Existence and uniqueness of solutions of nonlocal problems for hyperbolic equation $$u_{xt}=F(x,t,u,u_ x)$$.(English)Zbl 0725.35059

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.
Reviewer: T.Kakita (Tokyo)

MSC:

 35L70 Second-order nonlinear hyperbolic equations 35L20 Initial-boundary value problems for second-order hyperbolic equations 45K05 Integro-partial differential equations
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