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February  2020, 25(2): 545-554. doi: 10.3934/dcdsb.2019253

Nonnegative oscillations for a class of differential equations without uniqueness: A variational approach

José Ángel Cid 1,, and  Luís Sanchez 2,

1. 

Departamento de Matemáticas, Universidade de Vigo, 32004, Campus de Ourense, Spain
2. 

CMAFcIO - Faculdade de Ciências da Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal

* Corresponding author: José Ángel Cid

Dedicated to Professor Juan José Nieto on the occasion of his 60th birthday.

Received  January 2019 Revised  May 2019 Published  February 2020 Early access  November 2019

Fund Project: J. Á. Cid was partially supported by Ministerio de Educación y Ciencia, Spain, and FEDER, Project MTM2017-85054-C2-1-P and L. Sanchez was supported by Fundação para a Ciência e a Tecnologia, UID/MAT/04561/2019.

We deal with the existence of nonnegative and nontrivial $ T $–periodic solutions for the equation $ x'' = r(t)x^{\alpha}-s(t)x^{\beta} $ where $ r $ and $ s $ are continuous $ T $–periodic functions and $ 0<\alpha<\beta<1 $. This equation has been studied in connection with the valveless pumping phenomenon and we will take advantage of its variational structure in order to guarantee its solvability by means of the mountain pass theorem of Ambrosetti and Rabinowitz.

Keywords: Periodic solution, regular equation, critical point theory, mountain pass theorem, Liebau phenomenon.
Mathematics Subject Classification: Primary: 34C25; Secondary: 34B18.
Citation: José Ángel Cid, Luís Sanchez. Nonnegative oscillations for a class of differential equations without uniqueness: A variational approach. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 545-554. doi: 10.3934/dcdsb.2019253
References:
[1]

C. Chicone, The monotonicity of the period function for planar Hamiltonian vector fields, J. Differential Equations, 69 (1987), 310-321.  doi: 10.1016/0022-0396(87)90122-7.  Google Scholar

[2]

A. R. Chouikha, Monotonicity of the period function for some planar differential systems. Ⅰ. Conservative and quadratic systems, Appl. Math., 32 (2005), 305-325.  doi: 10.4064/am32-3-5.  Google Scholar

[3]

S.-N. Chow and D. Wang, On the monotonicity of the period function of some second order equations, Časopis Pěst. Mat., 111 (1986), 14–25, 89.  Google Scholar

[4]

J. A. Cid, G. Infante, M. Tvrdý and M. Zima, A topological approach to periodic oscillations related to the Liebau phenomenon, J. Math. Anal. Appl., 423 (2015), 1546-1556.  doi: 10.1016/j.jmaa.2014.10.054.  Google Scholar

[5]

J. A. Cid, G. Infante, M. Tvrdý and M. Zima, New results for the Liebau phenomenon via fixed point index, Nonlinear Anal. Real World Appl., 35 (2017), 457-469.  doi: 10.1016/j.nonrwa.2016.11.009.  Google Scholar

[6]

J. A. Cid, G. Propst and M. Tvrdý, On the pumping effect in a pipe/tank flow configuration with friction, Phys. D, 273/274 (2014), 28-33.  doi: 10.1016/j.physd.2014.01.010.  Google Scholar

[7]

I. Coelho and L. Sanchez, Travelling wave profiles in some models with nonlinear diffusion, Appl. Math. Comput., 235 (2014), 469-481.  doi: 10.1016/j.amc.2014.02.104.  Google Scholar

[8]

P. Drábek and J. Milota, Methods of Nonlinear Analysis. Applications to Differential Equations, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[9]

A. Gavioli and L. Sanchez, Positive homoclinic solutions to some Schrödinger type equations, Differential Integral Equations, 29 (2016), 665-682.   Google Scholar

[10]

G. Propst, Pumping effects in models of periodically forced flow configurations, Phys. D, 217 (2006), 193-201.  doi: 10.1016/j.physd.2006.04.007.  Google Scholar

[11]

R. Schaaf, Global Solution Branches of Two Point Boundary Value Problems, Lecture Notes in Mathematics, 1458. Springer-Verlag, Berlin, 1990. doi: 10.1007/BFb0098346.  Google Scholar

[12]

A. Sfecci, From isochronous potentials to isochronous systems, J. Differential Equations, 258 (2015), 1791-1800.  doi: 10.1016/j.jde.2014.11.013.  Google Scholar

[13] P. J. Torres, Mathematical Models with Singularities. A Zoo of Singular Creatures, Atlantis Briefs in Differential Equations, 1. Atlantis Press, Paris, 2015.  doi: 10.2991/978-94-6239-106-2.  Google Scholar
[14]

F. Wang, J. A. Cid and M. Zima, Lyapunov stability for regular equations and applications to the Liebau phenomenon, Discrete Contin. Dyn. Syst., 38 (2018), 4657-4674.  doi: 10.3934/dcds.2018204.  Google Scholar

[15]

W. S. Zhou, X. L. Qin, G. K. Xu and X. D. Wei, On the one-dimensional p-Laplacian with a singular nonlinearity, Nonlinear Anal., 75 (2012), 3994-4005.  doi: 10.1016/j.na.2012.02.015.  Google Scholar

show all references

Dedicated to Professor Juan José Nieto on the occasion of his 60th birthday.

References:
[1]

C. Chicone, The monotonicity of the period function for planar Hamiltonian vector fields, J. Differential Equations, 69 (1987), 310-321.  doi: 10.1016/0022-0396(87)90122-7.  Google Scholar

[2]

A. R. Chouikha, Monotonicity of the period function for some planar differential systems. Ⅰ. Conservative and quadratic systems, Appl. Math., 32 (2005), 305-325.  doi: 10.4064/am32-3-5.  Google Scholar

[3]

S.-N. Chow and D. Wang, On the monotonicity of the period function of some second order equations, Časopis Pěst. Mat., 111 (1986), 14–25, 89.  Google Scholar

[4]

J. A. Cid, G. Infante, M. Tvrdý and M. Zima, A topological approach to periodic oscillations related to the Liebau phenomenon, J. Math. Anal. Appl., 423 (2015), 1546-1556.  doi: 10.1016/j.jmaa.2014.10.054.  Google Scholar

[5]

J. A. Cid, G. Infante, M. Tvrdý and M. Zima, New results for the Liebau phenomenon via fixed point index, Nonlinear Anal. Real World Appl., 35 (2017), 457-469.  doi: 10.1016/j.nonrwa.2016.11.009.  Google Scholar

[6]

J. A. Cid, G. Propst and M. Tvrdý, On the pumping effect in a pipe/tank flow configuration with friction, Phys. D, 273/274 (2014), 28-33.  doi: 10.1016/j.physd.2014.01.010.  Google Scholar

[7]

I. Coelho and L. Sanchez, Travelling wave profiles in some models with nonlinear diffusion, Appl. Math. Comput., 235 (2014), 469-481.  doi: 10.1016/j.amc.2014.02.104.  Google Scholar

[8]

P. Drábek and J. Milota, Methods of Nonlinear Analysis. Applications to Differential Equations, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[9]

A. Gavioli and L. Sanchez, Positive homoclinic solutions to some Schrödinger type equations, Differential Integral Equations, 29 (2016), 665-682.   Google Scholar

[10]

G. Propst, Pumping effects in models of periodically forced flow configurations, Phys. D, 217 (2006), 193-201.  doi: 10.1016/j.physd.2006.04.007.  Google Scholar

[11]

R. Schaaf, Global Solution Branches of Two Point Boundary Value Problems, Lecture Notes in Mathematics, 1458. Springer-Verlag, Berlin, 1990. doi: 10.1007/BFb0098346.  Google Scholar

[12]

A. Sfecci, From isochronous potentials to isochronous systems, J. Differential Equations, 258 (2015), 1791-1800.  doi: 10.1016/j.jde.2014.11.013.  Google Scholar

[13] P. J. Torres, Mathematical Models with Singularities. A Zoo of Singular Creatures, Atlantis Briefs in Differential Equations, 1. Atlantis Press, Paris, 2015.  doi: 10.2991/978-94-6239-106-2.  Google Scholar
[14]

F. Wang, J. A. Cid and M. Zima, Lyapunov stability for regular equations and applications to the Liebau phenomenon, Discrete Contin. Dyn. Syst., 38 (2018), 4657-4674.  doi: 10.3934/dcds.2018204.  Google Scholar

[15]

W. S. Zhou, X. L. Qin, G. K. Xu and X. D. Wei, On the one-dimensional p-Laplacian with a singular nonlinearity, Nonlinear Anal., 75 (2012), 3994-4005.  doi: 10.1016/j.na.2012.02.015.  Google Scholar

Figure 1.  Graph of the potential $ V $ for the values $ r = 12, \, s = 8, \, \alpha = \frac 1 2 $ and $ \beta = \frac 3 4 $
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Figure 2.  Phase plane for (3) with the values $ r = 12, \, s = 8, \, \alpha = \frac 1 2 $ and $ \beta = \frac 3 4 $
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