Extremal solutions of nonlinear first-order impulsive integro-differential equations in Banach spaces
The object of the paper is presented in its title. A nonlinear first-order impulsive integro-differential initial value problem in a Banach space \((E,\|\cdot\|)\) is formulated as: \[ x'= f(t,x,Tx), \quad t\in J,\;t\neq t_k, \quad \Delta x|_{t=t_k}= I_k [x(t_k)], \quad (k=1,2,\dots, p) \text{ and }x(t_0)= x_0, \] where \(f\in C[J\times E\times E,E]\), \(J= [t_0, t_0+a]\), \(a>0\), \(I_k\in C[E,E]\), \(k=1,2,\dots, p\); \(t_0< t_1< \dots< t_k<\dots< t_p< t_0+a\), \(x_0\in E\), \(T(x)(t)= \int_{t_0}^t k(t,s) x(s) ds\), \(k\in C[ D,\mathbb{R}^+]\), \(D= \{(t,s)\in J\times J: t\geq s\}\), \(\mathbb{R}^+= [0,\infty)\), \(\Delta x|_{t=t_k}= x(t_k^+)- x(t_k^-)\), where \(x(t_k^+)\) and \(x(t_k^-)\) are the right and left limits of \(x(t)\) at \(t=t_k\), respectively. For the above formulated problem, the author proves a theorem giving the conditions in which it has minimal and maximal solutions, defined in the paper, very difficult to reproduce in such a review. His results improve and generalize some similar results obtained previously by other authors, quoted in the bibliography of the paper. It ends with an example illustrating his results.
nonlinear first-order impulsive integro-differential initial value problem
scientific article; zbMATH DE number 1375297