A unique solution of initial value problems for first order impulsive integro-differential equations of mixed type in Banach spaces
The authors investigate the following initial value problem for the first-order nonlinear impulsive integro-differential equation of mixed type in a real Banach space E: \[ x'=f(t,x,Tx,Sx) \quad\text{for} \quad t\in J,\;t\neq t_k, \tag{1} \] \[ \Delta x|_{t=t_k}= I_k\bigl(x(t_k)\bigr) \quad \text{for} \quad k=1,2,\dots,p, \tag{2} \] where \(f\in C(J\times E\times E \times E,E)\), \(J=[0,a]\), \(h\in C(E,E)\), \(0<t_1<t_2 \cdots <t_p<a\), \(x_0\in E\) and \[ (Tx)(t)= \int^t_0k(t,s) x(s)ds, \quad (Sx)(t)=\int^a_0 h(t,s)x(s)ds. \] Under some hypotheses formulated in terms of upper or lower solutions with help of an order determined by a cone in \(E\) a result on the existence and uniqueness of a solution of the problem (1)--(3) is obtained.
first-order nonlinear impulsive integro-differential equation
scientific article; zbMATH DE number 1866690