Existence and uniqueness of solutions of nonlocal problems for hyperbolic equation \(u_{xt}=F(x,t,u,u_ x)\) (Q2276669)

The initial-boundary value problem is considered in [0,a]\(\times [0,T]\) and in [0,a]\(\times [0,\infty)\) for the equation \(u_{x_ t}=F(x,t,u,u_ x).\) The boundary condition is given by \[ u(x,0)+\sum^{p}_{i=1}h_ i(x,T_ i)u(x,T_ i)=\phi (x) \] with the compatibility condition where \(0<T_ 1<...<T_ p\) (\(\leq T)\). The author proves the existence and uniqueness of classical solutions to both problems by solving the equivalent integro-differential integral equations in suitable Banach spaces.