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Laplace Transformation on Ordered Linear Space of Generalized Functions K World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:2, No:3, 2008 Laplace Transformation on Ordered Linear Space of Generalized Functions K. V. Geetha and N. R. Mangalambal Abstract—Aim. We have introduced the notion of order to multi- Let La,b denote the linear space of all complex valued normed spaces and countable union spaces and their duals. The smooth functions defined on R. Let topology of bounded convergence is assigned to the dual spaces. The aim of this paper is to develop the theory of ordered topological linear L (t)=eat, 0 ≤ t<∞ a,b spaces La,b, L (w, z), the dual spaces of ordered multinormed spaces = ebt, −∞ <t<0. La,b, ordered countable union spaces L(w, z), with the topology of bounded convergence assigned to the dual spaces. We apply Laplace (Km) be a sequence of compact subsets of R such that K1 ⊆ transformation to the ordered linear space of Laplace transformable ⊆ R generalized functions. We ultimately aim at finding solutions to non- K2 ... and such that each compact subset of is contained homogeneous nth order linear differential equations with constant in one Km. Define coefficients in terms of generalized functions and comparing different ( ) = sup | ( ) k ( )| =0 1 2 solutions evolved out of different initial conditions. γKm,k φ La,b t D φ t ,k , , ... ∈ Method. The above aim is achieved by t Km { }∞ L L • Defining the spaces La,b, L(w, z). γKm,k k=0 is a multinorm on a,b,Km where a,b,Km is • Assigning an order relation on these spaces by identifying a the subspace of La,b whose elements have their support in positive cone on them and studying the properties of the cone. K . The above multinorm generates the topology τ on L L(w, z) m a,b,Km • Defining an order relation on the dual spaces a,b, L L L L(w, z) a,b,Km . a,b is equipped with the inductive limit topology of a,b, and assigning a topology to these dual spaces L which makes the order dual and the topological dual the same. τa,b as Km varies over all compact sets K1,K2,.... a,b is −st • Defining the adjoint of a continuous map on these spaces complete for τa,b. For each fixed s, e ∈La,b if and only k −st and studying its behaviour when the topology of bounded if a ≤ Re s ≤ b. For each positive integer k, t e ∈La,b if convergence is assigned to the dual spaces. and only if a<Re s<b. • Applying the two-sided Laplace Transformation on the ordered We recall the notions of a positive cone, normal cone and linear space of generalized functions W and studying some properties of the transformation which are used in solving strict b-cone that have been defined in [1]. differential equations. Definition 1: Let V be a multinormed space whose field of Result. The above techniques are applied to solve non-homogeneous scalars is R. A subset C or C(V ) is a positive cone in V if n-th order linear differential equations with constant coefficients in (i) C + C ⊆ C terms of generalized functions and to compare different solutions of (ii) αC ⊆ C for all scalars α>0 the differential equation. (iii) C ∩ (−C)={0} Keywords —Laplace transformable generalized function, positive When the field of scalars is C, C + iC is the positive cone in cone, topology of bounded convergence. 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 I. THE SPACES La,b, L(w, z) AND THEIR DUALS with positive cone C. C is normal for the topology τ generated We have associated the notion of ‘order’ to multinormed by the multinorm S if there is a neighbourhood basis of 0 for spaces, countable union spaces (see [3], for details of multi- τ consisting of full sets. normed spaces, countable union spaces) and their duals, by Definition 3: Let G be a saturated class of τ-bounded identifying a positive cone on these spaces. Also, the topology subsets of an ordered multinormed space V (τ) such that of bounded convergence is assigned to the dual spaces so that V = ∪{s : s ∈G}. The positive cone C in V (τ) is a strict the order dual and the topological dual become identical [1]. G-cone if the class G = {(S ∩ C) − (S ∩ C):S ∈G}is a International Science Index, Mathematical and Computational Sciences Vol:2, No:3, 2008 waset.org/Publication/13650 L L( ) In this section we define the spaces a,b, w, z and apply fundamental system for G. A strict G-cone for the the class G the above ideas to these spaces and to their duals. 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 Manuscript received March 1, 2008. restricted to real-valued functions is the set of all non-negative K. V. Geetha is with the Department of Mathematics, St. Joseph’s L + L College, Irinjalakuda, Kerala, India. Pin.680 121 (Phone: Off. 0480- functions in a,b. Then C iC is the positive cone in a,b 2825358, Res. 0480-2824613, cell. 9447994135, fax. 0480-2830954, email. which is also denoted as C. kvgeethanand@yahoo.com) Now we prove that the cone of L is not normal but is a N. R. Mangalambal is with the Department of Mathematics, St. Joseph’s a,b College, Irinjalakuda, Kerala, India. Pin.680 121 (Phone: Off. 0480-2825358, strict b-cone. Res. 0480-2709858, cell. 9495246832, fax. 0480-2830954, email. thot- Theorem 1: The cone C of La,b is not normal. tuvai@sancharnet.in) Proof: Let La,b be restricted to real-valued functions. Let m be a fixed positive integer and (φi) be a sequence of International Scholarly and Scientific Research & Innovation 2(3) 2008 200 scholar.waset.org/1307-6892/13650 World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:2, No:3, 2008 L ∩ = sup{ ( ): ∈ } functions in a,b,Km C such that λi φi t t Km Proof: Every positive linear functional is continuous for ( ) L L converges to 0 but φi does not converge to 0 for a,b,Km . the Schwartz topology τa,b on a,b. Define L+ = (L R) − (L R) a,b C a,b, C a,b, ψi = λi,t∈ Km+1 where C(La,b, R) is the linear subspace consisting of all non- =0,t∈ K +1. m negative order bounded linear functionals of L(La,b, R), the L Let ξ be the regularization of ψ defined by linear space of all order bounded, linear functionals on a,b. i i L+ ⊆L It follows that a,b a,b. L ( )= ( − ) ( ) On the other hand, the space a,b equipped with the ξi t θα t t ψi t dt . R (L L ) topology of bounded convergence β a,b, a,b and ordered ∈L ∀ C C L Then ξi a,b,Km+2 , i. by the dual cone of the cone in a,b is a reflexive space Also, 0 ≤ φi ≤ ξi, ∀i and (ξi) converges to 0 for τa,b.In ordered by a closed normal cone. Hence if D is a directed Proposition 1.3, Chapter 1, [2], it has been proved that the set (≤) of generalized functions that is either majorized in L (L L ) sup positive cone in an ordered linear space E(τ) is normal if and a,b or contains a β a,b, a,b -bounded section, then D L F( ) only if for any two nets {xβ : β ∈ I} and {yβ : β ∈ I} exists in a,b and the filter D of sections of D converges to sup (L L ) L+ = L in E(τ) if 0 ≤ xβ ≤ yβ for all β ∈ I and if {yβ : β ∈ I} D for β a,b, a,b . Hence a,b a,b and we conclude L (L L ) converges to 0 for τ then {xβ : β ∈ I} converges to 0 for τ.So that a,b with respect to the topology β a,b, a,b is both we conclude that C is not normal for the Schwartz topology. order complete and topologically complete. (See Proposition It follows that C + iC is not normal for La,b, we conclude 1.8, Chapter 4, [2]). ≤ ≤ ∈ that the cone of La,b is not normal. Theorem 4: Let a c, d b. The restriction of any f L L L L L Theorem 2: The cone C is a strict b-cone in La,b. a,b to c,d is c,d when a,b, c,d are assigned the topology Proof: Assume that La,b is restricted to real-valued of bounded convergence. functions. Let B be the saturated class of all bounded cir- Proof: When a ≤ c, d ≤ b, Lc,d ⊆La,b. The topology L L cled subsets of La,b for τa,b. Then La,b = ∪B∈BBc where τc,d of c,d is stronger than the topology induced on c,d by L ∈L Bc = {(B ∩ C) − (B ∩ C):B ∈ B} is a fundamental system a,b. By Zemanian [3], the restriction of any f a,b to L L L L for B and C is a strict b-cone since B is the class of all τ- c,d is in c,d when c,d and a,b are assigned the topology bounded sets in La,b.
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