A nonlinear Neumann-type problem of a system of high order hyperbolic integro-differential equations (Q1315067)

Consider \(n,p \in N\), \(D=[0,A] \times [0,B]\), \(0<A\), \(B<+\infty\) and denote \(D^ r_ x=\partial^ r/ \partial x^ r\), \(D^ s_ y=\partial^ s/ \partial y^ s\), \(0 \leq r\), \(s \leq p\), \(u=(u^ k)_{1 \leq k \leq n}:D \to R^ n\), \(V=(v_ r)_ r\), \(W=(w_ r)_ r\), \(\varphi=(V,W)\), \(v_ r=D^ p_ x D^ r_ yu\), \(w_ r = D^ p_ y D^ r_ xu\), \(Z=(z_{rs})_{rs}\), \(z_{rs} = D^ s_ xD^ r_ yu\), \(0 \leq r\), \(s \leq p-1\), \(L=D^ 1_ x D^ 1_ y\). Consider the class \({\mathcal K}\) of functions \(u\) such that the derivatives \(D^ r_ x D^ s_ yu\) exist for \(0 \leq r\), \(s \leq p\) and are continuous, and the system \[ L^ pu(x,y)=F \bigl( x,y,Z(x,y),\;\varphi (x,y),\;\Omega(x,y) \bigr),\;(x,y) \in D \tag{1} \] where \[ \Omega (x,y)=\int^ x_ 0 \int^ y_ 0 g \bigl( x,y,t,\tau,Z (t,\tau),\;\varphi (t,\tau) \bigr) d \tau dt,\;(x,y) \in D, \] with \(F\) and \(g\) given functions. By a solution on \(D\) of the system (1) is meant a function \(u \in {\mathcal K}\) satisfying (1) at each point \((x,y) \in D\). Consider a system of \(2p\) curves, \(\Gamma_ i\), \(\widetilde \Gamma_ i\), \(0 \leq i \leq p-1\), of equations \(y=f_ i (x)\) and \(x=h_ i (y)\), respectively, where \(f_ i: [0,A] \to [0,B]\) and \(h_ i:[0,B] \times [0,A]\), \(0 \leq i \leq p-1\), are given functions of class \(C^ 1\). Denote by \(n_ i\), \(\widetilde n_ i\) the unit vectors normal to \(\Gamma_ i\) and \(\widetilde \Gamma_ i\), respectively. The authors prove a theorem on existence and uniqueness of solutions of the problem \(({\mathcal P})\): find a solution \(u\) of (1) satisfying the boundary conditions \[ {d^{p-i} \over dn^{p-i}} L^ iu \bigl( x,f_ i(x) \bigr) = M_ i \bigl( x,Z(x,f_ i(x) \bigr),\;\varphi(x,f_ i(x)) \] \[ {d^{p-i} \over d \widetilde n^{p-i}} L^ iu \bigl( h_ i(y),y \bigr) = N_ i \biggl( y,Z \bigl( h_ i(y),y \bigr),\;\varphi \bigl( h_ i(y),y \bigr) \biggr) \] where \((x,y) \in D\), \(0 \leq i \leq p- 1\). Under four assumptions and \(\max (A,B)\) sufficiently small, the problem \(({\mathcal P})\) has a unique solution \(u\) in the class \({\mathcal K}_ 1\) of functions \(u \in {\mathcal K}\) such that \(v_ i (0,0)=w_ i (0,0)=z_{ij} (0,0)=0\), \(0 \leq i\), \(j \leq p-1\). To this goal, the authors use the Banach fixed point theorem.