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), 3954
MSC:
Primary 34.75
MathSciNet review:
0287126
Fulltext 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 where 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 functionaldifferential
equations, Proc. U.S.Japan Seminar on Differential and Functional
Equations (Minneapolis, Minn., 1967) Benjamin, New York, 1967,
pp. 27–47. MR 0222410
(36 #5462)
 [2]
Rodney
D. Driver, Existence and stability of solutions of a
delaydifferential system, Arch. Rational Mech. Anal.
10 (1962), 401–426. MR 0141863
(25 #5260)
 [3]
W. Dubrovskiĭ, Sur certaines équations intégrales nonlinéaires, Uč. Zap. Moskov. Gos. Univ. Mat. 30 (1939), 4960.
 [4]
J.
Dugundji, An extension of Tietze’s theorem, Pacific J.
Math. 1 (1951), 353–367. MR 0044116
(13,373c)
 [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
(22 #8302)
 [6]
R.
E. Edwards, Functional analysis. Theory and applications,
Holt, Rinehart and Winston, New YorkTorontoLondon, 1965. MR 0221256
(36 #4308)
 [7]
Robert
E. Fennell, Periodic solutions of functional differential
equations, J. Math. Anal. Appl. 39 (1972),
198–201. MR 0308553
(46 #7667)
 [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
(38 #6185)
 [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
(26 #6743)
 [10]
A.
Halanay, Differential equations: Stability, oscillations, time
lags, Academic Press, New YorkLondon, 1966. MR 0216103
(35 #6938)
 [11]
J. K. Hale, Functional differential equations, Lectures, University of California, Los Angeles, 19681969.
 [12]
Daniel
Henry, The adjoint of a linear functional differential equation and
boundary value problems, J. Differential Equations 9
(1971), 55–66. MR 0274901
(43 #659)
 [13]
Junji
Kato, Asymptotic behaviors in functional differential
equations, Tôhoku Math. J. (2) 18 (1966),
174–215. MR 0206444
(34 #6263)
 [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
(38 #6416)
 [1]
 K. L. Cooke, Some recent work on functionaldifferential equations, Proc. U.S.Japan Seminar on Differential and Functional Equations (Minneapolis, Minn., 1967), Benjamin, New York, 1967, pp. 2747. MR 36 #5462. MR 0222410 (36:5462)
 [2]
 R. D. Driver, Existence and stability of solutions of a delaydifferential system, Arch. Rational Mech. Anal. 10 (1962), 401426. MR 25 #5260. MR 0141863 (25:5260)
 [3]
 W. Dubrovskiĭ, Sur certaines équations intégrales nonlinéaires, Uč. Zap. Moskov. Gos. Univ. Mat. 30 (1939), 4960.
 [4]
 J. Dugundji, An extension of Tietze's theorem, Pacific J. Math. 1 (1951), 353367. 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), 447453. 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, 19681969.
 [12]
 D. Henry, The adjoint linear functional equation and boundary value problems, J. Differential Equations 9 (1971), 5566. MR 0274901 (43:659)
 [13]
 J. Kato, Asymptotic behaviors in functional differential equations, Tôhoku Math. J. (2) 18 (1966), 174215. 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), 767777. MR 38 #6416. MR 0238140 (38:6416)
Similar Articles
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/S00029947197202871268
PII:
S 00029947(1972)02871268
Keywords:
Functional differential equations,
boundary value problems,
periodic solutions,
shooting methods,
Fredholm alternative
Article copyright:
© Copyright 1972
American Mathematical Society
