| ZHAO ZHEN.[J].数学年刊A辑,1981,2(1):91~100 |
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| THE SUFFICIENT AND NECESSARY CONDITIONS FORNOETHER’S SOLVABILITY OF SINGULAR INTEGRALEQUATIONS WITH TWO CARLEMAN’S SHIFTS |
| Received:December 28, 1979 |
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| In this paper we consider the problem of solvability of singular integral equtions with two Carleman's shifts
\[\begin{gathered}
(\mathcal{K}\varphi )(t) \equiv {a_0}(t)\varphi (t) + {a_1}(t)\varphi [\alpha (t)] + {a_2}(t)\varphi [\beta (t)] + {a_3}(t)\varphi [\gamma (t)] \hfill \ + \frac{{{b_0}(t)}}{{\pi i}}\int_\Gamma {\frac{{\varphi (\tau )}}{{\tau - t}}} d\tau + \frac{{{b_1}(t)}}{{\pi i}}\int_\Gamma {\frac{{\varphi (\tau )}}{{\tau - \alpha (t)}}} d\tau + \frac{{{b_2}(t)}}{{\pi i}}\int_\Gamma {\frac{{\varphi (\tau )}}{{\tau - \beta (t)}}d\tau } \hfill \ + \frac{{{b_s}(t)}}{{\pi i}}\int_\Gamma {\frac{{\varphi (\tau )}}{{\tau - \gamma (t)}}} d\tau + \int_\Gamma {K(t,\tau )\varphi (\tau )d\tau = g(t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (1,1)} \hfill \\
\end{gathered} \]
Suppose that Г is a closed simple Lyapunoff's curve and \[\alpha (t)\], \[\beta (t)\] which satisfy Carleman's. conditions and \[\alpha [\beta (t)] = \beta [\alpha (t)]\] are two different homeomorphisms of Г onto itself, and that \[{a_k}(t),{b_k}(t)\], k = 0, 1, 2, 3 belong to the,space \[{H_\mu }(\Gamma ),g(t)\] belongs to the space \[{L_p}(\Gamma ),p > 1\]), p>l and \[K(t,\tau )\] has only weak singularity.
The following main results are obtained:
1. Singular integral eqution (1.1) is solvable if and only if the Noether's conditions
\[det(p(t) \pm q(t)) \ne 0\]
are satisfied.
2. Index of sigular integral eqution (1.1) is calculated by the formula
\[Ind{\kern 1pt} {\kern 1pt} {\kern 1pt} \mathcal{K} = \frac{1}{{8\pi }}{\{ arg\frac{{\det (p(t) - q(t))}}{{\det (p(t) + q(t))}}\} _\Gamma }\]
where p(t) and q(t) are matrices of coeffioents of so-called corresponding system of equtions.
All these results have been generalized for systems of singular integral equtions with two Carleman's shifts and complex conjugate of unknown functions. |
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