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.
- E. Akin, Boundary value problems for a differential equation on a measure chain, PanAmerican Mathematical Journal 10 (2000) 3, 17-30.
- H. Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1971), 125-146.
- H. Amann, On the number of solutions of nonlinear equations in ordered Banach spaces, J. Funct. Anal. 11 (1972), 346-384.
- C. Bereanu, J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular \(\phi\)-Laplacian, J. Differential Equations 243 (2007), 536-557.
- C. Bereanu, J. Mawhin, Periodic solutions of nonlinear perturbations of \(\phi\)-Laplacians with possibly bounded \(\phi\), Nonlinear Anal. 68 (2008), 1668-1681.
- 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.
- M. Bohner, A. Peterson, Dynamic Equations on Time Scales, Birkhauser Boston, Massachusetts, 2001.
- M. Bohner, A. Peterson (eds.), Advances in Dynamic Equations on Time Scales, Birkhauser Boston, Massachusetts, 2003.
- 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.
- 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.
- K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
- M. Galewski, S. Glab, On the discrete boundary value problem for anisotropic equation, J. Math. Anal. Appl. 386 (2012), 956-965.
- 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.
- J. Henderson, C.C. Tisdell, Topological transversality and boundary value problems on time scales, J. Math. Anal. Appl. 289 (2004), 110-125.
- S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmanningfaltingkeiten, PhD thesis, Universität Würzburg, 1988.
- S. Hilger, Analysis on measure chains - a unified approach to continuous and discrete calculus, Results in Mathematics 18 (1990) 1-2, 18-56.
- Y. Kolesov, Periodic solutions of quasilinear parabolic equations of second order, Trans. Moscow Math. Soc. 21 (1970), 114-146.
- R. Manásevich, J. Mawhin, Boundary value problems for nonlinear perturbations of vector \(p\)-Laplacian-like operators, J. Korean Math. Soc. 37 (2000), 665-685.
- J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS series no. 40, American Math. Soc., Providence RI, 1979.
- P. Stehlík, Periodic boundary value problems on time scales, Adv. Difference Equ. 1 (2005), 81-92.
- 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.
- 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.
- Received: 2020-01-31.
- Revised: 2020-05-06.
- Accepted: 2020-05-11.
- Published online: 2020-07-09.