**Authors:**
K. V. Geetha,
N. R. Mangalambal

**Abstract:**

Aim. We have introduced the notion of order to multinormed spaces and countable union spaces and their duals. The topology of bounded convergence is assigned to the dual spaces. The aim of this paper is to develop the theory of ordered topological linear
spaces La,b, L(w, z), the dual spaces of ordered multinormed spaces
La,b, ordered countable union spaces L(w, z), with the topology of bounded convergence assigned to the dual spaces. We apply Laplace transformation to the ordered linear space of Laplace transformable
generalized functions. We ultimately aim at finding solutions to nonhomogeneous
nth order linear differential equations with constant
coefficients in terms of generalized functions and comparing different
solutions evolved out of different initial conditions.
Method. The above aim is achieved by
• Defining the spaces La,b, L(w, z).
• Assigning an order relation on these spaces by identifying a
positive cone on them and studying the properties of the cone.
• Defining an order relation on the dual spaces La,b, L(w, z) of La,b, L(w, z) and assigning a topology to these dual spaces which makes the order dual and the topological dual the same. • Defining the adjoint of a continuous map on these spaces
and studying its behaviour when the topology of bounded
convergence is assigned to the dual spaces.
• Applying the two-sided Laplace Transformation on the ordered
linear space of generalized functions W and studying some
properties of the transformation which are used in solving
differential equations.
Result. The above techniques are applied to solve non-homogeneous
n-th order linear differential equations with constant coefficients in
terms of generalized functions and to compare different solutions of the differential equation.

**Keywords:**
Laplace transformable generalized function,
positive cone,
topology of bounded convergence

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1334880

Procedia
APA
BibTeX
Chicago
EndNote
Harvard
JSON
MLA
RIS
XML
ISO 690
PDF
Downloads 1233
**References:**

[1] Geetha K. V. and Mangalambal N. R. : On Dual Spaces of Ordered
Multinormed Spaces and Countable Union Spaces, Bull. KMA, No.2, Vol.
4, Dec 2007, pp. 63-74.

[2] Perissini A. L.: Ordered Topological Vector Spaces, Harper and Row, New York, 1967.

[3] Zemanian A. H., Generalized Integral Transformations, Interscience, New York, 1968.