We show the existence of uniformly locally attractive solutions for a nonlinear Volterra integral equation of convolution type with a general kernel. We use methods and techniques of fixed point theorems and properties of measure of noncompactness. We extend earlier results obtained in the context of integral equations of fractional order. We give new insights about a new and striking relation between the size of data and the fractional order $\alpha >0$ of the kernel $k(t)=t^{\alpha -1}/\Gamma (\alpha )$.

In this paper, we introduce a class of mappings that is more general than the class of polynomials as well as the class of convex functions. We prove $L^{p}$ estimates of Marcinkiewicz Mintegral operators along surfaces generated by mappings belong to this class. Moreover, we establish the $L^{p}$ boundedness of the corresponding maximal functions. Our results extend as well as improve previously known results on Marcinkiewicz integral operators and maximal functions.

In this paper we study the classical Fredholm integral equation of second kind, in which the Lebesgue type integral is replaced by the more general Choquet integral with respect to a monotone, submodular and continuous from below set function, including the so-called distorted Lebesgue measures.

We consider a fractional diffusion equation with distributed-order Caputo derivative. We prove existence of a weak and regular solution for a general uniformly elliptic operator under the assumption that the weight function is only integrable.

We prove an existence and uniqueness theorem for a fractional-order Volterra population model via an efficient monotone iterative scheme. By coupling a spectral method with the proposed iterative scheme, the fractional-order integrodifferential equation is solved numerically. The numerical experiments show that the proposed iterative scheme is more efficient than an existing iterative scheme in the literature, the convergence of which is very sensitive to various parameters, including the fractional order of the derivative. The spectral method based on our proposed iterative scheme shows greater flexibility with respect to various parameters. Sufficient conditions are provided to select the initial guess that ensures the quadratic convergence of the quasilinearization scheme.

We present results for existence of global solutions and attractivity for multidimensional fractional differential equations involving Riemann-Liouville derivative. First, by using a Bielecki type norm and the Banach-fixed point theorem, we prove a Picard-Lindelof-type theorem on the existence and uniqueness of solutions. Then, applying the properties of Mittag-Leffler functions, we describe the attractivity of solutions to some classes of Riemann-Liouville linear fractional differential systems.

A necessary and sufficient condition for the core function $\varphi \in L^{1}(0,1)$ is established in order to define a bounded cordial Volterra integral operator $(V_{\varphi }u)(t)=\int _{0}^{t}t^{-1}\varphi (t^{-1}s)u(s)\,ds$ in a space $L^{p}(0,T)$ for a $p\in [1,\infty ]$. This condition implies the boundedness of $V_{\varphi }$ also in the Sobolev spaces $W^{m,p}(0,T)$, $m\geq 1$. The spectra of $V_{\varphi }$ in the spaces $L^{p}(0,T)$ and $W^{m,p}(0,T)$ are determined and the spectral properties of $V_{\varphi }$ are examined in $L^{p}(0,T)$.