On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces

Abstract

a b s t r a c tThe objective of this paper is to establish the existence of solutions of nonlinear impulsive fractional integrodifferential equations of Sobolev type with nonlocal condition. The results are obtained by using fractional calculus and fixed point techniques.

Perturbation method for nonlocal impulsive evolution equations

“…In 2009, Liang et al [13,14] combined impulsive conditions and nonlocal conditions, and investigated the nonlocal impulsive evolution equation in Banach spaces. Later, Fan and Li [15], Fan [16], Fan and Liang [17], Ji et al [18], Chang et al [19], Wang and Wei [20], Yan [21], Debbouche and Baleanu [22], Balachandran et al [23] studied the impulsive evolution equation with nonlocal conditions. Moreover, Fu and Cao [24], Chang et al [25], Abada et al [26], Ji and Li [27], Cardindi and Rubbioni [28] studied the impulsive evolution inclusion with nonlocal conditions.…”

“…On the other hand, the study of the abstract nonlocal Cauchy problem was initiated by Byszewski and Lakshmikantham [29]. Since it is demonstrated that nonlocal problems have better effects in applications than the traditional Cauchy problems, differential equations with nonlocal conditions were studied by many authors and some basic results on nonlocal problems have been obtained; see [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] and the references therein for more comments and citations. In 2009, Liang et al [13,14] combined impulsive conditions and nonlocal conditions, and investigated the nonlocal impulsive evolution equation in Banach spaces.…”

Fractional evolution equation nonlocal problems with noncompact semigroups

“…Since it is demonstrated that the nonlocal problems have are better in applications than the traditional Cauchy problems, differential equations with nonlocal conditions were studied by many authors and some basic results on nonlocal problems have been obtained, see [2,3,[6][7][8]10,13,17,21,22,29,31,32] and the references therein.…”

Existence of solutions for impulsive fractional partial neutral integro-differential inclusions with state-dependent delay in Banach spaces

“…Chauhan et al [CD] extended the results of [SL] to impulsive fractional order semilinear evolution equations with nonlocal conditions. Balachandran et al [BK1], [BK2] discussed some fractional-order impulsive integrodifferential equations. The existence of solutions of fractional differential equation of Sobolev type with impulse effect in Banach spaces was also considered in [BK2].…”

“…Balachandran et al [BK1], [BK2] discussed some fractional-order impulsive integrodifferential equations. The existence of solutions of fractional differential equation of Sobolev type with impulse effect in Banach spaces was also considered in [BK2]. Debbouche and Baleanu [DB] proved the controllability of a class of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems.…”