On Multi–Singular Integral Equations Involving n Weakly Singular Kernels

Abstract

We deal with some sources of Banach spaces which are closely related to an important issue in applied mathematics i.e. the problem of existence and uniqueness  of the solution for the very applicable  weakly singular integral equations. In the classical mode the uniform space $(C[a,b],\|.\|_\infty)$ is usually applied for the related discussion. Here, we apply some new types of Banach spaces, in order to extend the area of problems we could discuss. We consider a very general type of singular integral equations involving $n$ weakly singular kernels, for an arbitrary natural number $n$, without any restrictive assumption of differentiability or even continuity on engaged functions. We show that in appropriate conditions the following multi--singular integral equation of weakly singular type has got exactly a solution in a defined Banach space
  \begin{eqnarray*}
x(t)=\sum_{i=1}^p{{\gamma_i}\over\Gamma(\widehat{\alpha_i})}\int_0^{\hat{t}}{{f_i(s,x(s))}\over{(t_n-t_{n-1})^{1-\alpha_n^i}\cdots(t_1-s)^{1-\alpha_1^i}}}d{\hat{t}}+\phi(t).
\end{eqnarray*}
In particular we consider the famous fractional Langevin equation and by the method we could extend the the region of variations of parameter $\alpha+\beta$ from interval $[0,1)$ in earlier work to interval $[0,2)$.