We study the existence of solutions of a nonlinear Volterra integral equation in the space 1 [0, +∞). With the help of Krasnoselskii's fixed point theorem and the theory of measure of weak noncompactness, we prove an existence result for a functional integral equation which includes several classes on nonlinear integral equations. Our results extend and generalize some previous works. An example is given to support our results.

Given a bounded positive linear operator A on a Hilbert space H we consider the semi-Hilbertian space (H , | A ), where ξ | η A := Aξ | η . In this paper we introduce a class of operators on a semi Hilbertian space H with inner product | A . We call the elements of this class A-positive-normal or A-posinormal. An operator T ∈ B(H) is said to be A-posinormal if there exists a A-positive operator P ∈ B(H) (i.e., AP ≥ 0) such that T AT * = T * AP T. We study some basic properties of these operators. Also we study the relationship between a special case of this class with the other kinds of classes of operators in semi-Hilbertian spaces.Mathematics Subject Classification: Primary 47A05, 47A30 Secondary 47A62

In this paper, the concept of n-power quasi-normal operators on a Hilbert space defined by Sid Ahmed in [14] is generalized when an additional semi-inner product is considered. This new concept is described by means of oblique projections.

In this article, we introduce and study a new class of operators, larger than α , β − A -normal operators and different than α , β − A -normal operators, named m -quasi- α , β − A -normal operators. Considering the semi-inner product induced by a positive operator A , the m -quasi- α , β − A -normal operators turn into a generalization (for this new structure) of classical m -quasi- α , β -normal operators. Several results concerning properties of this kind of operators are presented in the paper. Several inequalities for the A -numerical radius and A -operator norm for members of this class are established.

The investigation of new operators belonging to some specific classes has been quite fashionable since the beginning of the century, and sometimes it is indeed relevant. In this study, we introduce and study a new class of operators called k -quasi- m , n -isosymmetric operators on Hilbert spaces. This new class of operators emerges as a generalization of the m , n -isosymmetric operators. We give a characterization for any operator to be k -quasi- m , n -isosymmetric operator. Using this characterization, we prove that any power of an k -quasi- m , n -isosymmetric operator is also an k -quasi- m , n -isosymmetric operator. Furthermore, we study the perturbation of an k -quasi- m , n -isosymmetric operator with a nilpotent operator. The product and tensor products of two k -quasi- m , n -isosymmetries are investigated.