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

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

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.

Citation: José Ángel Cid, Luís Sanchez. Nonnegative oscillations for a class of differential equations without uniqueness: A variational approach. Discrete and 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.

[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.

[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.

[4]

J. A. CidG. InfanteM. 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.

[5]

J. A. CidG. InfanteM. 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.

[6]

J. A. CidG. 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.

[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.

[8]

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

[9]

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

[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.

[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.

[12]

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

[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.
[14]

F. WangJ. 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.

[15]

W. S. ZhouX. L. QinG. 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.

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.

[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.

[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.

[4]

J. A. CidG. InfanteM. 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.

[5]

J. A. CidG. InfanteM. 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.

[6]

J. A. CidG. 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.

[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.

[8]

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

[9]

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

[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.

[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.

[12]

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

[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.
[14]

F. WangJ. 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.

[15]

W. S. ZhouX. L. QinG. 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.

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