Triple positive solutions of three-point boundary value problems for \(p\)-Laplacian dynamic equations on time scales. (English) Zbl 1120.39019

Hong, Shihuang

J. Comput. Appl. Math. 206, No. 2, 967-976 (2007).

The author deals with the \(p\)-Laplacian dynamic equation on a time scale \[ [\phi_p(u^\Delta(t))]^\nabla+a(t)f(u(t))=0 \] subject to the boundary condition \(u(0)-B_0(u^\Delta(\eta))=0\), \(u^\Delta(T)=0\) or \(u(T)+B_1(u^\Delta(\eta))=0\), \(u^\Delta(0)=0\), where \(\phi_p\) is the \(p\)-Laplacian operator given by \(\phi_p(u)=| u| ^{p-2}u\) for \(p>1\). Moreover, \(f:(0,\infty)\to(0,\infty)\) is supposed to be continuous, \(a:{\mathbb T}\to[0,\infty)\) does not vanish on a closed interval and \(B_0,B_1\) satisfy appropriate linear growth bounds.
Using a fixed point theorem on cones due to R. I. Avery and A. C. Peterson [Comput.Math.Appl.42, No.3–5, 313–322 (2001; Zbl 1005.47051)], the authors derive two sufficient conditions for the existence of at least three positive solutions of the above three-point boundary value problem. Two examples illustrate their results.

Reviewer: Christian Pötzsche (München)

MSC:

39A13	Difference equations, scaling (\(q\)-differences)
34B15	Nonlinear boundary value problems for ordinary differential equations
34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations
39A14	Partial difference equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations

Keywords:

time scale; positive solution; fixed point; completely continuous operator; three-point boundary value problem

Citations:

Zbl 1005.47051

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