Positive solutions of singular p-Laplacian dynamic equations with sign changing nonlinearity

Abstract Let T be a time scale such that 0 , T ∈ T . By the Schauder fixed-point theorem and the upper and lower solution method, we present some existence criteria of the positive solution of m-point singular p-Laplacian dynamic equation ( φ p ( u △ ( t ) ) ) ▽ + q ( t ) f ( t , u ( t ) ) = 0 , t ∈ ( 0 , T ) T with boundary conditions u ( 0 ) = 0 , ∑ i = 1 m - 1 ψ i ( u ( ξ i ) ) + u △ ( T ) = 0 , m ⩾ 2 , where φ p ( s ) = | s | p - 2 s with p > 1 , ψ i : R → R is continuous for i = 1 , 2 , … , m - 1 and nonincreasing if m ⩾ 3 , 0 ξ 1 ξ 2 ⋯ ξ m - 2 ξ m - 1 = T . The nonlinear term may be singular in its dependent variable and is allowed to change sign. Our results are new even for the corresponding differential ( T = R ) and difference equations ( T = Z ) . As an application, an example is given to illustrate our result.

[1]  Wan-Tong Li,et al.  Positive Solutions for Second–Order m–Point Boundary Value Problems on Time Scales , 2006 .

[2]  Ravi P. Agarwal,et al.  Existence theorems for the one-dimensional singular p--Laplacian equation with sign changing nonlinearities , 2003, Appl. Math. Comput..

[3]  B. Song,et al.  Controlling wound healing through debridement , 2004, Math. Comput. Model..

[4]  Wan-Tong Li,et al.  Eigenvalue problems for second-order nonlinear dynamic equations on time scales , 2006 .

[5]  S. Hilger.  Analysis on Measure Chains — A Unified Approach to Continuous and Discrete Calculus , 1990 .

[6]  Vanessa Speeding,et al.  Taming nature's numbers , 2003 .

[7]  D. O’Regan,et al.  A Generalized Upper and Lower Solution Method for Singular Discrete Boundary Value Problems for the One-Dimensional p-Laplacian , 2005 .

[8]  D. O’Regan,et al.  Existence theorems for the one-dimensional singular p-Laplacian equation with a nonlinear boundary condition , 2005 .

[9]  Hong‐Rui Sun,et al.  Existence of positive solutions to second-order time scale systems , 2005 .

[11]  Zhimin He,et al.  Double positive solutions of three-point boundary value problems for p-Laplacian dynamic equations on time scales , 2005 .

[12]  B. Aulbach,et al.  Integration on Measure Chains , 2003 .

[13]  Wan-Tong Li,et al.  Existence theory for positive solutions to one-dimensional p-Laplacian boundary value problems on time scales , 2007 .

[14]  A. Peterson,et al.  Dynamic Equations on Time Scales: An Introduction with Applications , 2001 .

[15]  Wan-Tong Li,et al.  Nonoscillation and Oscillation Theory for Functional Differential Equations , 2004 .

[16]  G. Guseinov,et al.  On Green's functions and positive solutions for boundary value problems on time scales , 2002 .

[17]  Johnny Henderson,et al.  Existence of Solutions for a One Dimensional p-Laplacian on Time-Scales , 2004 .

[18]  Diana M. Thomas,et al.  A mathematical evolution model for phytoremediation of metals , 2005 .

[19]  Wan-Tong Li,et al.  Multiple positive solutions for nonlinear dynamical systems on a measure chain , 2004 .

[20]  Amos Gilat,et al.  Matlab, An Introduction With Applications , 2003 .

[21]  V. Lakshmikantham,et al.  Dynamic systems on measure chains , 1996 .

[22]  Wan-Tong Li,et al.  Multiple positive solutions for p-Laplacian m-point boundary value problems on time scales , 2006, Appl. Math. Comput..

[23]  Ravi P. Agarwal,et al.  Positive solutions for singular p-Laplacian equations with sign changing nonlinearities using inequality theory , 2005, Appl. Math. Comput..