Geodesic
Dynamical systems method (DSM) for solving nonlinear operator equations in Banach spaces
Eurasian mathematical journal, Tome 3 (2012) no. 1, pp. 86-96.

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $F(u)=h$ be a solvable operator equation in a Banach space $X$ with a Gateaux differentiable norm. Under minimal smoothness assumptions on $F$, sufficient conditions are given for the validity of the Dynamical Systems Method (DSM) for solving the above operator equation. It is proved that the DSM (Dynamical Systems Method) $$ \dot u(t)=-A_{a(t)}^{-1}(u(t))[F(u(t))+a(t)u(t)-f)],\quad u(0)=u_0, $$ converges to $y$ as $t\to+\infty$, for $a(t)$ properly chosen. Here $F(y)=f$, and $\dot u$ denotes the time derivative. 
@article{EMJ_2012_3_1_a6,
     author = {A. G. Ramm},
     title = {Dynamical systems method {(DSM)} for solving nonlinear operator equations in {Banach} spaces},
     journal = {Eurasian mathematical journal},
     pages = {86--96},
     publisher = {mathdoc},
     volume = {3},
     number = {1},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2012_3_1_a6/}
}
TY  - JOUR
AU  - A. G. Ramm
TI  - Dynamical systems method (DSM) for solving nonlinear operator equations in Banach spaces
JO  - Eurasian mathematical journal
PY  - 2012
SP  - 86
EP  - 96
VL  - 3
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/EMJ_2012_3_1_a6/
LA  - en
ID  - EMJ_2012_3_1_a6
ER  - 
%0 Journal Article
%A A. G. Ramm
%T Dynamical systems method (DSM) for solving nonlinear operator equations in Banach spaces
%J Eurasian mathematical journal
%D 2012
%P 86-96
%V 3
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/EMJ_2012_3_1_a6/
%G en
%F EMJ_2012_3_1_a6
A. G. Ramm. Dynamical systems method (DSM) for solving nonlinear operator equations in Banach spaces. Eurasian mathematical journal, Tome 3 (2012) no. 1, pp. 86-96. http://geodesic.mathdoc.fr/item/EMJ_2012_3_1_a6/

[1] Yu.L. Daleckii, M.G. Krein, Stability of solutions of differential equations in Banach spaces, Amer. Math. Soc., Providence, RI, 1974 | MR

[2] M. Day, Normed linear spaces, Springer-Verlag, Berlin, 1958 | MR | Zbl

[3] N. Dunford, J. Schwartz, Linear operators, v. 1, General theory, Interscience, New York, 1958 | Zbl

[4] M. Gavurin, “Nonlinear functional equations and continuous analysis of iterative methods”, Izvestiya VUS'ov, Matem., 5 (1958), 18–31 (Russian) | MR | Zbl

[5] P. Hartman, Ordinary differential equations, J. Wiley, New York, 1964 | MR

[6] N.S. Hoang, A.G. Ramm, “Dynamical Systems Method for solving nonlinear equations with monotone operators”, Math. of Comput., 79:269 (2010), 239–258 | DOI | MR | Zbl

[7] N.S. Hoang, A.G. Ramm, “Nonlinear differential inequality”, Math. Inequalities Appl., 14:4 (2011), 967–976 | DOI | MR | Zbl

[8] N.S. Hoang, A.G. Ramm, “Some nonlinear inequalities and applications”, Journal of Abstract Diff. Equations and Applications, 2:1 (2011), 84–101 | MR

[9] N.S. Hoang, A.G. Ramm, Dynamical Systems Method and Applications. Theoretical Developments and Numerical Examples, Wiley, Hoboken, 2012 | MR

[10] A.G. Ramm, Dynamical systems method for solving operator equations, Elsevier, Amsterdam, 2007 | MR | Zbl

[11] A.G. Ramm, “Dynamical systems method (DSM) and nonlinear problems”, Spectral Theory and Nonlinear Analysis, World Scientific Publishers, Singapore, 2005, 201–228 | DOI | MR | Zbl

[12] A.G. Ramm, “On the DSM version of Newton's method”, Eurasian Math. J., 2:3 (2011), 89–97 | MR | Zbl

[13] A.G. Ramm, “Asymptotic stability of solutions to abstract differential equations”, Journal of Abstract Diff. Equations and Applications, 1:1 (2010), 27–34 | MR | Zbl

[14] A.G. Ramm, “Stability of solutions to some evolution problems”, Chaotic Modeling and Simulation, 1 (2011), 17–27 | MR

[15] A.G. Ramm, “On the DSM Newton-type method”, J. Appl. Math. and Comp., 38:1–2 (2012), 523–533 | DOI | MR | Zbl

[16] A.G. Ramm, “How large is the class of operator equations solvable by a DSM Newton-type method”, Appl. Math. Letters, 24:6 (2011), 860–865 | DOI | MR

[17] V. Shmulian, “On differentiability of the norm in Banach space”, Doklady Acad. Sci. USSR, 27 (1940), 643–648

[18] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Springer-Verlag, New York, 1997 | MR

[19] A.G. Ramm, “DSM for general nonlinear equations”, Applied Math. Letters http://dx.doi.org/10.1016/j.aml.2012.06.006 | MR