ω

½ ηsτr

·

Kr(y)∂fr

∂ys−Fr(y) ∂kr ∂ys ¸

+(Dγηs)τr ·

Kr(y)∂fr

∂ys γ

−Fr(y)∂kr

∂ys γ

¸¾ dv

≦−τr_ρ′

rb(x, y)kθ(x, y)k

2

.

From the domainsDand ∆ it follows

(4.4)

Z

Ω

[λα_i(t)X_αi(t, x, xγ) +µβ(t)Yβ(t, x, xγ)]dv≦

≦

Z

Ω

and according to c) from (4.4) we find

Summing now side by side implications (4.3) and (4.4) and taking into account c), we obtain

(4.6)

, it follows

Z

Using the divergence formula, we have

Z Then relation (4.7) becomes

(4.8)

Taking into account the first constraint of problem (WFD), we have

According to the hypothesis d) of the theorem, this inequality becomes 0 < 0, which is false. Then, from (4.6)

(4.9)

τr_{[Fr(x)Kr(y)−Kr(x)Fr(y)]+} Z

Ω

[λα_i(t)X_αi(t, x, xγ) +µβ(t)Yβ(t, x, xγ)]dv− −

Z

Ω

[λα_i(t)X_αi(t, y, yγ) +µβ(t)Yβ(t, y, yγ)]dv >0.

Taking into account relation (4.4), from (4.9) it resultsτr_{[Fr(x)Kr(y)−Kr(x)Fr(y)]>0,}

which contradict relation (4.3). Thereforeπ(x)≤δ(y, λ, µ, ν) is false.

Theorem 4.2 (Direct duality). Let x0 be a normal efficient solution of the pri-mal (VFP) and suppose satisfied the hypotheses of Theorem 4.1. Then there are vector τ0

∈Rp _{and the piecewise smooth functions} λ0

= (λα i)

0

: Ω → Rnm _and µ0

= (µ0

) : Ω → Rq _{such that (}x0

, λ0

, µ0

, ν0

) is an efficient solution of the dual (MWFD)andπ(x0

) =δ(x0

, λ0

, µ0

, ν0 ).

Proof. Because x0

is a normal efficient solution to (VFP), according to The-orem 3.3 there are vector τ0 _{= (}

τr₎0 _{∈ R}p _{and the piecewise smooth functions} λ0_{= (}

λα i)

0_{: Ω→R}nm _{and µ}0 _{= (}

µβ₎0_{: Ω →R}q _{which satisfy relations (MFJ)0. It}

follows that (x0

, τ0

, λ0

, µ0_{)∈∆ and}

π(x0_{) =}

δ(x0

, τ0

, λ0

, µ0_).

Theorem 4.3 (Converse duality). Let(x0

, τ0

, λ0

, µ0_{)∈∆be an efficient solution} of dual and(MWFD)and assume satisfied the next conditions

i) ¯xis a normal efficient solution of primal(VFP);

ii)the hypotheses of Theorem 4.1 are satisfied with(y, τ, λ, µ) = (x0

, τ0

, λ0

, µ0 ). Thenx¯=x0

andπ(x0

) =δ(x0

, τ0

, λ0

, µ0 ).

Proof. On the contrary, suppose that ¯x6=x0 _{and we will obtain a contradiction.} According to Theorem 3.3, because ¯xis normal efficient solution of (VFP), then there are vector ¯τ ∈Rp _{and vector functions ¯λ= (¯λ}α

i) : Ω→Rnm and ¯µ= (¯µβ) : Ω→Rq

that satisfy conditions (MFJ)0. It results ¯λα

i(t)Xαi(t,x,¯ xγ¯ ) + ¯µβ(t)Yβ(t,¯x,xγ¯ ) = 0,

therefore (¯x,τ ,¯ ¯λ,µ¯)∈∆. Moreover,π(¯x) =δ(¯x,τ ,¯ λ,¯ µ¯). According to Theorem 4.1 relationπ(¯x)≤δ(x0

, τ0

, λ0

, µ0_{) is false. Then the relation}

δ(¯x,τ ,¯ ¯λ,µ¯)≤δ(x0

, τ0

, λ0

, µ0₎ is false. Therefore, the maximal efficiency of (x0

, τ0

, λ0

, µ0_{) is contradicted. Then, it} results that the assumption ¯x6=x0, above made, is false. It follows ¯x=x0 and also ¯

τ=τ0

, ¯λ=λ0

, ¯µ=µ0

. Finally, we haveπ(¯x) =δ(x0

, λ0

, µ0

, ν0 ).

For other advances on this field, we address the reader to [1], [3], [7], [8], [16].

Acknowledgements. The authors are grateful to Professor Constantin Udri¸ste for his remarks and encouragements during the preparation of various versions of this work. This paper could not have been refined without his help.

References

[1] V. Chankong and Y. Y. Haimes,Multiobjective Decisions Making: Theory and Methodology, North Holland, New York, 1983.

[3] M. A. Hanson,Duality for variational problems, J. Math. Anal. Appl., 18(1967), 355-364.

[4] R. Jagannathan, Duality for nonlinear fractional programming, Z. Oper. Res., 17(1973), 1-3.

[5] S¸t. Mititelu,Invex sets and preinvex functions, J. Adv. Math. Stud. 2, 2 (2009), 41-52.

[6] S¸t. Mititelu,Efficiency conditions for multiobjective fractional variational prob-lems, Appl. Sci., 10 (2008), 162-175.

[7] S¸t. Mititelu, Optimality and duality for invex multi-time control problems with mixed constraints, J. Adv. Math. Stud., 2, 1 (2009), 25-34.

[8] S¸t. Mititelu and M. Postolache,Mond-Weir dualities with Lagrangians for multi-objective fractional and nonfractional variational problems, J. Adv. Math. Stud., 3, 1 (2010), 41-58.

[9] S¸t. Mititelu and I. M. Stancu-Minasian,Efficiency and duality for multiobjective fractional variational problems with (ρ, b)-quasiinvexity, Yugoslav J. Oper. Res., 19, 1 (2009), 85-99.

[10] B. Mond and T. Weir, Generalized concavity and duality, inGeneralized Con-cavity in Optimization and Economics (S. Schaible and W. I. Ziemba (Eds.)), Academic Press, 1981, 263-279.

[11] Ariana Pitea, C. Udri¸ste and S¸t. Mititelu,PDE-constrained optimization prob-lems with curvilinear functional quotients as objective vectors, Balkan J. Geom. Appl., 14, 2 (2009), 75-88.

[12] Ariana Pitea, C. Udri¸ste and S¸t. Mititelu,New type dualities in PDI and PDE constrained optimization problems, J. Adv. Math. Stud., 2, 1 (2009), 81-90. [13] V. Preda, On Mond-Weir duality for variational problems, Rev. Roum. Math.

Pures Appl., 28, 2 (1993), 155-164.

[14] C. Udri¸ste,Simplified multiple maximum principle, Balkan J. Geom. Appl., 14, 1 (2009), 102-119.

[15] C. Udri¸ste,Equivalence of multitime optimal control problems, Balkan J. Geom. Appl., 15, 1 (2010), 155-162.

[16] F. A. Valentine,The problem of Lagrange with differentiable inequality as added side conditions, in

”Contributions to the Calculus of Variations”, 1933-37, Univ. of Chicago Press, 1937, 407-448.

[17] G. J. Zalmai,Generalized(F, b, ϕ, ρ, θ)-univexn-set functions and semiparamet-ric duality models in multiobjective fractional subset programming, Int. J. Math. Sci. 7 (2006), 1109-1133.

Authors’ addresses:

S

¸tefan Mititelu

Technical University of Civil Engineering, 124 Lacul Tei, 020396 Bucharest (RO) E-mail: st mititelu@yahoo.com Mihai Postolache

University Politehnica of Bucharest,