Vol. 36, No. 2, 1971

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Spectral theory of monotone Hammerstein operators

Charles Vernon Coffman

Vol. 36 (1971), No. 2, 303–322

Consider the linear integral equation,

y(t) = μ  K (t,s)p(s)y(s)ds,

where K(s,t) is a real-valued symmetric positive definite kernel and p(s) is a positive function. Let L denote the inverse of the integral operator u Ω K(,s)u(s)ds, and for a function y in the domain of L,y0, (all functions are assumed to be real valued) define the Rayleigh quotient J(y) for (1) by,

       ∫             ∫
J (y) =   y(t)[Ly](t)dt∕  p(t)y2(t)dt.
Ω             Ω

If y10 and y1 is in the domain of L and if

y2 =   K (⋅,s)p(s)y1(s)ds,

Shen several applications of the Schwarz inequality show that,

J(y2) ≦ J(y1),

with equality only if y1 is an eigenfunction of (1). On the basis of this fact, when the integral operator in (1) is compact, one can develop the complete spectral theory of (1).

In this paper it is shown that the approach indicated above for the study of (1) has a simple and natural extension for the study of the nonlinear integral equation,

y(t) = μ   K(t,s)f(s,y(s))ds,

where K(t,s) is as above and f(t,y) is an odd function of y,

f (t,y) = − f(t,− y),

and satisfies,

yf(t,y) > 0,y ⁄= 0,


f(t,y2) ≧ f(t,y1),y2 ≧ y1.

Mathematical Subject Classification
Primary: 47.85
Secondary: 45.00
Received: 10 February 1970
Published: 1 February 1971
Charles Vernon Coffman