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

Ford, Wayne T.

doi:10.1090/S0002-9939-1972-0308506-3

On the first boundary value problem for $[h(x, x^{’} , t)]^{’} =$ $f(x, x^{’} , t)$
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by Wayne T. Ford PDF

Proc. Amer. Math. Soc. 35 (1972), 491-498 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.

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The first boundary value problem for