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The boundary value problem for      [ h ( x , x ′ , t ) ] ′ = f ( x , x ′ , t ) [h(x,x’,t)]’ = f(x,x’,t)       is studied with      x ( 0 ) = x ( 1 ) = 0 x(0) = x(1) = 0      . It is assumed that substitution of functions  u  and  v  in      L 2 ( 0 , 1 ) {L_2}(0,1)       into  h  and  f  produces the functions      h [ u ( ⋅  ) , v ( ⋅  ) , ⋅  ] h[u( \cdot ),v( \cdot ), \cdot ]       and      f [ u ( ⋅  ) , v ( ⋅  ) , ⋅  ] f[u(\cdot ),v(\cdot ),\cdot ]       in      L 2 ( 0 , 1 ) {L_2}(0,1)       such that this map from      L 2 ( 0 , 1 ) ×  L 2 ( 0 , 1 ) {L_2}(0,1) \times {L_2}(0,1)       into      L 2 ( 0 , 1 ) ×  L 2 ( 0 , 1 ) {L_2}(0,1) \times {L_2}(0,1)       is hemicontinuous. Existence and uniqueness are shown in      H 0 1 ( 0 , 1 ) H_0^1(0,1)       under the assumption that constants      λ  \lambda       and      η  \eta       exist such that  \[    [ ( V −  v ) [ h ( U , V , t ) −  h ( u , v , t ) ] + ( U −  u ) [ f ( U , V , t ) −  f ( u , v , t ) ] ] ≧  λ  ( V −  v ) 2 −  η  ( U −  u ) 2 [(V - v)[h(U,V,t) - h(u,v,t)] + (U - u)[f(U,V,t) - f(u,v,t)]] \geqq \lambda {(V - v)^2} - \eta {(U - u)^2}     \]  whenever  t  lies between zero and one while  u ,  v ,  U  and  V  are arbitrary. Also, it is assumed that      λ  \lambda       and      λ  π  2 −  η  \lambda {\pi ^2} - \eta       are positive.

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