The bifurcation of solutions in Banach spaces

Hall, William S.

doi:10.1090/S0002-9947-1971-0282267-2

The bifurcation of solutions in Banach spaces
HTML articles powered by AMS MathViewer

by William S. Hall PDF

Trans. Amer. Math. Soc. 161 (1971), 207-218 Request permission

Abstract:

Let $L:D \subset X \to D \subset {X^ \ast }$ be a densely defined linear map of a reflexive Banach space X to its conjugate ${X^\ast }$. Define M and ${M^\ast }$ to be the respective null spaces of L and its formal adjoint ${L^\ast }$. Let $f:X \to {X^\ast }$ be continuous. Under certain conditions on ${L^\ast }$ and f there exist weak solutions to $Lu = f(u)$ provided for each $w \in X,v(w) \in M$ can be found such that $f(v(w) + w)$ annihilates ${M^ \ast }$. Neither M and ${M^\ast }$ nor their annihilators need be the ranges of continuous linear projections. The results have applications to periodic solutions of partial differential equations.

References

Stephen Bancroft, Jack K. Hale, and Daniel Sweet, Alternative problems for nonlinear functional equations, J. Differential Equations 4 (1968), 40–56. MR 220118, DOI 10.1016/0022-0396(68)90047-8
Jean Mawhin, Degré topologique et solutions périodiques des systèmes différentiels non linéaires, Bull. Soc. Roy. Sci. Liège 38 (1969), 308–398 (French). MR 594965
Lamberto Cesari, Existence in the large of periodic solutions of hyperbolic partial differential equations, Arch. Rational Mech. Anal. 20 (1965), 170–190. MR 183976, DOI 10.1007/BF00276443
Jack K. Hale, Periodic solutions of a class of hyperbolic equations containing a small parameter, Arch. Rational Mech. Anal. 23 (1966), 380–398. MR 206503, DOI 10.1007/BF00276781
Otto Vejvoda, Periodic solutions of a linear and weakly nonlinear wave equation in one dimension. I, Czechoslovak Math. J. 14(89) (1964), 341–382 (English, with Russian summary). MR 174872, DOI 10.21136/CMJ.1964.100626
P. H. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, Comm. Pure Appl. Math. 20 (1967), 145–205. MR 206507, DOI 10.1002/cpa.3160200105
William S. Hall, On the existence of periodic solutions for the equations $D_{tt}u+(-1)^{p}\,D_{x}^{2p}u=\varepsilon f(\cdot ,\cdot ,u)$, J. Differential Equations 7 (1970), 509–526. MR 265738, DOI 10.1016/0022-0396(70)90098-7
William S. Hall, Periodic solutions of a class of weakly nonlinear evolution equations, Arch. Rational Mech. Anal. 39 (1970), 294–322. MR 274914, DOI 10.1007/BF00281367
José Luis Massera and Juan Jorge Schäffer, Linear differential equations and function spaces, Pure and Applied Mathematics, Vol. 21, Academic Press, New York-London, 1966. MR 0212324
Lawrence M. Graves, Some mapping theorems, Duke Math. J. 17 (1950), 111–114. MR 35398

Functional analysis

Walter A. Strauss, On the solutions of abstract nonlinear equations, Proc. Amer. Math. Soc. 18 (1967), 116–119. MR 217657, DOI 10.1090/S0002-9939-1967-0217657-1
Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210528