Solvability of operator equations involving nonlinear perturbations of Fredholm mappings of nonnegative index and applications

1982 ◽  
pp. 212-228
Author(s):  
P. S. Milojević
Keyword(s):  
Nonlinear Perturbations ◽  
Operator Equations

Lacunary bifurcation for operator equations and nonlinear boundary value problems on ℝN

1991 ◽  
Vol 118 (3-4) ◽  
pp. 237-270 ◽  
Author(s):  
Hans-Peter Heinz
Keyword(s):  
Nonlinear Boundary ◽  
Eigenvalue Problems ◽  
Real Hilbert Space ◽  
Point Theory ◽  
Elliptic Partial Differential Equations ◽  
Existence Results ◽  
Nonlinear Perturbations ◽  
Nonlinear Boundary Value Problems ◽  
Operator Equations ◽  
Nonlinear Eigenvalue Problems

SynopsisWe consider nonlinear eigenvalue problems of the form Lu + F(u) = λu in a real Hilbert space, where L is a positive self-adjoint linear operator and F is a nonlinearity vanishing to higher order at u = 0. We suppose that there are gaps in the essential spectrum of L and use critical point theory for strongly indefinite functionals to derive conditions for the existence of non-zero solutions for λ belonging to such a gap, and for the bifurcation of such solutions from the line of trivial solutions at the boundary points of a gap. The abstract results are applied to the L2-theory of semilinear elliptic partial differential equations on ℝN. We obtain existence results for the general case and bifurcation results for nonlinear perturbations of the periodic Schrödinger equation.

On Homogenization of Controlled Objects Described by Operator Equations of Hammerstein Type

2003 ◽  
Vol 35 (11) ◽  
pp. 19-27
Author(s):  
Victor N. Mizernyi ◽  
Peter I. Kogut ◽  
Tatyana N. Rudyanova
Keyword(s):  
Operator Equations
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