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Linear differential systems with infinitely many boundary points

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Summary

The article extends results previously known for boundary value problems involving only a finite number of boundary points to those which involve an infinite number of (possibly dense) boundary points. Specifically, the system\(Ly = y' + Py,\sum\limits_1^\infty {A_i } y(t_i ) = 0,0 \leqslant t_i \leqslant {\text{1,}}i = 1,...,\infty \), is discussed in the Hilbert space L2(0, 1). Suitable conditions for inverting the operator L are found, and the Green's function is exhibited. It is shown to have the standard properties as well as some which are new, when considered as a function of its second variable It is further shown to be the limit a.e. of Green's function for problems involving only a finite number of boundary points, as those points increase in number. Finally it is shown that L−1 is compact. By using the Green's function the domain of L is shown to be dense in L2(0, 1). and the adjoint L* and its domain are found. L is also shown to be closed. Lastly, by using some theorems concerning entire functions, the eigenvalues of L are shown to lie in a vertical strip with infinity as their only limit point. This in turn implies that if L−1 fails to exist, a slight perturbation in P will result in an invertible L, and the assumption made earlier concerning the existence of the Green's function is reasonable.

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Entrata in Redazione il 25 febbraio 1971.

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Green, G.B., Krall, A.M. Linear differential systems with infinitely many boundary points. Annali di Matematica 91, 53–67 (1971). https://doi.org/10.1007/BF02428813

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Keywords

  • Entire Function
  • Boundary Point
  • Infinite Number
  • Adjoint Operator
  • Vertical Strip