Spectral Properties of Nonhomogenous Differential Equations with Spectral Parameter in the Boundary Condition

In this paper, using the boundary properties of the analytic functions we investigate the structure of the discrete spectrum of the boundary value problem (0.1)iy1'+q1(x)y2−λy1=ϕ1(x) −iy2'+q2(x)y1−λy2=ϕ2(x),x∈R+$$\matrix{\hfill {iy_1^\prime + q_1 \left(x \right)y_2 - \lambda y_1 = \varphi _1 \left(x \right)\;\;} & \hfill {} \cr \hfill {- iy_2^\prime + q_2 \left(x \right)y_1 - \lambda y_2 = \varphi _2 \left(x \right),} & \hfill {x \in R_ + } \cr }$$ and the condition (0.2)(a1λ+b1)y2(0,λ)−(a2λ+b2)y1(0,λ)=0$$\left({a_1 \lambda + b_1 } \right)y_2 \left({0,\lambda } \right) - \left({a_2 \lambda + b_2 } \right)y_1 \left({0,\lambda } \right) = 0$$ where q_1,q₂, φ₁, φ₂ are complex valued functions, a_k ≠ 0, b_k ≠ 0, k = 1, 2 are complex constants and λ is a spectral parameter. In this article, we investigate the spectral singularities and eigenvalues of (0.1), (0.2) using the boundary uniqueness theorems of analytic functions. In particular, we prove that the boundary value problem (0.1), (0.2) has a finite number of spectral singularities and eigenvalues with finite multiplicities under the conditions, sup⁡x∈R+[|ϕk(x)|exp⁡(εxδ)]<∞, k=1.2sup⁡x∈R+[|qk(x)|exp⁡(εxδ)]<∞, k=1.2$$\matrix{{\mathop {\sup }\limits_{x \in R_ + } \left[ {\left| {\varphi _k \left(x \right)} \right|\exp \left({\varepsilon x^\delta } \right)} \right] < \infty ,\;\;\;k = 1.2} \hfill \cr {\mathop {\sup }\limits_{x \in R_ + } \left[ {\left| {q_k \left(x \right)} \right|\exp \left({\varepsilon x^\delta } \right)} \right] < \infty ,\;\;\;k = 1.2} \hfill \cr }$$ for some ε > 0, 12<δ<1${1 \over 2} < \delta < 1$