Non-well-ordered lower and upper solutions for semilinear systems of PDEs
Abstract
We prove existence results for systems of boundary value problems involving elliptic second-order differential operators. The assumptions involve lower and upper solutions, which may be either well-ordered, or not at all. The results are stated in an abstract framework, and can be translated also for systems of parabolic type.
References
- 1. , On the number of solutions of nonlinear equations in ordered Banach spaces, J. Funct. Anal. 11 (1972) 346–384. Crossref, Google Scholar
- 2. , Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976) 620–709. Crossref, ISI, Google Scholar
- 3. , Elliptic equations with noninvertible Fredholm linear part and bounded nonlinearities, Math. Z. 158 (1978) 179–194. Crossref, ISI, Google Scholar
- 4. , Periodic boundary value problems for systems of second order differential equations, J. Differential Equations 13 (1973) 32–47. Crossref, ISI, Google Scholar
- 5. , Invariant sets and the Hukuhara–Kneser property for systems of parabolic partial differential equations, Rocky Mountain J. Math. 7 (1977) 557–567. Crossref, Google Scholar
- 6. , Two-Point Boundary Value Problems, Lower and Upper Solutions (Elsevier, Amsterdam, 2006). Google Scholar
- 7. , Existence and localization of solution for elliptic problem in presence of lower and upper solutions without any order, J. Differential Equations 145 (1998) 420–452. Crossref, ISI, Google Scholar
- 8. ,
A qualitative analysis, via lower and upper solutions, of first order periodic evolutionary equations with lack of uniqueness , in Handbook of Differential Equations, ODE’s, eds. A. CanadaP. DrabekA. Fonda (Elsevier, Amsterdam, 2006), pp. 203–339. Crossref, Google Scholar - 9. , Stability and instability in periodic parabolic problems via lower and upper solutions, Quad. Mat. 539 (2003) 1–103. Google Scholar
- 10. A. Fonda, G. Klun and A. Sfecci, Periodic solutions of second order differential equations in Hilbert spaces, Mediterr. J. Math., to appear. Google Scholar
- 11. , Lower and upper solutions to semilinear boundary value problems: An abstract approach, Topol. Methods Nonlinear Anal. 38 (2011) 59–94. ISI, Google Scholar
- 12. , Elliptic Partial Differential Equations of Second Order, 2nd edn. (Springer, Berlin, 1983). Crossref, Google Scholar
- 13. , A necessary and sufficient condition of nonresonance for a semilinear Neumann problem, Proc. Amer. Math. Soc. 114 (1992) 433–442. Crossref, ISI, Google Scholar
- 14. , Existence and localization of solutions of second order elliptic problems using lower and upper solutions in the reversed order, Top. Methods Nonlinear Anal. 8 (1996) 25–56. Crossref, Google Scholar
- 15. ,
Qualitative methods for nonlinear diffusion equations , in Nonlinear Diffusion Problems (Montecatini Terme, 1985),Lecture Notes in Mathematics , Vol. 1224 (Springer, Berlin, 1986), pp. 47–118. Crossref, Google Scholar - 16. , Non-ordered lower and upper solutions and solvability of the periodic problem for the Liénard and the Rayleigh equations, Rend. Istit. Mat. Univ. Trieste 20 (1988) 54–64. Google Scholar
- 17. , Nonlinear Parabolic and Elliptic Equations (Plenum Press, New York, 1992). Google Scholar
- 18. , Elliptic Differential Equations and Obstacle Problems (Plenum Press, New York, 1987). Crossref, Google Scholar
Remember to check out the Most Cited Articles! |
---|
Be inspired by these NEW Mathematics books for inspirations & latest information in your research area! |