Eigenvalue Problems for Positive Monotonic Nonlinear Operators

Citation

Laetsch, Theodore Willis (1968) Eigenvalue Problems for Positive Monotonic Nonlinear Operators. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/R8T3-TJ18. https://resolver.caltech.edu/CaltechETD:etd-09252002-110658

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. The determination of the set [Lambda] of values of [lambda] for which a family of operators {A[subscript lambda]} on a real, partially-ordered Banach space has positive fixed points and the description of the behavior of the fixed points as functions of [lambda] are considered. The operators A[subscript lambda] are usually assumed to be compact monotonic operators which satisfy A[subscript lambda] 0 > 0, and the elements A[subscript lambda]u are assumed to be continuous increasing functions of [lambda] for every positive u. It is shown that [Lambda] is an interval, that for each [lambda] the operator A[subscript lambda] has a smallest positive fixed point u[?]([lambda]), and that u[?]([lambda]) is an increasing function of [lambda] which is continuous from the left in -[lambda]. Conditions are given which guarantee the uniqueness of the fixed point of A[subscript lambda] for each [lambda] and permit the precise determination of the set [Lambda]. When sup [Lambda][...][Lambda] and A[subscript lambda]u satisfies certain differentiability conditions, the behavior of u[...]([lambda]) for near sup [Lambda] is described and the existence of a second positive fixed point for [lambda] near sup [Lambda] is proved. The asymptotic behavior of A[subscript lambda]u for large positive u is used to determine the behavior of the fixed points of large norm and the existence and value of a number [mu subscript 1] such that the norms of a sequence of fixed points approach infinity as the corresponding values of [lambda] approach [mu subscript 1]. The existence of a second positive fixed point is proved under various conditions, including the case when the operators A[subscript lambda] are Frechet differentiable and 0 < [mu subscript 1] < sup [Lambda][member symbol][Lambda] . More precise results are obtained when the operators A[subscript lambda] are concave or convex. These results are used to study the eigenvalue problem for Hammerstein integral equations and nonlinear ordinary differential equations. For certain ordinary differential equations with convex nonlinearities, the existence of precisely two positive fixed points is proved. Finally, an independent treatment is given of the eigenvalue problem for the equation u''+[lambda]f(u)=0 with the boundary conditions u(0) = u(1) = 0; use is made of the first integral of the differential equation and a study of the equation in the phase plane.

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