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

MathSciNet review:
0287126

Full-text PDF Free Access

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]**Kenneth 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**0222410****[2]**Rodney D. Driver,*Existence and stability of solutions of a delay-differential system*, Arch. Rational Mech. Anal.**10**(1962), 401–426. MR**0141863****[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**0044116****[5]**Nelson Dunford and Jacob T. Schwartz,*Linear Operators. I. General Theory*, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR**0117523****[6]**R. E. Edwards,*Functional analysis. Theory and applications*, Holt, Rinehart and Winston, New York-Toronto-London, 1965. MR**0221256****[7]**Robert E. Fennell,*Periodic solutions of functional differential equations*, J. Math. Anal. Appl.**39**(1972), 198–201. MR**0308553****[8]**Robert Fennell and Paul Waltman,*A boundary value problem for a system of nonlinear functional differential equations*, J. Math. Anal. Appl.**26**(1969), 447–453. MR**0237908****[9]**A. Granas,*The theory of compact vector fields and some of its applications to topology of functional spaces. I*, Rozprawy Mat.**30**(1962), 93. MR**0149253****[10]**A. Halanay,*Differential equations: Stability, oscillations, time lags*, Academic Press, New York-London, 1966. MR**0216103****[11]**J. K. Hale,*Functional differential equations*, Lectures, University of California, Los Angeles, 1968-1969.**[12]**Daniel Henry,*The adjoint of a linear functional differential equation and boundary value problems*, J. Differential Equations**9**(1971), 55–66. MR**0274901****[13]**Junji Kato,*Asymptotic behaviors in functional differential equations*, Tôhoku Math. J. (2)**18**(1966), 174–215. MR**0206444****[14]**M. Z. Nashed and J. S. W. Wong,*Some varaints of a fixed point theorem of Krasnoselskii and applications to nonlinear integral equations*, J. Math. Mech.**18**(1969), 767–777. MR**0238140**

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:
http://dx.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