General theorems on the existence and uniqueness of solutions to impulsive differential equations (Q2722243)

Let \(E\) be a Banach space, \(\mathbb R_+:=[0,\infty)\), and let \(T\) be a countable subset of \((0,\infty)\). The author considers the following Cauchy problem for the equation with impulses: NEWLINE\[NEWLINE\dot x(t)=A(t,x(t)), t\in \mathbb R_+\setminus T,\quad \Delta x(t)=B(t,x(t-0)), t\in T,\quad x(0)=x_0,\tag{1}NEWLINE\]NEWLINE with \(x_0\in E\) and \(A:\mathbb (\mathbb R_+\setminus T)\times E \mapsto E, B:T\times E \mapsto E\) are continuous mappings. The case where the set of limit points for \(T\) is nonempty is not excluded. The problem can be reduced to the equationNEWLINE\[NEWLINEx(t)=x_0+\int_{0}^{t}A(s,x(s)) ds+\sum_{s\in T\cap(0,t)}B(s,x(s-0)).NEWLINE\]NEWLINE The existence and uniqueness results for this equation are derived from more general ones obtained for the abstract equation \(x(t)=(\mathfrak U x)(t)+h(t), t\geq 0\), where \(h(t)\in \mathfrak N(\mathbb R_+ \mapsto E)\) is a given vector function, \(\mathfrak U:\mathfrak M(\mathbb R_+ \mapsto E) \mapsto \mathfrak N(\mathbb R_+ \mapsto E)\) is a Lipschitzian operator, \(\mathfrak M(\mathbb R_+ \mapsto E)\) is the space of vector functions which are continuous at \(t=0\) and continuous from the left on \((0,\infty)\), \(\mathfrak N(\mathbb R_+ \mapsto E)\) is the subspace of \(\mathfrak M(\mathbb R_+ \mapsto E)\) containing vector functions with finite variation on \([0,b]\), \(b>0\) being arbitrary. NEWLINENEWLINENEWLINEThe author proves the existence and uniqueness of solutions to problem (1) under the following assumptions: NEWLINENEWLINENEWLINE1) \(\exists a(\cdot)\): \(\mathbb R_+\setminus T \mapsto \mathbb R_+, \forall t\in \mathbb R_+\setminus T, \forall \{x,y\}\in E\): \(\|A(t,x)-A(t,y)\|\leq a(t)\|x-y\|\); \(\exists b(\cdot):T \mapsto \mathbb R_+, \forall s\in T, \forall \{x,y\}\in E\): \(\|B(s,x)-B(s,y)\|\leq b(s)\|x-y\|\); NEWLINENEWLINENEWLINE2) \(\forall t\geq 0\): \(\int_{0}^{t}a(s) ds+\sum_{s\in T\cap(0,t)}b(s)<\infty \); NEWLINENEWLINENEWLINE3) \(\forall t\geq 0\): \(\int_{0}^{t}\|A(s,0)\|ds+\sum_{s\in T\cap(0,t)}\|B(s,0)\|<\infty \).NEWLINENEWLINENEWLINEA theorem on the existence of a positive solution with respect to a cone in \(\mathfrak N(\mathbb R_+ \mapsto E)\) is also established.