Abstract We study a problem arising in open-closed $$p $$ -adic string theory for a convolution type integral equation with a cubic nonlinearity on the real line $$\mathbb {R} $$ . We establish conditions on the function and the real parameter occurring in the equation under which the problem has an odd continuous solution on $$\mathbb {R}\setminus \{0\}$$ . The proof of the existence of such a solution is constructive. Examples of equations satisfying all assumptions of the theorem are given.

中文翻译：

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开闭$${\boldsymbol {p}}$$-Adic 弦理论中非线性问题的可解性

摘要 我们研究了在实线 $$\mathbb {R} $$ 上具有三次非线性的卷积类型积分方程的开闭 $$p $$ -adic 弦理论中出现的问题。我们建立了方程中出现的函数和实参数的条件，在该条件下，问题在 $$\mathbb {R}\setminus \{0\}$$ 上具有奇连续解。这种解存在的证明是有建设性的。给出了满足定理所有假设的方程的例子。