Two point boundary value problems for nonlinear functional differential equations

Authors:
Paul Waltman and James S. W. Wong

Journal:
Trans. Amer. Math. Soc. **164** (1972), 39-54

MSC:
Primary 34.75

DOI:
https://doi.org/10.1090/S0002-9947-1972-0287126-8

MathSciNet review:
0287126

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with the existence of solutions of two point boundary value problems for functional differential equations. Specifically, we consider

*M*and

*N*are linear operators on . Growth conditions are imposed on

*f*to obtain the existence of solutions. This result is then specialized to the case where , that is, when the reduced linear equation is an ordinary rather than a functional differential equation. Several examples are discussed to illustrate the results.

**[1]**K. L. Cooke,*Some recent work on functional-differential equations*, Proc. U.S.-Japan Seminar on Differential and Functional Equations (Minneapolis, Minn., 1967), Benjamin, New York, 1967, pp. 27-47. MR**36**#5462. MR**0222410 (36:5462)****[2]**R. D. Driver,*Existence and stability of solutions of a delay-differential system*, Arch. Rational Mech. Anal.**10**(1962), 401-426. MR**25**#5260. MR**0141863 (25:5260)****[3]**W. Dubrovskiĭ,*Sur certaines équations intégrales nonlinéaires*, Uč. Zap. Moskov. Gos. Univ. Mat.**30**(1939), 49-60.**[4]**J. Dugundji,*An extension of Tietze's theorem*, Pacific J. Math.**1**(1951), 353-367. MR**13**, 373. MR**0044116 (13:373c)****[5]**N. Dunford and J. T. Schwartz,*Linear operators*. I:*General theory*, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR**22**#8302. MR**0117523 (22:8302)****[6]**R. E. Edwards,*Functional analysis. Theory and applications*, Holt, Rinehart and Winston, New York, 1965. MR**36**#4308. MR**0221256 (36:4308)****[7]**R. E. Fennell,*Periodic solutions of functional differential equations*, J. Math. Anal. Appl. (to appear). MR**0308553 (46:7667)****[8]**R. Fennell and P. Waltman,*A boundary value problem for a system of nonlinear functional differential equations*, J. Math. Anal. Appl.**26**(1969), 447-453. MR**38**#6185. MR**0237908 (38:6185)****[9]**A. Granas,*The theory of compact vector fields and some of its applications to topology and functional spaces*. I, Rozprawy Mat.**30**(1962), 93 pp. MR**26**#6743. MR**0149253 (26:6743)****[10]**A. Halanay,*Differential equations*:*Stability, oscillations, time lags*, Academic Press, New York, 1966. MR**35**#6938. MR**0216103 (35:6938)****[11]**J. K. Hale,*Functional differential equations*, Lectures, University of California, Los Angeles, 1968-1969.**[12]**D. Henry,*The adjoint linear functional equation and boundary value problems*, J. Differential Equations**9**(1971), 55-66. MR**0274901 (43:659)****[13]**J. Kato,*Asymptotic behaviors in functional differential equations*, Tôhoku Math. J. (2)**18**(1966), 174-215. MR**34**#6263. MR**0206444 (34:6263)****[14]**M. Z. Nashed and J. S. W. Wong,*Some variants of a fixed point theorem of Krasnoselskii and applications to nonlinear integral equations*, J. Math. Mech.**18**(1969), 767-777. MR**38**#6416. MR**0238140 (38:6416)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
34.75

Retrieve articles in all journals with MSC: 34.75

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1972-0287126-8

Keywords:
Functional differential equations,
boundary value problems,
periodic solutions,
shooting methods,
Fredholm alternative

Article copyright:
© Copyright 1972
American Mathematical Society