Nonlinear vector Duffing inclusions with no growth restriction on the orientor field
Keywords
Duffing system, extremal solutions, nonlinear operator of monotone type, strong relaxationsAbstract
We consider nonlinear multivalued Dirichlet Duffing systems. We do not impose any growth condition on the multivalued perturbation. Using tools from the theory of nonlinear operators of monotone type, we prove existence theorems for the convex and the nonconvex problems. Also we show the existence of extremal trajectories and show that such solutions are $C_0^1(T,\mathbb{R}^N)$-dense in the solution set of the convex problem (strong relaxation theorem).References
J. Francu̇, Monotone operators. A survey directed to applications to differential equations, Appl. Math. 35 (1990), 257–301.
M. Galewski, On the Dirichlet problem for a Duffing type equation, Electron. J. Qual. Theory Differ. Equ. 2011 (2011), no. 15, 1–12.
L. Gasiński and N.S. Papageorgiou, Nonlinear Analysis, Ser. Math. Anal. Appl., vol. 9, Chapman and Hall/CRC Press, Boca Raton, Florida, 2006.
L. Gasiński and N.S. Papageorgiou, Exercises in Analysis, Part 1, Problem Books in Mathematics, Springer, Cham, 2014.
L. Gasiński and N.S. Papageorgiou, Exercises in Analysis, Part 2, Problem Books in Mathematics, Springer, Cham, 2016.
P. Hartman, On boundary value problems for systems of ordinary nonlinear second order differential equations, Trans. Amer. Math. Soc. 96 (1960), 493–509.
P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964.
S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I. Theory, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis. Vol. II. Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.
P. Kalita and P.M. Kowalski, On multivalued Duffing equation, J. Math. Anal. Appl. 462 (2018), 1130–1147.
P.M. Kowalski, Well-posed Dirichlet problems pertaining to the Duffing equation, Electron. J. Qual. Theory Differ. Equ. 2014 (2014), no. 59, 1–15.
R. Manásevich and J. Mawhin, Boundary value problems for nonlinear perturbations of vector p-Laplacian-like operators, J. Korean Math. Soc. 37 (2000), 665–685.
N.S. Papageorgiou, C. Vetro and F. Vetro, Extremal solutions and strong relaxation for nonlinear multivalued systems with maximal monotone terms, J. Math. Anal. Appl. 461 (2018), 401–421.
N.S. Papageorgiou, C. Vetro and F. Vetro, Nonlinear multivalued Duffing systems, J. Math. Anal. Appl. 468 (2018), 376–390.
P. Tomiczek, Remark on Duffing equation with Dirichlet boundary condition, Electron. J. Differential Equations 2007 (2007), no. 81, 1–3.
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