Geodesic
The Neumann problem for quasilinear differential equations
Archivum mathematicum, Tome 40 (2004) no. 4, pp. 321-333 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this note we prove the existence of extremal solutions of the quasilinear Neumann problem $-( \vert x^{^{\prime }}(t) \vert ^{p-2}x^{^{\prime }}(t))^{^{\prime }} = f(t,x(t),x ^{^{\prime }}(t))$, a.e. on $T$, $x^{^{\prime }}(0) = x^{^{\prime }}(b) =0$, $2\le p \infty $ in the order interval $[\psi ,\varphi ]$, where $\psi $ and $\varphi $ are respectively a lower and an upper solution of the Neumann problem.
In this note we prove the existence of extremal solutions of the quasilinear Neumann problem $-( \vert x^{^{\prime }}(t) \vert ^{p-2}x^{^{\prime }}(t))^{^{\prime }} = f(t,x(t),x ^{^{\prime }}(t))$, a.e. on $T$, $x^{^{\prime }}(0) = x^{^{\prime }}(b) =0$, $2\le p \infty $ in the order interval $[\psi ,\varphi ]$, where $\psi $ and $\varphi $ are respectively a lower and an upper solution of the Neumann problem.
Classification : 34B15, 35J25, 35J60, 35J65
Keywords: upper solution; lower solution; order interval; truncation function; penalty function; pseudomonotone operator; coercive operator; Leray-Schauder principle; maximal solution; minimal solution 
@article{ARM_2004_40_4_a0,
     author = {Cardinali, Tiziana and Papageorgiou, Nikolaos S. and Servadei, Raffaella},
     title = {The {Neumann} problem for quasilinear differential equations},
     journal = {Archivum mathematicum},
     pages = {321--333},
     year = {2004},
     volume = {40},
     number = {4},
     mrnumber = {2129954},
     zbl = {1122.35030},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2004_40_4_a0/}
}
TY  - JOUR
AU  - Cardinali, Tiziana
AU  - Papageorgiou, Nikolaos S.
AU  - Servadei, Raffaella
TI  - The Neumann problem for quasilinear differential equations
JO  - Archivum mathematicum
PY  - 2004
SP  - 321
EP  - 333
VL  - 40
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/ARM_2004_40_4_a0/
LA  - en
ID  - ARM_2004_40_4_a0
ER  - 
%0 Journal Article
%A Cardinali, Tiziana
%A Papageorgiou, Nikolaos S.
%A Servadei, Raffaella
%T The Neumann problem for quasilinear differential equations
%J Archivum mathematicum
%D 2004
%P 321-333
%V 40
%N 4
%U http://geodesic.mathdoc.fr/item/ARM_2004_40_4_a0/
%G en
%F ARM_2004_40_4_a0
Cardinali, Tiziana; Papageorgiou, Nikolaos S.; Servadei, Raffaella. The Neumann problem for quasilinear differential equations. Archivum mathematicum, Tome 40 (2004) no. 4, pp. 321-333. http://geodesic.mathdoc.fr/item/ARM_2004_40_4_a0/

[1] Boccardo L., Drábek P., Giachetti D., Kučera M.: Generalization of Fredholm alternative for nonlinear differential operators. Nonlinear Anal. 10 (1986), 1083–1103. | MR

[2] Brézis H.: Analyse functionelle: Théorie et applications. Masson, Paris 1983. | MR

[3] Del Pino M., Elgueta M., Manasevich R.: A homotopic deformation along p of a Leray-Schauder degree result and existence for $(\vert u^{^{\prime }}(t) \vert ^{p-2}u^{^{\prime }}(t))^{^{\prime }} + f(t,u(t)) = 0, u(0)=u(T)=0, p>1)$. J. Differential Equations 80 (1989), 1–13. | MR

[4] Drábek P.: Solvability of boundary value problems with homogeneous ordinary differential operator. Rend. Istit. Mat. Univ. Trieste 18 (1986), 105–125. | MR

[5] Dunford N., Schwartz J. T.: Linear operators. Part I: General theory. Interscience Publishers, New York 1958–1971. | MR

[6] Gao W., Wang J.: A nonlinear second order periodic boundary value problem with Carathéodory functions. Ann. Polon. Math. LXVII. 3 (1995), 283–291. | MR

[7] Gilbarg D., Trudinger N.: Elliptic partial differential equations of second order. Springer-Verlag, Berlin 1983. | MR | Zbl

[8] Dugundji J., Granas A.: Fixed point theory, Vol. I. Monogr. Mat. PWN, Warsaw 1992.

[9] Guo Z.: Boundary value problems of a class of quasilinear ordinary differential equations. Differential Integral Equations 6, No. 3 (1993), 705–719. | MR | Zbl

[10] Halidias N., Papageorgiou N. S.: Existence of solutions for nonlinear parabolic problems. Arch. Math. (Brno) 35 (1999), 255–274. | MR | Zbl

[11] Hu S., Papageorgiou N. S.: Handbook of multivalued analysis. Volume I: Theory. Kluwer, Dordrecht, The Netherlands 1997. | MR | Zbl

[12] Marcus M., Mizel V. J.: Absolute continuity on tracks and mapping of Sobolev spaces. Arch. Rational Mech. Anal. 45 (1972), 294–320. | MR

[13] O’Regan D.: Some General existence principles and results for $(\phi (y^{^{\prime }}))= qf(t,y,y^{^{\prime }}), 0. SIAM J. Math. Anal. 24 No. 30 (1993), 648–668. MR 1215430

[14] Pascali D., Sburlan S.: Nonlinear mapping of monotone type. Editura Academiei, Bucuresti, Romania 1978. | MR

[15] Peressini A. L.: Ordered topological vector spaces. Harper & Row, New York, Evanstone, London 1967. | MR | Zbl

[16] Wang J., Jiang D.: A unified approach to some two-point, three-point and four-point boundary value problems with Carathéodory functions. J. Math. Anal. Appl. 211 (1997), 223–232. | MR | Zbl

[17] Zeidler E.: Nonlinear functional analysis and its applications II. Springer-Verlag, New York 1990. | MR | Zbl