# On the topological transversality principle

## Abstract

We propose a generalization of the topological transversality theorem of Granas [1] (see also [2]). We show the implication of this theorem in the axiomatic theory of the topological degree due to Amann and Weiss [3] (see also [4]) and its connection with the fixed point theory. Finally, as an application, we prove a continuation theorem for equations of the form

$$L(x) = N(x)$$

where $$L$$ is a linear Fredholm mapping of index zero and $$N$$ is a nonlinear mapping satisfying a compactness-like condition of Monch type. This theorem extends some continuation results due to Mawhin [5] (see also [6]) and Volkmann [7] and in a particular case reduces to a fixed point theorem of Monch [8].

## Authors

Babeş-Bolyai University, Department of Mathematics, Cluj-Napoca, Romania

## Keywords

Topological transversality; essential mappings; topological degree; fixed point theory; continuation theorems

## Paper coordinates

R. Precup, On the topological transversality principle, Nonlinear Anal.: Theory, Meth. Appl., 20 (1993), 1-9. https://doi.org/10.1016/0362-546X(93)90181-Q

## PDF

##### Journal

Nonlinear Analysis: Theory, Methods & Applications

Pergamon

##### Print ISSN

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MR: 94a:58028.

## References

[1] GRANAS A., Homotopy extension theorem in Banach spaces and some of its applications to the theory of non-linear equations, Bull. Acad. pol. Sci. 7, 387-394 0959).
[2] DUGUNDII J. & GRANAS A., Fixed Point Theory I. Polish Scientific Publishers, Warszawa (1982).
[3] AMANN N. & WEISS S. A., On the uniqueness of the topological degree, Math. Z. 130, 39-54 (1973).
[4] LLOYD N. G., Degree Theory. Cambridge University Press, Cambridge (1978).
[5] MAWmN J., Compacit6, monotonie et convexit6 dans l’6tude de probl~mes aux limites semi-lin~aires, S~minaire d’Analyse Moderne No. 19, University Sherbrooke (1981).
[6] MAWmN J., Points fixes, points critiques et probl~mes aux limites, Les Presses de l’Universit6 de Montr6al (1985).
[7] VOLKMANN P., D6monstration d’un th6or~me de coincidence par la m6thode de Granas, Bull. Soc. math. Belgique, set. B 36, 235-242 (1984).
[8] M6Ncn H., Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Analysis 4, 985-999 (1980).
[9] PRECUP R., Generalized topological transversality and existence theorems, Libertas Mathematica 11, 65-79 (1991).
[10] PRECUP R., Measure of noncompactness and second order differential equations with deviating argument, Studia Univ. Babes-Bolyai 34(2), 25-35 (1989).