August  2021, 41(8): 3683-3708. doi: 10.3934/dcds.2021012

A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo"

1. 

Dipartimento di Matematica e Geoscienze, Università di Trieste, P.le Europa 1, Trieste, Italy

2. 

Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università di Udine, Via delle Scienze 206, Udine, Italy

* Corresponding author: Alessandro Fonda

Received  November 2020 Published  August 2021 Early access  January 2021

We prove the existence of bounded and periodic solutions for planar systems by introducing a notion of lower and upper solutions which generalizes the classical one for scalar second order equations. The proof relies on phase plane analysis; after suitably modifying the nonlinearities, the Ważewski theory provides a solution which remains bounded in the future. For the periodic problem, the Massera Theorem applies, yielding the existence result. We then show how our result generalizes some well known theorems on the existence of bounded and of periodic solutions. Finally, we provide some corollaries on the existence of almost periodic solutions for scalar second order equations.

Citation: Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3683-3708. doi: 10.3934/dcds.2021012
References:
[1]

S. Ahmad, A nonstandard resonance problem for ordinary differential equations, Trans. Amer. Math. Soc., 323 (1991), 857-875.  doi: 10.1090/S0002-9947-1991-1010407-9.

[2]

I. Barbălat, Applications du principe topologique de T. Ważewski aux équations différentielles du second ordre, Ann. Polon. Math., 5 (1958), 303-317. 

[3]

J. W. Bebernes and R. Wilhelmsen, A general boundary value problem technique, J. Differential Equations, 8 (1970), 404-415.  doi: 10.1016/0022-0396(70)90014-8.

[4]

C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular $\phi$-Laplacian, J. Differential Equations, 243 (2007), 536-557.  doi: 10.1016/j.jde.2007.05.014.

[5]

A. Cabada, An overview of the lower and upper solutions method with nonlinear boundary value conditions, Bound. Value Probl., 2011 (2011), Art. ID 893753, 18 pp. doi: 10.1155/2011/893753.

[6]

C. Corduneanu, Soluţii aproape periodice ale ecuaţiilor diferenţiale neliniare de ordinul al doilea, Com. Acad. R. P. Romîne, 5 (1955), 793–797.

[7]

C. De Coster and P. Habets, Two-Point Boundary Value Problems, Lower and Upper Solutions, Elsevier, Amsterdam, 2006.

[8]

A. M. Fink, Uniqueness theorems and almost periodic solutions to second order differential equations, J. Differential Equations, 4 (1968), 543-548.  doi: 10.1016/0022-0396(68)90004-1.

[9]

A. Fonda, G. Klun and A. Sfecci, Non-well-ordered lower and upper solutions for semilinear systems of PDEs, preprint, 2020.

[10]

A. Fonda and R. Toader, Lower and upper solutions to semilinear boundary value problems: An abstract approach, Topol. Methods Nonlinear Anal., 38 (2011), 59-93. 

[11]

A. Fonda and F. Zanolin, Bounded solutions of nonlinear second order ordinary differential equations, Discrete Contin. Dynam. Systems, 4 (1998), 91-98.  doi: 10.3934/dcds.1998.4.91.

[12]

P. Habets and R. L. Pouso, Examples of the nonexistence of a solution in the presence of upper and lower solutions, ANZIAM J., 44 (2003), 591-594.  doi: 10.1017/S1446181100012955.

[13]

P. Hartman, Ordinary Differential Equations, Wiley and Sons, New York, 1964.

[14]

L. K. Jackson and G. Klaasen, A variation of the topological method of Ważewski, SIAM J. Appl. Math., 20 (1971), 124-130.  doi: 10.1137/0120016.

[15]

J. L. KaplanA. Lasota and J. A. Yorke, An application of the Ważewski retract method to boundary value problems, Zeszyty Nauk. Uniw. Jagielloń. Prace Mat., 16 (1974), 7-14. 

[16]

H.-W. Knobloch, Zwei Kriterien für die Existenz periodischer Lösungen von Differentialgleichungen zweiter Ordnung, Arch. Math., 14 (1963), 182-185.  doi: 10.1007/BF01234941.

[17]

H.-W. Knobloch, Eine neue Methode zur Approximation periodischer Lösungen nicht-linearer Differentialgleichungen zweiter Ordnung, Math. Z., 82 (1963), 177-197.  doi: 10.1007/BF01111422.

[18]

J. L. Massera, The existence of periodic solutions of systems of differential equations, Duke Math. J., 17 (1950), 457-475. 

[19]

J. Mawhin and J. R. Ward, Bounded solutions of some second order nonlinear differential equations, J. London Math. Soc., (2) 58 (1998), 733–747. doi: 10.1112/S0024610798006784.

[20]

M. Nagumo, Über die Differentialgleichung $y" = f(t, y, y')$, Proc. Phys-Math. Soc. Japan, 19 (1937), 861-866. 

[21]

F. ObersnelP. Omari and S. Rivetti, Existence, regularity and stability properties of periodic solutions of a capillarity equation in the presence of lower and upper solutions, Nonlinear Anal. Real World Appl., 13 (2012), 2830-2852.  doi: 10.1016/j.nonrwa.2012.04.012.

[22]

R. Ortega, A boundedness result of Landesman-Lazer type, Differential Integral Equations, 8 (1995), 729-734. 

[23]

R. Ortega, Periodic Differential Equations in the Plane. A Topological Perspective, De Gruyter, Berlin, 2019. doi: 10.1515/9783110551167.

[24]

R. Ortega and M. Tarallo, Almost periodic upper and lower solutions, J. Differential Equations, 193 (2003), 343-358.  doi: 10.1016/S0022-0396(03)00130-X.

[25]

R. Ortega and A. Tineo, Resonance and non-resonance in a problem of boundedness, Proc. Amer. Math. Soc., 124 (1996), 2089-2096.  doi: 10.1090/S0002-9939-96-03457-0.

[26]

E. Picard, Sur l'application des méthodes d'approximations successives à l'étude de certaines équations différentielles ordinaires, J. Math. Pures Appl., 9 (1893), 217-271. 

[27]

F. Sadyrbaev, Ważewski method and upper and lower functions for higher order ordinary differential equations, Univ. Iagel. Acta Math., 36 (1998), 165-170. 

[28]

K. Schmitt, Bounded solutions of nonlinear second order differential equations, Duke Math. J., 36 (1969), 237-243. 

[29]

K. Schmitt and J. R. Ward, Almost periodic solutions of nonlinear second order differential equations, Results Math., 21 (1992), 190-199.  doi: 10.1007/BF03323078.

[30]

G. Scorza Dragoni, Il problema dei valori ai limiti studiato in grande per le equazioni differenziali del secondo ordine, Math. Ann., 105 (1931), 133-143.  doi: 10.1007/BF01455811.

[31]

R. Srzednicki, Ważewski method and Conley index, Handbook of Differential Equations (eds. A. Cañada, P. Drábek, A. Fonda) Elsevier/North-Holland, Amsterdam, 2004,591–684.

[32]

N. Soave and G. Verzini, Bounded solutions for a forced bounded oscillator without friction, J. Differential Equations, 256 (2014), 2526-2558.  doi: 10.1016/j.jde.2014.01.015.

[33]

M. Tarallo and Z. Zhou, Limit periodic upper and lower solutions in a generic sense, Discrete Contin. Dyn. Syst., 38 (2018), 293-309.  doi: 10.3934/dcds.2018014.

[34]

J. Y. Wang, W. J. Gao and Z. H. Lin, Boundary value problems for general second order equations and similarity solutions to the Rayleigh problem, Tohoku Math. J., (2) 47 (1995), 327–344. doi: 10.2748/tmj/1178225520.

[35]

T. Ważewski, Sur un principe topologique de l'examen de l'allure asymptotique des intégrales des équations différentielles ordinaires, Ann. Soc. Polon. Math., 20 (1947), 279-313. 

show all references

References:
[1]

S. Ahmad, A nonstandard resonance problem for ordinary differential equations, Trans. Amer. Math. Soc., 323 (1991), 857-875.  doi: 10.1090/S0002-9947-1991-1010407-9.

[2]

I. Barbălat, Applications du principe topologique de T. Ważewski aux équations différentielles du second ordre, Ann. Polon. Math., 5 (1958), 303-317. 

[3]

J. W. Bebernes and R. Wilhelmsen, A general boundary value problem technique, J. Differential Equations, 8 (1970), 404-415.  doi: 10.1016/0022-0396(70)90014-8.

[4]

C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular $\phi$-Laplacian, J. Differential Equations, 243 (2007), 536-557.  doi: 10.1016/j.jde.2007.05.014.

[5]

A. Cabada, An overview of the lower and upper solutions method with nonlinear boundary value conditions, Bound. Value Probl., 2011 (2011), Art. ID 893753, 18 pp. doi: 10.1155/2011/893753.

[6]

C. Corduneanu, Soluţii aproape periodice ale ecuaţiilor diferenţiale neliniare de ordinul al doilea, Com. Acad. R. P. Romîne, 5 (1955), 793–797.

[7]

C. De Coster and P. Habets, Two-Point Boundary Value Problems, Lower and Upper Solutions, Elsevier, Amsterdam, 2006.

[8]

A. M. Fink, Uniqueness theorems and almost periodic solutions to second order differential equations, J. Differential Equations, 4 (1968), 543-548.  doi: 10.1016/0022-0396(68)90004-1.

[9]

A. Fonda, G. Klun and A. Sfecci, Non-well-ordered lower and upper solutions for semilinear systems of PDEs, preprint, 2020.

[10]

A. Fonda and R. Toader, Lower and upper solutions to semilinear boundary value problems: An abstract approach, Topol. Methods Nonlinear Anal., 38 (2011), 59-93. 

[11]

A. Fonda and F. Zanolin, Bounded solutions of nonlinear second order ordinary differential equations, Discrete Contin. Dynam. Systems, 4 (1998), 91-98.  doi: 10.3934/dcds.1998.4.91.

[12]

P. Habets and R. L. Pouso, Examples of the nonexistence of a solution in the presence of upper and lower solutions, ANZIAM J., 44 (2003), 591-594.  doi: 10.1017/S1446181100012955.

[13]

P. Hartman, Ordinary Differential Equations, Wiley and Sons, New York, 1964.

[14]

L. K. Jackson and G. Klaasen, A variation of the topological method of Ważewski, SIAM J. Appl. Math., 20 (1971), 124-130.  doi: 10.1137/0120016.

[15]

J. L. KaplanA. Lasota and J. A. Yorke, An application of the Ważewski retract method to boundary value problems, Zeszyty Nauk. Uniw. Jagielloń. Prace Mat., 16 (1974), 7-14. 

[16]

H.-W. Knobloch, Zwei Kriterien für die Existenz periodischer Lösungen von Differentialgleichungen zweiter Ordnung, Arch. Math., 14 (1963), 182-185.  doi: 10.1007/BF01234941.

[17]

H.-W. Knobloch, Eine neue Methode zur Approximation periodischer Lösungen nicht-linearer Differentialgleichungen zweiter Ordnung, Math. Z., 82 (1963), 177-197.  doi: 10.1007/BF01111422.

[18]

J. L. Massera, The existence of periodic solutions of systems of differential equations, Duke Math. J., 17 (1950), 457-475. 

[19]

J. Mawhin and J. R. Ward, Bounded solutions of some second order nonlinear differential equations, J. London Math. Soc., (2) 58 (1998), 733–747. doi: 10.1112/S0024610798006784.

[20]

M. Nagumo, Über die Differentialgleichung $y" = f(t, y, y')$, Proc. Phys-Math. Soc. Japan, 19 (1937), 861-866. 

[21]

F. ObersnelP. Omari and S. Rivetti, Existence, regularity and stability properties of periodic solutions of a capillarity equation in the presence of lower and upper solutions, Nonlinear Anal. Real World Appl., 13 (2012), 2830-2852.  doi: 10.1016/j.nonrwa.2012.04.012.

[22]

R. Ortega, A boundedness result of Landesman-Lazer type, Differential Integral Equations, 8 (1995), 729-734. 

[23]

R. Ortega, Periodic Differential Equations in the Plane. A Topological Perspective, De Gruyter, Berlin, 2019. doi: 10.1515/9783110551167.

[24]

R. Ortega and M. Tarallo, Almost periodic upper and lower solutions, J. Differential Equations, 193 (2003), 343-358.  doi: 10.1016/S0022-0396(03)00130-X.

[25]

R. Ortega and A. Tineo, Resonance and non-resonance in a problem of boundedness, Proc. Amer. Math. Soc., 124 (1996), 2089-2096.  doi: 10.1090/S0002-9939-96-03457-0.

[26]

E. Picard, Sur l'application des méthodes d'approximations successives à l'étude de certaines équations différentielles ordinaires, J. Math. Pures Appl., 9 (1893), 217-271. 

[27]

F. Sadyrbaev, Ważewski method and upper and lower functions for higher order ordinary differential equations, Univ. Iagel. Acta Math., 36 (1998), 165-170. 

[28]

K. Schmitt, Bounded solutions of nonlinear second order differential equations, Duke Math. J., 36 (1969), 237-243. 

[29]

K. Schmitt and J. R. Ward, Almost periodic solutions of nonlinear second order differential equations, Results Math., 21 (1992), 190-199.  doi: 10.1007/BF03323078.

[30]

G. Scorza Dragoni, Il problema dei valori ai limiti studiato in grande per le equazioni differenziali del secondo ordine, Math. Ann., 105 (1931), 133-143.  doi: 10.1007/BF01455811.

[31]

R. Srzednicki, Ważewski method and Conley index, Handbook of Differential Equations (eds. A. Cañada, P. Drábek, A. Fonda) Elsevier/North-Holland, Amsterdam, 2004,591–684.

[32]

N. Soave and G. Verzini, Bounded solutions for a forced bounded oscillator without friction, J. Differential Equations, 256 (2014), 2526-2558.  doi: 10.1016/j.jde.2014.01.015.

[33]

M. Tarallo and Z. Zhou, Limit periodic upper and lower solutions in a generic sense, Discrete Contin. Dyn. Syst., 38 (2018), 293-309.  doi: 10.3934/dcds.2018014.

[34]

J. Y. Wang, W. J. Gao and Z. H. Lin, Boundary value problems for general second order equations and similarity solutions to the Rayleigh problem, Tohoku Math. J., (2) 47 (1995), 327–344. doi: 10.2748/tmj/1178225520.

[35]

T. Ważewski, Sur un principe topologique de l'examen de l'allure asymptotique des intégrales des équations différentielles ordinaires, Ann. Soc. Polon. Math., 20 (1947), 279-313. 

Figure 1.  In the red region $ f(t,x,y)>\alpha'(t) $, in the green one $ f(t,x,y)<\alpha'(t) $
Figure 2.  A section of the set $ V $ at time $ t $ with its egress points
Figure 3.  The grid made of quadrilaterals near the point $ (\alpha(t),y_\alpha(t)) $
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