caa a22 4500 606200819 CHVBK 20210128100945.0 cr unu---uuuuu 210128e20151101xx s 000 0 eng 10.1007/s00025-015-0454-2 doi (NATIONALLICENCE)springer-10.1007/s00025-015-0454-2 Spectral Properties of Fourth Order Differential Operators with Periodic and Antiperiodic Boundary Conditions [Elektronische Daten] [Hikmet Gunes, Nazim Kerimov, Ufuk Kaya] In this paper, we consider the following periodic and antiperiodic problem $$\begin{aligned} y^{\text{iv}}+p_{2}\left( x \right) y^{\prime \prime}+p_{1} \left( x\right) y^{\prime}+p_{0}\left( x\right) y=\lambda y, \quad 0 < x < 1,\\ y^{\left( s \right) }\left( 1 \right) -\left( -1 \right) ^{\sigma}y^{\left(s \right) } \left( 0 \right) =0,\quad s=\overline{0,3},\end{aligned}$$ y iv + p 2 x y ″ + p 1 x y ′ + p 0 x y = λ y , 0 < x < 1 , y s 1 - - 1 σ y s 0 = 0 , s = 0 , 3 ¯ , where λ is a spectral parameter; $${p_{j}(x) \in L_{1}(0,1), j=0,1, p_{2} (x) \in W_{1}^{1} (0,1)}$$ p j ( x ) ∈ L 1 ( 0 , 1 ) , j = 0 , 1 , p 2 ( x ) ∈ W 1 1 ( 0 , 1 ) with $${\int_{0}^{1} p_{2} (\xi)d\xi=0}$$ ∫ 0 1 p 2 ( ξ ) d ξ = 0 are complex-valued functions and σ =0,1. The boundary conditions of this problem are periodic-antiperiodic boundary conditions and it is well known that they are regular but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established. Under the condition $$\left( p_{2}\left( 1 \right)- p_{2}\left( 0 \right)-2c_{1}\right)\left( p_{2}\left( 1 \right)- p_{2}\left( 0 \right)+2c_{1} \right)\neq 0,$$ p 2 1 - p 2 0 - 2 c 1 p 2 1 - p 2 0 + 2 c 1 ≠ 0 , it is proved that all the eigenvalues (except for finite number) are simple, where $${c_{1}=\int_{0}^{1} p_{1} (\xi)d\xi}$$ c 1 = ∫ 0 1 p 1 ( ξ ) d ξ . Furthermore, we prove that the system of root functions of this spectral problem forms a basis in the space $${L_{p}( 0,1), 1 < p < \infty}$$ L p ( 0 , 1 ) , 1 < p < ∞ , when $${p_{1}( 1)= p_{1}( 0); p_{2}^{(s)}(1)= p_{2}^{(s)}(0), s=0,1; p_{j}(x) \in W_{1}^{j} (0,1), j=0,1,2; c_{1} \neq 0}$$ p 1 ( 1 ) = p 1 ( 0 ) ; p 2 ( s ) ( 1 ) = p 2 ( s ) ( 0 ) , s = 0 , 1 ; p j ( x ) ∈ W 1 j ( 0 , 1 ) , j = 0 , 1 , 2 ; c 1 ≠ 0 . Also, it is shown that this basis is unconditional for p=2. Springer Basel, 2015 Fourth order eigenvalue problem nationallicence periodic and antiperiodic boundary conditions nationallicence asymptotic behavior of eigenvalues and eigenfunctions nationallicence basis properties of the system of root functions nationallicence Gunes Hikmet Department of Mathematics, Mersin University, 33343, Mersin, Turkey aut Kerimov Nazim Department of Mathematics, Mersin University, 33343, Mersin, Turkey aut Kaya Ufuk Department of Mathematics, Bitlis Eren University, 13000, Bitlis, Turkey aut Results in Mathematics Springer Basel 68/3-4(2015-11-01), 501-518 1422-6383 68:3-4<501 2015 68 25 https://doi.org/10.1007/s00025-015-0454-2 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00025-015-0454-2 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Gunes Hikmet Department of Mathematics, Mersin University, 33343, Mersin, Turkey aut NATIONALLICENCE 700 1- Kerimov Nazim Department of Mathematics, Mersin University, 33343, Mersin, Turkey aut NATIONALLICENCE 700 1- Kaya Ufuk Department of Mathematics, Bitlis Eren University, 13000, Bitlis, Turkey aut NATIONALLICENCE 773 0- Results in Mathematics Springer Basel 68/3-4(2015-11-01), 501-518 1422-6383 68:3-4<501 2015 68 25