Two point boundary value problems for nonlinear functional differential equations

Waltman, Paul; Wong, James S. W.

doi:10.1090/S0002-9947-1972-0287126-8

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6850(online) ISSN 0002-9947(print)

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

$\displaystyle y'(t) = L(t,{y_t}) + f(t,{y_t}),\quad M{y_a} + N{y_b} = \psi ,$

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 $L(t,{y_t}) = A(t)y(t)$ , 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, (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 delay-differential 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), 49-60.
[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, 1958. MR 0117523 (22 #8302)
[6] R. E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart and Winston, New York, 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 York, 1966. MR 0216103 (35 #6938)
[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 (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)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|

Bibliography