The periodic boundary value problem for impulsive differential equations with variable time (Q2720026)

The authors consider a PBVP for impulse differential equations with variable time NEWLINE\[NEWLINEx'=f(t,x),\quad t\neq\tau(x),\;t\in J,\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\Delta x=I(x),\quad t=\tau(x),\tag{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINEx(0)=x(T),\tag{3}NEWLINE\]NEWLINE over the interval \(J=[0,T]\) without the strongly restriction of monotonicity of \(t=\tau(x)\). Sufficient conditions for the validity of the comparison principle between the inequalities NEWLINE\[NEWLINEw'\geq f(t,w),\;t\neq\tau(x), \quad\Delta w\geq I(w),\;t=\tau(x), \quad w(0)\geq x_0,NEWLINE\]NEWLINE and NEWLINE\[NEWLINEv'\leq f(t,v),\;t\neq\tau(x), \quad\Delta v\leq I(v),\;t=\tau(x), \quad v(0)\leq x_0,NEWLINE\]NEWLINE i.e. for \(w(t)\geq v(t)\), \(t\in J,\) are derived. By help of monotonic iterative technique, sufficient conditions for the existence of extremal solutions to (1)--(3) are derived.