An approximation theory for generalized Fredholm quadratic forms and integral-differential equations

Authors:
J. Gregory and G. C. Lopez

Journal:
Trans. Amer. Math. Soc. **222** (1976), 319-335

MSC:
Primary 45J05; Secondary 34C10

DOI:
https://doi.org/10.1090/S0002-9947-1976-0423024-8

MathSciNet review:
0423024

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Abstract | References | Similar Articles | Additional Information

Abstract: An approximation theory is given for a very general class of elliptic quadratic forms which includes the study of 2*n*th order (usually in integrated form), selfadjoint, integral-differential equations. These ideas follows in a broad sense from the quadratic form theory of Hestenes, applied to integral-differential equations by Lopez, and extended with applications for approximation problems by Gregory.

The application of this theory to a variety of approximation problem areas in this setting is given. These include focal point and focal interval problems in the calculus of variations/optimal control theory, oscillation problems for differential equations, eigenvalue problems for compact operators, numerical approximation problems, and finally the intersection of these problem areas.

In the final part of our paper our ideas are specifically applied to the construction and counting of negative vectors in two important areas of current applied mathematics: In the first case we derive comparison theorems for generalized oscillation problems of differential equations. The reader may also observe the essential ideas for oscillation of many nonsymmetric (indeed odd order) ordinary differential equation problems which will not be pursued here. In the second case our methods are applied to obtain the ``Euler-Lagrange equations'' for symmetric tridiagonal matrices. In this significant new result (which will allow us to reexamine both the theory and applications of symmetric banded matrices) we can construct in a meaningful way, negative vectors, oscillation vectors, eigenvectors, and extremal solutions of classical problems as well as faster more efficient algorithms for the numerical solution of differential equations.

In conclusion it appears that many physical problems which involve symmetric differential equations are more meaningful presented as integral differential equations (effects of friction on physical processes, etc.). It is hoped that this paper will provide the general theory and present examples and methods to study integral differential equations.

**[1]**John Gregory,*An approximation theory for elliptic quadratic forms on Hilbert spaces: Application to the eigenvalue problem for compact quadratic forms*, Pacific J. Math.**37**(1971), 383–395. MR**0303311****[2]**John Gregory,*A theory of focal points and focal intervals for an elliptic quadratic form on a Hilbert space*, Trans. Amer. Math. Soc.**157**(1971), 119–128. MR**0278147**, https://doi.org/10.1090/S0002-9947-1971-0278147-9**[3]**John Gregory,*An approximation theory for focal points and focal intervals*, Proc. Amer. Math. Soc.**32**(1972), 477–483. MR**0296788**, https://doi.org/10.1090/S0002-9939-1972-0296788-6**[4]**John Gregory,*A theory of numerical approximation for elliptic forms associated with second order differential systems. Application to eigenvalue problems*, J. Math. Anal. Appl.**38**(1972), 416–426. MR**0322652**, https://doi.org/10.1016/0022-247X(72)90099-6**[5]**John Gregory and Franklin Richards,*Numerical approximation for 2𝑚th order differential systems via splines*, Rocky Mountain J. Math.**5**(1975), 107–116. MR**0388790**, https://doi.org/10.1216/RMJ-1975-5-1-107**[6]**Magnus R. Hestenes,*Applications of the theory of quadratic forms in Hilbert space to the calculus of variations*, Pacific J. Math.**1**(1951), 525–581. MR**0046590****[7]**Magnus R. Hestenes,*Calculus of variations and optimal control theory*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0203540****[8]**G. C. Lopez,*Quadratic variational problems involving higher order ordinary derivates*, Dissertation, University of California, Los Angeles, 1961.**[9]**E. Y. Mikami,*Focal points in a control problem*, Pacific J. Math.**35**(1970), 473–485. MR**0281081****[10]**Marston Morse,*The calculus of variations in the large*, American Mathematical Society Colloquium Publications, vol. 18, American Mathematical Society, Providence, RI, 1996. Reprint of the 1932 original. MR**1451874**

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DOI:
https://doi.org/10.1090/S0002-9947-1976-0423024-8

Keywords:
Approximation theory,
conjugate points,
oscillations,
calculus of variations,
Fredholm integral differential equations,
spline approximations

Article copyright:
© Copyright 1976
American Mathematical Society