Nonresonance and existence for systems of nonlinear operator equations


An existence theory for systems of two non-linear operator equations in Hilbert spaces is presented under non-resonance conditions with respect to two spectra and in terms of matrices convergent to zero. The theory is then applied to elliptic systems.


Dezideriu Muzsi
Department of Applied Mathematics , Babeş–Bolyai University , Cluj, Romania

Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania


non-linear operator equation; non-linear system; fixed point; non resonance; eigenvalues; energetic norm; elliptic system.

Paper coordinates

D. Muzsi, R. Precup, Nonresonance and existence for systems of nonlinear operator equations, Appl. Anal. 87 (2008), no. 9, 1005-1018,


About this paper


Applicable Analysis

Publisher Name

Taylor and Francis

Print ISSN
Online ISSN

google scholar link

[1] Cardoulis, L2002Existence of solutions for some semilinear elliptic systemsRostock. Math. Kolloq., 56: 2938.  [Google Scholar]
Clément, Phde Figueiredo, DG and Mitidieri, EE1992Positive solutions of semilinear elliptic systemsComm. P.D.E., 17: 923940.  [Taylor & Francis Online][Web of Science ®][Google Scholar]
Dall’Acqua, A2003Positive solutions for a class of reaction-diffusion systemsComm. Pure Appl. Anal., 2: 6576.  [Google Scholar]
de Figueiredo, DG1998. “Semilinear elliptic systems”. In in Nonlinear Functional Analysis and Applications to Differential Equations (Trieste, 1997)122152River Edge, NJWorld Sci. Publ..  [Google Scholar]
Fleckinger, JHernandez, J and de Thélin, F1995On maximum principles and existence of positive solutions for some cooperative systemsDiff. Int. Eq., 8: 6885.  [Google Scholar]
Rothe, F1981Global existence of branches of stationary solutions for a system of reaction diffusion equations from biologyNonlinear Anal., 5: 487498.  [Crossref][Google Scholar]
Souto, MAS1995A priori estimates and existence of positive solutions of non-linear cooperative elliptic systemsDiff. Int. Eq., 8: 12451258.  [Google Scholar]
Muzsi, DA theory of semilinear operator equations under nonresonance conditionsNonlinear Funct. Anal. Appl., to appear [Google Scholar]
Mawhin, J and Ward, J Jr1981Nonresonance and existence for non-linear elliptic boundary value problemsNonlinear Anal., 6: 677684.  [Google Scholar]
Mawhin, J and Ward, JR1982Nonuniform nonresonance conditions at the first two eigenvalues for periodic solutions forced Lienard and Duffing equationsRocky M. J. Math., 12: 643654.  [Google Scholar]
Perov, AI and Kibenko, AV1966O a certain general method for investigation of boundary value problemsIzv. Akad. Nauk SSSR, 30: 249264. (Russian) [Google Scholar]
Precup, R2007A vector version of Krasnoselskii’s fixed point theorem in cones and positive periodic solutions of non-linear systemsJ. Fixed Point Theor. Appl., 2: 141151.  [Crossref][Web of Science ®][Google Scholar]
Mihlin, SG1977Linear Partial Differential EquationsMoscowVysshaya Shkola. (Russian) [Google Scholar]
Brezis, H1983Analyse fonctionelle. Theorie et applicationsParisDunod.  [Google Scholar]
Precup, R2004Lectures on Partial Differential EquationsCluj-NapocaCluj University Press. (Romanian) [Google Scholar]
Precup, R1995. “Existence results for non-linear boundary value problems under nonresonance conditions”. In Qualitative Problems for Differential Equations and Control Theory, Edited by: Corduneanu, C263273SingaporeWorld Scientific.  [Google Scholar]
O’Regan, D and Precup, R2001Theorems of Leray-Schauder Type and ApplicationsAmsterdamGordon and Breach.  [Google Scholar]
Precup, R2002Methods in Nonlinear Integral EquationsDordrechtKluwer.  [Crossref][Google Scholar]
Gilbarg, D and Trudinger, NS1983Elliptic Partial Differential Equations of Second OrderBerlinSpringer.  [Crossref][Google Scholar]

Related Posts