Periodic boundary value problem for the second order impulsive functional differential equations

Positive Solutions for a Second-Orderp-Laplacian Boundary Value Problem with Impulsive Effects and Two Parameters

“…Motivated by the papers mentioned above, we will extend the results of [11,14,23,31,33,47,48] to problem (7). We remark that on impulsive differential equations with a parameter only a few results have been obtained, not to mention impulsive differential equations with two parameters; see, for instance, [12,18,19,45].…”

“…Therefore, the study of this class of impulsive differential equations has gained prominence, and it is a rapidly growing field. For the general theory of impulsive differential equations, we refer the reader to [1][2][3], whereas the applications of impulsive differential equations can be found in [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. In particular, we would like to mention some results of Lin and Jiang [8] and Feng and Xie [10].…”

“…Some special cases of (7) have been investigated. For example, Ding and Wang [14] considered problem (7) with = 2, = 1, = 1, and ( ) ≡ 1, ∈ [0, 1].…”

“…For example, Ding and Wang [14] considered problem (7) with = 2, = 1, = 1, and ( ) ≡ 1, ∈ [0, 1]. By using Krasnoselskii's fixed point theorem, they proved the existence results of positive solutions of problem (7). However, the authors only obtained that problem (7) has at least one positive solution.…”

“…These results only dealt with the case that = 2 and = 1. Many difficulties occur when we study problem (7); for example, it is difficult to construct the cone and the operator because its state is discontinuous. It is also difficult to deal with and because of with onedimensional -Laplacian, and without one-dimensional -Laplacian in the same equation (21).…”