Representing solutions of singularly perturbed equations as convergent series (Q757696)

The following singularly perturbed Cauchy problem is considered: \[ dy/dt=F_ 1(x,t,\mu),\quad \mu (dz/dt)=F_ 2(x,t,\mu);\quad y(0,\mu)=y^ 0(\mu),\quad z(0,\mu)=z^ 0(\mu). \] This problem is investigated under the conditions: 1) \(F_ i(0,t,0)=0\), \(i=1,2\); \(y^ 0(0)=0\). 2) The eigenvalues of the matrix \(F_{2z}(0,0,0)\) are in the half-plane Re \(\lambda\) \(<0\). 3) \(z^ 0(0)\in D_ z\). 4) The vector- functions \(F_ i(x,t,\mu)\) are analytic for \(x\in D\), \(t\in D_ t\), \(| \mu | <{\bar \mu}\), \(i=1,2\). 5) The vector-function \(x^ 0(\mu)\) is analytic for \(| \mu | <{\bar \mu}\). 6) There exists a constant \(C_ 1\), which does not depend on x,t, such that \(\| F^{- 1}_{2z}(x,t,0)\| \leq C_ 1\) for \(x\in D\), \(t\in D_ t\). 7) There exist constants \(C_ 2,\kappa >0\), not depending on t,s,\(\chi\), and such that \(\| V_ 1(t,s,\chi)\| \leq C_ 2 \exp (-\kappa (t-s)/\chi)\) for \(0\leq s\leq t\), \(t\in D_ t\), \(s\in D_ t\), \(0<\chi <{\bar \mu}\). Here \(V_ 1\) is the Cauchy matrix for the equation \(\chi dq/dt=F_{2z}(0,t,0)q.\) If \(D_ t=[0,T]\) it is proved that there exists a constant \(\mu_*>0\), such that for \(0\leq t\leq T\), \(0<\mu \leq \mu_*\) there exists a unique solution of the singularly perturbed problem, which can be represented by a series which is uniformly convergent in (t,\(\mu\)).