Positive solutions of a three-point  boundary-value problem for $\mathcal {p}$-Laplacian   dynamic equation  on time scales

A. Dogan

doi:10.37863/umzh.v72i6.646

A. Dogan Dep. Appl. Math., Abdullah Gul Univ., Kayseri, Turkey

DOI: https://doi.org/10.37863/umzh.v72i6.646

Keywords: Time scales, Boundary value problem, p-Laplacian, Positive solutions, Fixed point theorem

Abstract

UDC 517.9

We consider a three-point boundary-value problem for p-Laplacian dynamic equation on time scales. We show the existence at least three positive solutions of the boundary-value problem by using the Avery and Peterson fixed point theorem. The conditions we used here differ from those in the majority of papers as we know. The interesting point is that the nonlinear term $ f$ involves the first derivative of the unknown function. As an application, an example is given to illustrate our results.

References

R. P. Agarwal, D. O’Regan, Triple solutions to boundary value problems on time scales, Appl. Math. Lett., 44, 527 – 535 (2001) https://doi.org/10.1016/S0362-546X(99)00290-4 DOI: https://doi.org/10.1016/S0362-546X(99)00290-4

R. P. Agarwal, D. O’Regan, Nonlinear boundary value problems on time scales, Appl. Math. Lett., 13, 7 – 11 (2000) https://doi.org/10.1016/S0893-9659(99)00200-1 DOI: https://doi.org/10.1016/S0893-9659(99)00200-1

D. Anderson, Solutions to second-order three-point problems on time scales, J. Difference Equat. and Appl., 8, 673 – 688 (2002)https://doi.org/10.1080/1023619021000000717 DOI: https://doi.org/10.1080/1023619021000000717

D. R. Anderson, R. Avery, J. Henderson, Existence of solutions for a one-dimensional p-Laplacian on time scales, J. Difference Equat. and Appl., 10, 889 – 896 (2004) https://doi.org/10.1080/10236190410001731416 DOI: https://doi.org/10.1080/10236190410001731416

R. I. Avery, C. J. Chyan, J. Henderson, Twin solutions of a boundary value problems for ordinary differential equations and nite difference equations, Comput. Math. Appl., 42, 695 – 704 (2001) https://doi.org/10.1016/S0898-1221(01)00188-2 DOI: https://doi.org/10.1016/S0898-1221(01)00188-2

R. I. Avery, A. Peterson, Three positive xed points of nonlinear operators on ordered Banach spaces, Comput. Math. Appl., 42, 313 – 322 (2001) https://doi.org/10.1016/S0898-1221(01)00156-0 DOI: https://doi.org/10.1016/S0898-1221(01)00156-0

M. Bohner, A. Peterson, Dynamic equations on time scales: An introduction with applications, Birkha ̈user, Boston, Cambridge, MA (2001) MA, 2001. x+358 pp. ISBN: 0-8176-4225-0 https://doi.org/10.1007/978-1-4612-0201-1

M. Bohner, A. Peterson, Advances in dynamic equations on time scales, Birkha ̈user, Boston, Cambridge, MA (2003) 2001. x+358 pp. ISBN: 0-8176-4225-0 https://doi.org/10.1007/978-1-4612-0201-1 DOI: https://doi.org/10.1007/978-1-4612-0201-1

A. Dogan, On the existence of positive solutions for the one-dimensional $p$-Laplacian boundary value problems on time scales, Dyn. Systems and Appl., 24, 295 – 304 (2015) https://acadsol.eu/dsa/24/1-4/23

A. Dogan, Three positive solutions of a three-point boundary value problem for the $p$-Laplacian dynamic equation on time scales, Commun. Optim. Theory, 2018, 1 – 13 (2018) ISSN: 1072-6691. URL: http://ejde.math.txstate.edu DOI: https://doi.org/10.23952/cot.2018.1

M. Guo, Existence of positive solutions for $p$-Laplacian three-point boundary value problems on time scales, Math. Comput. Modelling, 50, 248 – 253 (2009) https://doi.org/10.1016/j.mcm.2009.03.001 DOI: https://doi.org/10.1016/j.mcm.2009.03.001

Z. He, Double positive solutions of three-point boundary value problems for $p$-Laplacian dynamic equations on time scales, J. Comput. and Appl. Math., 182, 304 – 315 (2005) https://doi.org/10.1016/j.cam.2004.12.012 DOI: https://doi.org/10.1016/j.cam.2004.12.012

Z. He, X. Jiang, Triple positive solutions of boundary value problems for p-Laplacian dynamic equations on time scales, J. Math. Anal. and Appl., 321, 911 – 920 (2006) https://doi.org/10.1016/j.jmaa.2005.08.090 DOI: https://doi.org/10.1016/j.jmaa.2005.08.090

Z. He, L. Li, Multiple positive solutions for the one-dimensional $p$-Laplacian dynamic equations on time scales, Math. Comput. Modelling, 45, 68 – 79 (2007) https://doi.org/10.1016/j.mcm.2006.03.021 DOI: https://doi.org/10.1016/j.mcm.2006.03.021

S. Hong, Triple positive solutions of three-point boundary value problems for $p$-Laplacian dynamic equations on time scales, J. Comput. and Appl. Math., 206, 967 – 976 (2007) https://doi.org/10.1016/j.cam.2006.09.002 DOI: https://doi.org/10.1016/j.cam.2006.09.002

H. Luo, Q. Z. Ma, Positive solutions to a generalized second-order three-point boundary value problem on time scales, Electron. J. Different. Equat., 17, 1 – 14 (2005) ISSN: 1072-6691. URL: http://ejde.math.txstate.edu

D. O’Regan, Existence theory for nonlinear ordinary differential equations, Kluwer Acad. Publ. Group, Dordrecht (1997) vi+196 pp. ISBN: 0-7923-4511-8 https://doi.org/10.1007/978-94-017-1517-1 DOI: https://doi.org/10.1007/978-94-017-1517-1

H. Su, B. Wang, Z. Wei, Positive solutions of four-point boundary value problems for four-order $p$-Laplacian dynamic equations on time scales, Electron. J. Different. Equat., 78, 1 – 13 (2006) https://www.researchgate.net/publication/26436685_Positive_solutions_of_four-point_boundary-value_problems_for_four-order_p-Laplacian_dynamic_equations_on_time_scales

H. R. Sun, W. T. Li, Positive solutions for nonlinear three-point boundary value problems on time scales, J. Math. Anal. and Appl., 299, 508 – 524 (2004) https://doi.org/10.1016/j.jmaa.2004.03.079 DOI: https://doi.org/10.1016/j.jmaa.2004.03.079

H. R. Sun, L. T. Tang, Y. H. Wang, Eigenvalue problem for p-Laplacian three-point boundary value problems ontime scales, J. Math. Anal. and Appl., 331, 248 – 262 (2007) https://doi.org/10.1016/j.jmaa.2006.08.080 DOI: https://doi.org/10.1016/j.jmaa.2006.08.080

H. R. Sun, W. T. Li, Existence theory for positive solutions to one-dimensional $p$-Laplacian boundary value problems on time scales, J. Different. Equat., 240, 217 – 248 (2007) https://doi.org/10.1016/j.jde.2007.06.004 DOI: https://doi.org/10.1016/j.jde.2007.06.004

H. R. Sun, Y. H. Wang, Existence of positive solutions for $p$-Laplacian three-point boundary value problems on time scales, Electron. J. Different. Equat., 92, 1 – 14 (2008) ISSN: 1072-6691. URL: http://ejde.math.txstate.edu

D. B. Wang, Three positive solutions of three-point boundary value problems for $p$-Laplacian dynamic equations on time scales, Nonlinear Anal., 68, 2172 – 2180 (2008) https://doi.org/10.1016/j.na.2007.01.037 DOI: https://doi.org/10.1016/j.na.2007.01.037