Existence and approximation of a solution for a two point nonlinear Dirichlet problem

Pasquale Candito; Roberto Livrea; Luís Sanchez

doi:10.3934/dcdss.2024094

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2024. Doi: 10.3934/dcdss.2024094

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Existence and approximation of a solution for a two point nonlinear Dirichlet problem

1.
Department DICEAM, University of Reggio Calabria, 89122 Reggio Calabria, Italy
2.
Department of Mathematics and Informatics, University of Palermo, Via Archirafi, Palermo, Italy
3.
CMAFcIO – Faculdade de Ciências, Universidade de Lisboa, Portugal

^*Corresponding author
^*Corresponding author

Received: March 2024

Revised: May 2024

Early access: June 2024

This work was started while the first-named author was visiting CMAFcIO, and he is grateful for the kind hospitality of the host institute

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Abstract

The existence of at least one positive solution to a second-order nonlinear two-point boundary value problem, is established. Combining difference methods with Brouwer fixed point and Ascolì-Arzelà theorems, we get a solution as the limit of an appropriate sequence of piecewise linear interpolations. Furthermore, a priori bounds on the infinite norm of a solution and its derivatives are pointed out. Some examples are also discussed to illustrate our results.

Keywords:

Difference methods,
a priori bounds,
fixed point methods,
nonlinear two point boundary value problems.

Mathematics Subject Classification: Primary: 39A27; Secondary: 34B15.

Citation:

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References

[1]	R. P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York, Basel, 2000.
[2]	E. Amoroso, P. Candito and J. Mawhin, Existence of a priori bounded solutions for discrete two-point boundary value problems, J. Math. Anal. Appl., 519 (2023), Paper No. 126807, 18 pp. doi: 10.1016/j.jmaa.2022.126807.
[3]	P. Amster and J. Haddad, A Hartman-Nagumo type condition for a class of contractible domains, Topol. Methods Nonlinear Anal., 41 (2013), 287-304.
[4]	G. Bonanno, P. Candito and G. D'Aguì, Variational methods on finite dimensional Banach spaces and discrete problems, Adv. Nonlinear Stud., 14 (2014), 915-939. doi: 10.1515/ans-2014-0406.
[5]	G. Bonanno, P. Candito and D. Motreanu, A coincidence point theorem for sequentially continuous mappings, J. Math. Anal. Appl., 435 (2016), 606-615. doi: 10.1016/j.jmaa.2015.10.039.
[6]	A. Cabada, D. O'Regan and R. L. Pouso, Second order problems with functional conditions including Sturm-Liouville and multipoint conditions, Math. Nachr., 281 (2008), 1254-1263. doi: 10.1002/mana.200510675.
[7]	P. Candito and R. Livrea, An existence result for a Neumann problem, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 22 (2015), 481-488.
[8]	P. Candito and R. Livrea, Differential and difference equations with applications, Existence Results for Periodic Boundary Value Problems with a Convection Term, Springer Proc. Math. Stat., 333, Springer, Cham, 2020, 593-602
[9]	G. Dinca and J. Mawhin, Brouwer Degree. The Core of Nonlinear Analysis, Birkhäuser, Basel, 2021. doi: 10.1007/978-3-030-63230-4.
[10]	G. Feltrin and F. Zanolin, Bound sets for a class of $\phi-$Laplacian operators, J. Differential Equations, 297 (2021), 508-535. doi: 10.1016/j.jde.2021.06.034.
[11]	M. Fukuhara, Quelques recherches sur les équations différentielles du second ordre, Japanese J. Math., 5 (1928), 351-367.
[12]	R. Gaines, Difference equations associated with boundary value problems for second order nonlinear ordinary differential equations, SIAM J. Numer. Anal., 11 (1974), 411-434. doi: 10.1137/0711035.
[13]	M. R. Grossinho, F. Minhós and A. I. Santos, A note on a class of problems for a higher-order fully nonlinear equation under one-sided Nagumo-type condition, Nonlinear Anal., 70 (2009), 4027-4038. doi: 10.1016/j.na.2008.08.011.
[14]	P. Hartman, Ordinary Differential Equations, The Society for Industrial and Applied Mathematics, 2$^{nd}$, Philadelphia, 2002. doi: 10.1137/1.9780898719222.
[15]	W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, Inc., Boston, MA, 1991.
[16]	J. Mahwin, Boundary value problems for nonlinear ordinary differential equations: From successive approximations to topology, in Development of Mathematics 1900-1950, Birkhäuser, Basel, 1994, 443-477.
[17]	S. A. Marano, Existence theorems for a multivalued boundary value problem, Bull. Austral. Math. Soc., 45 (1992), 249-260. doi: 10.1017/S0004972700030112.
[18]	C. Marcelli and F. Papalini, Boundary value problems for strongly nonlinear equations under a Wintner-Nagumo growth condition, Bound. Value Probl., (2017), Paper No. 183, 15 pp. doi: 10.1186/s13661-017-0913-7.
[19]	J. Mawhin, The Bernstein-Nagumo problem and two-point boundary value problems for ordinary differential equations, Qualitative Theory of Differential Equations, I, II Szeged, (1979), 709-740, Colloq. Math. Soc. János Bolyai, 30, North-Holland, Amsterdam-New York, 1981.
[20]	J. Mawhin and H. B. Thompson, Nagumo conditions and second-order quasilinear equations with compatible nonlinear functional boundary conditions, Rocky Mountain J. Math., 41 (2011), 573-596. doi: 10.1216/RMJ-2011-41-2-573.
[21]	I. Rachůnková and C. C. Tisdell, Existence of non-spurious solutions to discrete Dirichlet problems with lower and upper solutions, Nonlinear Anal., 67 (2007), 1236-1245. doi: 10.1016/j.na.2006.07.010.
[22]	C. C. Tisdell, A discrete approach to continuous second-order boundary value problems via monotone iterative techniques, Int. J. Difference Equ., 12 (2017), 145-160.
[23]	C. C. Tisdell, Y. Liu and Z. Liu, Existence of solutions to discrete and continuous second-order boundary value problems via Lyapunov functions and a priori bounds, Electron. J. Qual. Theory Differ. Equ., (2019), Paper No. 42, 11 pp.