## On the first boundary value problem for $[h(x, x^{’} , t)]^{’} =$ $f(x, x^{’} , t)$

HTML articles powered by AMS MathViewer

- by Wayne T. Ford
- Proc. Amer. Math. Soc.
**35**(1972), 491-498 - DOI: https://doi.org/10.1090/S0002-9939-1972-0308506-3
- PDF | Request permission

## Abstract:

The boundary value problem for $[h(x,x’,t)]’ = f(x,x’,t)$ is studied with $x(0) = x(1) = 0$. It is assumed that substitution of functions*u*and

*v*in ${L_2}(0,1)$ into

*h*and

*f*produces the functions $h[u( \cdot ),v( \cdot ), \cdot ]$ and $f[u(\cdot ),v(\cdot ),\cdot ]$ in ${L_2}(0,1)$ such that this map from ${L_2}(0,1) \times {L_2}(0,1)$ into ${L_2}(0,1) \times {L_2}(0,1)$ is hemicontinuous. Existence and uniqueness are shown in $H_0^1(0,1)$ under the assumption that constants $\lambda$ and $\eta$ exist such that \[ [(V - v)[h(U,V,t) - h(u,v,t)] + (U - u)[f(U,V,t) - f(u,v,t)]] \geqq \lambda {(V - v)^2} - \eta {(U - u)^2}\] whenever

*t*lies between zero and one while

*u*,

*v*,

*U*and

*V*are arbitrary. Also, it is assumed that $\lambda$ and $\lambda {\pi ^2} - \eta$ are positive.

## References

- Paul B. Bailey, Lawrence F. Shampine, and Paul E. Waltman,
*Nonlinear two point boundary value problems*, Mathematics in Science and Engineering, Vol. 44, Academic Press, New York-London, 1968. MR**0230967** - Felix E. Browder,
*Problèmes nonlinéaires*, Séminaire de Mathématiques Supérieures, No. 15 (Été, vol. 1965, Les Presses de l’Université de Montréal, Montreal, Que., 1966 (French). MR**0250140** - Jim Douglas Jr. and Todd Dupont,
*Galerkin methods for parabolic equations*, SIAM J. Numer. Anal.**7**(1970), 575–626. MR**277126**, DOI 10.1137/0707048 - Herbert Goldstein,
*Classical Mechanics*, Addison-Wesley Press, Inc., Cambridge, Mass., 1951. MR**0043608** - G. H. Hardy, J. E. Littlewood, and G. Pólya,
*Inequalities*, Cambridge, at the University Press, 1952. 2d ed. MR**0046395** - Philip Hartman,
*Ordinary differential equations*, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR**0171038** - V. Komkov,
*On boundedness and oscillation of the differential equation $\textbf {x}^{\prime \prime }+A(t)\textbf {g}(\textbf {x})=\textbf {f}(t)$ in $R^{n}$*, SIAM J. Appl. Math.**22**(1972), 561–568. MR**311992**, DOI 10.1137/0122051
—, - Milton Lees,
*Discrete methods for nonlinear two-point boundary value problems*, Numerical Solution of Partial Differential Equations (Proc. Sympos. Univ. Maryland, 1965) Academic Press, New York, 1966, pp. 59–72. MR**0202323** - Thomas L. Saaty,
*Modern nonlinear equations*, McGraw-Hill Book Co., New York-Toronto-London, 1967. MR**0218160** - J. T. Schwartz,
*Nonlinear functional analysis*, Notes on Mathematics and its Applications, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher. MR**0433481**
J. M. Tippett,

*Existence, continuability and estimates of solutions of*$(a(t)\psi (x)x’)’ + c(t)f(x) = 0$ (to appear).

*The first boundary value problem for*$x'' = F(x,x’,t)$, M.S. Thesis, Texas Tech University, 1971.

## Bibliographic Information

- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**35**(1972), 491-498 - MSC: Primary 34B15
- DOI: https://doi.org/10.1090/S0002-9939-1972-0308506-3
- MathSciNet review: 0308506