Solvability of Second Order Delta-Nabla p-Laplacian m-point Eigenvalue Problem on Time Scales

Year 2015, Volume: 5 Issue: 1, 98 - 109, 01.06.2015

S. N. Rao

Abstract

In this paper, we are concerned with the following eigenvalue problem of m-point boundary value problem for p-Laplacian dynamic equation on time scales, ϕp u ∆ t ∇ + λh t f u t = 0, t ∈ [a, b]T , u a − u ∆ a = m∑−2 i=1 u ∆ ξi , u ∆ b = 0, m ≥ 3, where ϕp u = |u| p−2u, p > 1 and λ > 0 is a real parameter. Under certain assumptions, some new results on existence of one or two positive solutions and nonexistence are obtained for λ evaluated in different intervals by using Guo-Krasnosel’skii fixed point theorem.

Keywords

eigenvalue, time scale, p-Laplacian, positive solution, Şxed point, cone

References

• Agarwal, R. P., O’Regan, D. and Wong, P. J. J., (1999), Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic, Dordrecht, The Netherlands.
• Agarwal, R. P., Bohner, M. and Rehak, P., (2003), Half-Linear Dynamic Equation, Nonlinear Analysis and Applications: To V. Lakshmikantham on his 80thbirthday, Kluwer Acad. Publ. Dordrecht, 1, pp. 1-57.
• Agarwal, R. P. and L¨u, H. and O’Regan, D., (2002), Eigenvalues and the one-dimensional p-Laplacian, J. Math. Anal. Appl., 266, pp. 383-400.
• Anderson, D. R., (2002), Eigenvalue intervals for a second-order mixed-conditions problem on time scale, Int. J. Nonlinear Diff. Eqns., 7, pp. 97-104.
• Anderson, D. R., (2002), Eigenvalue intervals for a two-point boundary value problem on a measure chain, J. Comp. Appl. Math., 141, (1-2), pp. 57-64.
• Anderson, D. R., Avery, R. and Henderson, J., (2004), Existence of solutions for a one-dimensional p-Laplacian on time scales, J. Diff. Eqns. Appl., 10, pp. 889-896.
• Aulbach, B. and Neidhart, L., (2004), Integration on measure chains, in: Proceedings of the Sixth International Conference on Difference Equations, CRC, Boca Raton, FL., pp. 239-252.
• Bohner, M. and Peterson, A., (2001), Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, Mass, USA.
• Chyan, C. J. and Henderson, J., (2000), Eigenvalue problems for nonlinear differential equations on a measure chain, J. Math. Anal. Appl., 24, (2), pp. 547-559.
• Davis, J. M., Henderson, J., Prasad, K. R. and Yin, W. K. C., (2000), Eigenvalue intervals for nonlinear right focal problem, Appl. Anal., 74, pp. 215-231.
• Davis, J. M., Henderson, J., Prasad, K. R. and Yin, W. K. C., (2000), Solvability of a nonlinear second order conjugate eigenvalue problems on time scales, Abstract and Applied Analysis., 5, pp. 91-100.
• Fan, J., and Li, L., (2013), Existence of positive solutions for p-Laplacian dynamic equations with derivative on time scales, J. Appl. Math., 2013, pp. 1-7.
• Guo, D. J. and Lakshmikantham, V., (1988), Nonlinear Problems in Abstract Cones., vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA.
• Guo, M. and Sun, H. R., (2009), Eigenvalue problems for p-Laplacian dynamic equations on time scales, Bull. Korean Math. Soc., 46, (5), pp. 999-1011.
• Goodrich, C. S., (2011), Existence of a positive solutions to a Şrst-order p-Laplacian BVP on a time scale, Nonlinear Anal., 74, pp. 1926-1936.
• Goodrich, C. S., (2012), The existence of a positive solution to a second order delta nabla p-Laplacian BVP on a time scale, Appl. Math. Lett., 25, pp. 157-162.
• He, Z., (2005), Double positive solutions of boundary value problems for p-Laplacian dynamic equa- tions on time scales, Appl. Anal., 84, (4), pp. 377-390.
• Hilger, S., (1990), Analysis on measure chains–a uniŞed approach to continuous and discrete calculus, Results Math., 18, No.1-2, pp. 18-56.
• Infante, G., (2003), Eigenvalues of some non-local boundary-value problems, Proc. Edinburgh Math. Soc, 46, (1), pp. 75-86.
• Krasnosel’skii, M. A., (1964), Positive solutions of operator equations, P. Noordhoff Ltd, Groningen, The Netherlands.
• Nageswararao, S., (2011), Solvability of a nonlinear general third order four point eigenvalue problem on time scales, Creat. Math. Inform., 20, (2), pp. 171-182.
• Prasad, K. R., Nageswararao, S, and Murali, P., (2009), Solvability of a nonlinear general third order two-point eigenvalue problem on time scales, Diff. Eqns. Dyn. Syst., 17, (3), pp. 269-282.
• Sun, H. R. and Li, W. T., (2006), Positive solutions for nonlinear m-point boundary value problems on time scales, Acta Math. Sinica., 49, (2), pp. 369-380.
• Sun, H. R. and Li, W. T., (2006), Positive solutions p-Laplacian m-point boundary value problems on time scales, Appl. Math. Comput., 182, (1), pp. 478-491.
• Sun, H. R., Tang, L. T. and Y. H. Wang., (2007), Eigenvalue problem for p-Laplacian three-point boundary value problems on time scales, J. Math. Anal. Appl., 331, pp. 248-262.