Some problems of differential equations could be reduced to fixed points of mappings between Banach spaces of different dimensions, in which case the classical Leray-Schauder degree does not apply. So, I am wondering whether there exists topological degree of this type? Could someone tell me some references on this topic?
Best wishes,
Liangping Qi
Perhaps you should look for some papers by Wojciech Kryszewski. ( Fredholm theory, linking arguments? )
it is working for Fredholm maps between infinite dimensional Hilbert manifolds..
Look to books by J.Mawhin " he is the father of degree theory now" also try the L compact If you need he is books I can send some of them to you just let me know thanks
In the late 1960's Browder and Petryshyn - constructed a degree theory for a class of mappings between two different Banach spaces.
F. E, Browder and W. V. Petryshyn, Approximation methods and the generalized topological degree for nonlinear mappings in Banach spaces, J. Funct. Anal. 3 (1969)
Also in the late 1960's Elworthy and Tromba developed a degree theory for some types of nonlinear proper Fredholm operators (C^1 mapping whose differential at each point is linear Fredholm operator) of index zero between Banach manifolds.
K. D, Elworthy and A. J. Tromba, Differential structures and Fredholm maps on Banach manifolds, Global Analysis, vol 14-16, Proc. Sympos. Pure Math, American Math Soc., 1970.
A pretty good overview of the history of index theory can be found in
J. Mawhin, Lerary-Schauder Degree: A half century of extensions and applications, Topological Methods in Norlin. Anal, (14), 1999.
The question could be more clear. There are two possible meaning of the problem: 1) degree of a map between two finite dimensional spaces - it is the homotopical type of this map; 2) degree for the pair (LN) where L is a linear Fredholm mapping of any positive index. This second theory generalizes the coincidence degree for L-compact maps where L has index 0 and it has origin in a paper by Nirenberg (1972 as far as I remember).
Thanks very much for your help.
This question arises from applications of coincidence degree theory to differential equations of the type Lx=Nx, where L is the differential operator, also a Fredholm operator of index zero. It happens that the operator N can not be well-defined on the whole function space (say Y). It is the case that the image N(Y) is not contained in Y (some point of N(Y) is not in Y). So my idea is to restrict the domain of N to a subspace of Y, such as
dom N={ x in Y; N(x) is in Y},
and consider a degree for mappings between spaces of different dimensions. I do not know whethe it is possible to do in this way.
I'm happy to receive so many valuble advices from all of you.
Best wishes,
Liangping Qi
- Badges
- Science topic
- Similar topics
- Mathematics
- Applied Mathematics
- Differential Equations