This paper considers the existence of $L^{p}$-solutions for a class of fractional integral equations involving abstract Volterra operators in a separable Banach space. Some applications for the existence of $L^{p}$-solutions for different classes of fractional differential equations are given.

This work addresses the solvability and solution of volume integrodifferential equations (VIEs) associated with 3D free-space transmission problems (FSTPs) involving elastic or conductive inhomogeneities. A modified version of the singular volume integral equation (SVIE) associated with the VIE is introduced and shown to be of second kind involving a contraction operator, i.e., solvable by Neumann series, implying the well-posedness of the initial VIE. Then, the solvability of VIEs for frequency-domain FSTPs (modelling the scattering of waves by compactly-supported inhomogeneities) follows by a compact perturbation argument. This approach extends work by Potthast \cite {potthast:99} on 2D electromagnetic problems (transverse-electric polarization conditions) involving orthotropic inhomogeneities in a isotropic background and contains recent results on the solvability of Eshelby's equivalent inclusion problem as special cases. The proposed modified SVIE is also useful for iterative solution methods, as Neumannn series converge (i)~unconditionally for static problems and (ii)~on some inhomogeneity configurations for which divergence occurs with the usual SVIE for wave scattering problems.

We study the well-posedness of degenerate fractional differential equations with infinite delay $(P_\alpha ): D^\alpha (Mu)(t) =Au(t)+\int _{-\infty }^t a(t-s)Au(s)\,ds+f(t)$, $0\leq t\leq 2\pi $, in Lebesgue-Bochner spaces $L^p(\mathbb {T}; X)$ and Besov spaces $B_{p,q}^s(\mathbb {T}; X)$, where $A$ and $M$ are closed linear operators on a Banach space~$X$ satisfying $D(A)\subset D(M)$, $\alpha >0$ and $a\in L^1(\mathbb {R}_+)$ are fixed. Using well known operator-valued Fourier multiplier theorems, we completely characterize the well-posedness of $(P_\alpha )$ in the above vector-valued function spaces on $\mathbb {T}$.

The aim of this paper is to discuss the global existence, uniqueness and asymptotic stability of mild solutions for a class of semilinear evolution equations with nonlocal initial conditions on infinite interval. A sufficient condition is given for judging the relative compactness of a class of abstract continuous family of functions on infinite intervals. With the aid of this criteria the compactness of the solution operator for the problem studied on the half line is obtained. The theorems proved in this paper improve and extend some related results in this direction. Discussions are based on stochastic analysis theory, analytic semigroup theory, relevant fixed point theory and the well known Gronwall-Bellman type inequality. An example to illustrate the feasibility of our main results is also given.

By use of the so-called max-product method, in this paper we associate to the truncated linear sampling operators based on the Fej\'er-type kernel, nonlinear sampling operators of Kantorovich type, for which we prove convergence results in the $L^{p}$-norm, $1\le p\le +\infty $, with quantitative estimates.