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BY 4.0 license Open Access Published by De Gruyter Open Access September 26, 2022

Generalized invexity and duality in multiobjective variational problems involving non-singular fractional derivative

  • Ved Prakash Dubey , Devendra Kumar EMAIL logo , Hashim M. Alshehri , Jagdev Singh and Dumitru Baleanu
From the journal Open Physics

Abstract

In this article, we extend the generalized invexity and duality results for multiobjective variational problems with fractional derivative pertaining to an exponential kernel by using the concept of weak minima. Multiobjective variational problems find their applications in economic planning, flight control design, industrial process control, control of space structures, control of production and inventory, advertising investment, impulsive control problems, mechanics, and several other engineering and scientific problems. The proposed work considers the newly derived Caputo–Fabrizio (CF) fractional derivative operator. It is actually a convolution of the exponential function and the first-order derivative. The significant characteristic of this fractional derivative operator is that it provides a non-singular exponential kernel, which describes the dynamics of a system in a better way. Moreover, the proposed work also presents various weak, strong, and converse duality theorems under the diverse generalized invexity conditions in view of the CF fractional derivative operator.

1 Introduction

The present scenario indicates that the fractional differential equations (FDEs) and fractional variational problems (FVPs) are being used to delineate the physical models and engineering processes in a better way. The clear reason is that the standard mathematical models of integer-order derivatives incorporating models of non-linear nature do not perform efficiently in many instances according to desired results. Recently, the field of fractional calculus has portrayed a significant part in various areas of knowledge such as chemistry [1], biology [2,3], mechanics [4,5], and finance [6]. The application area of fractional modelling and fractional operators encompasses anomalous diffusion [7], physics [8], heat conduction [9], geophysics [10], epidemiology [11], fractals and fractional derivative [12], computational fractional derivative equations [13], fractional predator–prey system [14], and porous media [15]. The models related to these fields utilize fractional derivative operators frequently.

There are various types of fractional derivative operators in the literature of fractional calculus founded by so many famous mathematicians. But the most popular definitions of them are Riemann–Liouville (RL) fractional derivative and Riesz fractional derivative described in the studies of Samko et al. [16] and Podlubny [17], Caputo fractional derivative in refs. [18,19], Weyl fractional derivative [20], Hadamard fractional derivative [16,17], Jumarie’s fractional derivative propounded in works of Jumarie [21,22], Atangana–Baleanu derivative proposed in ref. [23], and Liouville–Caputo derivative described in ref. [24]. The interesting fact is that these definitions of fractional derivative operators have their own significance and their uses vary according to the structure and behaviour of particular models along with initial conditions. A wide literature is available on different perceptions of fractional derivatives. But the most celebrated fractional calculi are the Caputo fractional derivative and the RL derivative. The Caputo fractional derivative handles initial value problems efficiently in comparison to the RL derivative. The newly introduced Caputo–Fabrizio (CF) fractional derivative operator propounded by Caputo and Fabrizio in ref. [25] is actually a convolution of an exponential function and the first-order derivative. In this definition, the derivative of a constant is equal to zero like the usual Liouville–Caputo definition but it also provides the non-singular kernel which was not a characteristic of the Liouville–Caputo fractional derivative. The main purpose of the CF definition was to introduce a new fractional derivative with an exponential kernel to describe even better the dynamics of systems with memory effect.

Recently, some authors presented a new analysis on fractional modelling of real-world problems and application of fractional order Lagrangian approach towards study of problems arising in physical sciences and engineering. Some recent works related to these fields are necessary to be cited here. Jajarmi et al. [26] suggested a general fractional formulation for immunogenic tumour dynamics. Baleanu et al. [27] presented a new study on the general fractional model of COVID-19 with isolation and quarantine effects. Erturk et al. [28] utilized a new fractional-order Lagrangian to describe the dynamics of a beam on nanowire. Jajarmi et al. [29] implemented a new fractional Lagrangian approach to study the case of capacitor microphone. Dubey et al. [30] solved the fractional model of Phytoplankton–Toxic Phytoplankton–Zooplankton system with convergence analysis. Moreover, a fractional model of atmospheric dynamics of carbon dioxide gas [31] and a fractional-order hepatitis E virus model [32] were also recently investigated with efficient computational methods.

Multiobjective variational problems proficiently handle the problems of science, engineering, logistics, and economics where optimal decisions have to be decided between two or more clashing objectives. To derive the optimality conditions it is necessary to study the behaviour of functions and their derivatives at that point. In the theory of mathematical optimization, the duality principle indicates two perspectives of optimization problems: the primal problem and the dual problem. If the primal is a minimization problem, then the dual is a maximization problem, and if the primal is a maximization problem, then the dual is a minimization problem. The concept of duality considers a problem with less number of variables and constraints and so it is much advantageous regarding computational procedure. Duality results play a major role in construction of numerical algorithms for solving some specific types of optimization problems. The duality theory is applied mainly in economics, management, physics, etc. On the other hand, calculus of variations significantly deals with the solution of several problems arising in theory of variations, optimization of orbits, dynamics of rigid bodies, etc. It is closely related to optimization of functional and is expressed in terms of definite integrals pertaining to functions and their derivatives. In the past few years, a number of contributions have been made towards the duality results for multiobjective variational problems. For the first time, Hanson [33] established and developed the linkage between classical calculus of variation and mathematical programming. After that, Mond and Hanson [34] derived optimality and duality results for scalar valued variational problems in view of convexity assumptions. Chandra et al. [35] studied optimality and duality for a class of non-differentiable variational problems. In this sequence, Bector and Husain [36] investigated duality for multiobjective variational problems. Nahak and Nanda [37] and Chen [38] constructed duality results for multiobjective variational problems with invexity. Some years later, Bhatia and Mehra [39] extended further the results of Mond et al. [40] and explored the optimality conditions and duality results for multiobjective variational problems with generalized B-invexity.

The concept of invexity is of great significance in variational problems and mathematical programming. Hanson [41] introduced the notion of invexity to mathematical programming. Mishra and Mukherjee [42] presented duality results for multiobjective FVPs. Furthermore, Mond and Husain [43] also investigated sufficient optimality criteria and duality for variational problems with generalized invexity. It is clearly observed that the duality results derived for variational problems presented in refs. [40,42,43] that hold for convex functions are also well-fitted for the wide range of invex functions. Weir and Mond [44] considered the concept of weak minima to derive the duality results for multiobjective programming problems. Different scalar duality results have also been extended for multiobjective programming problems by Weir and Mond [44]. Mukherjee and Mishra [45] have considered the concept of weak minima in the continuous case and have delivered a complete generalization of the results of Weir and Mond [44] to multiobjective variational problems. Moreover, they also relaxed the generalized convexity conditions to generalized invexity conditions.

Recently, Kumar [46] extended the invexity for continuous functions to invexity of order m. They further generalized the invexity of order m to ρ-pseudoinvexity type-I of order m, ρ-pseudoinvexity type-II of order m, as well as ρ-quasi-invexity type-I and type-II of order m. In 2016, Kumar et al. [47] also analysed the multiobjective FVP under F-Kuhn–Tucker (KT) pseudoinvexity conditions. Hachimi and Aghezzaf [48] established the mixed duality results and the sufficient optimality conditions concerning multiobjective variational problems under generalized (F,α,ρ,d)-type I functions which assimilate the several concepts of generalized type-I functions successfully. Later on, Mishra et al. [49] extended the generalized type-I invexity and duality for non-differentiable multiobjective variational problems. In 2014, Wolfe-type and Mond–Weir-type duality results were formulated for multiobjective variational control models under (ϕ,ρ)-invexity conditions by Antczak [50]. More recently, Upadhyay et al. [51] presented optimality conditions and duality for multiobjective semi-infinite programming problems on Hadamard manifolds utilizing generalized geodesic convexity. Moreover, Upadhyay et al. [52] also investigated Minty’s variational principle for non-smooth multiobjective optimization problems on Hadamard manifolds. Guo et al. [53] showed applications of symmetric gH-derivative to dual interval-valued optimization problems in a very efficient way. Furthermore, optimality conditions and duality for a class of generalized convex interval-valued optimization problems are recently investigated in works of Guo et al. [54].

The main purpose of this study is to derive the weak and strong duality results for multiobjective variational problems pertaining to a CF fractional derivative operator with exponential kernel. The CF fractional derivative possesses the non-singular kernel and so is better than the Caputo and RL fractional derivative operators. The proposed work presents the derivation of duality as well as strict converse duality theorems for variational problems with CF fractional derivative by employing some propositions and theorems of fractional calculus. In this article, we propounded first the optimality conditions for the variational problem. Furthermore, we present Theorem 1 which proves that a minimizer of the variational problem is a solution of the fractional Euler–Lagrange equation containing the CF fractional derivative. Now, we derived the formula for integration by parts for the CF fractional derivative in Proposition 1. The extended invexity definitions in view of CF fractional derivative operator along with Proposition 1 and Theorem 1 have been the key motivation behind the study of the variational problems with fractional calculus approach. Theorem 2 proves the fact that if a function is convex, the solution of the fractional Euler–Lagrange equation containing the CF fractional derivative will be a minimizer of the variational problem. Theorems 3 and 4 present the results for primal variational problems having a weak minimum. Furthermore, Theorems 5–10 are concerned with weak and strong duality results depending on the CF fractional derivative. Finally, Theorems 11 and 12 provide the strict converse duality results in view of the CF fractional derivative.

In the present work, the concept of weak minima has been considered and the generalizations of weak, strong, and strict converse duality results of Mukherjee and Mishra [45] have been extended to multiobjective variational problems pertaining to the CF fractional derivative operator. The remaining part of the article is organized as follows: In Section 2, we present the elemental definitions, formulae, and theorems regarding invexity and fractional derivative operators. Section 3 derives weak and strong duality results. Section 4 presents a strict converse duality result. Finally, Section 5 records the epilogue for the proposed work.

2 Basic definitions, theorems, and symbols

We follow these definitions and symbols in the present article.

Definition 1

[55]: The left and the right RL fractional derivatives of order α are defined by

D ξ α a y ( ξ ) = 1 Γ ( 1 α ) d d ξ a ξ ( ξ τ ) α y ( τ ) d τ ,

D b α ξ y ( ξ ) = 1 Γ ( 1 α ) d d ξ ξ b ( τ ξ ) α y ( τ ) d τ , α ( 0 , 1 ) .

Definition 2

[56]: The Caputo fractional derivative of y ( ξ ) : [ a , b ] of order α ∈ (0, 1) is stated as:

D a + α C y ( ξ ) = 1 Γ ( 1 α ) d d ξ a ξ 1 ( ξ τ ) α [ y ( τ ) y ( a ) ] d τ .

If yC 1, then

D a + α C y ( ξ ) = 1 Γ ( 1 α ) a ξ 1 ( ξ τ ) α y ( τ ) d τ .

As α → 1, D a + α C y ( ξ ) approaches to y′(ξ).

Definition 3

[25]: The new CF fractional derivative operator is described as follows:

D a + α CF y ( ξ ) = K ( α ) ( 1 α ) a ξ exp α ( ξ τ ) 1 α y ( τ ) d τ , α ( 0 , 1 ) ,

where K(α) signifies the normalization function with the property K(0) = K(1) = 1. Clearly, D a + α CF y ( ξ ) = 0 if y(ξ) is a constant function, i.e. the CF derivative of a constant function vanishes to zero same as the Caputo derivative but kernel of CF derivative does not have singularity for ξ = τ like the Caputo fractional derivative. It is remarkable that the CF fractional derivative has an exponential kernel.

Remark 1

Here we consider the value of K(α) as ( 1 α ) + α Γ ( α ) .

Remark 2

As α → 1, D a + α CF y ( ξ ) approaches to y′(ξ) and as α → 0, D a + α CF y ( ξ ) approaches to y(ξ) − y(a).

Definition 4

Abdeljawad and Baleanu [57] have defined the right CF fractional derivative as

D b α CF y ( ξ ) = K ( α ) ( 1 α ) ξ b exp α ( τ ξ ) 1 α y ( τ ) d τ , α ( 0 , 1 ) .

Definition 5

The first-order Sobolev space defined in the interval (a, b) is stated as H 1(a, b) = {xL 2(a, b) |x′ ∈ L 2(a, b)}, where x′ denotes the weak derivative of x.

Definition 6

[25]: Let yH 1(a, b), b > a, 0 < α < 1, then the CF fractional derivative is stated as in Definition (3), where K(α) specifies the normalization function with characteristic K(0) = K(1) = 1. If the function yH 1(a, b), then the derivative is formulated as follows:

D a + α CF y ( ξ ) = α K ( α ) ( 1 α ) a ξ exp α ( ξ τ ) 1 α [ y ( ξ ) y ( τ ) ] d τ .

Here, the CF fractional derivative has an exponential kernel.

Definition 7

[57]: Let x be a function in such a way that xH 1(a, b) a < b. The left Riemann fractional derivative of order α in the CF sense is given by

D a + α CFR y ( ξ ) = K ( α ) 1 α d d ξ a ξ exp α 1 α ( ξ τ ) y ( τ ) d τ ,

where a ξ , α ( 0 < α < 1 ) is a real number and K ( α ) is a normalization function depending on α with K ( 0 ) = K ( 1 ) = 1 .

Similarly, the right Riemann fractional derivative of order α in the CF sense can be written as follows:

D b α CFR y ( ξ ) = K ( α ) 1 α d d ξ ξ b exp α 1 α ( τ ξ ) y ( τ ) d τ ,

where ξ b .

Remark 3

When α 0 , lim α 0 D a + α CFR y ( ξ ) = d d ξ a ξ y ( τ ) d τ = y ( ξ ) .

Proposition 1

Let α ( 0 , 1 ) and y , z : [ a , b ] be two continuous functions of class C 1 [ a , b ] . Then the following formula for integration by parts holds:

a b y ( ξ ) D a + α CF z ( ξ ) d ξ = [ z ( ξ ) I b 1 α y ( ξ ) ] ξ = a ξ = b + a b z ( ξ ) D b α CFR y ( ξ ) d ξ .

Proof

We define the left and right auxiliary fractional integrals as

(1) I a + 1 α z ( ξ ) = K ( α ) ( 1 α ) a ξ exp α 1 α ( ξ τ ) z ( τ ) d τ ,

(2) I b 1 α z ( ξ ) = K ( α ) ( 1 α ) ξ b exp α 1 α ( τ ξ ) z ( τ ) d τ .

Now in view of Definition (3) and Eq. (1), it is concluded that

(3) D a + α CF z ( ξ ) = I a + 1 α d d ξ z ( ξ ) .

In the next step, we evaluate the integral a b y ( ξ ) D a + α CF z ( ξ ) d ξ as follows.

Using Eq. (3) along with further utilization of Theorem 1 of ref. [57] and integration by parts for classical derivatives, we obtain

a b y ( ξ ) D a + α CF z ( ξ ) d ξ = a b y ( ξ ) I a + 1 α d d ξ z ( ξ ) d ξ = a b d d ξ z ( ξ ) I b 1 α y ( ξ ) d ξ = a b I b 1 α y ( ξ ) d d ξ z ( ξ ) d ξ = [ z ( ξ ) I b 1 α y ( ξ ) ] ξ = a ξ = b a b z ( ξ ) × K ( α ) ( 1 α ) d d ξ ξ b exp α 1 α ( ξ τ ) y ( τ ) d τ d ξ .

Now in view of Definition 7, we obtain

a b y ( ξ ) D a + α CF z ( ξ ) d ξ = [ z ( ξ ) I b 1 α y ( ξ ) ] ξ = a ξ = b + a b z ( ξ ) D b α CFR y ( ξ ) d ξ .

Definition 8

(Optimality conditions for variational problems):

The following variational problem with the CF fractional derivative is considered here for given y C 1 ( a , b ) ,

(4) min V ( y ) = a b Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) d ξ ,

with y ( a ) = y a and y ( b ) = y b , where y ( a ) , y ( b ) . The assumptions are as follows:

  1. Q : [ a , b ] × 2 is continuously differentiable w.r.t. the second and third arguments.

  2. Given any x , the map ξ D b α CFR ( 3 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) ) = 0 is continuous.

Here, we denote i g ( y 1 , y 2 , y n ) = g y i ( y 1 , y 2 , y n ) for a function g : T n .

Theorem 1

Let y be a minimizer of the variational V(y) defined on E = { y C 1 ( a , b ) : y ( a ) = y a , y ( b ) = y b } , where y a , y b are fixed. Then y is a solution of the following fractional Euler–Lagrange equation:

(5) 2 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) + D b α CFR ( 3 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) ) = 0 , ξ [ a , b ] .

Proof

Let y be a solution for the functional V ( y ) . Assume y + δ ω be a variation of y with δ 1 , and ω : [ a , b ] R be a function of class C 1 [ a , b ] in such a way that the conditions ω ( a ) = ω ( b ) = 0 hold. Let ϑ ( δ ) = V ( y + δ ω ) . Since y satisfies Eq. (4) as a solution, the first variation of V must vanish, and hence ϑ ( 0 ) = 0 . Now, computing ϑ ( δ ) δ = 0 , equating to zero, and further employing Proposition 1, we have

a b 2 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) ω ( ξ ) d ξ + a b 3 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) D a + α CF ω ( ξ ) d ξ = a b 2 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) ω ( ξ ) d ξ + a b ω ( ξ ) D b α CFR ( 3 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) ) d ξ + [ ω ( ξ ) I b 1 α ( 3 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) ) ] ξ = a ξ = b = a b [ 2 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) + D b α CFR ( 3 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) ) ] ω ( ξ ) d ξ , ( ω ( a ) = ω ( b ) = 0 ) = 0 .

Now utilizing the boundary conditions ω ( a ) = ω ( b ) = 0 along with the assumption of arbitrariness of ω , we obtain the desired equation as:

2 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) + D b α CFR ( 3 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) ) = 0 ξ [ a , b ] .

Remark 4

Eq. (5) is called the Euler–Lagrange equation associated with the variational V ( y ) and the solutions for this equation are termed as extremals.

Remark 5

It is notable that Eq. (5) provides necessary criterion only. Now to obtain sufficient criterion, the concept of convex function is necessary to recall.

Definition 9

A function Q ( ξ , , ) is said to be convex in T R 3 if Q possesses continuous derivatives in respect of the second and third arguments and also satisfies the following inequality:

Q ( ξ , + 1 , + 1 ) Q ( ξ , , ) 2 Q ( ξ , , ) 1 + 3 Q ( ξ , , ) 1 , ( ξ , , ) , ( ξ , + 1 , + 1 ) T .

Theorem 2

If the function Q as described in Eq. ( 4 ) is convex in [ a , b ] × 2 , then each solution of the fractional Euler–Lagrange Eq. (5) minimizes V in E.

Proof

Let y be a solution for the fractional Euler–Lagrange Eq. (5). Assume y + δ ω to be a variation of y with δ 1 , and ω : [ a , b ] is a function that belongs to C 1 [ a , b ] such that the boundary conditions ω ( a ) = ω ( b ) = 0 hold. Now, we compute V ( y + δ ω ) V ( y ) in view of Definition 9 and Proposition 1 as follows:

(6) V ( y + δ ω ) V ( y ) = a b [ Q ( ξ , y ( ξ ) + δ ω ( ξ ) , D a + α CF y ( ξ ) + δ D a + α CF ω ( ξ ) ) Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) ] d ξ a b [ 2 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) δ ω ( ξ ) + 3 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) δ D a + α CF ω ( ξ ) ] d ξ = a b 2 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) δ ω ( ξ ) d ξ + a b 3 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) δ D a + α CF ω ( ξ ) d ξ = a b 2 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) δ ω ( ξ ) d ξ + δ [ ω ( ξ ) I b 1 α 3 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) ] ξ = a ξ = b + a b δ ω ( ξ ) D b α CFR 3 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) d ξ = a b 2 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) δ ω ( ξ ) d ξ + a b δ ω ( ξ ) D b α CFR 3 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) d ξ ( ω ( a ) = ω ( b ) = 0 ) , = a b [ 2 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) + D b α CFR 3 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) ] δ ω ( ξ ) d ξ .

Since y is a solution of the fractional Euler–Lagrange Eq. (5), we have

(7) 2 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) + D b α CFR 3 Q ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) = 0 .

Hence in view of Eq. (7), Eq. (6) provides the inequality as follows:

(8) V ( y + δ ω ) V ( y ) 0 .

Consequently, V ( y + δ ω ) V ( y ) , which implies that y is a local minimizer of V. □

Definition 10

Invexity definitions

Let Ω = [ a , b ] be a real interval. Let g : Ω × n × n be a continuously differentiable function. Consider g ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) , where y : Ω n is a function of class C 1 [ a , b ] and D a + α CF y represents the CF derivative of order 0 < α < 1 of a function y. We denote the partial derivatives of g by

g ξ = g ξ , g y = g y 1 , g y 2 , g y 3 , , g y n ,

g D a + α CF y = g ( D a + α CF y 1 ) , g ( D a + α CF y 2 ) , g ( D a + α CF y 3 ) , , g ( D a + α CF y n ) .

Let Y be the space of piecewise smooth functions y : Ω n along with the norm y = y + D y , where the differential operator D is described as follows:

v = D y y ( ξ ) = y 0 + a ξ v ( s ) d s ,

where y 0 signifies the boundary value.

Let G : Y defined by G ( y ) = a b g ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) d ξ be Fréchet differentiable. For notational convenience g ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) will be written as g ( ξ , y , D a + α CF y ) . Here, it is assumed that g ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) is convex in 3 if 2 g and 3 g exist and are continuous, and the condition

g ( ξ , y + y 1 , D a + α CF y + D a + α CF y 2 ) g ( ξ , y , D a + α CF y ) g y ( ξ , y , D a + α CF y ) y 1 + g D a + α CF y ( ξ , y , D a + α CF y ) D a + α CF y 2 ,

holds for every ( ξ , y , D a + α CF y ) , ( ξ , y + y 1 , D a + α CF y + D a + α CF y 2 ) 3 . Here, 2 g and 3 g denote the partial derivatives of g with respect to y and D a + α CF y , respectively.

Let y ¯ be a solution of the variational functional G ( y ¯ ) = a b g ( ξ , y ¯ ( ξ ) , D a + α CF y ¯ ( ξ ) ) d ξ y ¯ Y and ξ [ a , b ] . Let y = y ¯ + η ( ξ , y , y ¯ ) and η C 1 [ a , b ] with η ( ξ , y , y ¯ ) ξ = a = η ( ξ , y , y ¯ ) ξ = b . Clearly also η ( ξ , y , y ) = 0 .

Now utilizing the linearity property of the CF derivative operator and further the convexity assumption of g ( ξ , y ¯ ( ξ ) , D a + α CF y ¯ ( ξ ) ) , we have

G ( y ) G ( y ¯ ) = G ( y ¯ + η ) G ( y ¯ ) = a b [ g ( ξ , y ¯ ( ξ ) + η ( ξ , y , y ¯ ) , D a + α CF y ¯ ( ξ ) + D a + α CF η ( ξ , y , y ¯ ) ) g ( ξ , y ¯ ( ξ ) , D a + α CF y ¯ ( ξ ) ) ] d ξ a b [ g y ¯ ( ξ , y ¯ ( ξ ) , D a + α CF y ¯ ( ξ ) ) η ( ξ , y , y ¯ ) + g D a + α CF y ¯ ( ξ , y ¯ ( ξ ) , D a + α CF y ¯ ( ξ ) ) D a + α CF η ( ξ , y , y ¯ ) ] d ξ = a b [ η ( ξ , y , y ¯ ) g y ¯ ( ξ , y ¯ ( ξ ) , D a + α CF y ¯ ( ξ ) ) + ( D a + α CF η ( ξ , y , y ¯ ) ) g D a + α CF y ¯ ( ξ , y ¯ ( ξ ) , D a + α CF y ¯ ( ξ ) ) ] d ξ .

Clearly,

G ( y ) G ( y ¯ ) a b { η ( ξ , y , y ¯ ) g y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) + ( D a + α CF η ( ξ , y , y ¯ ) ) g D a + α CF y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) } d ξ , y , y ¯ Y .

Clearly as α 1 , the above obtained inequality reduces to

a b g ξ , y ( ξ ) , d d ξ y ( ξ ) d ξ a b g ξ , y ¯ ( ξ ) , d d ξ y ¯ ( ξ ) d ξ a b η ( ξ , y , y ¯ ) g y ¯ ξ , y ¯ , d d ξ y ¯ + d d ξ η ( ξ , y , y ¯ ) g d d ξ y ¯ ξ , y ¯ , d d ξ y ¯ d ξ ,

which is the definition of invexity in the continuous case extended by Mond et al. [40]. It is notable that if the function g is independent of ξ, the above given definition of invexity transforms to the inequality g ( y ) g ( y ¯ ) η ( y , y ¯ ) g y ¯ ( y ¯ ) , which is the fundamental interpretation of invexity prescribed by Hanson [41].

Example 1

The proposed inequality which is derived earlier is given as follows:

(9) G ( y ) G ( y ¯ ) a b { η ( ξ , y , y ¯ ) g y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) + ( D a + α CF η ( ξ , y , y ¯ ) ) g D a + α CF y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) } d ξ .

Let y ¯ = ξ , y = 2 ξ , η ( ξ , y , y ¯ ) = y y ¯ , g ( ξ , y ¯ , D a + α CF y ¯ ) = ξ + y ¯ + D a + α CF y ¯ , and g ( ξ , y , D a + α CF y ) = ξ + y + D a + α CF y .

Then

(10) η ( ξ , y , y ¯ ) = y y ¯ = ξ , D a + α CF η ( ξ , y , y ¯ ) = D a + α CF ξ ,

(11) g ( ξ , y ¯ , D a + α CF y ¯ ) = ξ + y ¯ + D a + α CF y ¯ = 2 ξ + D a + α CF ξ .

Now utilizing the formula of CF derivative, we obtain

(12) D a + α CF ξ = K ( α ) ( 1 α ) a ξ exp α 1 α ( ξ τ ) d τ = K ( α ) α 1 exp α 1 α ( ξ a ) ,

where K ( α ) = ( 1 α ) + α Γ ( α ) signifies the normalization function with the property K ( 0 ) = K ( 1 ) = 1 .

Thus,

(13) D a + α CF η ( ξ , y , y ¯ ) = K ( α ) α 1 exp α 1 α ( ξ a ) ,

and

(14) g ( ξ , y ¯ , D a + α CF y ¯ ) = 2 ξ + K ( α ) α 1 exp α 1 α ( ξ a ) .

Now

(15) g y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) = 2 + K ( α ) ( 1 α ) exp α 1 α ( ξ a ) , g D a + α CF y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) = 1 .

Now, we evaluate the term G ( y ) G ( y ¯ ) of the aforementioned proposed inequality (9) in view of Eq. (14) as follows:

(16) G ( y ) G ( y ¯ ) = a b g ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) d ξ a b g ( ξ , y ¯ ( ξ ) , D a + α CF y ¯ ( ξ ) ) d ξ = 1 2 ( b 2 a 2 ) + K ( α ) α ( b a ) + K ( α ) ( 1 α ) α 2 exp α ( b a ) 1 α K ( α ) ( 1 α ) α 2 .

In this sequence, we also evaluate the integral term of the proposed inequality (9) in view of Eqs. (10) and (15) as follows:

(17) a b { η ( ξ , y , y ¯ ) g y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) + ( D a + α CF η ( ξ , y , y ¯ ) ) g D a + α CF y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) } d ξ = ( b 2 a 2 ) b K ( α ) α exp α ( b a ) 1 α + 2 K ( α ) ( 1 α ) α 2 exp α ( b a ) 1 α 2 K ( α ) ( 1 α ) α 2 + b K ( α ) α .

Case I: For a = 0 , b = 1 , α = 0.5 , K ( α ) = ( 1 α ) + α Γ ( α ) , the proposed inequality is satisfied.

Case II: For a = 0 , b = 1 , α = 0.2 , K ( α ) = ( 1 α ) + α Γ ( α ) , the proposed inequality is also satisfied.

Consequently, it is concluded that the proposed inequality (9) with CF fractional derivative holds well for the function g ( ξ , y ¯ , D a + α CF y ¯ ) = 2 ξ + 1 α α + 1 Γ ( α ) 1 exp α 1 α ξ , where 0 < α < 1 .

Now, we extend the definitions of invex, pseudoinvex (PIX), strictly pseudoinvex (SPIX), and quasi-invex (QIX) as described in ref. [58] with the CF fractional derivative of order 0 < α < 1 in the following way:

Definition 11

Invex

The functional G is stated as invex with respect to η if there exists a differentiable vector function η ( ξ , y , y ¯ ) C 1 [ a , b ] with η ( ξ , y , y ) = 0 such that y , y ¯ Y ,

G ( y ) G ( y ¯ ) a b { η ( ξ , y , y ¯ ) g y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) + ( D a + α CF η ( ξ , y , y ¯ ) ) g D a + α CF y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) } d ξ .

Definition 12

PIX

The functional G is stated as PIX w.r.t. η if a differentiable vector function η ( ξ , y , y ¯ ) C 1 [ a , b ] with η ( ξ , y , y ) = 0 such that y , y ¯ Y ,

a b { η ( ξ , y , y ¯ ) g y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) + ( D a + α CF η ( ξ , y , y ¯ ) ) g D a + α CF y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) } d ξ 0 G ( y ) G ( y ¯ ) ,

or equivalently,

G ( y ) < G ( y ¯ ) a b { η ( ξ , y , y ¯ ) g y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) + ( D a + α CF η ( ξ , y , y ¯ ) ) g D a + α CF y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) } d ξ < 0 .

Definition 13

SPIX

The functional G is stated as SPIX w.r.t. η if a differentiable vector function η ( ξ , y , y ¯ ) C 1 [ a , b ] with η ( ξ , y , y ) = 0 such that y , y ¯ Y ,

a b { η ( ξ , y , y ¯ ) g y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) + ( D a + α CF η ( ξ , y , y ¯ ) ) g D a + α CF y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) } d ξ 0 G ( y ) > G ( y ¯ ) ,

or equivalently,

G ( y ) G ( y ¯ ) a b { η ( ξ , y , y ¯ ) g y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) + ( D a + α CF η ( ξ , y , y ¯ ) ) g D a + α CF y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) } d ξ < 0 .

Definition 14

QIX

The functional G is stated as QIX w.r.t. η if a differentiable vector function η ( ξ , y , y ¯ ) C 1 [ a , b ] with η ( ξ , y , y ) = 0 such that y , y ¯ Y ,

a b { η ( ξ , y , y ¯ ) g y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) + ( D a + α CF η ( ξ , y , y ¯ ) ) g D a + α CF y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) } d ξ > 0 G ( y ) > G ( y ¯ ) ,

or equivalently,

G ( y ) G ( y ¯ ) a b { η ( ξ , y , y ¯ ) g y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) + ( D a + α CF η ( ξ , y , y ¯ ) ) g D a + α CF y ¯ ( ξ , y ¯ , D a + α CF y ¯ ) } d ξ 0 .

In the aforementioned definitions, D a + α CF η ( ξ , y , y ¯ ) is the vector whose ith component is ( d α / d ξ α ) η i ( ξ , y , y ¯ ) . Let g ( ξ , y , D a + α CF y ( ξ ) ) be a real scalar function and h ( ξ , y , D a + α CF y ( ξ ) ) be an m-dimensional function with continuous derivatives up to the second order with respect to each of its arguments. Here, y is an n-dimensional function of ξ and D a + α CF y ( ξ ) denotes the CF fractional derivative of order α with respect to ξ where 0 < α < 1 .

Now we deal with the multiobjective variational primal problem, as discussed in the work of Mukherjee and Mishra [45], with the CF fractional derivative operator in the following way:

(P) Minimize

a b g ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) d ξ ,

subject to y ( a ) = y 0 , y ( b ) = y 1 ,

h ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) 0 ,

where g : [ a , b ] × n × n p and h : [ a , b ] × n × n m , 0 < α < 1 .

For the primal problem (P), a point y 0 is referred to as a weak minimum if there exists no other feasible point y for which the following inequality will hold

(18) a b g ( ξ , y 0 ( ξ ) , D a + α CF y 0 ( ξ ) ) d ξ > a b g ( ξ , y ( ξ ) , D a + α CF y ( ξ ) ) d ξ .

Now, we frame the continuous versions of Theorems 2.1 and 2.2, as described in the work of Weir and Mond [44], involving fractional derivative operators with exponential kernel in the following way:

Theorem 3

Let y = y 0 be a weak minimum for the primal problem (P). Then λ p , z m such that

(19) λ T g y ( ξ , y 0 ( ξ ) , D a + α CF y 0 ( ξ ) ) + z ( ξ ) T h y ( ξ , y 0 ( ξ ) , D a + α CF y 0 ( ξ ) ) = D b α CFR [ λ T g D a + α CF y ( ξ ) ( ξ , y 0 ( ξ ) , D a + α CF y 0 ( ξ ) ) + z ( ξ ) T h D a + α CF y ( ξ ) ( ξ , y 0 ( ξ ) , D a + α CF y 0 ( ξ ) ) ] ,

(20) z ( ξ ) T h ( ξ , y 0 ( ξ ) , D a + α CF y 0 ( ξ ) ) = 0 ,

(21) ( λ , y ) 0 .

Proof

The proof is easily established through Theorem 1.□

Theorem 4

Let the primal problem (P) have a weak minimum at a point y 0 , which satisfies the KT constraint qualification. Then λ p , z m such that

(22) λ T g y ( ξ , y 0 ( ξ ) , D a + α CF y 0 ( ξ ) ) + z ( ξ ) T h y ( ξ , y 0 ( ξ ) , D a + α CF y 0 ( ξ ) ) = D b α CFR [ λ T g D a + α CF y ( ξ ) ( ξ , y 0 ( ξ ) , D a + α CF y 0 ( ξ ) ) + z ( ξ ) T h D a + α CF y ( ξ ) ( ξ , y 0 ( ξ ) , D a + α CF y 0 ( ξ ) ) ] ,

(23) z ( ξ ) T h ( ξ , y 0 ( ξ ) , D a + α CF y 0 ( ξ ) ) = 0 ,

(24) z ( ξ ) 0 ,

(25) λ ( ξ ) 0 , λ T e = 1 ,

where e = ( 1 , .... . , 1 ) p .

Proof

The proof is easily established through Theorem 1.□

3 Duality

In relation to the primal problem (P), the dual problem (D) as discussed in ref. [45] is considered with fractional derivative operators pertaining to exponential kernel in the following way:

(D) Maximize

a b { g ( ξ , v ( ξ ) , D a + α CF v ( ξ ) ) + z ( ξ ) T h ( ξ , v ( ξ ) , D a + α CF v ( ξ ) ) e } d ξ ,

subject to

(26) y ( a ) = y 0 , y ( b ) = y 1 ,

(27) g v ( ξ , v ( ξ ) , D a + α CF v ( ξ ) ) + z ( ξ ) T h v ( ξ , v ( ξ ) , D a + α CF v ( ξ ) ) e = D b α CFR [ g D a + α CF v ( ξ ) ( ξ , v ( ξ ) , D a + α CF v ( ξ ) ) + z ( ξ ) T h D a + α CF v ( ξ ) ( ξ , v ( ξ ) , D a + α CF v ( ξ ) ) e ] ,

(28) z 0 .

λ Λ ,

where

(29) Λ = { λ p : λ 0 , λ T e = 1 } .

In upcoming steps, we discuss the duality theorems, as discussed in the work of Mukherjee and Mishra [45], with the CF fractional derivative operator of order 0 < α < 1 as follows:

Theorem 5

(Weak