Geodesic
Factorization of ordinary and hyperbolic integro-differential equations with integral boundary conditions in a Banach space
Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 27 (2021) no. 1, pp. 29-43

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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 $$ B_1u=\mathcal{A}u-S\Phi(A_0u)-GF(\mathcal{A}u)=f ,\quad u\in D(B_1), $$ where $\mathcal{A}, A_0$ are linear abstract operators, $G, S$ are linear vectors and $\Phi, F$ are linear functional vectors is investigagted. This class is useful for solving Boundary Value Problems (BVPs) with Integro-Differential Equations (IDEs), where $\mathcal{A}, A_0$ are differential operators and $F(\mathcal{A}u), \Phi(A_0u)$ are Fredholm integrals. It was shown that the operators of the type $B_1$ can be factorized in the some cases in the product of two more simple operators $B_G$, $B_{G_0}$ of special form, which are derived analytically. Further the solvability condition and the unique exact solution for $B_1u=f$ easily follow from the solvability condition and the unique exact solutions for the equations $B_G v=f$ and $B_{G_0}u=v$.
Keywords: correct operator, factorization (decomposition) method, Fredholm integro-differential equations, initial problem, nonlocal boundary value problem with integral boundary conditions. 
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     author = {E. Providas and L. S. Pulkina and I. N. Parasidis},
     title = {Factorization of ordinary and hyperbolic integro-differential equations with integral boundary conditions in a {Banach} space},
     journal = {Vestnik Samarskogo universiteta. Estestvennonau\v{c}na\^a seri\^a},
     pages = {29--43},
     publisher = {mathdoc},
     volume = {27},
     number = {1},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VSGU_2021_27_1_a2/}
}
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E. Providas; L. S. Pulkina; I. N. Parasidis. Factorization of ordinary and hyperbolic integro-differential equations with integral boundary conditions in a Banach space. Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 27 (2021) no. 1, pp. 29-43. http://geodesic.mathdoc.fr/item/VSGU_2021_27_1_a2/