Generalized Inverse Operator for an Integrodifferential Operator in the Banach Space
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GENERALIZED INVERSE OPERATOR FOR AN INTEGRODIFFERENTIAL OPERATOR IN THE BANACH SPACE V. F. Zhuravlev
UDC 517.983
By using the theory of generalized inversion of operators in Banach spaces, we establish conditions for the generalized invertibility of integrodifferential operators with degenerate kernels in Banach spaces. We obtain formulas for the construction of bounded projectors and the bounded generalized inverse operator for an integrodifferential operator with degenerate kernel in a Banach space. The results of investigations are illustrated by an example.
In the recent decades, the theory of generalized inversion of operators [1–6] is extensively used for the investigation of the solvability of various types of functional-differential equations and boundary-value problems for these equations. In view of the specific features of each analyzed problem, this approach enables one to use all advantages of the “operator technique” for its solution. The methods of generalized inversion of Noetherian operators in Euclidean spaces [2, 4] and topologically Noetherian [5], n- and d -normal [7], evolutionary [8], and integral operators [9] in Banach spaces are well developed and extensively applied. A specific feature of the problem of investigation of solvability and construction of the solutions of integrodifferential equations is connected with the fact that the corresponding integrodifferential operators do not have inverse operators. Equations of this kind in Euclidean spaces were considered, e.g., in [10–12]. In the present work, by using the theory of generalized inversion of operators in Banach spaces [4, 13] and the theory of generalized inversion of integral operators in Banach spaces [9], we establish the conditions of generalized invertibility of integrodifferential operators with degenerate kernels in Banach spaces and construct the corresponding generalized inverse operators and bounded projectors. Statement of the Problem Let z.t/ be a continuously differentiable function acting from a segment Œa; bç into a real Banach space º
Ω
B1 W z.t / 2 C.Œa; bç; B1 / WD z.�/W Œa; bç ! B1 ; jjjzjjj D sup kz.t /k ; t 2Œa;bç
and let C.Œa; bç; B1 / be a Banach space of vector functions continuous on Œa; bç with values in B1 : Also let z.t / 2 P / 2 C1 .Œa; bç; B1 /; where C1 .Œa; bç; B1 / is a Banach space of continuously differentiable C.Œa; bç; B1 / and z.t vector functions with the norm jjjzjjj D
1 X
� � � � sup �z .k/ .t /�
kD0 t 2Œa;bç
Zhytomyr National Agroecological University, Staryi Blvd., 7, Zhytomyr, 10008, Ukraine; e-mail: [email protected]. Translated from Neliniini Kolyvannya, Vol. 22, No. 2, pp. 202–219, April–June, 2019. Original article submitted February 28, 2019; revision submitted May 2, 2019. 1072-3374/20/2494–0609
c 2020 �
Springer Science+Business Media, LLC
609
V. F. Z HURAVLEV
610
and z .k/ .t/ is the kth derivative of z.t /: The derivative z.t P / is understood in a sense of [14]. Consider an integrodifferential operator
.Lz/.t / WD z.t P / � M.t /
Zb a
⇥
⇤ W .s/z.s/ C V .s/Pz
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