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

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.

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

Zbl 1005.47051
Full Text:

References:

 [1] Agarwal, R.P.; Bohner, M., Basic calculus on time scales and its applications, Results math., 35, 3-22, (1999) · Zbl 0927.39003 [2] Agarwal, R.P.; O’Regan, D., Nonlinear boundary value problems on time scales, Nonlinear anal., 44, 527-535, (2001) · Zbl 0995.34016 [3] Anderson, D., Solutions to second-order three-point problems on time scales, J. difference equations appl., 8, 673-688, (2002) · Zbl 1021.34011 [4] Atici, F.M.; Guseinov, G.Sh., On Green’s functions and positive solutions for boundary-value problems on time scales, J. comput. appl. math., 141, 75-99, (2002) · Zbl 1007.34025 [5] Avery, R.I.; Chyan, C.J.; Henderson, J., Twin solutions of a boundary value problem for ordinary differential equations and finite difference equations, Comput. math. appl., 42, 695-704, (2001) · Zbl 1006.34022 [6] Avery, R.I.; Peterson, A.C., Three positive fixed points of nonlinear operators on ordered Banach spaces, Comput. math. appl., 42, 313-322, (2001) · Zbl 1005.47051 [7] Bai, Z.; Wang, Y.; Ge, W., Triple positive solutions for a class of two-point boundary value problems, Electronic J. differential equations, 6, 1-8, (2004) [8] Bohner, M.; Peterson, A., Dynamic equations on time scales, an introduction with applications, (2001), Birkhäuser Boston · Zbl 0978.39001 [9] He, Z., Double positive solutions of three-point boundary value problems for $$p$$-Laplacian dynamic equations on time scales, J. comput. appl. math., 182, 304-315, (2005) · Zbl 1075.39011 [10] Hilger, S., Analysis on measure chains- a unified approach to continuous and discrete calculus, Results math., 18, 18-56, (1990) · Zbl 0722.39001 [11] Kaufmann, E.R., Positive solutions of a three-point boundary value problem on a time scale, Electronic J. differential equations, 82, 1-11, (2003) · Zbl 1047.34015 [12] Kaymakcalan, B.; Lakshmikantham, V.; Sivasundaram, S., Dynamical systems on measure chains, (1996), Kluwer Academic Publishers Boston · Zbl 0869.34039 [13] Leggett, R.W.; Williams, L.R., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana univ. math. J., 28, 673-688, (1979) · Zbl 0421.47033 [14] Shu, X.; Xu, Y., Triple positive solutions for a class of boundary value problem of second-order functional differential equations, Nonlinear anal., 61, 1401-1411, (2005) · Zbl 1069.34090
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.