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<p>In this paper, we examine the Cauchy problem for an integro-differential equation with a finite-dimensional operator acting on the leading derivative and a convolutional integral term. The operator acting on the leading derivative has an infinite-dimensional kernel. Problems of this type are applicable to modeling dynamic processes with memory. The research was conducted using methods from the theory of generalized functions in Banach spaces. A&#xa0;fundamental operator function corresponding to the equation under consideration was constructed, and using it, a unique solution to the Cauchy problem was constructed in the space of distributions with left-bounded support. Based on the analysis of the structure of the generalized solution, theorems on the solvability of the Cauchy problem in spaces of functions of finite smoothness are obtained. An illustrative example is given.</p>

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ON THE SOLVABILITY OF INTEGRO-DIFFERENTIAL EQUATIONS WITH DERIVATIVES OF FUNCTIONALS IN BANACH SPACES

  • 期刊论文

  • 发表日期：

    2026年8月4日

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  • M. V. Falaleev,
  • E. Yu. Grazhdantseva

摘要

In this paper, we examine the Cauchy problem for an integro-differential equation with a finite-dimensional operator acting on the leading derivative and a convolutional integral term. The operator acting on the leading derivative has an infinite-dimensional kernel. Problems of this type are applicable to modeling dynamic processes with memory. The research was conducted using methods from the theory of generalized functions in Banach spaces. A fundamental operator function corresponding to the equation under consideration was constructed, and using it, a unique solution to the Cauchy problem was constructed in the space of distributions with left-bounded support. Based on the analysis of the structure of the generalized solution, theorems on the solvability of the Cauchy problem in spaces of functions of finite smoothness are obtained. An illustrative example is given.