# The nonlocal problem with multi- point perturbations of the boundary conditions of the Sturm-type for an ordinary differential equation with involution of even order

### Abstract

The spectral properties of the nonself-adjoint problem with multipoint perturbations of the Dirichlet conditions for differential operator of order $2n$ with involution are investigated. The system of eigenfunctions of a multipoint problem is constructed. Sufficient conditions have been established, under which this system is complete and, under some additional assumptions, forms the Riesz basis. The research is structured as follows. In section 2 we investigate the properties of the Sturm-type conditions and nonlocal problem with self-adjoint boundary conditions for the equation $$(-1)^ny^{(2n)}(x)+ a_{0}y^{(2n-1)}(x)+ a_{1}y^{(2n-1)}(1-x)=f(x),\,x\in (0,1).$$ In section 3 we study the spectral properties for nonlocal problem with nonself-adjoint boundary conditions for this equation. In sections 4 we construct a commutative group of transformation operators. Using spectral properties of multipoint problem and conditions for completeness the basis properties of the systems of eigenfunctions are established in section 5. In section 6 some analogous results are obtained for multipoint problems generated by differential equations with an involution and are proved the main theorems.

### References

A. Ashyralyev, A.M. Sarsenbi, Well-posedness of an elliptic equations with an involution, Electr. J. Diff. Eq., 284 (2015), 1–8.

Ya.O. Baranetskij, P.I. Kalenyuk, L.I. Kolyasa, Boundary-value problem for abstract differential equations with operator involution, Bukov. Math. J., 4 (2016), №3-4, 22–29. (in Ukrainian)

Ya.O. Baranetskij, P.I. Kalenyuk, L.I. Kolyasa, M.I. Kopach, The nonlocal problem for the differentialoperator equation of the even order with the involution, Carpathian Math. Publ., 9 (2017), №2, 109–119.

Ya.O. Baranetskij, P.I. Kalenyuk, L.I. Kolyasa, M.I. Kopach, The nonlocal multipoint problem for an ordinary differential equations of even order with the involution, Mat. Stud., 49 (2018), №1, 80–94.

Ya.O. Baranetskij, P.I. Kalenyuk, L.I. Kolyasa, Spectral properties of nonself-adjoint nonlocal boundaryvalue problems for the operator of differentiation of even order, Ukr. Mat. Zh., 70 (2018), №6, 851–865.

Ya.O. Baranetskij, I.Ya. Ivasiuk, P.I. Kalenyuk, A.V. Solomko, The nonlocal boundary problem with perturbations of antiperiodicity conditions for the elliptic equation with constant coefficients, Carpathian Math. Publ., 10 (2018), №2, 215–234.

Ya.O. Baranetskij, I.I. Demkiv, I.Ya. Ivasiuk, M.I. Kopach, The nonlocal problem for the 2n order differential equations with unbounded operator coefficients and the involution, Carpathian Math. Publ., 10 (2018), №1, 1–17.

Ya.O. Baranetskij, Dirichlet problem for even differential equations with operator coefficients containing involution, Precarpathian Bulletin of the NTSh, 46 (2018), 26–37. (in Ukrainian)

Ya.O. Baranetskij, P.I. Kalenyuk, A nonlocal problem with multipoint perturbations of the boundary conditions of the Sturm-type for an ordinary differential equation of even order, Mathematical methods and physicomechanical fields, 62 (2019), №1, 25–37. (in Ukrainian)

Ya.O. Baranetskij, P.I. Kalenyuk, M.I. Kopach, A.V. Solomko, The nonlocal boundary value problem with perturbations of mixed boundary conditions for an elliptic equation with constant coefficients, I, Carpathian Math. Publ., 11 (2019), №2, 228–239.

C.M. Bender, S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett., 80 (1998), 5243-5246.

C.M. Bender, Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys., 70 (2007), 947–1018.

Ju.M. Berezanskii, Expansions in eigenfunctions of self-adjoint operators, Hardcover, 1968: 809.

M.Sh. Burlutskaya, A.P. Khromov, Initial-boundary value problems for first-order hyperbolic equations with involution, Dokl. Math., 84 (2011), №3, 783–786.

A. Cabada, F.A.F. Tojo, Existence results for a linear equation with reflection, non-constant coefficient and periodic boundary conditions, J. Math. Anal. Appl., 412 (2014), №1, 529–546.

G. Freiling, Irregular boundary value problems revisited, Results Math., 62 (2012), №3-4, 265–294.

I.C. Gohberg, M.G. Krein, Introduction to the theory of linear nonself-adjoint operators, Providence: Amer. Math. Soc., 1969, 378 p.

C.P. Gupta, Two-point boundary value problems involving reflection of the argument, Int. J. Math. Math. Sci., 10 (1987), №2, 361–371.

G.M. Kessel’man, Unconditional convergence of the eigenvalues expansions of some differential operators, Izvestija Vysshih Uchebnyh Zavedenii. Matematika, 39 (1964), №92, 82–93. (in Russian)

M. Kirane, N. Al-Salti, Inverse problems for a nonlocal wave equation with an involution perturbation, J. Nonlinear Sci. Appl., 9 (2016), 1243–1251.

L.V. Kritskov, A.M. Sarsenbi, Spectral properties of a nonlocal problem for the differential equation with involution, Differ. Eq., 51 (2015), №8, 984–990.

V.P. Kurdyumov, On Riesz bases of eigenfunction of 2-nd order differential operator with involution and integral boundary conditions, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 15 (2015), №4, 392–405. (in Russian)

V.P. Mikhailov, Riesz basis in L2[0; 1], Doklady AN SSSR, 144 (1962), №5, 981–984.

A.M. Minkin, Odd and even cases of Birkhoff-regularity, Math. Nachr., 174 (1995), №1, 219–230.

E.I. Moiseev, V.E. Ambartsumyan, On the basis property of the eigenfunctions of the Frankl problem with nonlocal evenness and oddness conditions of the second kind, Dokl. Math., 432 (2010), №4, 451–455. (in Russian)

M.A. Naimark, Linear differential operators., Frederick Ungar Publ. Co., New York, 1967.

D. O’Regan, Existence results for differential equations with reflection of the argument, J. Aust. Math. Soc., 57 (1994), №2, 237–260.

M.A. Sadybekov, A.M. Sarsenbi, Mixed problem for a differential equation with involution under boundary conditions of general form, AIP Conf. Proc., 1470 (2012).

A.A. Shkalikov, On the basis problem of the eigenfunctions of an ordinary differential operator, Rus. Math. Surveys, 34 (1979), №5, 249–250.

A.A. Shkalikov, The completeness of eigenfunctions and associated functions of an ordinary differential operator with irregular-separated boundary conditions, Funct. Anal. Appl., 10 (1976), №4, 305–316.

V.E. Vladykina, A.A. Shkalikov, Spectral properties of ordinary differential operators with involution, Doklady Mathematics, 99 (2019), №1, 5–10.

V.E. Vladykina, A.A. Shkalikov, Regular ordinary differential operators with involution, Math. Notes, 106 (2019), №5, 674–687.

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