scholarly journals A Class of Bridges of Iterated Integrals of Brownian Motion Related to Various Boundary Value Problems Involving the One-Dimensional Polyharmonic Operator

2011 ◽  
Vol 2011 ◽  
pp. 1-32
Author(s):  
Aimé Lachal
Keyword(s):  
Brownian Motion ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Classical Problem ◽  
Polyharmonic Operator ◽  
One Dimensional ◽  
The One ◽  
Interpolation Polynomials ◽  
Derivatives Of ◽  
Hermite Interpolation Polynomials

Let be the linear Brownian motion and the -fold integral of Brownian motion, with being a positive integer: for any In this paper we construct several bridges between times and of the process involving conditions on the successive derivatives of at times and . For this family of bridges, we make a correspondence with certain boundary value problems related to the one-dimensional polyharmonic operator. We also study the classical problem of prediction. Our results involve various Hermite interpolation polynomials.

scholarly journals Positive Solutions of the One-Dimensional p-Laplacian with Nonlinearity Defined on a Finite Interval

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Ruyun Ma ◽  
Chunjie Xie ◽  
Abubaker Ahmed
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Finite Interval ◽  
Boundary Value ◽  
Quadrature Method ◽  
One Dimensional ◽  
Existence And Multiplicity ◽  
The One ◽  
Multiplicity Of Positive Solutions

We use the quadrature method to show the existence and multiplicity of positive solutions of the boundary value problems involving one-dimensional p-Laplacian u′t|p−2u′t′+λfut=0, t∈0,1, u(0)=u(1)=0, where p∈(1,2], λ∈(0,∞) is a parameter, f∈C1([0,r),[0,∞)) for some constant r>0, f(s)>0 in (0,r), and lims→r-(r-s)p-1f(s)=+∞.

scholarly journals The existence of three positive solutions to integral type BVPs for singular second ODEs with one-dimensional p-Laplacian

2011 ◽  
Vol 27 (2) ◽  
pp. 239-248
Author(s):  
YUJI LIU ◽  
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Integral Type ◽  
Singular Differential Equations ◽  
One Dimensional ◽  
Type Boundary ◽  
The One

This paper is concerned with the integral type boundary value problems of the second order singular differential equations with one-dimensional p-Laplacian. Sufficient conditions to guarantee the existence of at least three positive solutions are established. An example is presented to illustrate the main results. The emphasis is put on the one-dimensional p-Laplacian term [ρ(t)Φ(x 0 (t))]0 involved with the function ρ, which makes the solutions un-concave. Furthermore, f, g, h and ρ may be singular at t = 0 or t = 1.

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