Uniqueness and existence results for implicit impulsive differential equations
The authors deal with the implicit differential equation with impulses of the form \[ F(t,u(t),u'(t)+p_{1}(t)u(t))=0\;\text{ a.e. on } [t_0,t_1]:=J,\tag{*} \] with the conditions \(D(u(t_0),x_0)=0, u(\alpha+0)-u(\alpha)=Iu(\alpha), \alpha \in \Lambda,\) where \(\Lambda\) is a well-ordered subset of the interval \((t_0,t_1).\) The values of the functions are in a Banach space. First, the authors provide conditions which guarrantee the fact that equation (*) admits unique solutions in a prespecified set \(V.\) Next, the authors show the existence by transforming (*) into an infinite functional system of the form \(Lu_{n+1}=Nu_{n}, \) where the operators \(L,N\) are known. It is shown that the sequence \(u_n\) with \(u_0=0\) converges on \(J\) pointwise to a solution \(u\in V\) to problem (*). A special case is discussed in the last section of the paper.
scientific article; zbMATH DE number 1491635