Annali di Matematica Pura ed Applicata

, Volume 105, Issue 1, pp 141–170

# Duality theory forN-th order differential operators under stieltjes boundary ponditions. II: Nonsmooth coefficients and nonsingular measures

• R. C. Brown
Article

## Summary

Adjoint relations are characterized for an n-th order vector valued differential system with nonsmooth coefficients and with boundary conditions represented by Stieltjes measures of bounded variation when the system is viewed as an operator with domain and range in a space of Lp integrable functions. This is done by developing an abstract theory of « normally sovable » linear relations and by constructing a compact partial inverse (generalized Green’s matrix) for the operator.

## Keywords

Boundary Condition Differential Operator Linear Relation Integrable Function Differential System
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
E. Arghiriade,Sur l’inverse généralisée d’un opérateur linéaire dans les espaces de Hilbert, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., ser 8,45 (1968), pp. 471–477.
2. [2]
R. Arens,Operational calculus of linear relations, Pacific J. Math.,11 (1961), pp. 9–23.
3. [3]
A. Ben-Israel -T. N. E. Greville,Generalised Inverses: Theory and Applications, Interscience, New York, 1974.Google Scholar
4. [4]
R. C. Brown,The adjoint and Fredholm index of a linear system with general boundary conditions, Univ. of Wisconsin-Madison, Mathematics Research Center TSR no. 1287.Google Scholar
5. [5]
R. C. Brown,Generalised Green’s functions and generalised inverses for linear differential systems with Stieltjes boundary conditions, J. Differential Equations,16 (1974), pp. 335–351.
6. [6]
R. C. Brown,Duality theory for n-thorder differential operators under Stieltjes boundary conditions, University of Wisconsin-Madison, Mathematics Research Center TSR no. 1329 (1973).Google Scholar
7. [7]
R. C. Brown,Adjoint domains and generalized splines, Czech. Math. J.25 (100) (1975), pp. 134–147, University of Wisconsin-Madison, Mathematics Research Center TSR no. 1341 (1973).
8. [8]
R. C. Brown -A. M. Krall,Ordinary differential operators under Stieltjes boundary conditions, Trans. Amer. Math. Soc.,198 (1974), pp. 73–91.
9. [9]
H. Chitwood,Generalized Green’s matrices for linear differential systems, SIAM J. Math. Anal.,4 (1973), pp. 104–109.
10. [10]
H. Chitwood,Generalized Green’s matrices for linear differential system, University of Tennessee Ph. D. dissertation (1971).Google Scholar
11. [11]
N. Dunford -J. Schwartz,Linear Operators, Part I, Interscience, New York (1957).Google Scholar
12. [12]
S. Goldberg,Unbounded Linear Operators, McGraw-Hill, New York (1966).Google Scholar
13. [13]
J. Locker,Self-adjointness for multi-point differential operators, Pacific J. Math.,45 (1973), pp. 561–570.
14. [14]
M. A. Naimark,Linear Differential Operators, Part I, Ungar, New York (1968).Google Scholar
15. [15]
M. A. Naimark,Linear Differential Operators, Part II, Ungar, New York (1968).Google Scholar
16. [16]
M. Z. Nashed,Generalized Inverses, Normal Solvability and Iteration for Singular Operator Equations, «Nonlinear Functional Analysis », L. B. Rall Editor, Academic Press, New York (1971), pp. 311–359.Google Scholar
17. [17]
W. T. Reid,Generalized Green’s matrices for compatible systems of differential equations, Amer. J. Math.,53 (1931), pp. 443–459.
18. [18]
W. T. Reid,Generalized Green’s matrices for two-point boundary problems, SIAM J. Appl. Math.,15 (1967), pp. 856–870.
19. [19]
W. T. Reid,Generalized inverses of differential and integral operators, «Theory and Application of Generalized Inverses of Matrices », Symposium Proceedings, Texas Technological College Mathematics Series, no. 4, Lubbock, Texas (1968), pp. 1–25.Google Scholar
20. [20]
A. Smogorshewsky,Les fonctions de Green des systèmes différentials linéaires dans un domaine à une seule dimension, Recueil Math.,7 (49) (1940), pp. 179–196.
21. [21]
J. von Neumann,Functional Operators, vol. II:The Geometry of Orthogonal Spaces, Annals of Mathematical Studies, no. 22, Princeton (1950).Google Scholar