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Asymptotics for Eigenvalues of a Non-Linear Integral System

D.E. EdmundsJ. Lang — 2008

Bollettino dell'Unione Matematica Italiana

Let I = [ a , b ] , let 1 < q , p < , let u and v be positive functions with u L p ( I ) e v L q ( I ) and let T : L p ( I ) L q ( I ) be the Hardy-type operator given by ( T f ) ( x ) = v ( x ) a x f ( t ) u ( t ) d t , x I . We show that the asymptotic behavior of the eigenvalues λ of the non-linear integral system g ( x ) = ( T F ) ( x ) ( f ( x ) ) ( p ) = λ ( T * g ( p ) ) ) ( x ) (where, for example, t ( p ) = | t | p - 1 sgn ( t ) is given by lim n n λ ^ n ( T ) = c p , q I ( u v ) r ) 1 / r d t 1 / r , for 1 < p < q < lim n n λ ˇ n ( T ) = c p , q I ( u v ) r d t 1 / r for 1 < q < p < Here r = 1 p + 1 p , c p , q is an explicit constant depending only on p and q , λ ^ ( T ) = max ( s p n ( T , p , q ) ) , λ ˇ n ( T ) = min ( s p n ( T , p , q ) ) where s p n ( T , p , q ) stands for the set of all eigenvalues λ corresponding to eigenfunctions g with n zeros.

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