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Proceedings of the American Mathematical Society

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On the first boundary value problem for $ [h(x,\,x\sp{'} ,\,t)]\sp{'} =$ $ f(x,\,x\sp{'} ,\,t)$


Author: Wayne T. Ford
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
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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

$\displaystyle [(V - v)[h(U,V,t) - h(u,v,t)] + (U - u)[f(U,V,t) - f(u,v,t)]] \geqq \lambda {(V - v)^2} - \eta {(U - u)^2}$

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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DOI: https://doi.org/10.1090/S0002-9939-1972-0308506-3
Article copyright: © Copyright 1972 American Mathematical Society