## 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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### References:

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