The aim of this paper is to solve a class of  auto-convolution Volterra integral equations by the well-known differential transform method. The analytic property of solution and convergence of the method under some assumptions are discussed and some illustrative examples are given to clarify the theoretical results, accuracy and performance of the proposed method.

The weighted semigroup algebra Mb (S, w) is studied via its identification with Mb (S) together with a weighted algebra product *w so that (Mb (S, w), *) is isometrically isomorphic to (Mb (S), *w). This identification enables us to study the relation between regularity and amenability of Mb (S, w) and Mb (S), and improve some old results from discrete to general case.

Let G be a locally compact Hausdorff topological group and H be a compact subgroup of G. Then, the homogeneous space G/H possesses a specific Radon measure, which is called a relatively invariant measure. We show that the concepts of convolution and involution can be extended to the integrable functions defined on this homogeneous space. We study the properties of convolution and prove that the space of integrable functions is an involutive Banach algebra with an approximate identity. We also find a necessary and sufficient condition on a closed subspace of this Banach algebra to make it a left ideal.

We show that the second-order Hochschild cohomology groups of measure algebra M(G) and group algebra L-1(G) with coefficients in C-phi are Banach spaces where G is a locally compact group and phi is the augmentation character. 

Let G be a locally compact group or a discrete group and let w be a weight function on G. We define a new convolution on the Beurling algebra L1(G, w) and show that it is also a Banach algebra with this new convolution. We show that there is a one to one correspondence between the unitary representations of G and the non-degenerate ?-representations of L1(G, w). We show that L1(G, w) is amenable if and only if G is amenable. This improves earlier results which require w to be diagonally bounded.