Calderon's remarkable

duality theorem [3, Section 12.

lambda]]) for (MWD) follows from weak

duality theorem.

Recall the

duality theorem [4] which states that the Laplace transform is a topological isomorphism from [F'.

The graduate textbook illustrates how Cohen-Macaulay rings arise naturally, develops the Hartshorne-Lichtenbaum vanishing theorem, applies two classes of rings to polyhedral geometry, explains Grothendieck's

duality theorem, and defines D-modules over rings of differential operators.

Chapters discuss duality, linear mappings, matrices, determinant and trace, spectral theory, Euclidean structure, calculus of vector- and matrix-valued functions, matrix inequalities, kinematics and dynamics, convexity, the

duality theorem, normed liner spaces, linear mappings between normed linear spaces, positive matrices, and solutions of systems of linear equations.

Teo: A converse

duality theorem on higher-order dual models in nondifferentiable mathematical programming, Optim.

Wolfe: A

duality theorem for nonlinear programming Quaterly Appl.

Beginning chapters are devoted to the abstract structure of finite dimensional vector spaces, and subsequent chapters address convexity and the

duality theorem as well as describe the basics of normed linear spaces and linear maps between normed spaces.

Lee: On

duality theorems for nonsmooth Lipschitz optimization problems, J.

Kuk [13] then proved the weir type

duality theorems and schaible type

duality theorems under V-[rho]--invexity assumptions.

It was the aim of the present paper to utilize these conditions, in order to establish new duality conditions of Mond-Weir-Zalmai type for the fractional problem (VFP) through weak, direct and converse

duality theorems.

In this paper, we define a duality of Mond-Weir-Zalmai type for problem (MFP) through weak, direct and converse

duality theorems.