| Abstrakt: |
In this paper, we consider the following periodic and antiperiodic problem where λ is a spectral parameter; $${p_{j}(x) \in L_{1}(0,1), j=0,1, p_{2} (x) \in W_{1}^{1} (0,1)}$$ with $${\int_{0}^{1} p_{2} (\xi)d\xi=0}$$ 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 it is proved that all the eigenvalues (except for finite number) are simple, where $${c_{1}=\int_{0}^{1} p_{1} (\xi)d\xi}$$ . Furthermore, we prove that the system of root functions of this spectral problem forms a basis in the space $${L_{p}( 0,1), 1 < p < \infty}$$ , when $${p_{1}( 1)= p_{1}( 0); p_{2}^{(s)}(1)= p_{2}^{(s)}(0), s=0,1; p_{j}(x) \in W_{1}^{j} (0,1), j=0,1,2; c_{1} \neq 0}$$ . Also, it is shown that this basis is unconditional for p = 2. [ABSTRACT FROM AUTHOR] |
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