Existence of nonlinear neutral integrodifferential equations of Sobolev type with nonlocal condition
The purpose of this paper is to prove the existence of mild solutions for a nonlinear impulsive neutral delay integrodifferential equation of Sobolev type with time varying delays using semigroup theory and the fixed point approach. More precisely, it is used the Mönch type fixed point theorem and the Hausdorff measure of noncompactness. Since operators of Mönch type generalize compact operators, the obtained result is a generalization of other results from the literature. As an application of the abstract result, in the last part of this work the authors establish conditions which ensure the existence of a mild solution for the partial differential equation NEWLINE\[NEWLINE\begin{aligned} \frac{\partial }{\partial t}\left[z\left(t,x\right)-z_{xx}\left(t,x\right)+\int_{-\infty}^{t} a_1\left(s-t\right)z_t\left(s,t\right) ds\right]-z_{xx}\left(t,x\right) \\ =\rho\left(t,z\left(t,x\right)\right)+\int_{0}^{t}\sigma\left(t,s,z\left(t,x\right)\right)ds, \, 0\leq x\leq \pi,\\ z\left(t,0\right)=z\left(t,\pi\right)=0, \, t\geq 0,\\ z\left(0,x\right) + \sum_{k=1}^{p}z\left(t_k,x\right) = z_0\left(x\right), 0<t_1 <t_2 < \ldots < t_p<a; \, x\in \left[0,\pi\right],\\ \Delta z|_{t=t_i} = I_i\left(z\left(x\right)\right)=\left(\gamma_i \left(z\left(x\right)\right)+t_i\right)^{-1}, \, z\in X, \, 1\geq i \geq p. \end{aligned}NEWLINE\]
scientific article; zbMATH DE number 6365589