In this paper, we are concerned with the controllability for a class of impulsive fractional integro-differential evolution equation in a Banach space. Sufficient conditions of the existence of mild solutions and approximate controllability for the concern problem are presented by considering the term \(u'(\cdot )\) and finding a control \(v\) such that the mild solution satisfies \(u(b)=u_{b}\) and \(u'(b)=u'_{b}\). The discussions are based on Mönch fixed point theorem as well as the theory of fractional calculus and \((\alpha ,\beta )\)-resolvent operator. Finally, an example is given to illustrate the feasibility of our results.

中文翻译：

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脉冲分数阶积分-微分演化方程的可控性

在本文中，我们关注的是 Banach 空间中一类脉冲分数阶积分微分演化方程的可控性。通过考虑项\(u'(\cdot )\)并找到控制\(v\)使得温和解满足\(u( b)=u_{b}\)和\(u'(b)=u'_{b}\)。讨论基于 Mönch 不动点定理以及分数阶微积分理论和\((\alpha ,\beta )\) -resolvent operator。最后，给出一个例子来说明我们的结果的可行性。