Variational approach to impulsive differential equations

Existence of solutions for a class of second-order impulsive Hamiltonian system with indefinite linear part

“…Since impulsive problems are important in real world, many researchers have extensively studied the theory and applications of impulsive differential equations, see [2,4,5,8,11] for more details. In recent years, variational methods have been widely used to study Hamiltonian system and impulsive problems, see [1,3,6,7,9,[12][13][14][15][16][17] and references therein. In [7], Kyritsi and Papageorgion investigated problem (1.2) and obtained an existence result by using Morse critical groups.…”

Impulsive boundary value problems for p(t)-Laplacian’s via critical point theory

“…a one-dimensional counterpart of the p(x)-Laplacian, subject to some impulsive changes. In our research we mainly follow the approach applied in [10] with one significant difference. The existence results for problems with a fixed right hand side in [10] were proved via the Lax-Milgram Lemma and in our paper we apply a direct method of the calculus of variations together with the Fundamental Lemma of the calculus of variations which we prove in the case of functions from relevant Orlicz-Sobolev spaces.…”

“…In our research we mainly follow the approach applied in [10] with one significant difference. The existence results for problems with a fixed right hand side in [10] were proved via the Lax-Milgram Lemma and in our paper we apply a direct method of the calculus of variations together with the Fundamental Lemma of the calculus of variations which we prove in the case of functions from relevant Orlicz-Sobolev spaces. It is the variational approach for boundary value problems with a p(x)-Laplacian that prevails in the literature, see again [7], while for impulsive problems, the variational approach has only recently begun and most results have been obtained by other methods, see [5], [8].…”

“…The variational investigation of impulsive problems inspired by [10] has received a lot of attention recently. In [6] another variational framework for the SturmLiuville boundary value problem is developed in the case of second order impulsive ordinary differential equation of p-Laplacian type.…”

On the new concept of solutions and existence results for impulsive fractional evolution equations

“…For a wide bibliography and exposition on this object see for instance the monographs of [1,2,3,4] and the papers [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19].…”