Abstract
In this paper, the conditions of correct solvability are found for a class of the third order operator-differential equations whose principal part has multiple characteristics in the Sobolev type space with exponential weight. The estimations of the norms of intermediate derivative operators closely connected with the solvability conditions are carried out. Moreover, the connection between the weight exponent and the lower bound of the spectrum of the main operator involved in the principal part of the equation is determined in the results of the paper.
References
[1] ALIEV, A. R.: On the solvability of a class of operator differential equations of the second order on the real axis, Zh. Vychisl. Mat. Mat. Fiz. 2 (2006), 347-357.Search in Google Scholar
[2] ALIEV, A. R.: Solvability of a class of boundary value problems for second-order operatordifferential equations with a discontinuous coefficient in a weighted space, Differ. Equ. 43 (2007), 1459-1463, [Transl. of: Differ. Uravn. 43 (2007), 1423-1426].Search in Google Scholar
[3] ALIEV, A. R.: On a class of third-order operator-differential equations in a weighted space, Dokl. Nats. Akad. Nauk Azerb. 63 (2007), 15-18 (Russian).Search in Google Scholar
[4] ALIEV, A. R.: Solvability of boundary-value problems for a class of third-order operatordifferential equations in a weighted space, Russian Math. Surveys 60 (2005), 791-793 [Transl. of: Uspekhi Mat. Nauk 60 (2005), (364), 215-216].Search in Google Scholar
[5] ALIEV, A. R.: On a boundary-value problem for one class of differential equations of the fourth order with operator coefficients, Azerb. J. Math. 1 (2011), 145-156. Search in Google Scholar
[6] ALIEV, A. R.: On the solvability of initial boundary-value problems for a class of operator-differential equations of third order, Math. Notes 90 (2011), 307-321 [Transl. of: Mat. Zametki 90 (2011), 323-339].Search in Google Scholar
[7] ALIEV, A. R.-GASYMOV, A. A.: On the correct solvability of the boundary-value problem for one class operator-differential equations of the fourth order with complex characteristics, Bound. Value Probl. 2009 (2009), Article ID 710386, 1-20.Search in Google Scholar
[8] ALIEV, A. R.-MIRZOEV, S. S.: On boundary value problem solvability theory for a class of high-order operator-differential equations, Funct. Anal. Appl. 44 (2010), 209-211 [Transl. of: Funktsional. Anal. i Prilozhen. 44 (2010), 63-65].Search in Google Scholar
[9] DUBINSKII, YU. A.: On some differential-operator equations of arbitrary order, Sb. Math. 19 (1973), 1-21 [Transl. of: Mat. Sb. 90(132) (1973), 3-22].Search in Google Scholar
[10] FAVINI, A.-SHAKHMUROV, V.-YAKUBOV, YA.: Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces, Semigroup Forum 79 (2009), 22-54.10.1007/s00233-009-9138-0Search in Google Scholar
[11] FAVINI, A.-YAKUBOV, Y.: Higher order ordinary differential-operator equations on the whole axis in UMD Banach spaces, Differential Integral Equations 21 (2008), 497-512.10.57262/die/1356038630Search in Google Scholar
[12] FAVINI, A.-YAKUBOV, YA.: Irregular boundary value problems for second order elliptic differential-operator equations in UMD Banach spaces, Math. Ann. 348 (2010), 601-632.10.1007/s00208-010-0491-9Search in Google Scholar
[13] GOLDSTEIN, J. A.: Semigroups of Linear Operators and Applications, Oxford University Press, New York, 1985.Search in Google Scholar
[14] GORBACHUK, V. I.-GORBACHUK, M. L.: Boundary Value Problems for Operator- Differential Equations, Naukova Dumka, Kiev, 1984 (Russian).Search in Google Scholar
[15] HILLE, E.-FILLIPS, R. S.: Functional Analysis and Semi-Groups. Amer. Math. Soc. Colloq. Publ. 31, Amer. Math. Soc., Providence, RI, 1957; Izdat. Inostran. Lit., Moscow, 1962.Search in Google Scholar
[16] KREIN, S. G.: Linear Differential Equations in a Banach Space, Nauka, Moscow, 1967 (Russian).Search in Google Scholar
[17] LIONS, J. L.-MAGENES, E: Non-Homogeneous Boundary Value Problems and Applications, Dunod, Paris, 1968; Mir, Moscow, 1971, Springer, Berlin, 1972.10.1007/978-3-642-65161-8Search in Google Scholar
[18] MIRZOEV, S. S.: On solvability of boundary-value problems for operator-differential equations of the second order in spaces with weight. In: Linear Operators and Their Applications, Baku, 1989, pp. 46-49 (Russian).Search in Google Scholar
[19] MIRZOEV, S. S.-GULIEVA, F. A.: On the completeness of elementary solutions of a class of second-order differential equations with operator coefficients, Math. Notes 86 (2009), 747-750 [Transl. of: Mat. Zametki 86 (2009), 797-800].Search in Google Scholar
[20] MIRZOEV, S. S.-SALIMOV, M. YU.: Completeness of elementary solutions to a class of second order operator-differential equations, Sib. Math. J. 51 (2010), 648-659 [Transl. of: Sibirsk. Mat. Zh. 51 (2010), 815-828].Search in Google Scholar
[21] PILIPCHUK, V. N.: On essentially nonlinear dynamics of arches and rings, J. Appl. Math. Mech. 46 (1982), 461-466 (Russian).10.1016/0021-8928(82)90112-5Search in Google Scholar
[22] SHKALIKOV, A. A.: Elliptic equations in Hilbert space and associated spectral problems, J. Soviet Math. 51 (1990), 2399-2467 [Transl. of: Trudy Sem. Petrovsk. 14 (1989), 140-224].Search in Google Scholar
[23] YAKUBOV, S. YA.: Linear Differential-Operator Equations and their Applications, Elm, Baku, 1985 (Russian).Search in Google Scholar
[24] YAKUBOV, S.: Completeness of Root Functions of Regular Differential Operators, Longman, Scientific and Technical, New York, 1994.Search in Google Scholar
[25] YAKUBOV, S.-YAKUBOV, YA: Differential-Operator Equations. Ordinary and Partial Differential Equations, Chapman and Hall/CRC, Boca Raton, FL, 2000.Search in Google Scholar
© Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Fuzzy Pseudo Subalgebras and Ideals of Pseudo D-Algebras
- Five Curious Congruences Modulo p2
- On the Hybrid Mean Value Involving Dedekind Sums and Kloosterman Sums
- Constructing New Crossed Group Categories Over Weak Hopf Group Algebras
- Generalization of Hilbert Inequality with Some Parameters
- Some Applications of Differential Subordination of p-Valent Functions
- Coefficient Estimates and Quasi-Subordination Properties Associated with Certain Subclasses of Analytic and Bi-Univalent Functions
- On Properties of Entire Solutions of Difference Equations and Difference Polynomials
- On the Ψ-Strong Stability of Nonlinear Lyapunov Matrix Differential Equations
- On the Existence of Solutions of Ordinary Differential Equations in Banach Spaces
- Global Dynamics of a Delayed n + m-Species Competition Predator-Prey System on Time Scales
- Comparison Of (G′/G)-Methods for Finding Exact Solutions of the Drinfeld-Sokolov System
- A Generalization of Cyclic Amenability of Banach Algebras
- n-Weak Module Amenability of Triangular Banach Algebras
- On a Class of Operator-Differential Equations of the Third Order With Multiple Characteristics on the Whole Axis in the Weighted Space
- Optimality Conditions for Vector Equilibrium Problems with Set-Valued Mappings and Cone Constraints
- Classification of the Blaschke Isoparametric Hypersurfaces in Lorentzian Space Forms
- A New Proof of a Theorem on Long Cycles
Articles in the same Issue
- Fuzzy Pseudo Subalgebras and Ideals of Pseudo D-Algebras
- Five Curious Congruences Modulo p2
- On the Hybrid Mean Value Involving Dedekind Sums and Kloosterman Sums
- Constructing New Crossed Group Categories Over Weak Hopf Group Algebras
- Generalization of Hilbert Inequality with Some Parameters
- Some Applications of Differential Subordination of p-Valent Functions
- Coefficient Estimates and Quasi-Subordination Properties Associated with Certain Subclasses of Analytic and Bi-Univalent Functions
- On Properties of Entire Solutions of Difference Equations and Difference Polynomials
- On the Ψ-Strong Stability of Nonlinear Lyapunov Matrix Differential Equations
- On the Existence of Solutions of Ordinary Differential Equations in Banach Spaces
- Global Dynamics of a Delayed n + m-Species Competition Predator-Prey System on Time Scales
- Comparison Of (G′/G)-Methods for Finding Exact Solutions of the Drinfeld-Sokolov System
- A Generalization of Cyclic Amenability of Banach Algebras
- n-Weak Module Amenability of Triangular Banach Algebras
- On a Class of Operator-Differential Equations of the Third Order With Multiple Characteristics on the Whole Axis in the Weighted Space
- Optimality Conditions for Vector Equilibrium Problems with Set-Valued Mappings and Cone Constraints
- Classification of the Blaschke Isoparametric Hypersurfaces in Lorentzian Space Forms
- A New Proof of a Theorem on Long Cycles