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On a representation of the Jost solution for ordinary differential equations. (English. Russian original) Zbl 0947.34003

Funct. Anal. Appl. 33, No. 3, 222-224 (1999); translation from Funkts. Anal. Prilozh. 33, No. 3, 75-77 (1999).
The author begins by introducing the accepted definitions of Riemann-Liouville fractional-order differential and integral operators of order \(\alpha\), with \(0<\alpha<1\). The equation studied here is the \(n\)th-order ordinary differential equation: \[ y^{(n)}+ \sum^{n-2}_0 p_j(x)y^{(j)}= \lambda^n y,\quad x\in\mathbb{R}, \tag{1} \] where \(p_j(x)\) are \(j\)-continuously differentiable functions satisfying the condition \[ \int^\infty_{-\infty} \bigl(1+|x|\bigr)^{n-1-j+ s}|p_j|^s dx<+ \infty \tag{2} \] with \(0\leq s\leq j\leq n-2\).
The author introduces the notation: \[ q_j(t)=(-1)^m \sum_{m=j}^{n-2} (C_m)^j (p_m)^{(m-j)} (t),\quad J\psi(x)= \int^x_{-\infty} \psi(t)dt, \quad \sigma(x)= \sum^{n-2}_{m=j} \bigl(J^{n-j} |q_j |\bigr) (x), \] and proves the following theorem: If condition (2) holds, then for all \(\lambda\) such that \(\arg (\lambda)\leq \pi/n\) equation (1) has a unique solution \(g(x, \lambda)\) satisfying: \(\lim_{x\to- \infty} g(x,\lambda) \exp(-\lambda x)=1\), which is of the form: \[ g(x,\lambda)= e^{\lambda x}\left\{1+ \int^\infty_0 G(x,t) \exp(-\lambda^{n/2}) dt\right\},\tag{3} \] with the kernel \(G(x,.)\in L_1\) \((0,\infty)\), \(\int^\infty_0 |G(x,t) |dt\leq \exp(C \sigma(x))-1\), where \(C\) is a positive constant.
The author establishes the existence of certain orders of derivatives for \(G(x,t)\), integrability of these derivatives, and their asymptotic behaviour as \(x\) approaches \(-\infty\). Moreover, the kernel function \(G(x,t)\) is shown to satisfy a fractional-order ordinary differential equation. To prove it, he uses a Paley-Wiener type theorem, derives estimates and applies Mittag-Leffler functions in evaluation of fractional integrals.

MSC:

34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
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