# Asymptotics for Solutions of Linear Differential Equations Having Turning Points with Applications

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*S. Strelitz*

Asymptotics are built for the solutions \(y_j(x,\lambda)\),
\(y_j^{(k)}(0,\lambda)=\delta_{j\,n-k}\), \(0\le j,k+1\le n\) of
the equation \[L(y)=\lambda p(x)y,\quad x\in [0,1],
\qquad\qquad\qquad(1)\] where
\(L(y)\) is a linear differential operator of whatever order \(n\ge
2\) and \(p(x)\) is assumed to possess a finite number of turning
points. The established asymptotics are afterwards applied to the study of:

1) the existence of infinite eigenvalue sequences for various multipoint
boundary problems posed on Equation (1),
especially as \(n=2\) and \(n=3\) (let us be aware that the same
method can be successfully applied on many occasions in case \(n>3\)
too) and

2) asymptotical distribution of the corresponding eigenvalue sequences
on the complex plane.

#### Table of Contents

# Table of Contents

## Asymptotics for Solutions of Linear Differential Equations Having Turning Points with Applications

- Table of Contents vii8 free
- Chapter 1: The Construction of Asymptotics 110 free
- §1 Introduction 110
- §2 Formulation of the main result 615
- §3 The main auxiliary lemma 918
- §4 The equation Y[sup(n)] = λp[sub(0)]X[sup(αY)] 2130
- §5 Asymptotics in [0, x[sub(0)] 3039
- §6 Asymptotics in [x*, l] 3544
- §7 Proof of Theorem 1 3847
- §8 Completion of the proof of Theorem 1 and of Theorem 2 4756

- Chapter 2: Application: Existence and Asymptotics of Eigenvalues 5160
- References 8897