Localization of critical points via mountain pass type theorems


REZUMAT: Metode itera\U{21b}iior monotone pentru problema cu valori ini\U{21b}iale relativ\u{a} la o ecua\U{21b}ie integral\u{a} din biomatematic\u{a}. \^{I}n lucrare este prezentat\u{a} o metod\u{a} constructiv\u{a} de rezolvare a problemei (1)-(2) \^{\i}n ipotezele (i)-(iv) presupun\^{a}nd c\u{a} func\U{21b}ia \(f(t,x))\ este monoton\u{a} \^{\i}n raport cu \(x)\.
Un aspect nou con\U{21b}inut \^{\i}n acest articol \^{\i}l constituie adaptarea metodei itera\U{21b}iilor monotone la cazul operatorilor anti-izotoni, \^{\i}n particular, la cazul c\^{a}nd \(f(t,x))\ este o func\U{21b}ie
necresc\u{a}toare \^{\i}n \(x)\.


REZUMAT: Metode iterative \U{21b}iior monotone pentru problema cu valori inițiale relative la o ecuație integrală din biomatematică. În lucrare este prezentată o metodă constructivă de rezolvare a problemei (1)-(2) și în ipotezele (i)-(iv) presupunând că funcția (f(t,x)) este monotonă și în raport cu (x). Un aspect nou conținut și în acest articol îl constituie adaptarea metodei iterațiilor monotone la cazul operatorilor anti-izotoni, și în particular, la cazul când (f(t,x)) este o funcție necrescătoare și în (x).


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



Paper coordinates

R. Precup, Localization of critical points via mountain pass type theorems, in Critical Point Theory and Its Applications, Proceedings of the International Summer School on Critical Point Theory and Applications Cluj-Napoca, July 9th-July 13th 2007, Cs. Varga, A. Kristaly and P.A. Blaga eds., Casa Cartii de Stiinta, Cluj-Napoca, 2007, 53-67.


About this paper

Publisher Name
Print ISSN
Online ISSN


google scholar link

[1] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381.
[2] K. Deimling, Nonlinear Functional Analysis, Springer, 1985.
[3] M. Frigon, On a new notion of linking and application to elliptic problems at resonance, J. Differential Equations 153 (1999), 96-120.
[4] A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, 2003.
[5] N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points, Ann. Inst. Poincare Anal. Nonlineaire 6 (1989), 321-330.
[6] D. Guo, J. Sun and G. Qi, Some extensions of the mountain pass lemma, Differential Integral Equations 1 (1988), 351-358.
[7] M.A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.
[8] Z. Liu and J. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations 172 (2001), 257-299.
[9] L. Ma, Mountain pass on a closed convex set, J. Math. Anal. Appl. 205 (1997), 531-536.
[10] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989.
[11] D. O’Regan and R. Precup, Theorems of Leray-Schauder Type and Applications, Gordon and Breach, Amsterdam, 2001.
[12] R. Precup, On the Palais-Smale condition for Hammerstein integral equations in Hilbert spaces, Nonlinear Anal. 47 (2001), 1233-1244.
[13] R. Precup, Nontrivial solvability of Hammerstein integral equations in Hilbert spaces. In: Seminaire de la Theorie de la Meilleure Approximation, Convexite et Optimisation (E. Popoviciu ed.), Srima, Cluj–Napoca, 2000, 255–265.
[14] R. Precup, Methods in Nonlinear Integral Equations, Kluwer, Dordrecht, 2002.
[15] R. Precup, A compression type mountain pass theorem in conical shells, J. Math. Anal. Appl., to appear.
[16] P. Pucci and J. Serrin, A mountain pass theorem, J. Differential Equations 60 (1985), 142-149.
[17] P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. in Math., Amer. Math. Soc., Providence, RI, vol. 65, 1986.
[18] M. Schechter, A bounded mountain pass lemma without the (PS) condition and applications, Trans. Amer. Math. Soc. 331 (1992), 681-703.
[19] M. Schechter, Linking Methods in Critical Point Theory, Birkhauser, Basel, 1999.
[20] M. Struwe, Variational Methods, Springer, Berlin, 1990.


Related Posts