Studies on the Product of Integro Quasi-Differential Equations and Their Solutions in Direct Sum Spaces

Abstract

In this present chapter, we consider a product of quasi-differential expressions \({\tau}_1, {\tau}_2,... {\tau}_n\) each of order n with complex coefficients and their formal adjoints \({\tau}_1^+, {\tau}_2^+,... {\tau}_n^+\) on [0,b) respectively. We show in the direct sum spaces \(L_w^2 (I_p), p = 1,2, ... , N\) of functions defined on each of the separate intervals in the case of one singular end-points and under suitable conditions on the function F that all solutions of the product integro quasi-differential equations \({\prod}_{j=1}^n {\tau}_j, - {\lambda}I]y(t)= wF\) are bounded and \(L_w^2 -\) bounded on [0,b).