A general nonresonance theory of semilinear operator equations under regularity conditions is developed. Existence of weak solutions (in the energetic space) is established by means of several fixed point principles. Typical applications to elliptic equations with convection terms are presented.
Department of Applied Mathematics Babes–Bolyai University
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
nonlinear operator equation, fixed point, nonresonance, eigenvalues, energetic norm, elliptic equation.
D. Muzsi, R. Precup, Nonresonance theory for semilinear operator equations under regularity conditions, Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity, 6 (2008), 75-89.
Annals of the Tiberiu Popoviciu Seminar
of Functional Equations, Approximation and Convexity
google scholar link
 R. P. Agarwal, D. O’Regan and V. Lakshmikantham, Nonuniform nonresonance at the first eigenvalue for singular boundary value problems with sign changing nonlinearities, J. Math. Anal. Appl. 274 (2002), no. 1, 404–423.
 H. Brezis, Analyse fonctionelle. Theorie et applications, Dunod, Paris, 1983.
 D.D. Hai and K. Schmitt, Existence and uniqueness results for nonlinear boundary value problems, Rocky Mountain J. Math. 24 (1994), 77-91.
 L.V. Kantorovich and G.P. Akilov, Functional Analysis, Pergamon Press, Oxford-Elmsford, N.Y., 1982.
 J. Mawhin and J. Ward Jr., Nonresonance and existence for nonlinear elliptic boundary value problems, Nonlinear Anal. 6 (1981), 677-684.
 J. Mawhin and J. R. Ward, Nonuniform nonresonance conditions at the first two eigenvalues for periodic solutions forced Lienard and Duffing equations, Rocky M. J. Math. 12 (1982), 643-654.
 M. Meehan and D. O’Regan, Existence Theory for Nonlinear Integral and Integrodifferential Equations, Kluwer Academic Publishers, Dordrecht-Boston-London, 1998.
 S.G. Mihlin, Linear Partial Differential Equations (Russian), Vysshaya Shkola, Moscow, 197
 D. Muzsi, A theory of semilinear operator equations under nonresonance conditions, J. Nonlinear Funct. Anal. Appl., to appear.
 D. O’Regan, Nonresonant nonlinear singular problems in the limit circle case, J. Math. Anal. Applic., 197(1996), 708-725.
 D. O’Regan, Caratheodory theory of nonresonant second order boundary value problems, Differential Equations and Dynamical Systems, 4 (1996), 57-77.
 D. O’Regan and R. Precup, Theorems of Leray-Schauder Type and Applications, Gordon and Breach, Amsterdam, 2001.
 R. Precup, Existence Results for Nonlinear Boundary Value Problems Under Nonresonance Conditions, in: Qualitative Problems for Differential Equations and Control Theory, C. Corduneanu (ed.), World Scientific, Singapore, 1995, 263-273.
 R. Precup, Lectures on Partial Differential Equations (Romanian), Cluj University Press, Cluj-Napoca, 2004.
 R. Precup, Methods in Nonlinear Integral Equations, Kluwer, Dordrecht, 2002.