Opuscula Math. 40, no. 4 (2020), 405-425
https://doi.org/10.7494/OpMath.2020.40.4.405

Opuscula Mathematica

# Multiple solutions of boundary value problems on time scales for a φ-Laplacian operator

Pablo Amster
Mariel Paula Kuna
Dionicio Pastor Santos

Abstract. We establish the existence and multiplicity of solutions for some boundary value problems on time scales with a $$\varphi$$-Laplacian operator. For this purpose, we employ the concept of lower and upper solutions and the Leray-Schauder degree. The results extend and improve known results for analogous problems with discrete $$p$$-Laplacian as well as those for boundary value problems on time scales.

Keywords: dynamic equations on time scales, nonlinear boundary value problems, upper and lower solutions, Leray-Schauder degree, multiple solutions.

Mathematics Subject Classification: 34N05, 34K10, 47H11.

Full text (pdf)

1. E. Akin, Boundary value problems for a differential equation on a measure chain, PanAmerican Mathematical Journal 10 (2000) 3, 17-30.
2. H. Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1971), 125-146.
3. H. Amann, On the number of solutions of nonlinear equations in ordered Banach spaces, J. Funct. Anal. 11 (1972), 346-384.
4. C. Bereanu, J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular $$\phi$$-Laplacian, J. Differential Equations 243 (2007), 536-557.
5. C. Bereanu, J. Mawhin, Periodic solutions of nonlinear perturbations of $$\phi$$-Laplacians with possibly bounded $$\phi$$, Nonlinear Anal. 68 (2008), 1668-1681.
6. C. Bereanu, H.B. Thompson, Periodic solutions of second order nonlinear difference equations with discrete $$\phi$$-Laplacian, J. Math. Anal. Appl. 330 (2007), 1002-1015.
7. M. Bohner, A. Peterson, Dynamic Equations on Time Scales, Birkhauser Boston, Massachusetts, 2001.
8. M. Bohner, A. Peterson (eds.), Advances in Dynamic Equations on Time Scales, Birkhauser Boston, Massachusetts, 2003.
9. A. Cabada, Extremal solutions and Green's functions of higher order periodic boundary value problems in time scales, J. Math. Anal. Appl 290 (2004), 35-54.
10. C. De Coster, P. Habets, The lower and upper solutions method for boundary value problems, [in:] Handbook of Differential Equations, Elsevier/North-Holland, Amsterdam, 2004, 69-160.
11. K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
12. M. Galewski, S. Glab, On the discrete boundary value problem for anisotropic equation, J. Math. Anal. Appl. 386 (2012), 956-965.
13. A. Guiro, I. Nyanquini, S. Ouaro, On the solvability of discrete nonlinear Neumann problems involving the $$p(x)$$-Laplacian, Adv. Difference Equ. 2011 (2011), Article no. 32.
14. J. Henderson, C.C. Tisdell, Topological transversality and boundary value problems on time scales, J. Math. Anal. Appl. 289 (2004), 110-125.
15. S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmanningfaltingkeiten, PhD thesis, Universität Würzburg, 1988.
16. S. Hilger, Analysis on measure chains - a unified approach to continuous and discrete calculus, Results in Mathematics 18 (1990) 1-2, 18-56.
17. Y. Kolesov, Periodic solutions of quasilinear parabolic equations of second order, Trans. Moscow Math. Soc. 21 (1970), 114-146.
18. R. Manásevich, J. Mawhin, Boundary value problems for nonlinear perturbations of vector $$p$$-Laplacian-like operators, J. Korean Math. Soc. 37 (2000), 665-685.
19. J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS series no. 40, American Math. Soc., Providence RI, 1979.
20. P. Stehlík, Periodic boundary value problems on time scales, Adv. Difference Equ. 1 (2005), 81-92.
21. Y. Tian, W. Ge, Existence of multiple positive solutions for discrete problems with $$p$$-Laplacian via variational methods, Electron. J. Differential Equations 45 (2011), 1-8.
22. S.G. Topal, Second-order periodic boundary value problems on time scales, J. Comput. Appl. Math. 48 (2004), 637-648.
• Pablo Amster (corresponding author)
• https://orcid.org/0000-0003-2829-7072
• Universidad de Buenos Aires & IMAS-CONICET, Facultad de Ciencias Exactas y Naturales, Departamento de Matemática, Ciudad Universitaria, Pabellón I, Buenos Aires (1428), Argentina
• Mariel Paula Kuna
• https://orcid.org/0000-0001-6466-973X
• Universidad de Buenos Aires & IMAS-CONICET, Facultad de Ciencias Exactas y Naturales, Departamento de Matemática, Ciudad Universitaria, Pabellón I, Buenos Aires (1428), Argentina
• Dionicio Pastor Santos
• https://orcid.org/0000-0001-5574-6254
• Universidad de Buenos Aires & IMAS-CONICET, Facultad de Ciencias Exactas y Naturales, Departamento de Matemática, Ciudad Universitaria, Pabellón I, Buenos Aires (1428), Argentina
• Communicated by Petr Stehlík.
• Revised: 2020-05-06.
• Accepted: 2020-05-11.
• Published online: 2020-07-09.