Infinitely many positive solutions for an iterative system of singular BVP on time scales

K. Rajendra Prasad; Mahammad Khuddush; K. V. Vidyasagar

doi:10.4067/S0719-06462022000100021

DOI:

https://doi.org/10.4067/S0719-06462022000100021

Abstract

In this paper, we consider an iterative system of singular two-point boundary value problems on time scales. By applying Hölder‘s inequality and Krasnoselskii‘s cone fixed point theorem in a Banach space, we derive sufficient conditions for the existence of infinitely many positive solutions. Finally, we provide an example to check the validity of our obtained results.

Keywords

Iterative system , time scales , singularity , cone , Krasnoselskii‘s fixed point theorem , positive solutions

K. Rajendra Prasad Department of Applied Mathematics, College of Science and Technology, Andhra University, Visakhapatnam, 530003, India.
Mahammad Khuddush Department of Mathematics, Dr. Lankapalli Bullayya College, Resapuvanipalem, Visakhapatnam, 530013, India.
K. V. Vidyasagar Department of Mathematics, S. V. L. N. S. Government Degree College, Bheemunipatnam, Bheemili, 531163, India.

Pages: 21–35
Date Published: 2022-04-04
Vol. 24 No. 1 (2022)

R. P. Agarwal and M. Bohner, “Basic calculus on time scales and some of its applications”, Results Math., vol. 35, no. 1–2, pp. 3–22, 1999.

R. P. Agarwal, M. Bohner, D. O‘Regan and A. Peterson, “Dynamic equations on time scales: a survey”, J. Comput. Appl. Math., vol. 141, no. 1-2, pp. 1–26, 2002.

R. P. Agarwal, V. Otero-Espinar, K. Perera and D. R. Vivero, “Basic properties of Sobolev‘s spaces on time scales”, Adv. Difference. Equ., Art. ID 38121, 14 pages, 2006.

G. A. Anastassiou, Intelligent mathematics: computational analysis, Intelligent Systems Reference Library, vol. 5, Heidelberg: Springer, 2011.

M. Bohner and H. Luo, “Singular second-order multipoint dynamic boundary value problems with mixed derivatives”, Adv. Difference Equ., Art. ID 54989, 15 pages, 2006.

M. Bohner and A. Peterson, Dynamic equations on time scales: An introduction with applications, Boston: Birkh Ìˆauser Boston, Inc., 2001.

M. Bohner and A. Peterson, Advances in dynamic equations on time scales, Boston: Birkhäuser Boston, Inc., 2003.

A. Dogan, “Positive solutions of the p-Laplacian dynamic equations on time scales with sign changing nonlinearity”, Electron. J. Differential Equations, Paper No. 39, 17 pages, 2018.

A. Dogan, “Positive solutions of a three-point boundary-value problem for p-Laplacian dynamic equation on time scales”.‘ Ukraïn. Mat. Zh., vol. 72, no. 6, pp. 790–805, 2020.

D. Guo and V. Lakshmikantham, Nonlinear problems in abstract cones, Boston: Academic Press, 1988.

G. S. Guseinov, “Integration on time scales”, J. Math. Anal. Appl., vol. 285, no.1, pp. 107–127, 2003.

M. Khuddush, K. R. Prasad and K. V. Vidyasagar, “Infinitely many positive solutions for an iterative system of singular multipoint boundary value problems on time scales”, Rend. Circ. Mat. Palermo, ll Ser., 2021. doi: 10.1007/s12215-021-00650-6

C. Kunkel, “Positive solutions to singular second-order boundary value problems on time scales”, Adv. Dyn. Syst. Appl., vol. 4, no. 2, pp. 201–211, 2019.

S. Liang and J. Zhang, “The existence of countably many positive solutions for nonlinear singular m-point boundary value problems on time scales”, J. Comput. Appl. Math., vol. 223, no. 1, pp. 291–303, 2009.

U. M. Özkan, M. Z. Sarikaya and H. Yildirim, “Extensions of certain integral inequalities on time scales”, Appl. Math. Lett., vol. 21, no. 10, pp. 993–1000, 2008.

K. R. Prasad and M. Khuddush, “Countably infinitely many positive solutions for even order boundary value problems with Sturm-Liouville type integral boundary conditions on time scales”, Int. J. Anal. Appl., vol. 15, no. 2, pp. 198–210, 2017.

K. R. Prasad and M. Khuddush, “Existence of countably many symmetric positive solutions for system of even order time scale boundary value problems in Banach spaces”, Creat. Math. Inform., vol. 28, no. 2, pp. 163–182, 2019.

K. R. Prasad, M. Khuddush and K. V. Vidyasagar, “Denumerably many positive solutions for iterative systems of singular two-point boundary value problems on time scales”, Int. J. Difference Equ., vol. 15, no. 1, pp. 153–172, 2020.

S. Tikare and C. C. Tisdell, “Nonlinear dynamic equations on time scales with impulses and nonlocal conditions”, J. Class. Anal., vol. 16, no. 2, pp. 125–140, 2020.

P. A. Williams, “Unifying fractional calculus with time scales”, Ph.D. thesis, University of Melbourne, Melbourne, Australia, 2012.