Laplace Transformation on Ordered Linear Space of Generalized Functions

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. I. THE SPACES La,b , L(w, z) AND THEIR DUALS We have associated the notion of ‘order’ to multinormed spaces, countable union spaces (see [3], for details of multinormed spaces, countable union spaces) and their duals, by identifying a positive cone on these spaces. Also, the topology of bounded convergence is assigned to the dual spaces so that the order dual and the topological dual become identical [1]. In this section we define the spaces La,b, L(w, z) and apply the above ideas to these spaces and to their duals. Manuscript received March 1, 2008. K. V. Geetha is with the Department of Mathematics, St. Joseph’s College, Irinjalakuda, Kerala, India. Pin.680 121 (Phone: Off. 04802825358, Res. 0480-2824613, cell. 9447994135, fax. 0480-2830954, email. [email protected]) N. R. Mangalambal is with the Department of Mathematics, St. Joseph’s College, Irinjalakuda, Kerala, India. Pin.680 121 (Phone: Off. 0480-2825358, Res. 0480-2709858, cell. 9495246832, fax. 0480-2830954, email. [email protected]) Let La,b denote the linear space of all complex valued smooth functions defined on R. Let La,b(t) = e, 0 ≤ t < ∞ = e, −∞ < t < 0. (Km) be a sequence of compact subsets of R such that K1 ⊆ K2 ⊆ . . . and such that each compact subset of R is contained in one Km. Define γKm,k(φ) = sup t∈Km |La,b(t)Dφ(t)|, k = 0, 1, 2 . . . {γKm,k}k=0 is a multinorm on La,b,Km where La,b,Km is the subspace of La,b whose elements have their support in Km. The above multinorm generates the topology τa,b,Km on La,b,Km . La,b is equipped with the inductive limit topology τa,b as Km varies over all compact sets K1,K2, . . . . La,b is complete for τa,b. For each fixed s, e−st ∈ La,b if and only if a ≤ Re s ≤ b. For each positive integer k, tke−st ∈ La,b if and only if a < Re s < b. We recall the notions of a positive cone, normal cone and strict b-cone that have been defined in [1]. Definition 1: Let V be a multinormed space whose field of scalars is R. A subset C or C(V ) is a positive cone in V if (i) C + C ⊆ C (ii) αC ⊆ C for all scalars α > 0 (iii) C ∩ (−C) = {0} When the field of scalars is C, C + iC is the positive cone in V which is also denoted as C. C defines an order relation on V , φ ≤ ψ if ψ − φ ∈ C. Definition 2: Let V (τ) be an ordered multinormed space with positive cone C. C is normal for the topology τ generated by the multinorm S if there is a neighbourhood basis of 0 for τ consisting of full sets. Definition 3: Let G be a saturated class of τ -bounded subsets of an ordered multinormed space V (τ) such that V = ∪{s : s ∈ G}. The positive cone C in V (τ) is a strict G-cone if the class G = {(S ∩ C) − (S ∩ C) : S ∈ G} is a fundamental system for G. A strict G-cone for the the class G of all τ -bounded sets in V (τ) is called a strict b-cone. Definition 4: The positive cone C of La,b when La,b is restricted to real-valued functions is the set of all non-negative functions in La,b. Then C + iC is the positive cone in La,b which is also denoted as C. Now we prove that the cone of La,b is not normal but is a strict b-cone. Theorem 1: The cone C of La,b is not normal. Proof: Let La,b be restricted to real-valued functions. Let m be a fixed positive integer and (φi) be a sequence of World Academy of Science, Engineering and Technology 39 2008