Periodic and homoclinic solutions generated by impulses

The multiplicity of solutions for fourth-order equations generated from a boundary condition

“…However, our aim in this work is to study problem (P µ ) and its multiple solutions generated from a boundary condition g. Here, a solution for problem (P µ ) is said to be generated from a boundary condition g if this solution emerges when g is not zero, but disappears when g is zero. For example, if problem (P µ ) possesses at most one solution when g = 0, but possesses three solutions when boundary condition g is not zero, then problem (P µ ) has at least two solutions generated from a boundary condition g. Our work is motivated by results in [1,5]. Zhang and Li [5] studied a class of second-order impulsive differential systems and obtained the existence of a periodic solution generated from impulses by using the Mountain Pass Lemma.…”

“…For example, if problem (P µ ) possesses at most one solution when g = 0, but possesses three solutions when boundary condition g is not zero, then problem (P µ ) has at least two solutions generated from a boundary condition g. Our work is motivated by results in [1,5]. Zhang and Li [5] studied a class of second-order impulsive differential systems and obtained the existence of a periodic solution generated from impulses by using the Mountain Pass Lemma. To the best of our knowledge, for fourthorder differential equations, there has so far been no paper concerning its solutions generated from a boundary condition.…”

Solitary Wave and Periodic Wave Solutions for a Class of Singular p-Laplacian Systems with Impulsive Effeects

“…Some classical tools have been widely used to get the solutions of impulsive differential equations, such as fixed point theorems in cones, topological degree theory (including continuation method and coincidence degree theory), the method of lower and upper solutions, and the critical point theory. For the theory and classical results, we refer the readers to the references, [4], [17], [19], [26], [32] and books [2], [22], [31].…”

“…See, to name a few, [8], [29], [32]. For example, in [32], by applying the variational methods, Zhang and Li established the existence result of homoclinic solutions of the following second order impulsive differential equations q + V q (t, q) = f (t), for t ∈ (s k−1 , s k ),…”

Periodic Solutions Generated by Impulses for State-Dependent Impulsive Differential Equation

“…By using a variational method and a variant fountain theorem, Dai and Zhang [10] considered the existence and multiplicity of solutions for a class of nonlinear impulsive problem on the half-line. For more related work, the reader is referred to [11][12][13] and the references therein. As we know, state-dependent IDEs have become a hot topic in recent years due to their extensive application space, but it is also a difficult research field because of their essential properties: uncertainties for impulsive time and collision times.…”