Optimality Conditions and Duality for Multiobjective Variational Problems with Generalized 𝝆−(𝜼,𝜽)-B-Type-I Functions

Abstract

We use 𝜌−(𝜂,𝜃)-𝐵-type-I and generalized 𝜌−(𝜂,𝜃)-𝐵-type-I functions to establish sufficient optimality conditions and duality results for multiobjective variational problems. Some of the related problems are also discussed.

1. Introduction

The relationship between mathematical programming and classical calculus of variation was explored and extended by Hanson [1]. Thereafter variational programming problems have some attention in the literature. Mond and Hanson [2] obtained optimality conditions and duality results for scalar valued variational problems under convexity assumptions. Motivated by the approach by Bector and Husain [3], Nahak and Nanda [4] and later Bhatia and Mehra [5] extended the results of Mond et al. [6] to multiobjective variational problems involving invex functions and generalized 𝐵-invex functions, respectively. Analogous results were developed by Zalmai [7] for fractional variational programming problem containing arbitrary norms and by Liu [8] for generalized fractional case involving (𝐹,𝜌)-convex functions.

Type-I functions were first introduced by Hanson and Mond [9], and Rueda and Hanson [10] defined a class of pseudo-type-I and quasi-type-I functions as generalization of type-I functions. Bhatia and Mehra [5] studied the optimality conditions and duality results for multiobjective variational problems involving generalized 𝐵-invexity. We use 𝜌−(𝜂,𝜃)-𝐵-type-I and generalized 𝜌−(𝜂,𝜃)-𝐵-type-I functions to establish sufficient optimality conditions and duality results for multiobjective variational problems.

In Section 2 of this paper, we introduce 𝜌−(𝜂,𝜃)-𝐵-type-I and generalized 𝜌−(𝜂,𝜃)-𝐵-type-I functions for the continuous case. Using these new classes of functions, we establish various sufficient optimality conditions in Section 3 of this paper. Section 4 is devoted to the duality results. Some related problems are discussed in Section 5. The results obtained in this paper are more general than those obtained in [4, 5, 11].

Let 𝐼=[𝑎,𝑏] be a real interval, let 𝑓∶𝐼×ℝ^𝑛×ℝ^𝑛→ℝ^𝑝, and let ℎ∶𝐼×ℝ^𝑛×ℝ^𝑛→ℝ^𝑚 be continuously differentiable functions with respect to their arguments and 𝑥∶𝐼→ℝ^𝑛, 𝑥̇ denotes the derivative of 𝑥 with respect 𝑡. Denote the partial derivative of scalar valued function 𝑔∶𝐼×ℝ^𝑛×ℝ^𝑛→ℝ, with respect to 𝑡, 𝑥, and 𝑥̇, respectively, by 𝑔_𝑡, 𝑔_𝑥, and 𝑔_̇𝑥 such that 𝑔_𝑥=[𝜕𝑔/𝜕𝑥₁,…,𝜕𝑔/𝜕𝑥_𝑛], 𝑔_̇𝑥=[𝜕𝑔/𝜕𝑥̇₁,…,𝜕𝑔/𝜕𝑥̇_𝑛]. Similarly we write the partial derivative of the vector functions 𝑓 and ℎ using matrices with 𝑝 and 𝑚 rows, respectively, instead of one. Let 𝐶(𝐼,ℝ^𝑛) denote the space of piecewise smooth functions 𝑥 with norm ‖𝑥‖=‖𝑥‖_∞+‖𝐷𝑥‖_∞, where the differential operator 𝐷 is given by 𝑢=𝐷𝑥⟺𝑥(𝑡)=𝑥(𝑢)+^𝑡𝑎𝑢(𝑠)𝑑𝑠.(1) Therefore, 𝐷=𝑑/𝑑𝑡 except at discontinuities.

Consider the following multiobjective variational problem: (𝑉𝑃)minimize^𝑏𝑎=𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡,_𝑏𝑎𝑓₁(𝑡,𝑥,𝑥̇),…,^𝑏𝑎𝑓_𝑝(𝑡,𝑥,𝑥̇)subjecttoℎ𝑥((𝑎𝑡,)𝑥=,𝑥𝛼̇),≦0,𝑥(𝑡𝑏∈)=𝐼.𝛽,(2) Let 𝐾={𝑥∈𝐶(𝐼,ℝ^𝑛)∣𝑥(𝑎)=𝛼,𝑥(𝑏)=𝛽,ℎ(𝑡,𝑥,𝑥̇)≦0,𝑡∈𝐼}.

Efficiency and proper efficiency are defined in their usual sense as defined in [4].

In relation to (𝑉𝑃), we introduce the following multiobjective problem (𝑃^∗𝑘), for each 𝑘=1,…,𝑝, each problem with single objective: 𝑃^∗𝑘Minimize^𝑏𝑎𝑓_𝑘(𝑡,𝑥,𝑥̇)𝑑𝑡Subjectto𝑥(𝑎)=𝛼,𝑥(𝑏)=𝛽,_𝑏𝑎𝑓_𝑖(𝑡,𝑥,𝑥̇)𝑑𝑡≦^𝑏𝑎𝑓_𝑖𝑡,^∗ℎ𝑖=1,2,…,𝑝,𝑖≠𝑥,𝑘𝑥^̇∗,_𝑗𝑑𝑡,(𝑡,𝑥,𝑥̇)≦0,𝑗=1,…,𝑚,𝑡∈𝐼.(3)

The following lemma can be established on the lines of Chankong and Haimes [12].

Definition 1 (see [4]). A point 𝑥^∗ in 𝐾 is said to be an efficient solution of (𝑉𝑃) if for all 𝑥∈𝐾^𝑏𝑎𝑓_𝑖𝑡,^∗⟹𝑥,𝑥^̇∗𝑑𝑡≥^𝑏𝑎𝑓_𝑖(𝑡,𝑥,𝑥̇)𝑑𝑡,∀𝑖∈{1,…,𝑝},_𝑏𝑎𝑓_𝑖𝑡,^∗𝑥,𝑥^̇∗𝑑𝑡=^𝑏𝑎𝑓_𝑖(𝑡,𝑥,𝑥̇)𝑑𝑡,∀𝑖∈{1,…,𝑝}.(4)

Definition 2 (see [13]). A point 𝑥^∗ in 𝐾 is said to be a properly efficient solution of (𝑉𝑃) if for all 𝑥∈𝐾 there exists a scalar 𝑀>0 such that for all 𝑖∈{1,…,𝑝}^𝑏𝑎𝑓_𝑖𝑡,^∗≤𝑀𝑥,𝑥^̇∗_𝑏𝑎𝑑𝑡−^𝑏𝑎𝑓_𝑖(𝑡,𝑥,𝑥̇)𝑑𝑡,∀𝑖∈{1,…,𝑝},𝑓_𝑗(𝑡,𝑥,𝑥̇)𝑑𝑡−^𝑏𝑎𝑓_𝑗𝑡,^∗𝑥,𝑥^̇∗𝑑𝑡(5) for some 𝑗 such that ^𝑏𝑎𝑓_𝑗(𝑡,𝑥,𝑥̇)𝑑𝑡>^𝑏𝑎𝑓_𝑗𝑡,^∗𝑥,𝑥^̇∗𝑑𝑡(6) whenever 𝑥 is in 𝐾 and ^𝑏𝑎𝑓_𝑗(𝑡,𝑥,𝑥̇)𝑑𝑡<^𝑏𝑎𝑓_𝑗𝑡,^∗𝑥,𝑥^̇∗𝑑𝑡.(7)

Lemma 3. 𝑥^∗ is an efficient solution of (𝑉𝑃) if and only if 𝑥^∗ is an optimal solution of (𝑃^∗𝑘) for each 𝑘=1,2,…,𝑝.

Theorem 4 (see [2]). For every optimal normal solution of (𝑃^∗𝑘), for each 𝑘=1,2,…,𝑝, there exist real numbers 𝜆_1𝑘,…,𝜆_𝑝𝑘 with 𝜆_𝑘𝑘=1 and piecewise smooth function 𝑦_𝑘∶𝐼→ℝ^𝑚 such that 𝑓_𝑘𝑥𝑡,^∗𝑥,𝑥^̇∗+^𝑝_𝑖_{=1𝑖≠𝑘}𝜆_𝑖𝑘𝑓_𝑖𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑦_𝑘(𝑡)^𝑇ℎ_𝑥𝑡,^∗=𝑑𝑥,𝑥^̇∗𝑑𝑡⎛⎜⎜⎜⎝𝑓^𝑘̇𝑥𝑡,^∗𝑥,𝑥^̇∗+^𝑝_𝑖_{=1𝑖≠𝑘}𝜆_𝑖𝑘𝑓_𝑖̇𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑦_𝑘(𝑡)^𝑇ℎ_̇𝑥𝑡,^∗𝑦_𝑘𝑥,𝑥^̇∗⎞⎟⎟⎟⎠,(𝑡)^𝑇𝑦ℎ𝑡,𝑥^∗,𝑥^̇∗=0,𝑡∈𝐼,_𝑘𝜆(𝑡)≧0,𝑡∈𝐼,_𝑖𝑘≧0,𝑖=1,2,…,𝑝,𝑖≠𝑘.(8)

2. 𝜌−(𝜂,𝜃)-𝐵-Type-I and Generalized 𝜌−(𝜂,𝜃)-𝐵-Type-I Functions

Definition 5. Let  𝑓 and ℎ be real valued functions. A pair  (𝑓,ℎ) is said to be (𝜌₀,𝜌₁)−(𝜂,𝜃)-𝐵-type-I at 𝑢∈𝐶(𝐼,ℝ^𝑛) with respect to 𝑏₀, 𝑏₁, 𝜂, and 𝜃 if there exist functions 𝑏₀, 𝑏₁∶𝐶(𝐼,ℝ^𝑛)×𝐶(𝐼,ℝ^𝑛)→ℝ₊ and 𝜂, 𝜃∶𝐼×ℝ^𝑛×ℝ^𝑛→ℝ^𝑛 with 𝜂(𝑡,𝑥,𝑥)=0, 𝜌₀,𝜌₁∈ℝ such that for all 𝑥∈𝐾𝑏₀(𝑥≧,𝑢)^𝑏𝑎𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡−^𝑏𝑎𝑓(𝑡,𝑢,𝑢)̇𝑑𝑡_𝑏𝑎+𝜂𝜌(𝑡₀,𝑥,𝑢)^𝑇𝑓_𝑥(𝑡,𝑢,𝑢)̇+𝑑𝑑𝑡𝜂(𝑡,𝑥,𝑢)^𝑇𝑓_̇𝑥(𝑡,𝑢,𝑢)̇‖𝜃(𝑡,𝑥,𝑢)‖²𝑑𝑡,(9)−𝑏₁≧(𝑥,𝑢)^𝑏𝑎ℎ(𝑡,𝑢,𝑢)̇𝑑𝑡_𝑏𝑎+𝜂𝜌(𝑡₁,𝑥,𝑢)^𝑇ℎ_𝑥(𝑡,𝑢,𝑢)̇+𝑑𝑑𝑡𝜂(𝑡,𝑥,𝑢)^𝑇ℎ_̇𝑥(𝑡,𝑢,𝑢)̇‖𝜃(𝑡,𝑥,𝑢)‖²𝑑𝑡.(10) If, in the previous definition, (9) is satisfied as a strict inequality, then we say that a pair (𝑓,ℎ) is (𝜌₀,𝜌₁)−(𝜂,𝜃)−𝐵-semistrictly-type-I at 𝑢∈𝐶(𝐼,ℝ^𝑛) with respect to 𝑏₀, 𝑏₁, 𝜂, and 𝜃 and 𝜌₀,𝜌₁∈ℝ.

Definition 6. A pair (𝑓,ℎ) is said to be (𝜌₀,𝜌₁)−(𝜂,𝜃)-𝐵-quasi-type-I at 𝑢∈𝐶(𝐼,ℝ^𝑛) with respect to 𝑏₀, 𝑏₁, 𝜂, 𝜃 if there exist functions 𝑏₀, 𝑏₁∶𝐶(𝐼,ℝ^𝑛)×𝐶(𝐼,ℝ^𝑛)→ℝ₊ and 𝜂, 𝜃∶𝐼×ℝ^𝑛×ℝ^𝑛→ℝ^𝑛 with 𝜂(𝑡,𝑥,𝑥)=0, 𝜌₀,𝜌₁∈ℝ such that for all 𝑥∈𝐾𝑏₀(𝑥⟹,𝑢)^𝑏𝑎𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡−^𝑏𝑎𝑓(𝑡,𝑢,𝑢)̇𝑑𝑡≦0_𝑏𝑎+𝜌𝜂₀(𝑡,𝑥,𝑢)^𝑇𝑓_𝑥(𝑡,𝑢,𝑢)̇+𝑑𝑑𝑡𝜂(𝑡,𝑥,𝑢)^𝑇𝑓_̇𝑥(𝑡,𝑢,𝑢)̇−𝑏₁‖𝜃(𝑡,𝑥,𝑢)‖²𝑑𝑡≦0,⟹(𝑥,𝑢)^𝑏𝑎ℎ(𝑡,𝑢,𝑢)̇𝑑𝑡≦0,_𝑏𝑎+𝜌𝜂₁(𝑡,𝑥,𝑢)^𝑇ℎ_𝑥(𝑡,𝑢,𝑢)̇+𝑑𝑑𝑡𝜂(𝑡,𝑥,𝑢)^𝑇ℎ_̇𝑥(𝑡,𝑢,𝑢)̇‖𝜃(t,𝑥,𝑢)‖²𝑑𝑡≦0.(11)

Definition 7. A pair (𝑓,ℎ) is said to be (𝜌₀,𝜌₁)−(𝜂,𝜃)-𝐵-strongly pseudo-type-I at  𝑢∈𝐶(𝐼,ℝ^𝑛) with respect to 𝑏₀, 𝑏₁, 𝜂, and 𝜃 if there exist functions 𝑏₀, 𝑏₁∶𝐶(𝐼,ℝ^𝑛)×𝐶(𝐼,ℝ^𝑛)→ℝ₊ and 𝜂, 𝜃∶𝐼×ℝ^𝑛×ℝ^𝑛→ℝ^𝑛 with 𝜂(𝑡,𝑥,𝑥)=0, 𝜌₀,𝜌₁∈ℝ such that for all 𝑥∈𝐾, ^𝑏𝑎𝜂(𝑡,𝑥,𝑢)^𝑇𝑓_𝑥(𝑡,𝑢,𝑢)̇+𝑑𝑑𝑡𝜂(𝑡,𝑥,𝑢)^𝑇+𝜌₀𝑓_̇𝑥(𝑡,𝑢,𝑢)̇⟹‖𝜃𝑏₀(𝑡,𝑥,𝑢)‖²𝑑𝑡≧0(𝑥,𝑢)^𝑏𝑎𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡−^𝑏𝑎𝑓(𝑡,𝑢,𝑢)̇𝑑𝑡≧0,_𝑏𝑎+𝜂(𝜌𝑡₁,𝑥,𝑢)^𝑇ℎ_𝑥(𝑡,𝑢,𝑢)̇+𝑑𝑑𝑡𝜂(𝑡,𝑥,𝑢)^𝑇ℎ_̇𝑥(𝑡,𝑢,𝑢)̇⟹‖𝜃−(𝑡𝑏,₁𝑥,𝑢)‖²𝑑𝑡≧0(𝑥,𝑢)^𝑏𝑎ℎ(𝑡,𝑢,𝑢)̇𝑑𝑡≧0.(12) Clearly the class of (𝜌₀,𝜌₁)−(𝜂,𝜃)-𝐵-quasi-type-I functions and the class of (𝜌₀,𝜌₁)−(𝜂,𝜃)-𝐵-strongly pseudo-type-I functions are more general than the class of (𝜌₀,𝜌₁)−(𝜂,𝜃)−𝐵-type-I functions.

Definition 8. A pair (𝑓,ℎ) is said to be 𝐵−𝜌₀−(𝜂,𝜃)-quasi-𝜌₁−(𝜂,𝜃)-pseudo-type-I at 𝑢∈𝐶(𝐼,ℝ^𝑛) with respect to 𝑏₀, 𝑏₁, 𝜂, and 𝜃 if there exist functions 𝑏₀, 𝑏₁∶𝐶(𝐼,ℝ^𝑛)×𝐶(𝐼,ℝ^𝑛)→ℝ₊ and 𝜂, 𝜃∶𝐼×ℝ^𝑛×ℝ^𝑛→ℝ^𝑛 with 𝜂(𝑡,𝑥,𝑥)=0, 𝜌₀,𝜌₁∈ℝ such that for all 𝑥∈𝐾𝑏₀(𝑥⟹,𝑢)^𝑏𝑎𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡−^𝑏𝑎𝑓(𝑡,𝑢,𝑢)̇𝑑𝑡≦0_𝑏𝑎+𝜌𝜂₀(𝑡,𝑥,𝑢)^T𝑓_𝑥(𝑡,𝑢,𝑢)̇+𝑑𝑑𝑡𝜂(𝑡,𝑥,𝑢)^𝑇𝑓_̇𝑥(𝑡,𝑢,𝑢)̇‖𝜃(𝑡,𝑥,𝑢)‖²𝑑𝑡≦0,(13)^𝑏𝑎𝜂(𝑡,𝑥,𝑢)^𝑇ℎ_𝑥(𝑡,𝑢,𝑢)̇+𝑑𝑑𝑡𝜂(𝑡,𝑥,𝑢)^𝑇+𝜌₁ℎ_̇𝑥(𝑡,𝑢,𝑢)̇⟹‖−𝜃(𝑏𝑡,₁𝑥,𝑢)‖²𝑑𝑡≧0(𝑥,𝑢)^𝑏𝑎ℎ(𝑡,𝑢,𝑢)̇𝑑𝑡≧0.(14) If, in the above definition, inequality (14) is satisfied as ^𝑏𝑎𝜂(𝑡,𝑥,𝑢)^𝑇ℎ_𝑥(𝑡,𝑢,𝑢)̇+𝑑𝑑𝑡𝜂(𝑡,𝑥,𝑢)^𝑇+𝜌₁ℎ_̇𝑥(𝑡,𝑢,𝑢)̇⟹‖−𝜃(𝑏𝑡,₁𝑥,𝑢)‖²𝑑𝑡≧0(𝑥,𝑢)^𝑏𝑎ℎ(𝑡,𝑢,𝑢)̇𝑑𝑡>0,(15) then we say that a pair (𝑓,ℎ) is 𝐵−𝜌₀−(𝜂,𝜃)-quasi-𝜌₁−(𝜂,𝜃)-strictly pseudo-type-I at 𝑢∈𝐶(𝐼,ℝ^𝑛) with respect to 𝑏₀, 𝑏₁, 𝜂, and 𝜃 and 𝜌₀,𝜌₁∈ℝ.

Definition 9. A pair (𝑓,ℎ) is said to be 𝐵−𝜌₀−(𝜂,𝜃)-strongly pseudo-𝜌₁−(𝜂,𝜃)-quasi-type-I at 𝑢∈𝐶(𝐼,ℝ^𝑛) with respect to 𝑏₀, 𝑏₁, 𝜂, 𝜃 if there exist functions 𝑏₀, 𝑏₁∶𝐶(𝐼,ℝ^𝑛)×𝐶(𝐼,ℝ^𝑛)→ℝ₊ and 𝜂, 𝜃∶𝐼×ℝ^𝑛×ℝ^𝑛→ℝ^𝑛 with 𝜂(𝑡,𝑥,𝑥)=0, 𝜌₀,𝜌₁∈ℝ such that for all 𝑥∈𝐾^𝑏𝑎𝜂(𝑡,𝑥,𝑢)^𝑇𝑓_𝑥(𝑡,𝑢,𝑢)̇+𝑑𝑑𝑡𝜂(𝑡,𝑥,𝑢)^𝑇+𝜌₀𝑓_̇𝑥(𝑡,𝑢,𝑢)̇⟹‖𝑏𝜃₀(𝑡,𝑥,𝑢)‖²𝑑𝑡≧0(𝑥,𝑢)^𝑏𝑎𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡−^𝑏𝑎𝑓(𝑡,𝑢,𝑢)̇𝑑𝑡≧0,(16)−𝑏₁⟹(𝑥,𝑢)^𝑏𝑎ℎ(𝑡,𝑢,𝑢)̇𝑑𝑡≦0_𝑏𝑎+𝜌𝜂₁(𝑡,𝑥,𝑢)^𝑇ℎ_𝑥(𝑡,𝑢,𝑢)̇+𝑑𝑑𝑡𝜂(𝑡,𝑥,𝑢)^𝑇ℎ_̇𝑥(𝑡,𝑢,𝑢)̇‖𝜃(𝑡,𝑥,𝑢)‖²𝑑𝑡≦0.(17) If, in the above definition, inequality (16) is satisfied as ^𝑏𝑎𝜂(𝑡,𝑥,𝑢)^𝑇𝑓_𝑥(𝑡,𝑢,𝑢)̇+𝑑𝑑𝑡𝜂(𝑡,𝑥,𝑢)^𝑇+𝜌₀𝑓_̇𝑥(𝑡,𝑢,𝑢)̇⟹‖𝑏𝜃₀(𝑡,𝑥,𝑢)‖²𝑑𝑡≧0(𝑥,𝑢)^𝑏𝑎𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡−^𝑏𝑎𝑓(𝑡,𝑢,𝑢)̇𝑑𝑡>0(18) then we say that a pair (𝑓,ℎ) is 𝐵−𝜌₀−(𝜂,𝜃)-strictly pseudo-𝜌₁−(𝜂,𝜃)-quasi-type-I at 𝑢∈𝐶(𝐼,ℝ^𝑛) with respect to 𝑏₀, 𝑏₁, 𝜂, and 𝜃 and 𝜌₀,𝜌₁∈ℝ.

Remark 10. Let 𝜓∶𝐼×ℝ^𝑛×ℝ^𝑛→ℝ be a continuously differentiable function with respect to each of the arguments. Let 𝑥,𝑢∶𝐼→ℝ^𝑛 be differentiable with 𝑥(𝑎)=𝑢(𝑎)=𝛼 and 𝑥(𝑏)=𝑢(𝑏)=𝛽. Then, ^𝑏𝑎𝑑𝑑𝑡𝜂(𝑡,𝑥,𝑢)^𝑇𝜓_̇𝑥(𝑡,𝑢,𝑢)̇𝑑𝑡=−^𝑏𝑎𝜂(𝑡,𝑥,𝑢)𝑑𝑑𝑡𝜓^̇𝑥(𝑡,𝑢,𝑢)̇𝑑𝑡.(19)

Lemma 11. Every (𝜌₀,𝜌₁)−(𝜂,𝜃)-𝐵-type-I function is (𝜌₀,𝜌₁)−(𝜂,𝜃)-𝐵-strongly pseudo-type-I function but the converse is not true.

Proof. It is clear that the statement every (𝜌₀,𝜌₁)−(𝜂,𝜃)-𝐵-type-I function is (𝜌₀,𝜌₁)−(𝜂,𝜃)-𝐵-strongly pseudo-type-I function but the converse is not true follows from the following counter example. Example 12. Let 𝑓∶𝐼×[0,1]×[0,1]→ℝ be defined by 𝑓(𝑡,𝑥(𝑡),𝑥̇(𝑡))=−𝑥³(𝑡)−𝑥(𝑡).(20) Let ℎ∶𝐼×[0,1]×[0,1]→ℝ be defined by ℎ(𝑡,𝑥(𝑡),𝑥̇(𝑡))=3𝑥²(𝑡)+𝑥(𝑡)−4.(21) Let the functions 𝜂∶𝐼×[0,1]×[0,1]→ℝ and 𝜃∶𝐼×[0,1]×[0,1]→ℝ be defined by 𝜂(𝑡,𝑥(𝑡),𝑢(𝑡))=𝑥(𝑡)−5𝑢(𝑡)0if𝑢(𝑡)<𝑥(𝑡),𝜃(𝑡,𝑥(𝑡),𝑢(𝑡))=⎧⎪⎨⎪⎩if𝑢(𝑡)≥𝑥(t),𝑥(𝑡)−𝑢(𝑡)50if𝑢(𝑡)<𝑥(𝑡),if𝑢(𝑡)≥𝑥(𝑡).(22) Let the functions 𝑏₀∶[0,1]×[0,1]→ℝ₊ and 𝑏₁∶[0,1]×[0,1]→ℝ₊ be defined by 𝑏₀(𝑥(𝑡),𝑢(𝑡))=𝑥(𝑡)+𝑢(𝑡)+2=𝑏₁(𝑥(𝑡),𝑢(𝑡)).(23) Taking 𝜌₀=−1 and 𝜌₁=−1, we have to show that the pair (𝑓,ℎ) is (𝜌₀,𝜌₁)−(𝜂,𝜃)-𝐵-strongly pseudo-type-I but not (𝜌₀,𝜌₁)−(𝜂,𝜃)-𝐵-type-I with respect to 𝑏₀, 𝑏₁, 𝜂, and 𝜃. First we verify that (𝑓,ℎ) is (𝜌₀,𝜌₁)−(𝜂,𝜃)-𝐵-strongly pseudo-type-I function. Case 1. When 𝑢(𝑡)<𝑥(𝑡), it follows that 𝑏₀(=𝑥((𝑥𝑡)(,𝑡𝑢)(+𝑡)𝑢)(𝑡)+^𝑏𝑎𝑓2)(𝑡,𝑥_𝑏𝑎,𝑥̇)𝑑𝑡−^𝑏𝑎𝑓(𝑡,𝑢,𝑢)̇𝑑𝑡=(𝑥(𝑡)+𝑢(𝑡)+2)_𝑏𝑎−𝑥³(𝑡)−𝑥(𝑡)+𝑢³(𝑡)+𝑢(𝑡)𝑑𝑡=×(𝑥(𝑡)+𝑢(𝑡)+2)_𝑏𝑎𝑢³(𝑡)−𝑥³(𝑡)+(𝑢(𝑡)−𝑥(𝑡))𝑑𝑡<0((𝑢(𝑡)−𝑥(𝑡))×𝑢²(𝑡)+𝑥²(𝑡)+𝑢(t)𝑥(𝑡)+1𝑑𝑡⟹since𝑥(𝑡),𝑢(𝑡)∈[0,1_𝑏𝑎])𝜂(𝑡,𝑥(𝑡),𝑢(𝑡))^𝑇𝑓_𝑥(𝑡,𝑢(𝑡),𝑢̇(𝑡))+𝑑𝑑𝑡𝜂(𝑡,𝑥(𝑡),𝑢(𝑡))^𝑇=×𝑓_̇𝑥(𝑡,𝑢(𝑡),𝑢̇(𝑡)+𝜌₀‖𝜃(𝑡,𝑥(𝑡),𝑢(𝑡))‖²𝑑𝑡_𝑏𝑎𝑥(𝑡)−5𝑢(𝑡)−3𝑢²(𝑡)−1−𝑥(𝑡)−𝑢(𝑡)=5𝑑𝑡_𝑏𝑎𝑥(𝑡)−5𝑢(𝑡)−3𝑢²<0.(𝑡)−2𝑑𝑡(24) Again ^𝑏𝑎𝜂(𝑡,𝑥(𝑡),𝑢(𝑡))^𝑇ℎ_𝑥(𝑡,𝑢(𝑡),𝑢̇(𝑡))+𝑑𝑑𝑡𝜂(𝑡,𝑥(𝑡),𝑢(𝑡))^𝑇ℎ_̇𝑥=×(𝑡,𝑢(𝑡),𝑢̇(𝑡))+𝜌₁‖𝜃(𝑡,𝑥(𝑡),𝑢(𝑡))‖²𝑑𝑡_𝑏𝑎=𝑥(𝑡)−5𝑢(𝑡)(6𝑢(𝑡)+1)−𝑥(𝑡)−5𝑢(𝑡)𝑑𝑡_𝑏𝑎≧⟹0−𝑏𝑥₁(𝑡)−5𝑢(𝑡)6𝑢(𝑡)𝑑𝑡=−(𝑥(𝑡)(+𝑥𝑢(𝑡()𝑡,)𝑢+(𝑡2)))^𝑏𝑎_𝑏𝑎ℎ(𝑡,𝑢(𝑡),𝑢̇(𝑡))𝑑𝑡≧0.3𝑢²(𝑡)+𝑢(𝑡)−4𝑑𝑡(25)Case 2. When 𝑢(𝑡)≥𝑥(𝑡), it is clear that ^𝑏𝑎𝜂(𝑡,𝑥(𝑡),𝑢(𝑡))^𝑇𝑓_𝑥(𝑡,𝑢(𝑡),𝑢̇(𝑡))+𝑑𝑑𝑡𝜂(𝑡,𝑥(𝑡),𝑢(𝑡))^𝑇⟹×𝑓_̇𝑥𝑏(₀𝑡,𝑢(𝑡),𝑢̇(𝑡))+𝜌₀‖𝜃(𝑡,𝑥(𝑡),𝑢(𝑡))‖²𝑑𝑡≧0×(𝑥(𝑡),𝑢(𝑡))_𝑏𝑎𝑓(𝑡,𝑥(𝑡),𝑥̇(𝑡))𝑑𝑡−^𝑏𝑎𝑓(𝑡,𝑢(𝑡),𝑢̇(𝑡))𝑑𝑡≧0,_𝑏𝑎𝜂(𝑡,𝑥(𝑡),𝑢(𝑡))^𝑇ℎ_𝑥(𝑡,𝑢(𝑡),𝑢̇(𝑡))+𝑑𝑑𝑡𝜂(𝑡,𝑥(𝑡),𝑢(𝑡))^𝑇⟹×ℎ_̇𝑥−(𝑡,𝑏𝑢₁(𝑡),𝑢̇(𝑡))+𝜌₁‖𝜃(𝑡,𝑥(𝑡),𝑢(𝑡))‖²𝑑𝑡≧0(𝑥(𝑡),𝑢(𝑡))^𝑏𝑎ℎ(𝑡,𝑢(𝑡),𝑢̇(𝑡))𝑑𝑡≧0.(26) From Cases 1 and 2, it is clear that the pair (𝑓,ℎ) is (𝜌₀,𝜌₁)−(𝜂,𝜃)-𝐵-strongly pseudo-type-I.
But the pair (𝑓,ℎ) is not (𝜌₀,𝜌₁)−(𝜂,𝜃)-𝐵-type-I for 𝑢(𝑡)<𝑥(𝑡) because𝑏−₀(𝑥(𝑡),𝑢(𝑡))^𝑏𝑎𝑓(𝑡,𝑥(𝑡),𝑥̇(𝑡))𝑑𝑡−^𝑏𝑎𝑓(𝑡,𝑢(𝑡),𝑢̇(𝑡))𝑑𝑡_𝑏𝑎𝜂(𝑡,𝑥(𝑡),𝑢(𝑡))^𝑇𝑓_𝑥(𝑡,𝑢(𝑡),𝑢̇(𝑡))+𝑑𝑑𝑡𝜂(𝑡,𝑥(𝑡),𝑢(𝑡))^𝑇=(𝑥(𝑡)×+𝑓_̇𝑥𝑢((𝑡𝑡,)𝑢+(𝑡2)),𝑢_𝑏𝑎̇(𝑡))+𝜌₀‖𝜃(𝑡,𝑥(𝑡),𝑢(𝑡))‖²𝑑𝑡𝑢³−𝑥(𝑡)(−𝑡)𝑢−(𝑡𝑥)³(𝑡)+(𝑢(𝑡)−𝑥(𝑡))5−3𝑢²=×(𝑥(𝑡)+𝑢(𝑡)+2)_𝑏𝑎(𝑡)−2𝑑𝑡×𝑢𝑥(𝑡)−5𝑢(𝑡)(𝑢(𝑡)−𝑥(𝑡))₂+𝑥((𝑡𝑡))−+𝑢𝑥²(𝑡()𝑡)+𝑢(𝑡)𝑥(𝑡)+153𝑢²<0.(𝑡)+2𝑑𝑡(27)

3. Optimality Conditions

Theorem 13 (necessary optimality conditions). Let 𝑥^∗∈𝐾 be a properly efficient solution for (𝑉𝑃) which is assumed to be normal solution for (𝑃^∗𝑘), for 𝑘=1,2,…,𝑝. Then, there exist 𝜆^∗∈ℝ^𝑝 and a piecewise smooth function 𝑦^∗∶𝐼→ℝ^𝑚 such that 𝜆^∗𝑇𝑓_𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑦^∗(𝑡)^𝑇ℎ_𝑥𝑡,^∗=𝑑𝑥,𝑥^̇∗𝑑𝑡^∗𝜆^𝑇𝑓_̇𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑦^∗(𝑡)^𝑇ℎ_̇𝑥𝑡,^∗𝑥,𝑥^̇∗,_∗𝑦(𝑡)^𝑇ℎ𝑡,𝑥^∗,𝑥^̇∗=0,𝑡∈𝐼,_∗𝑦(𝑡)≧0,𝑡∈𝐼,_∗𝜆≧0.(28)

Proof. See [5].

Theorem 14. Let 𝑥^∗ be a feasible solution of (VP), and let there exist 𝜆^∗∈ℝ^𝑝, 𝜆^∗>0, and a piecewise smooth function 𝑦^∗∶𝐼→ℝ^𝑚 such that for all 𝑡∈𝐼𝜆^∗𝑇𝑓_𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑦^∗(𝑡)^𝑇ℎ_𝑥𝑡,^∗=𝑑𝑥,𝑥^̇∗𝑑𝑡^∗𝜆^𝑇𝑓_̇𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑦^∗(𝑡)^𝑇ℎ_̇𝑥𝑡,^∗𝑥,𝑥^̇∗,(29)_∗𝑦(𝑡)^𝑇ℎ𝑡,𝑥^∗,𝑥^̇∗=0,(30)_∗𝑦(𝑡)≧0.(31) Further, if (𝜆^∗𝑇𝑓,𝑦^∗(𝑡)^𝑇ℎ) is (𝜌₀,𝜌₁)−(𝜂,𝜃)-𝐵-type-I at 𝑥^∗ with respect to functions 𝑏₀, 𝑏₁, 𝜂, 𝜃, 𝜌₀,𝜌₁∈ℝ, and 𝜌₀+𝜌₁≧0, with 𝑏₀(𝑥,𝑥^∗)>0forall𝑥∈𝐾, then 𝑥^∗ is a properly efficient solution of (𝑉𝑃).

Proof. As (𝜆^∗𝑇𝑓,𝑦^∗(𝑡)^𝑇ℎ) is (𝜌₀,𝜌₁)−(𝜂,𝜃)-𝐵-type-I at 𝑥^∗, for all 𝑥∈𝐾, we have 𝑏₀𝑥,^∗≧𝑥𝜆^{𝑏𝑎∗𝑇}𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡−𝜆^{𝑏𝑎∗𝑇}𝑓𝑡,𝑥^∗,𝑥^̇∗𝑑𝑡_𝑏𝑎𝜂𝑡,𝑥,^∗𝑥𝜆^𝑇∗𝑇𝑓_𝑥𝑡,^∗+𝑑𝑥,𝑥^̇∗𝑑𝑡𝜂𝑡,𝑥,^∗𝑥𝜆^𝑇∗𝑇𝑓_̇𝑥𝑡,^∗𝑥,𝑥^̇∗+𝜌₀‖‖‖𝜃𝑡,𝑥,^∗𝑥‖‖‖²𝑑𝑡,(32)−𝑏₁𝑥,^∗≧𝑥𝑦^𝑏𝑎∗(𝑡)^𝑇ℎ𝑡,𝑥^∗,𝑥^̇∗𝑑𝑡_𝑏𝑎𝜂𝑡,𝑥,^∗𝑥𝑦^𝑇∗(𝑡)^𝑇ℎ_𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑑𝑑𝑡𝜂𝑡,𝑥,^∗𝑥^𝑇×𝑦^∗(𝑡)^𝑇ℎ_̇𝑥𝑡,^∗𝑥,𝑥^̇∗+𝜌₁‖‖‖𝜃𝑡,𝑥,^∗𝑥‖‖‖²𝑑𝑡.(33) Since 𝑦^∗(𝑡)^𝑇ℎ(𝑡,𝑥^∗,𝑥^̇∗)=0, we write (33) as 0≧^𝑏𝑎𝜂𝑡,𝑥,^∗𝑥𝑦^𝑇∗(𝑡)^𝑇ℎ_𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑑𝑑𝑡𝜂𝑡,𝑥,^∗𝑥^𝑇×𝑦^∗(𝑡)^𝑇ℎ_̇𝑥𝑡,^∗𝑥,𝑥^̇∗+𝜌₁‖‖‖𝜃𝑡,𝑥,^∗𝑥‖‖‖²𝑑𝑡.(34) Adding (32) and (34), we have 𝑏₀𝑥,^∗≧𝑥𝜆^{𝑏𝑎∗𝑇}𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡−𝜆^{𝑏𝑎∗𝑇}𝑓𝑡,𝑥^∗,𝑥^̇∗𝑑𝑡_𝑏𝑎𝜂𝑡,𝑥,^∗𝑥^𝑇^∗𝜆^𝑇𝑓_𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑦^∗(𝑡)^𝑇ℎ_𝑥𝑡,^∗+𝑑𝑥,𝑥^̇∗𝑑𝑡𝜂𝑡,𝑥,^∗𝑥^𝑇×𝜆^∗𝑇𝑓_̇𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑦^∗(𝑡)^𝑇ℎ_̇𝑥𝑡,^∗+𝜌₀𝑥,𝑥^̇∗+𝜌₁‖‖‖𝜃𝑡,𝑥,𝑥^∗‖‖‖²𝑑𝑡.(35) Using Remark 10, (35) becomes 𝑏₀𝑥,^∗≧𝑥𝜆^{𝑏𝑎∗𝑇}𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡−𝜆^{𝑏𝑎∗𝑇}𝑓𝑡,𝑥^∗,𝑥^̇∗𝑑𝑡_𝑏𝑎𝜂𝑡,𝑥,^∗𝑥^𝑇^∗𝜆^𝑇𝑓_𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑦^∗(𝑡)^𝑇ℎ_𝑥𝑡,^∗−𝑑𝑥,𝑥^̇∗𝑑𝑡^∗𝜆^𝑇𝑓_̇𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑦^∗(𝑡)^𝑇ℎ_̇𝑥𝑡,^∗+𝜌𝑥,𝑥^̇∗₀+𝜌₁‖‖‖𝜃𝑡,𝑥,𝑥^∗‖‖‖²𝑑𝑡.(36) By (29) and 𝜌₀+𝜌₁≧0, we have 𝑏₀𝑥,^∗𝑥𝜆^{𝑏𝑎∗𝑇}𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡−𝜆^{𝑏𝑎∗𝑇}𝑓𝑡,𝑥^∗,𝑥^̇∗𝑑𝑡≧0.(37) As 𝑏₀(𝑥,𝑥^∗)>0, it follows that ^𝑏𝑎∗𝜆^𝑇𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡−𝜆^{𝑏𝑎∗𝑇}𝑓𝑡,𝑥^∗,𝑥^̇∗𝑑𝑡≧0,(38) that is, ^𝑏𝑎∗𝜆^𝑇𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡≧𝜆^{𝑏𝑎∗𝑇}𝑓𝑡,𝑥^∗,𝑥^̇∗𝑑𝑡,(39) which implies that 𝑥^∗ minimizes ∫𝜆^{𝑏𝑎∗𝑇}𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡 over 𝐾 with 𝜆^∗>0. Hence 𝑥^∗ is a properly efficient solution of (𝑉𝑃) on account of Theorem 1 of Bector and Husain [3].

Theorem 15. Let 𝑥^∗ be a feasible solution of (𝑉𝑃), and let there exist 𝜆^∗∈ℝ^𝑝, 𝜆^∗>0, and a piecewise smooth function 𝑦^∗∶𝐼→ℝ^𝑚 such that, for every 𝑡∈𝐼, conditions, (29)–(31) of Theorem 14 are satisfied. Further, if (𝜆^∗𝑇𝑓,𝑦^∗(𝑡)^𝑇ℎ) is (𝜌₀,𝜌₁)−(𝜂,𝜃)-𝐵-semistrictly-type-I at 𝑥^∗ with respect to functions 𝑏₀, 𝑏₁, 𝜂, 𝜃, 𝜌₀,𝜌₁∈ℝ, and 𝜌₀+𝜌₁≧0, with 𝑏₀(𝑥,𝑥^∗)>0  for all  𝑥∈𝐾, then 𝑥^∗ is a properly efficient solution of (𝑉𝑃).

Proof. Let 𝑥^∗ be not; an efficient solution of (𝑉𝑃), then there exist 𝑥∈𝐾 and an index 1≦𝑟≦𝑃 such that the following inequalities ^𝑏𝑎𝑓_𝑗(𝑡,𝑥,𝑥̇)𝑑𝑡≦^𝑏𝑎𝑓_𝑗𝑡,^∗𝑥,𝑥^̇∗𝑑𝑡,∀𝑗≠𝑟,𝑗=1,2,…𝑝,_𝑏𝑎𝑓_𝑟(𝑡,𝑥,𝑥̇)𝑑𝑡<^𝑏𝑎𝑓_𝑟𝑡,^∗𝑥,𝑥^̇∗𝑑𝑡(40) hold. Because 𝜆^∗>0 and 𝑏₀(𝑥,𝑥^∗)>0, (40) imply 𝑏₀𝑥,^∗𝑥𝜆^{𝑏𝑎∗𝑇}𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡−𝜆^{𝑏𝑎∗𝑇}𝑓𝑡,𝑥^∗,𝑥^̇∗𝑑𝑡≦0.(41) Further in view of (30) 𝑏₁𝑥,^∗𝑥𝑦^𝑏𝑎∗(𝑡)^𝑇ℎ𝑡,𝑥^∗,𝑥^̇∗𝑑𝑡=0.(42) As (𝜆^∗𝑇𝑓,𝑦^∗(𝑡)^𝑇ℎ) is (𝜌₀,𝜌₁)−(𝜂,𝜃)-𝐵-semistrictly-type-I at 𝑥^∗ with respect to functions 𝑏₀, 𝑏₁, 𝜂, and 𝜃, we have ^𝑏𝑎𝜂𝑡,𝑥,^∗𝑥𝜆^𝑇∗𝑇𝑓_𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑑𝑑𝑡𝜂𝑡,𝑥,^∗𝑥𝜆^𝑇∗𝑇𝑓_̇𝑥𝑡,^∗+𝜌₀𝑥,𝑥^̇∗‖‖‖𝜃𝑡,𝑥,^∗𝑥‖‖‖²𝑑𝑡<0,_𝑏𝑎𝜂𝑡,𝑥,^∗𝑥𝑦^𝑇∗(𝑡)^𝑇ℎ_𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑑𝑑𝑡𝜂𝑡,𝑥,^∗𝑥^𝑇×𝑦^∗(𝑡)^𝑇ℎ_̇𝑥𝑡,^∗𝑥,𝑥^̇∗+𝜌₁‖‖‖𝜃𝑡,𝑥,^∗𝑥‖‖‖²𝑑𝑡≦0.(43) Adding (43), we have ^𝑏𝑎𝜂𝑡,𝑥,^∗𝑥^𝑇^∗𝜆^𝑇𝑓_𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑦^∗(𝑡)^𝑇ℎ_𝑥𝑡,^∗+𝑑𝑥,𝑥^̇∗𝑑𝑡𝜂𝑡,𝑥,^∗𝑥^𝑇^∗𝜆^𝑇𝑓_̇𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑦^∗(𝑡)^𝑇ℎ_̇𝑥𝑡,^∗+𝜌₀𝑥,𝑥^̇∗+𝜌₁‖‖‖𝜃𝑡,𝑥,𝑥^∗‖‖‖²𝑑𝑡<0.(44) By Remark 10, we have ^𝑏𝑎𝜂𝑡,𝑥,^∗𝑥^𝑇^∗𝜆^𝑇𝑓_𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑦^∗(𝑡)^𝑇ℎ_𝑥𝑡,^∗−𝑑𝑥,𝑥^̇∗𝑑𝑡^∗𝜆^𝑇𝑓_̇𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑦^∗(𝑡)^𝑇ℎ_̇𝑥𝑡,^∗+𝜌𝑥,𝑥^̇∗₀+𝜌₁‖‖‖𝜃𝑡,𝑥,𝑥^∗‖‖‖²𝑑𝑡<0,(45) which contradicts (29) and 𝜌₀+𝜌₁≥0. Hence 𝑥^∗ is a properly efficient solution of (𝑉𝑃).

Theorem 16. Let 𝑥^∗ be a feasible solution of (VP), and let there exist 𝜆^∗∈ℝ^𝑝, 𝜆^∗>0, and a piecewise smooth function 𝑦^∗∶𝐼→ℝ^𝑚 such that conditions (29) of Theorem 14 are satisfied by (𝑥^∗,𝜆^∗,𝑦^∗). Further, if (𝜆^∗𝑇𝑓,𝑦^∗(𝑡)^𝑇ℎ) is 𝐵−𝜌₀−(𝜂,𝜃)-strongly pseudo-𝜌₁−(𝜂,𝜃)-quasi-type-I at 𝑥^∗ with respect to functions 𝑏₀, 𝑏₁, 𝜂, 𝜃, 𝜌₀,𝜌₁∈ℝ and 𝜌₀+𝜌₁≧0, 𝑏₁(𝑥,𝑥^∗)>0𝑓𝑜𝑟𝑎𝑙𝑙𝑥∈𝐾, then 𝑥^∗ is a properly efficient solution of (VP).

Proof. From (30), we have 𝑏₁𝑥,^∗𝑥𝑦^𝑏𝑎∗(𝑡)^𝑇ℎ𝑡,𝑥^∗,𝑥^̇∗=0.(46) As (𝜆^∗𝑇𝑓,𝑦^∗(𝑡)^𝑇ℎ) is 𝐵−𝜌₀−(𝜂,𝜃)-strongly pseudo-𝜌₁−(𝜂,𝜃)-quasi-type-I at 𝑥^∗ with respect to functions 𝑏₀, 𝑏₁, 𝜂, and 𝜃, ^𝑏𝑎𝜂𝑡,𝑥,^∗𝑥𝑦^𝑇∗(𝑡)^𝑇ℎ_𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑑𝑑𝑡𝜂𝑡,𝑥,^∗𝑥𝑦^𝑇∗(𝑡)^𝑇×ℎ_̇𝑥𝑡,^∗𝑥,𝑥^̇∗+𝜌₁‖‖‖𝜃𝑡,𝑥,^∗𝑥‖‖‖²𝑑𝑡≦0.(47) Using Remark 10, (45) becomes ^𝑏𝑎𝜂𝑡,𝑥,^∗𝑥^𝑇^∗𝑦(𝑡)^𝑇ℎ_𝑥𝑡,^∗𝑥,𝑥^̇∗−𝑑𝑑𝑡^∗𝑦(𝑡)^𝑇ℎ_̇𝑥𝑡,^∗+𝜌₁𝑥,𝑥^̇∗‖‖‖𝜃𝑡,𝑥,^∗𝑥‖‖‖²𝑑𝑡≦0.(48) Equation (48) along with (29) gives ^𝑏𝑎𝜂𝑡,𝑥,^∗𝑥^𝑇^∗𝜆^𝑇𝑓_𝑥𝑡,^∗𝑥,𝑥^̇∗−𝑑𝑑𝑡^∗𝜆^𝑇𝑓_̇𝑥𝑡,^∗−𝜌₁𝑥,𝑥^̇∗‖‖‖𝜃𝑡,𝑥,^∗𝑥‖‖‖²𝑑𝑡≧0.(49) Using Remark 10, for all 𝑥∈𝐾, (48) becomes ^𝑏𝑎𝜂𝑡,𝑥,^∗𝑥𝜆^𝑇∗𝑇𝑓_𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑑𝑑𝑡𝜂𝑡,𝑥,^∗𝑥𝜆^𝑇∗𝑇𝑓_̇𝑥𝑡,^∗−𝜌₁𝑥,𝑥^̇∗‖‖‖𝜃𝑡,𝑥,^∗𝑥‖‖‖²𝑑𝑡≧0∀𝑥∈𝐾.(50) As 𝜌₀+𝜌₁≧0, we have ^𝑏𝑎𝜂𝑡,𝑥,^∗𝑥𝜆^𝑇∗𝑇𝑓_𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑑𝑑𝑡𝜂𝑡,𝑥,^∗𝑥𝜆^𝑇∗𝑇𝑓_̇𝑥𝑡,^∗+𝜌₀𝑥,𝑥^̇∗‖‖‖𝜃𝑡,𝑥,^∗𝑥‖‖‖²𝑑𝑡≧0.(51) Equation (51) along with the fact that (𝜆^∗𝑇𝑓,𝑦^∗(𝑡)^𝑇ℎ) is 𝐵−𝜌₀−(𝜂,𝜃)-strongly pseudo-𝜌₁−(𝜂,𝜃)-quasi-type-I at 𝑥^∗ gives 𝑏₀𝑥,^∗𝑥𝜆^{𝑏𝑎∗𝑇}𝑓(𝑡,𝑥,𝑥̇)−𝜆^{𝑏𝑎∗𝑇}𝑓𝑡,𝑥^∗,𝑥^̇∗≧0.(52) Since 𝑏₀(𝑥,𝑥^∗)>0, it follows that ^𝑏𝑎∗𝜆^𝑇𝑓(𝑡,𝑥,𝑥̇)≧𝜆^{𝑏𝑎∗𝑇}𝑓𝑡,𝑥^∗,𝑥^̇∗∀𝑥∈𝐾.(53) Hence 𝑥^∗ minimizes ∫𝜆^{𝑏𝑎∗𝑇}𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡 over 𝐾 with 𝜆^∗>0.
Therefore 𝑥^∗ is a properly efficient solution of (𝑉𝑃) ([3, Theorem 1]).

Theorem 17. Let 𝑥^∗ be a feasible solution of (𝑉𝑃), and let there exist 𝜆^∗∈ℝ^𝑝, 𝜆^∗>0, and a piecewise smooth function 𝑦^∗∶𝐼→ℝ^𝑚 such that the conditions (29)–(31) of Theorem 14 are satisfied by (𝑥^∗,𝜆^∗,𝑦^∗). Further, if (𝜆^∗𝑇𝑓,𝑦^∗(𝑡)^𝑇ℎ) is 𝐵−𝜌₀−(𝜂,𝜃)-quasi-𝜌₁−(𝜂,𝜃)-strictly pseudo-type-I at 𝑥^∗ with respect to functions 𝑏₀, 𝑏₁, 𝜂, 𝜃, 𝜌₀,𝜌₁∈ℝ, and 𝜌₀+𝜌₁≧0, with 𝑏₀(𝑥,𝑥^∗)>0 for all 𝑥∈𝐾, then 𝑥^∗ is a properly efficient solution of (𝑉𝑃).

Proof. If 𝑥^∗ is not an efficient solution of (𝑉𝑃), then there exist 𝑥∈𝐾 and an index 𝑟, 1≦𝑟≦𝑝, such that ^𝑏𝑎𝑓_𝑗(𝑡,𝑥,𝑥̇)𝑑𝑡≦^𝑏𝑎𝑓_𝑗𝑡,^∗𝑥,𝑥^̇∗𝑑𝑡∀𝑗≠𝑟,𝑗=1,2,…𝑝,_𝑏𝑎𝑓_𝑟(𝑡,𝑥,𝑥̇)𝑑𝑡<^𝑏𝑎𝑓_𝑟𝑡,^∗𝑥,𝑥^̇∗𝑑𝑡.(54) Because 𝜆^∗>0 and 𝑏₀(𝑥,𝑥^∗)>0, the previous relations give 𝑏₀𝑥,^∗𝑥𝜆^{𝑏𝑎∗𝑇}𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡−𝜆^{𝑏𝑎∗𝑇}𝑓𝑡,𝑥^∗,𝑥^̇∗𝑑𝑡≦0,(55) which along with the fact that (𝜆^∗𝑇𝑓,𝑦^∗(𝑡)^𝑇ℎ) is 𝐵−𝜌₀−(𝜂,𝜃)-quasi-𝜌₁−(𝜂,𝜃)-strictly pseudo-type-I at 𝑥^∗ with respect to functions 𝑏₀, 𝑏₁, 𝜂, and 𝜃 gives ^𝑏𝑎𝜂𝑡,𝑥,^∗𝑥𝜆^𝑇∗𝑇𝑓_𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑑𝑑𝑡𝜂𝑡,𝑥,^∗𝑥𝜆^𝑇∗𝑇𝑓_̇𝑥𝑡,^∗+𝜌₀𝑥,𝑥^̇∗‖‖‖𝜃𝑡,𝑥,^∗𝑥‖‖‖²𝑑𝑡≦0.(56) Using Remark 10, (56) becomes ^𝑏𝑎𝜂𝑡,𝑥,^∗𝑥𝜆^𝑇∗𝑇𝑓_𝑥𝑡,^∗𝑥,𝑥^̇∗−𝜂𝑡,𝑥,𝑥^∗^𝑇𝑑𝑑𝑡^∗𝜆^𝑇𝑓_̇𝑥𝑡,^∗+𝜌₀𝑥,𝑥^̇∗‖‖‖𝜃𝑡,𝑥,^∗𝑥‖‖‖²𝑑𝑡≦0.(57) that is, ^𝑏𝑎𝜂𝑡,𝑥,^∗𝑥^𝑇^∗𝜆^𝑇𝑓_𝑥𝑡,^∗𝑥,𝑥^̇∗−𝑑𝑑𝑡^∗𝜆^𝑇𝑓_̇𝑥𝑡,^∗+𝜌₀𝑥,𝑥^̇∗‖‖‖𝜃𝑡,𝑥,^∗𝑥‖‖‖²𝑑𝑡≦0.(58) Now using (29) in (58), we have ^𝑏𝑎−𝜂𝑡,𝑥,^∗𝑥^𝑇×𝑦^∗(𝑡)^𝑇ℎ_𝑥𝑡,^∗𝑥,𝑥^̇∗−𝑑𝑑𝑡^∗𝑦(𝑡)^𝑇ℎ_̇𝑥𝑡,^∗+𝜌₀𝑥,𝑥^̇∗𝑑𝑡‖‖‖𝜃𝑡,𝑥,^∗𝑥‖‖‖²≦0.(59) It follows that ^𝑏𝑎𝜂𝑡,𝑥,^∗𝑥^𝑇^∗𝑦(𝑡)^𝑇ℎ_𝑥𝑡,^∗𝑥,𝑥^̇∗−𝑑𝑑𝑡^∗𝑦(𝑡)^𝑇ℎ_̇𝑥𝑡,^∗𝑥,x^̇∗−𝜌₀‖‖‖𝜃𝑡,𝑥,^∗𝑥‖‖‖²⟹𝑑𝑡≧0,_𝑏𝑎𝜂𝑡,𝑥,𝑥^∗^𝑇^∗𝑦(𝑡)^𝑇ℎ_𝑥𝑡,^∗𝑥,𝑥^̇∗−𝑑𝑑𝑡^∗𝑦(𝑡)^𝑇ℎ_̇𝑥𝑡,^∗≧𝜌₀𝑥,𝑥^̇∗𝑑𝑡‖‖‖𝜃𝑡,𝑥,^∗𝑥‖‖‖².(60) As 𝜌₀+𝜌₁≧0, we have ^𝑏𝑎𝜂𝑡,𝑥,^∗𝑥^𝑇^∗𝑦(𝑡)^𝑇ℎ_𝑥𝑡,^∗𝑥,𝑥^̇∗−𝑑𝑑𝑡^∗𝑦(𝑡)^𝑇ℎ_̇𝑥𝑡,^∗+𝜌₁𝑥,𝑥^̇∗‖‖‖𝜃𝑡,𝑥,^∗𝑥‖‖‖²𝑑𝑡≧0.(61) Again using Remark 10, we obtain ^𝑏𝑎𝜂𝑡,𝑥,^∗𝑥𝑦^𝑇∗(𝑡)^𝑇ℎ_𝑥𝑡,^∗𝑥,𝑥^̇∗+𝑑𝑑𝑡𝜂𝑡,𝑥,^∗𝑥𝑦^𝑇∗(𝑡)^𝑇×ℎ_̇𝑥𝑡,^∗𝑥,𝑥^̇∗+𝜌₁‖‖‖𝜃𝑡,𝑥,^∗𝑥‖‖‖²𝑑𝑡≧0,(62) which in view of a given hypothesis implies that −𝑏₁𝑥,^∗𝑥𝑦^𝑏𝑎∗(𝑡)^𝑇ℎ𝑡,𝑥^∗,𝑥^̇∗>0.(63) Since 𝑏₁(𝑥,𝑥^∗)>0, we obtain −𝑦^𝑏𝑎∗(𝑡)^𝑇ℎ𝑡,𝑥^∗,𝑥^̇∗>0,(64) which contradicts (30). Hence 𝑥^∗ is a properly efficient solution of (𝑉𝑃).

4. Duality

The Mond-Weir-type dual problem associated with (𝑉𝑃) is given by (𝑉𝐷)maximize^𝑏𝑎=𝑓(𝑡,𝑢,𝑢)̇𝑑𝑡_𝑏𝑎𝑓₁(𝑡,𝑢,𝑢)̇𝑑𝑡,…,^𝑏𝑎𝑓_𝑝(𝑡,𝑢,𝑢)̇𝑑𝑡,subjectto𝑢(𝑎)=𝛼,𝑢(𝑏)=𝛽,(65)𝜆^𝑇𝑓=_𝑥(𝑡𝑑,𝑢,𝑢)̇+𝑦(𝑡)^𝑇ℎ_𝑥(𝑡,𝑢,𝑢)̇𝑑𝑡𝜆^𝑇𝑓_̇𝑥(𝑡,𝑢,𝑢)̇+𝑦(𝑡)^𝑇ℎ_̇𝑥(𝑡,𝑢,𝑢)̇,(66)𝑦(𝑡)^𝑇ℎ(𝑡,𝑢,𝑢)̇≧0,𝑡∈𝐼,(67)𝑦(𝑡)≧0,𝑡∈𝐼,(68)𝜆∈ℝ^𝑝,𝑒=(𝜆1,≧1,0…,,1)𝜆∈^𝑇𝑒ℝ=_𝑝1,.(69) We now establish duality results between (𝑉𝑃) and (𝑉𝐷) under generalized 𝐵-type-I conditions.

Theorem 18 (weak duality). Let 𝑥 be a feasible solution of (𝑉𝑃), and let (𝑢,𝜆,𝑦) be a feasible solution of (VD ). Let either the following conditions hold. (i)(𝜆^𝑇𝑓,𝑦(𝑡)^𝑇ℎ) is (𝜌₀,𝜌₁)−(𝜂,𝜃)−𝐵-semi-strictly-type-I at 𝑢 with respect to functions 𝑏₀, 𝑏₁, 𝜂, 𝜃, 𝜌₀,𝜌₁∈ℝ and 𝜌₀+𝜌₁≧0.(ii)𝜆>0 and (𝜆^𝑇𝑓,𝑦(𝑡)^𝑇ℎ) is 𝐵−𝜌₀−(𝜂,𝜃)-strongly pseudo-𝜌₁−(𝜂,𝜃)-quasi-type-I at 𝑢 with respect to functions 𝑏₀, 𝑏₁, 𝜂, and 𝜃 with 𝑏₁(𝑥,𝑢)>0, 𝜌₀,𝜌₁∈ℝ and 𝜌₀+𝜌₁≧0.(iii)(𝜆^𝑇𝑓,𝑦(𝑡)^𝑇ℎ) is 𝐵−𝜌₀−(𝜂,𝜃)-quasi-𝜌₁−(𝜂,𝜃)-strictly pseudo-type-I at 𝑢 with respect to functions 𝑏₀, 𝑏₁, 𝜂, and 𝜃, with 𝑏₀(𝑥,𝑢)>0, 𝜌₀,𝜌₁∈ℝ, and 𝜌₀+𝜌₁≧0, Then ^𝑏𝑎𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡^𝑏𝑎𝑓(𝑡,𝑢,𝑢)̇𝑑𝑡.(70)

Proof. (i) Let, if possible, ^𝑏𝑎𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡≤^𝑏𝑎𝑓(𝑡,𝑢,𝑢)̇𝑑𝑡.(71) Then there exists an index 𝑟, 1≦𝑟≦𝑝, such that ^𝑏𝑎𝑓_𝑗(𝑡,𝑥,𝑥̇)𝑑𝑡≦^𝑏𝑎𝑓_𝑗𝑡,^∗𝑥,𝑥^̇∗𝑑𝑡∀𝑗≠𝑟,𝑗=1,2,…𝑝,_𝑏𝑎𝑓_𝑟(𝑡,𝑥,𝑥̇)𝑑𝑡<^𝑏𝑎𝑓_𝑟𝑡,^∗𝑥,𝑥^̇∗𝑑𝑡.(72) Because 𝜆>0 and 𝑏₀(𝑥,𝑢)>0, the above relations give 𝑏₀(𝑥,𝑢)^𝑏𝑎𝜆^𝑇𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡≦𝑏₀(𝑥,𝑢)^𝑏𝑎𝜆^𝑇𝑓(𝑡,𝑢,𝑢)̇𝑑𝑡,(73) that is, 𝑏₀(𝑥,𝑢)^𝑏𝑎𝜆^𝑇𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡−^𝑏𝑎𝜆^𝑇𝑓(𝑡,𝑢,𝑢)̇𝑑𝑡≦0(74) which is the same as (41) with 𝑥^∗ replaced by 𝑢 and 𝜆^∗ replaced by 𝜆. The rest of the proof runs on the same line of that of Theorem 15 and hence is omitted.
(ii) From (67), we get−^𝑏𝑎𝑦(𝑡)^𝑇ℎ(𝑡,𝑢,𝑢)̇≦0.(75) Since 𝑏₁(𝑥,𝑢)>0, we have −𝑏₁(𝑥,𝑢)^𝑏𝑎𝑦(𝑡)^𝑇ℎ(𝑡,𝑢,𝑢)̇≦0,(76) which implies that ^𝑏𝑎𝜂(𝑡,𝑥,𝑢)^𝑇𝑦(𝑡)^𝑇ℎ_𝑥(𝑡,𝑢,𝑢)̇+𝑑𝑑𝑡𝜂(𝑡,𝑥,𝑢)^𝑇𝑦(𝑡)^𝑇×ℎ_̇𝑥(𝑡,𝑢,𝑢)̇+𝜌₁‖𝜃(𝑡,𝑥,𝑢)‖²𝑑𝑡≦0,(77) on account of hypothesis (ii). This is the same as (45) with 𝑥^∗ replaced by 𝑢 and 𝑦^∗ replaced by 𝑦. Again proceeding on the lines of Theorem 16, we get the result.
(iii) Proof of this part follows on the lines of Theorem 17 and hence is omitted.

Theorem 19 (strong duality). Let 𝑥^∗ be a properly efficient solution of (𝑉𝑃). Assume that 𝑥^∗ is normal for each (𝑃^∗𝑘), 𝑘=1,2,…,𝑝. Then there exist 𝜆^∗∈ℝ^𝑝 and a piecewise smooth function 𝑦^∗∶𝐼→ℝ^𝑚 such that (𝑥^∗,𝜆^∗,𝑦^∗) is feasible for (𝑉𝐷). Further, if, for each feasible (𝑢,𝜆,𝑦) of (𝑉𝐷), any of the conditions of Theorem 18 hold, then (𝑥^∗,𝜆^∗,𝑦^∗) is a properly efficient solution of (𝑉𝐷).

Proof. Because 𝑥^∗ is a properly efficient solution of (𝑉𝑃), it follows that, from Theorem 13 there exist 𝜆^∗∈ℝ^𝑝 and a piecewise smooth function 𝑦^∗∶𝐼→ℝ^𝑚 such that (28) hold. Moreover, 𝑥^∗∈𝐾, and hence the feasibility of (𝑥^∗,𝜆^∗,𝑦^∗) for (𝑉𝐷) follows. As the weak duality (Theorem 18) holds between (𝑉𝑃) and (𝑉𝐷), (𝑥^∗,𝜆^∗,𝑦^∗) is an efficient solution of (𝑉𝐷). If (𝑥^∗,𝜆^∗,𝑦^∗) is not a properly efficient solution of (𝑉𝐷), then, proceeding on the lines to that of [3, Theorem 1], we get a contradiction to the weak duality.

Theorem 20 (strict converse duality). Let 𝑥^∗ be feasible for (𝑉𝑃), and let (𝑢^∗,𝜆^∗,𝑦^∗) be feasible for (VD ) such that ^𝑏𝑎∗𝜆^𝑇𝑓𝑡,𝑥^∗,𝑥^̇∗𝑑𝑡=𝜆^{𝑏𝑎∗𝑇}𝑓𝑡,𝑢^∗,𝑢^̇∗𝑑𝑡.(78) Further, let (𝜆^∗𝑇𝑓,𝑦^∗(𝑡)^𝑇ℎ) be 𝐵−𝜌₀−(𝜂,𝜃)-strictly pseudo-𝜌₁−(𝜂,𝜃)-quasi-type-I at 𝑢^∗ with respect to functions 𝑏₀, 𝑏₁, 𝜂, 𝜃, 𝜌₀,𝜌₁∈ℝ, and 𝜌₀+𝜌₁≧0, 𝑏₀(𝑥^∗,𝑢^∗)>0, 𝑏₁(𝑥^∗,𝑢^∗)>0 for all 𝑥^∗∈𝐾. Then 𝑥^∗ is a properly efficient solution of (𝑉𝑃).

Proof. We assume that 𝑥^∗≠𝑢^∗ and get a contradiction. Feasibility of (𝑢^∗,𝜆^∗,𝑦^∗) for (𝑉𝐷) implies that −𝑦^𝑏𝑎∗(𝑡)^𝑇ℎ𝑡,𝑢^∗,𝑢^̇∗𝑑𝑡≦0.(79) As 𝑏₁(𝑥^∗,𝑢^∗)>0, we have −𝑏₁^∗𝑥,𝑢^∗𝑦^𝑏𝑎∗(𝑡)^𝑇ℎ𝑡,𝑢^∗,𝑢^̇∗𝑑𝑡≦0.(80) Equation (80) along with the fact that (𝜆^∗𝑇𝑓,𝑦^∗(𝑡)^𝑇ℎ) is 𝐵−𝜌₀−(𝜂,𝜃)-strictly pseudo-𝜌₁−(𝜂,𝜃)-quasi-type-I at 𝑢^∗ implies that ^𝑏𝑎𝜂𝑡,^∗𝑥,𝑢^∗𝑦^𝑇∗(𝑡)^𝑇ℎ_𝑥𝑡,^∗+𝑑𝑢,𝑢^̇∗𝑑𝑡𝜂𝑡,^∗𝑥,𝑢^∗𝑦^𝑇∗(𝑡)^𝑇ℎ_̇𝑥𝑡,^∗𝑢,𝑢^̇∗+𝜌₁‖‖‖𝜃𝑡,^∗𝑥,𝑢^∗‖‖‖²≦0.𝑑𝑡(81) Using Remark 10 (with 𝜓 replaced by 𝑦^∗(𝑡)^𝑇ℎ) in (81), we get ^𝑏𝑎𝜂𝑡,^∗𝑥,𝑢^∗^𝑇^∗𝑦(𝑡)^𝑇ℎ_𝑥𝑡,^∗𝑢,𝑢^̇∗−𝑑𝑑𝑡^∗𝑦(𝑡)^𝑇ℎ_̇𝑥𝑡,^∗+𝜌₁𝑢,𝑢^̇∗‖‖‖𝜃𝑡,^∗𝑥,𝑢^∗‖‖‖²≦0.(82) Equation (82) along with (66) gives ^𝑏𝑎𝜂𝑡,^∗𝑥,𝑢^∗^𝑇^∗𝜆^𝑇𝑓_𝑥𝑡,^∗𝑢,𝑢^̇∗−𝑑𝑑𝑡^∗𝜆^𝑇𝑓_̇𝑥𝑡,^∗−𝜌₁𝑢,𝑢^̇∗‖‖‖𝜃𝑡,^∗𝑥,𝑢^∗‖‖‖²≧0.(83) Since 𝜌₀+𝜌₁≧0, we have ^𝑏𝑎𝜂𝑡,^∗𝑥,𝑢^∗^𝑇^∗𝜆^𝑇𝑓_𝑥𝑡,^∗𝑢,𝑢^̇∗−𝑑𝑑𝑡^∗𝜆^𝑇𝑓_̇𝑥𝑡,^∗+𝜌₀𝑢,𝑢^̇∗‖‖‖𝜃𝑡,^∗𝑥,𝑢^∗‖‖‖²𝑑𝑡≧0.(84) Again using Remark 10 (with 𝜓 replaced by 𝜆^∗𝑇𝑓) in (84), we obtain ^𝑏𝑎𝜂𝑡,^∗𝑥,𝑢^∗𝜆^𝑇∗𝑇𝑓_𝑥𝑡,^∗𝑢,𝑢^̇∗+𝑑𝑑𝑡𝜂𝑡,^∗𝑥,𝑢^∗𝜆^𝑇∗𝑇𝑓_̇𝑥𝑡,^∗+𝜌₀𝑢,𝑢^̇∗𝑑𝑡‖‖‖𝜃𝑡,^∗𝑥,𝑢^∗‖‖‖²≧0,(85) which in view of the given hypothesis implies that 𝑏₀^∗𝑥,𝑢^∗𝜆^{𝑏𝑎∗𝑇}𝑓𝑡,𝑥^∗,𝑥^̇∗𝑑𝑡−𝜆^{𝑏𝑎∗𝑇}𝑓𝑡,𝑢^∗,𝑢^̇∗𝑑𝑡>0.(86) As 𝑏₀(𝑥^∗,𝑢^∗)>0, it follows that ^𝑏𝑎∗𝜆^𝑇𝑓𝑡,𝑥^∗,𝑥^̇∗𝑑𝑡>𝜆^{𝑏𝑎∗𝑇}𝑓𝑡,𝑢^∗,𝑢^̇∗.(87) But this contradicts (78), hence the result.

5. Some Related Problems

All the optimality conditions and duality results developed for (𝑉𝑃) in the previous sections can easily be modified for several other classes of variational problems.

5.1. Natural Boundary Value Problems

Omitting the boundary conditions for the fixed end points as was done by Mond and Hanson [2], we obtain𝑉𝑃₀minimize^𝑏𝑎=𝑓(𝑡,𝑥,𝑥̇)𝑑𝑡_𝑏𝑎𝑓₁(𝑡,𝑥,𝑥̇)𝑑𝑡,…,^𝑏𝑎𝑓_𝑝(𝑡,𝑥,𝑥̇)𝑑𝑡subjecttoℎ(𝑡,𝑥,𝑥̇)≦0𝑡∈𝐼.(88) The corresponding Mond-Weir type dual is given by 𝑉𝐷₀maximize^𝑏𝑎=𝑓(𝑡,𝑢,𝑢)̇𝑑𝑡_𝑏𝑎𝑓₁(𝑡,𝑢,𝑢)̇𝑑𝑡,…,^𝑏𝑎𝑓_𝑝(𝑡,𝑢,𝑢)̇𝑑𝑡subjectto𝜆^T𝑓=_𝑥(𝑑𝑡,𝑢,𝑢)̇+𝑦(𝑡)^𝑇ℎ_𝑥(𝑡,𝑢,𝑢)̇𝑑𝑡𝜆^𝑇𝑦𝑦((𝑡𝑡))_𝑇≥0,𝑡𝑓∈_̇𝑥(𝐼𝑡,,𝑢,𝑢)̇+𝑦(𝑡)^𝑇ℎ_̇𝑥(𝑡,𝑢,𝑢)̇,𝑦(𝑡)_𝑇ℎ_𝑥(𝑡,𝑢,𝑢)̇_𝑡=𝑎=0,𝜆ℎ_̇𝑥(𝑡,𝑢,𝑢)̇_𝑡=𝑏=0,_𝑇𝜆𝑓_𝑥(𝑡,𝑢,𝑢)̇_𝑡=𝑎=0,_𝑇𝜆≧𝑓0_̇𝑥,(𝑡,𝑢,𝜆𝑢_𝑇)̇_𝑡=𝑏=0,𝑒=(1,1,…,𝑒1)=∈1ℝ,_𝑝.(89) If the problems (𝑉𝑃₀) and (𝑉𝐷₀) are independent of 𝑡, then these problems essentially reduce to the static case (𝑉𝑃) and (𝑉𝐷) of multiobjective nonlinear programs: 𝑉𝑃minimize𝑓(𝑥)=𝑓₁(𝑥),𝑓₂(𝑥),…,𝑓_𝑝(𝑥)𝑉subjecttoℎ(𝑥)≤0,𝐷maximize𝑓(𝑢)=𝑓₁(𝑢),𝑓₂(𝑢),…,𝑓_𝑝(𝑢),subjectto𝜆𝑦^𝑇≧𝑓_𝑥0,+𝑦(𝑡𝜆)^𝑇≧ℎ_𝑥0=,0,𝜆_𝑇𝑒=(1,1,…,1)𝑒∈=ℝ1_𝑝,.(90)

Acknowledgments

The authors wish to thank the referees and the Editor for their valuable suggestions which improved the presentation of the paper.