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Eigenvalues of a system of Fredholm integral equations. (English) Zbl 1068.45001

The authors consider two systems of Fredhlom integral equations, one on a finite interval \([0,1]\) and the other on the half-line \([0,\infty)\). A solution \(u= (u_1,u_2,\dots, u_n)\) of the former system is sought in \(C[0,1]^n\) and a solution \(u= (u_1,u_2,\dots, u_n)\) for the latter system is sought in a subset of \((BC[0,\infty))^n\), where \(BC[0,\infty)\) denotes the space of functions that are bounded and continuous on \([0,\infty)\). In both cases, \(u\) is called a solution of constant sign. For each of the system of equations, those values of \(\lambda\) are characterized for which the system has a constant-sign solution. \(\lambda\) is called an eigenvalue and \(u\) the corresponding eigenfunction of the system, if for a particular \(\lambda\) the system has a constant-sign solution.
The paper consists of five sections dealing with the Krasnosel’skij’s fixed point theorem, establishment of the condition for the set of eigenvalues \(E\) to contain an interval and for \(E\) to be an interval (bounded or unbounded), explicit sub-intervals are derived, and applications to boundary value problems, respectively.

MSC:

45C05 Eigenvalue problems for integral equations
45B05 Fredholm integral equations
45F05 Systems of nonsingular linear integral equations
45F15 Systems of singular linear integral equations
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