Factorization of Ordinary and Hyperbolic Integro-Differential Equations With Integral Boundary Conditions in a Banach Space

Abstract

The solvability condition and the unique exact solution by the universal factorization (decomposition) method for a class of the abstract operator equations of the type B1u = Au S(A0u) GF(Au) = f, u D(B1),where A,A0 are linear abstract operators, G, S are linear vectors and , F are linear functional vectors is investigagted. This class is useful for solving Boundary Value Problems (BVPs) with Integro-Differential Equations (IDEs), where A,A0 are differential operators and F(Au), (A0u) are Fredholm integrals. It was shown that the operators of the type B1 can be factorized in the some cases in the product of two moresimple operators BG, BG0 of special form, which are derived analytically. Further the solvability condition and the unique exact solution for B1u = f easily follow from the solvability condition and the unique exact solutions for the equations BGv = f and BG0u = v.

Factorization of abstract operators into two second degree operators and its applications to integro-differential equations

“…Furthermore by Theorem 3 (iv), since (24) and bijectivity of A 0 , the operator B 1 is bijective if (9) holds. The unique solution to (22), (23), by Theorem 3 (iv), is given by ( 13), (14). The theorem is proved.…”

“…Proof. Substituting A = AA 0 into (22) we obtain the operator B 1 in the form (6). Construct the operators B 0 and B by using ( 4) and (5), respectively, where for B we take the elements G, S, Ψ and A from (22) and (23), and for B 0 the elements A 0 , Φ and S 0 , G 0 from (22) and (25), (26).…”

“…on a Banach space was found in [22] by factorization of B 1 in two simple operators, and then the corresponding theory was applied for solving of Hyperbolic integro-differential equations with integral boundary conditions. The exact solution to the abstract operator equation…”