Calderon's remarkable duality theorem
[3, Section 12.
lambda]]) for (MWD) follows from weak duality theorem
Recall the duality theorem
 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  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