Decay integral solutions for neutral fractional differential equations with infinite delays

Abstract

Communicated by M. KiraneOur aim in this work is to find decay integral solutions for a class of neutral fractional differential equations in Banach spaces involving unbounded delays. By constructing a suitable measure of noncompactness on the space of solutions and establishing new estimates for fractional resolvent operators, we prove the existence of a compact set of decay integral solutions to the mentioned problem.deriving new estimates for fractional resolvent operators, constructing a satisfactory space of solutions and find on this space a regular measure of noncompactness (MNC).These features enable us to utilize the fixed point theory for condensing maps. Consequently, we obtain the existence of decay integral solutions x with kx.t/k D O .t ˛/ as t ! 1. Because the case˛2 .0, 1/ is more involved than the case˛D 1, we focus on the former one and make a note that our technique can be applied to the latter case with the same manner. (B*) The phase space B satisfies (B) with K 2 BC R C ; R C and M being such that t M.t/ D O.1/ as t ! 1. (H*) The function h verifies (H) for all T > 0 and for Á 2 BC R C , R C . In addition, h.t, 0/ D 0 for all t 2 R C . (F*) f satisfies (F) for all T > 0. Put Á 1 D sup t 0 Á.t/, K 1 D sup t 0 K.t/, and M 1 D sup t 0 M.t/.

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