Existence theory for positive solutions of p-laplacian multi-point BVPs on time scales

You-hui SU

Article Details

Existence theory for positive solutions of p-laplacian multi-point BVPs on time scales

Authors:

You-hui SU

Publisher:

TUBITAK

Journal:

Turkish Journal of Mathematics 2011 Vol.35 Issue 2, pp.219-248

Document Type:

Article

Publication Date:

1/2011

Document Language:

Turkish

Document ID (Org No.):

2000747972

Frequency:

Bimonthly

eISSN:

1303-6149

pISSN:

1300-0098

Keywords:

Time scales; boundary value problem; positive solutions; p-Laplacian; fixed-point theorem

Abstract:

 

This paper is concerned with the one-dimensional p-Laplacian  multi-point boundary value problem on time scales T:     (jp(uD))\nabla + h(t)f(u) = 0, t \in [0,T]T,    subject to multi-point boundary conditions    u(0) - B0(\sumi=1m-2ai uD(xi)) = 0, uD(T) = 0,    or    uD(0) = 0, u(T) + B1(\sumi=1m-2biuD(x'i)) = 0,    where jp(u) is p-Laplacian operator, i.e., jp(u = |u|p-2u,  p>1, xi,x'i\in [0,T]T, m \geq 3 and satisfy 0 \leq x1 <   x2 < ... < xm-2 < r(T), s(0) < x'1 < x'2 <   ... < x'm-2 \leq T, ai, bi\in [0,\infty) (i=1,2,..., m-2).  Some new sufficient conditions are obtained for the existence of at  least one positive solution by using Krasnosel'skii's fixed-point  theorem and new sufficient conditions are obtained for the existence  of twin, triple or arbitrary odd positive solutions by using  generalized Avery and Henderson fixed-point theorem and Avery-Peterson  fixed-point theorem.  Our results include and extend some known  results.  As applications, two examples are given to illustrate the  main results and their differences.  These results are new even for  the special cases of continuous and discrete equations, as well as in  the general time scale setting.

 

This paper is concerned with the one-dimensional p-Laplacian  multi-point boundary value problem on time scales T:     (jp(uD))\nabla + h(t)f(u) = 0, t \in [0,T]T,    subject to multi-point boundary conditions    u(0) - B0(\sumi=1m-2ai uD(xi)) = 0, uD(T) = 0,    or    uD(0) = 0, u(T) + B1(\sumi=1m-2biuD(x'i)) = 0,    where jp(u) is p-Laplacian operator, i.e., jp(u = |u|p-2u,  p>1, xi,x'i\in [0,T]T, m \geq 3 and satisfy 0 \leq x1 <   x2 < ... < xm-2 < r(T), s(0) < x'1 < x'2 <   ... < x'm-2 \leq T, ai, bi\in [0,\infty) (i=1,2,..., m-2).  Some new sufficient conditions are obtained for the existence of at  least one positive solution by using Krasnosel'skii's fixed-point  theorem and new sufficient conditions are obtained for the existence  of twin, triple or arbitrary odd positive solutions by using  generalized Avery and Henderson fixed-point theorem and Avery-Peterson  fixed-point theorem.  Our results include and extend some known  results.  As applications, two examples are given to illustrate the  main results and their differences.  These results are new even for  the special cases of continuous and discrete equations, as well as in  the general time scale setting.

Article Details

Existence theory for positive solutions of p-laplacian multi-point BVPs on time scales

Authors: You-hui SU

Publisher: TUBITAK

Journal: Turkish Journal of Mathematics 2011 Vol.35 Issue 2, pp.219-248

Document Type: Article

Publication Date: 1/2011

Document Language: Turkish

Document ID (Org No.): 2000747972

Frequency: Bimonthly

eISSN: 1303-6149

pISSN: 1300-0098

Time scales; boundary value problem; positive solutions; p-Laplacian; fixed-point theorem

Issues

Authors:

You-hui SU

Publisher:

TUBITAK

Journal:

Turkish Journal of Mathematics 2011 Vol.35 Issue 2, pp.219-248

Document Type:

Article

Publication Date:

1/2011

Document Language:

Turkish

Document ID (Org No.):

2000747972

Frequency:

Bimonthly

eISSN:

1303-6149

pISSN:

1300-0098