Abstract

In a weighted Sobolev space on the whole real axis, we obtain the sufficient conditions for the well-posed and unique solvability of m,n order operator-differential equations. These conditions were formulated only by the operator coefficients of the considered equation. According to the values of m,n the operator-differential equation has complicated and multiple characteristics. In addition, by using the main part of the equation, the norms of the operators of intermediate derivative were estimated. We deduce the relationship between the exponent of the weight and the lower bound of the spectrum of the operator of the main part of the equation. As an applied result of this paper, we formulated a problem for higher-order partial differential equations and we provided an alternative method for obtaining the regular solvability of operator pencil.

Citation details of the article

Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 34
Issue: 1
Year: 2021

DOI: 10.12732/ijam.v34i1.8

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References

1. [1] A.B.I. Ahmed, M.A. Labeeb, Solvability of a class of operator-differential equations of third order with complicated characteristic on the whole real axis, Open Access Library J., 5 (2018), 1-5.
2. [2] A.B.I. Ahmed, Solvability of a non-homogeneous boundary value problem for a class of higher-order operator-differential equations in a weighted Space, Technology Reports of Kansai Univ., 62 (2020), 5551-5561.
3. [3] A.B.I. Ahmed, Solvability of initial-boundary value problem for parabolic n th-order operator-differential equations in Hilbert space, Technology Reports of Kansai Univ., 62 (2020), 373-381.
4. [4] N. Faried, L. Rashed, A.B.I. Ahmed, M.A. Labeeb, Solvability of initialboundary value problem of a multiple characteristic fifth-order operatordifferential equation, J. Egypt. Math. Soc., 27 (2019), 1-7.
5. [5] N. Faried, A.B.I. Ahmed, M.A. Labeeb, Sufficient conditions for regular solvability of an arbitrary order operator-differential equation with initialboundary conditions, Adv. in Difference Equations, 2020, No 104 (2020), 1-14.
6. [6] E.G. Gamidov, On a boundary value problem for second order operator differential equations in space of smooth vector functions, Trans. of NAS of Azerbaijan, XXXIII (2013), 72-84.
7. [7] A.A. Gasymov, On solvability of a class of complicated characteristic operator differential equations of fourth order, Trans. of NAS of Azerbaijan, 1, No 28 (2007), 49-54.
8. [8] M.G. Gasymov, The multiple completeness of part of the eigen- and associated vectors of polynomial operator bundles, Izv. Akad. Nauk Arm. SSR, Ser. Matem, 6, No 2-3 (1971), 131-147.
9. [9] V.I. Gorbachuk, M.L. Gorbachuk, Boundary Value Problems for Differential-Operator Equations, Kiev, Naukova Dumka (1984) (In Russian).
10. [10] E. Hille, R. Phillips, Functional Analysis and Semi-Group, Revis. Ed., Amer. Math. Soc. Colloq. Publ., 31, Amer. Math. Soc., Providence, R.I. (1957); Russian transl., IL, Moscow (1962).
11. [11] R.Z. Humbataliev, On regular solvability of boundary value problems in weight space, Intern. J. of Math. Analysis, 1 (2007), 1209-1216.
12. [12] V. Jabrayilova, On double completeness of part of root vectors of a class of ploynomial operator bundles of the fourth order, Trans. of NAS of Azerbaijan, 7, No 25 (2005), 55-60.
13. [13] S.G. Krein, Linear Equations in Banach Spaces, Birkh¨auser, Boston (1982).
14. [14] E. Kreyszig, Introductory Functional Analysis with Applications, USA (1989).
15. [15] N.V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, American Mathematical Society, 96 (2008).
16. [16] M.A. Labeeb, A.B.I. Ahmed, L. Rashed, Conditional solvability of the boundary value problem of a self-adjoint operator-differential equations in a Sobolev-type space, International Journal of Applied Mathematics, 32, No 3 (2019), 469-478; doi:10.12732/ijam.v32i3.8.
17. [17] F.S. Lachinova, Solvability of a class of parabolic operator-differential equations of third order, Proc. of IMM of NAS of Azerbaijan, XXXIX (2013), 77-86.
18. [18] J.L. Lions, E. Majenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, Berlin-Heidelberg-New York (1972).
19. [19] S.S. Mirzoev, On solvability of boundary value problems for operator differential equations of the second order in spaces with weight, In: Linear Operators and Their Applications, ASU, Baku (1989), 46-49 (In Russian).
20. [20] S.S. Mirzoev, F.A. Gulieva, On the completeness of elementary solutions of a class of second order differential equations with operator coefficients, Mathematical Notes, 86, No 5 (2009), 747-750.
21. [21] S.S. Mirzoev, M.D. Karaaslan, R.Z. Gumbataliev, To the theory of operator differential equations of the second order, Doklady Mathematics, 88 (2013), 741-743.
22. [22] V.N. Pilipchuk, On essentially nonlinear dynamics of arches and rings, J. of Appl. Math. and Mech., 46, No 3 (1982), 360-364.
23. [23] M.A. Soylemez, Solvability conditions for one inverse parabolic operator differential equation in sobolev-type space, J. of Qafqaz Univ.-Mathematics and Computer Sci., 4, No 1 (2016), 95-100.
24. [24] G.A. Teters, Complex Loading and Stability of the Covers from Polymeric Materials, Latvia, Riga, Zinatne Press (In Russian) (1969).
25. [25] S. Yakubov, Y. Yakubov, Differential Operator Equations: Ordinary and Partial Differential Equations, Ser. Pure and Applied Mathematics, Chapman and Hall/CRC, USA (1999).