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<p>We consider linear inhomogeneous equations in Banach spaces with a Gerasimov–Caputo derivative and a generator of a strongly continuous resolving family of the corresponding homogeneous equation. We obtained conditions on the right-hand side of the equation, which are sufficient for the existence of a unique solution. The notions of classical and mild solutions are considered. The possibility of generating a resolving family by a self-adjoint high-order elliptic operator depending on the argument of a multiplier at the operator is studied. In the case of the existence of a resolving family, theorems on the existence of a unique classical and mild solution to an initial boundary value problem for an equation with a fractional Gerasimov–Caputo derivative in time and an elliptic operator in spatial variables are proved.</p>

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LINEAR INHOMOGENEOUS EQUATIONS WITH GERASIMOV–CAPUTO DERIVATIVES AND GENERATORS OF STRONGLY CONTINUOUS FAMILIES

  • 期刊论文

  • 发表日期：

    2025年9月22日

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  • Vladimir E. Fedorov,
  • Anton S. Skorynin

摘要

We consider linear inhomogeneous equations in Banach spaces with a Gerasimov–Caputo derivative and a generator of a strongly continuous resolving family of the corresponding homogeneous equation. We obtained conditions on the right-hand side of the equation, which are sufficient for the existence of a unique solution. The notions of classical and mild solutions are considered. The possibility of generating a resolving family by a self-adjoint high-order elliptic operator depending on the argument of a multiplier at the operator is studied. In the case of the existence of a resolving family, theorems on the existence of a unique classical and mild solution to an initial boundary value problem for an equation with a fractional Gerasimov–Caputo derivative in time and an elliptic operator in spatial variables are proved.