Abstract
LetE be a Banach lattice which consists of some functionsu:Ω→ℂ over a fixed domain Ω. This article is concerned with the local stability of a non-zero positive solutionu *∈E to the Hammerstein equationu(x)=\( = \int\limits_\Omega {k(x,y)f(y,u(y))d\mu } (y),\) x∈Ω, wherek≥0, andf:Ω×[0, ∞)→[0, ∞) is not necessarily increasing in the second variable. It is assumed thatf(x, 0)=0 and\(\left| {\frac{{\partial f(x,u)}}{{\partial u}}} \right|< \frac{{f(x,u)}}{u}\) forx∈Ω,u>0. Under some mild additional hypotheses onE, k, μ andf it is proved that the spectral radius of the Fréchet derivative atu * of the Hammerstein operator is less than one. Also the impact of local stability on secondary bifurcations is investigated. The proof of the main result is based on the spectral theory for completely continuous and irreducible positive operators on Banach lattices.
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References
Amann, H.: On the number of solutions of nonlinear equations in ordered Banach spaces. J. Funct. Anal.11, 346–384 (1972).
Amann, H.: Supersolutions, monotone iterations, and stability. J. Diff. Equat.21, 363–377 (1976).
Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev.18, 620–709 (1976).
Chow, S.-N., Hale, J. K.: Methods of Bifurcation Theory. New York: Springer. 1982.
Crandall, M. G., Rabinowitz, P. H.: Bifurcation, perturbation of simple eigenvalues, and linearized stability. Arch. Rat. Mech. Anal.52, 161–180 (1973).
Dancer, E. N.: Global solution branches for positive mappings. Arch. Rat. Mech. Anal.52, 181–192 (1973).
Dancer, E. N.: On the structure of solutions of non-linear eigenvalue problems. Indiana Univ. Math. J.23, 1069–1076 (1974).
Dancer, E. N.: Solution branches for mappings in cones, and applications. Bull. Austr. Math. Soc.11, 131–143 (1974).
Deimling, K.: Nonlinear Functional Analysis. Berlin-Heidelberg-New York: Springer. 1985.
Hardin, D. P., Takáč, P., Webb, G. F.: A comparison of dispersal strategies for survival of spatially heterogeneous populations. SIAM J. Appl. Math. (1988). (In print.)
Jörgens, K.: Lineare Integraloperatoren. Stuttgart: Teubner. 1970.
Krasnoselskii, M. A.: Positive Solutions of Operator Equations. Groningen: Noordhoff. 1964.
Krasnoselskii, M. A., Zabreiko, P. P.: Geometrical Methods of Nonlinear Analysis. Berlin-Heidelberg-New York: Springer. 1984.
Laetsch, T. W.: The number of solutions of a nonlinear two point boundary value problem. Indiana Univ. Math. J.20, 1–13 (1970).
Laetsch, T. W.: Uniqueness for sublinear boundary value problems. J. Diff. Equat.13, 13–23 (1973).
Leggett, R. W., Williams, L. R.: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J.28, 673–688 (1979).
Nussbaum, R. D.: A global bifurcation theorem with applications to functional differential equations. J. Funct. Anal.19, 319–339 (1975).
Nussbaum, R. D.: A periodicity threshold theorem for some nonlinear integral equations. SIAM. J. Math. Anal.9, 356–376 (1978).
Nussbaum, R. D.: Periodic solutions of some nonlinear integral equations. In: Dynamical Systems; pp. 221–249. (A. Bednarek andL. Cesari, eds.) New York: Academic Press. 1977.
de Pagter, B.: Irreducible compact operators. Math. Z.192, 149–153 (1986).
Rabinowitz, P. H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal.7, 487–513 (1971).
Schaefer, H. H.: Topological Vector Spaces. Berlin-Heidelberg-New York: Springer. 1971.
Schaefer, H. H.: Banach Lattices and Positive Operators. Berlin-Heidelberg-New York: Springer. 1974.
Thieme, H. R.: On a class of Hammerstein integral equations. Manuscripta Math.29, 49–84 (1979).
Turner, R. E. L.: Transversality and cone maps. Arch. Rat. Mech. Anal.58, 151–179 (1975).
Williams, L. R., Leggett, R. W.: Multiple fixed point theorems for problems in chemical reactor theory. J. Math. Anal. Appl.69, 180–193 (1979).
Williams, L. R., Leggett, R. W.: Unique and multiple solutions of a family of differential equations modeling chemical reactions. SIAM J. Math. Anal.13, 122–123 (1982).
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Takáč, P. The local stability of positive solutions to the Hammerstein equation with a nonmonotonic nemytskii operator. Monatshefte für Mathematik 106, 313–335 (1988). https://doi.org/10.1007/BF01295289
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DOI: https://doi.org/10.1007/BF01295289