We study a non Lipschitz integral equation in the framework of generalized functions. By means of suitable regularizations, we define an associated generalized problem. We obtain a generalized solution in an appropriate $(\C,\E,\P)$ algebra. We compare the generalized solution with the classical one when it exists.

Consider a nonlinear operator equation $ x - K ( x )=f$, where $K$ is a Urysohn integral operator with a smooth kernel. Using the orthogonal projection onto a space of discontinuous piecewise polynomials of degree $\leq r$, previous authors have established an order $r + 1$ convergence for the Galerkin solution and $2 r + 2$ for the iterated Galerkin solution. Equivalent results have also been established for the interpolatory projection at Gauss points. In this paper, a modified projection method is shown to have convergence of order $ 3 r + 3$ and one step of iteration is shown to improve the order of convergence to $ 4 r + 4$. The size of the system of equations that must be solved, in implementing this method, remains the same as for the Galerkin method.

We consider a nonautonomous, semilinear, hyperbolic-parabolic equation subject to a dynamical boundary condition of memory type. First we prove the existence and uniqueness of global bounded solutions having relatively compact range in the natural energy space. Under the assumption that the nonlinear term $f$ is real analytic, we then derive an appropriate Lyapunov energy and we use the {\L}ojasiewicz-Simon inequality to show the convergence of global weak solutions to single steady states as time tends to infinity. Finally, we provide an estimate for the convergence rate.

In this paper, we study the nonlocal Cauchy problems of fractional evolution equations with Riemann-Liouville derivative by considering an integral equation which is given in terms of probability density. By using the theory of Hausdorff measure of noncompactness, we establish various existence theorems of mild solutions for the Cauchy problems in the cases $C_0$ semigroup is compact or noncompact.

In this paper, we use different methods to investigate continuous solutions of the nonlinear Volterra integral inclusions. The results obtained here improve and generalize many known results.