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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"

Alessandro Fonda 1,, and  Rodica Toader 2,

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.

Keywords: Lower and upper solutions, periodic systems, Ważewski method, Massera Theorem, bounded solutions, almost periodic solutions.
Mathematics Subject Classification: Primary: 34C11, 34C25; Secondary: 34C27.
Citation: Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[9]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

J. L. Kaplan, A. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[21]

F. Obersnel, P. 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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[27]

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

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[9]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

J. L. Kaplan, A. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[21]

F. Obersnel, P. 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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[27]

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

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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