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], ||.||?) is
usually applied to 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 x(t) = ?p,i=1 ?i/?(^?i) ?t,0 fi(s,x(s))
(tn-tn-1)1-?i,n...(t1-s)1-?i,1 dt + ?(t). In particular we
consider the famous fractional Langevin equation and by the method we could
extend the region of variations of parameter ?+ ? from interval [0,1) in
the earlier works to interval [0,2).
Estimation of Unknown Function of a Class of Nonlinear Weakly Singular Integral Inequality