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Résolution des équations semilinéaires avec la partie linéaire à noyau de dimension infinie via des applications A-propres. (Solving semilinear equations with infinite dimensional linear kernel via A proper mappings). (French) Zbl 0722.47053
The semilinear equation at resonance $(1)\quad Lx+f(x)=y$ (x$$\in E$$, $$y\in F$$, E and F are Banach spaces) for the case of semi-Fredholm operator L (for the infinite dimensional kernel of the operator L) is investigated. The notion of A-proper mapping, generalizing the corresponding notion of Petryshyn and Browder, is systematically used [see F. E. Browder, Arch. Rational Mech. Anal. 26, 33-42 (1967; Zbl 0166.12603), Nonlinear Functional Anal., Proc. Sym. Pure Math. 28, Part 2, Am. Math. Soc. (1976; Zbl 0327.47022); W. V. Petryshyn, Arch. Rational Mech. Anal. 26, 43-49 (1967; Zbl 0166.12701), Arch. Rational Mech. Anal. 30, 270-284 (1968; Zbl 0176.45902)]. This notion allows to replace the equation (1) by certain sequence of equations $L_ nx_ n+f_ n(x_ n)=Q_ ny,$ $$x_ n\in E_ n$$, $$Q_ ny\in F_ n$$, $$E=\overline{\cup E_ n}$$, $$F=\overline{\cup F_ n}$$ and to employ for the investigation of (1) an analogue of Galerkin’s method [see W. V. Petryshyn, Bull. Am. Math. Soc. 81, 223-312 (1975; Zbl 0303.47038); J. Mawhin and M. Willem, C. R. Acad. Sci. Paris, Ser. A 287, 319- 322 (1978; Zbl 0397.47032)] and the coincidence degree theory [R. E. Gaines and J. Mawhin, Coincidence degree and nonlinear differential equations, Lect. Notes Math. 568 (1977; Zbl 0339.47031), W. V. Petryshyn, Nonlinear Anal. Theory, Meth. Appl. 4, 259-281 (1980; Zbl 0444.47046)].
##### MSC:
 47J05 Equations involving nonlinear operators (general) 47B38 Linear operators on function spaces (general) 65J15 Numerical solutions to equations with nonlinear operators 58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces