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A fixed point theorem for a class of differentiable stable operators in banach spaces
Fixed Point Theory and Applications volume 2006, Article number: 92429 (2006)
Abstract
We study Frèchet differentiable stable operators in real Banach spaces. We present the theory of linear and nonlinear stable operators in a systematic way and prove solvability theorems for operator equations with differentiable expanding operators. In addition, some relations to the theory of monotone operators in Hilbert spaces are discussed. Using the obtained solvability results, we formulate the corresponding fixed point theorem for a class of nonlinear expanding operators.
References
Adams RA: Sobolev Spaces, Pure and Applied Mathematics. Volume 65. Academic Press, New York; 1973.
Alekseev VM, Tichomirov VM, Fomin SV: Optimal Control, Contemporary Soviet Mathematics. Plenum Press, New York; 1987:xiv+309.
Aliprantis CD, Border KC: Infinite-Dimensional Analysis. 2nd edition. Springer, Berlin; 1999:xx+672.
Angermann L: A posteriori error estimates for approximate solutions of nonlinear equations with weakly stable operators. Numerical Functional Analysis and Optimization 1997,18(5–6):447–459.
Azhmyakov V: Stable Operators in Analysis and Optimization. Peter Lang, Berlin; 2005.
Baiocchi C, Capelo A: Variational and Quasivariational Inequalities. Applications to Free Boundary Problems, A Wiley-Interscience Publication. John Wiley & Sons, New York; 1984:ix+452.
Bebendorf M: A note on the Poincaré inequality for convex domains. Zeitschrift für Analysis und ihre Anwendungen 2003,22(4):751–756.
Berberian SK: Fundamentals of Real Analysis, Universitext. Springer, New York; 1999:xii+479.
Gajewski H, Gröger K, Zacharias K: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Mathematische Lehrbücher und Monographien, II. Abteilung, Mathematische Monographien. Volume 38. Akademie, Berlin; 1974:ix+281.
Ioffe AD, Tihomirov VM: Theory of Extremal Problems, Studies in Mathematics and Its Applications. Volume 6. North-Holland, Amsterdam; 1979:xii+460.
Kolmogorov AN, Fomin SV: Introductory Real Analysis. Prentice-Hall, New York; 1970:xii+403. Revised English edition, translated from the Russian and edited by R. A. Silverman
Kolmogorov AN, Fomin SV: Elements of the Theory of Functions and Functional Analysis. Nauka, Moscow; 1972.
Köthe G: Topological Vector Spaces I. Springer, Berlin; 1983.
Lax PD, Milgram AN: Parabolic equations. In Contributions to the Theory of Partial Differential Equations. Proceedings of the Conference on Partial Differential Equations, Annals of Mathematics Studies, no. 33. Edited by: Bers L, Bochner S, John F. Princeton University Press, New York; 1954:167–190.
Lions J-L, Magenes E: Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften. Volume 181. Springer, New York; 1972:xvi+357.
Lions J-L, Magenes E: Non-Homogeneous Boundary Value Problems and Applications. Vol. II., Die Grundlehren der mathematischen Wissenschaften. Volume 182. Springer, New York; 1972:xi+242.
Lions J-L, Magenes E: Non-Homogeneous Boundary Value Problems and Applications. Vol. III, Die Grundlehren der mathematischen Wissenschaften. Volume 183. Springer, New York; 1973:xii+308.
Minty GJ: Monotone (nonlinear) operators in Hilbert space. Duke Mathematical Journal 1962, 29: 341–346. 10.1215/S0012-7094-62-02933-2
Petryshyn WV: On the approximation-solvability of equations involving -proper and psuedo- -proper mappings. Bulletin of the American Mathematical Society 1975, 81: 223–312. 10.1090/S0002-9904-1975-13728-1
Petryshyn WV: Solvability of linear and quasilinear elliptic boundary value problems via the -proper mapping theory. Numerical Functional Analysis and Optimization 1980,2(7–8):591–635. 10.1080/01630563.1980.10120629
Phelps RR: Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics. Volume 1364. Springer, Berlin; 1993:xii+117.
Robertson AP, Robertson W: Topological Vector Spaces. Cambridge University Press, Cambridge; 1966.
Rudin W: Functional Analysis, McGraw-Hill Series in Higher Mathematics. McGraw-Hill, New York; 1973:xiii+397.
Ruzicka M: Nichtlineare Funktionalanalysis. Springer, Berlin; 2004.
Stetter HJ: Analysis of Discretization Methods for Ordinary Differential Equations, Springer Tracts in Natural Philosophy. Volume 23. Springer, New York; 1973:xvi+388.
Takahashi W: Nonlinear Functional Analysis. Yokohama, Yokohama; 2000:iv+276.
Zarantonello EH: Solving functional equations by contractive averaging. In Tech. Rep. 160. Mathematics Research Centre, University of Wisconsin, Madison; 1960.
Zeidler E: Nonlinear Functional Analysis and Its Applications. II/A. Linear Monotone Operators. Springer, New York; 1990:xviii+467.
Zeidler E: Nonlinear Functional Analysis and Its Applications. II/B. Nonlinear Monotone Operators. Springer, New York; 1990.
Zeidler E: Nonlinear Functional Analysis and Its Applications. III. Variational Methods and Optimization. Springer, New York; 1990:xxii+662.
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Azhmyakov, V. A fixed point theorem for a class of differentiable stable operators in banach spaces. Fixed Point Theory Appl 2006, 92429 (2006). https://doi.org/10.1155/FPTA/2006/92429
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DOI: https://doi.org/10.1155/FPTA/2006/92429