By making use of the Leray-Schauder fixed point theorem we prove the global existence of solutions to some integro-differential equations with delay subject to non-local conditions, and this problem is considered in an arbitrary Banach space.

Abstract. New bounds for solutions of an integro-differential inequality with weakly singular kernel are established using a weighted version of the Hardy-Littlewood-Sobolev inequality. Besides, some applications to real world problems are presented.Mathematics subject classification (2010): 42B20, 26D07, 26D15.

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This paper is concerned with the existence, uniqueness, and stability of the solution of some impulsive fractional problem in a Banach space subjected to a nonlocal condition. Meanwhile, we give a new concept of a solution to impulsive fractional equations of multiorders. The derived results are based on Banach's contraction theorem as well as Schaefer's fixed point theorem.

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