We study a class of abstract nonlinear integral equations of convolution type defined on a Banach space. We prove the existence of a unique, locally mild solution and an extension property when the nonlinear term satisfies a local Lipschitz condition. Moreover, we guarantee the existence of the global mild solution and blow up profiles for a large class of kernels and nonlinearities. If the nonlinearity has critical growth, we prove the existence of the local $\epsilon $-mild solution. Our results improve and extend recent results for special classes of kernels corresponding to nonlocal in time equations. We give an example to illustrate the application of the theorems so obtained.

This paper mainly concerns the existence of a mild solution for a neutral stochastic fractional integro-differential inclusion of order $1\lt \beta \lt 2$ with a nonlocal con\-dition in a separable Hilbert space. Utilizing the fixed point theorem for multi-valued operators due to O' Regan, we establish an existence result involving a $\beta $-resolvent operator. An illustrative example is provided to show the effectiveness of the established results.

The inverse scattering problem under consideration is to reconstruct both the shape and the impedance function of an impenetrable two-dimensional obstacle from the far field pattern for scattering of time-harmonic acoustic or E-polarized electromagnetic plane waves. We propose an inverse algorithm that is based on a system of nonlinear boundary integral equations associated with a single-layer potential approach to solve the forward scattering problem. This extends the approach we suggested for an inverse boundary value problem for harmonic functions in Kress and Rundell(2005) and is a counterpart of our earlier work on inverse scattering for shape and impedance in Kress and Rundell(2001). We present the mathematical foundation of the method and exhibit its feasibility by numerical examples.