On singular nonlinear distributional and impulsive initial and boundary value problems

Heikkilä

Description

Purpose To derive existence and comparison results for extremal solutions of nonlinear singular distributional initial value problems and boundary value problems. Main methods Fixed point results in ordered function spaces and recently introduced concepts of regulated and continuous primitive integrals of distributions. Maple programming is used to determine solutions of examples. Results New existence results are derived for the smallest and greatest solutions of considered problems. Novel results are derived for the dependence of solutions on the data. The obtained results are applied to impulsive differential equations. Concrete examples are presented and solved to illustrate the obtained results. MSC: 26A24, 26A39, 26A48, 34A12, 34A36, 37A37, 39B12, 39B22, 47B38, 47J25, 47H07, 47H10, 58D25

Sujets

• Distribution
• Primitive
• Intégral
• Regulation
• Synergy (logiciel)
• Initial value problem
• Boundary value problem
• Singular

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 6
• Langue: English

Extrait

HeikkiläBoundary Value Problems2011,2011:24 http://www.boundaryvalueproblems.com/content/2011/1/24

R E S E A R C HOpen Access On singular nonlinear distributional and impulsive initial and boundary value problems

Seppo Heikkilä

Correspondence: [e-mail] Department of Mathematical Sciences, University of Oulu, BOX 3000, FIN90014, Oulu, Finland

Abstract Purpose:To derive existence and comparison results for extremal solutions of nonlinear singular distributional initial value problems and boundary value problems. Main methods:Fixed point results in ordered function spaces and recently introduced concepts of regulated and continuous primitive integrals of distributions. Maple programming is used to determine solutions of examples. Results:New existence results are derived for the smallest and greatest solutions of considered problems. Novel results are derived for the dependence of solutions on the data. The obtained results are applied to impulsive differential equations. Concrete examples are presented and solved to illustrate the obtained results. MSC:26A24, 26A39, 26A48, 34A12, 34A36, 37A37, 39B12, 39B22, 47B38, 47J25, 47H07, 47H10, 58D25 Keywords:distribution; primitive, integral; regulated, continuous; initial value problem, boundary value problem, singular, distributional

1 Introduction In this paper, existence and comparison results are derived for the smallest and great est solutions of first and second order singular nonlinear initial value problems as well as second order boundary value problems. Recently, similar problems are studied in ordered Banach spaces, e.g., in [14], by con verting problems into systems of integral equations, integrals in these systems being BochnerLebesgue or HenstockKurzweil integrals. A novel feature in the present study is that the righthand sides of the considered differential equations comprise distribu tions on a compact real interval [a,b]. Every distribution is assumed to have a primitive in the space R a,b of those functions from [a,b] toℝwhich are leftcontinuous on (a,b], rightcontinuous ata, and which have right limits at every point of (a,b). With this presupposition, the considered problems can be transformed into integral equations which include the regulated primitive integral of distributions introduced recently in [5]. The paper is organized as follows. Distributions on [a,b], their primitives, regulated primitive integrals and some of their properties, as well as a fixed point lemma are pre sented in Section 2. In Section 3, existence and comparison results are derived for the smallest and greatest solutions of first order initial value problems. A fact that makes the solution space R a,b important in applications is that it con tains primitives of Dirac delta distributionsδ l ,lÎ(a,b). This fact is exploited in

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