Existence and multiplicity of solutions for some second-order systems on time scales with impulsive effects

Abstract

AbstractIn this paper, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for the nonautonomous second-order system on time scales with impulsive effects{uΔ2(t)+A(σ(t))u(σ(t))+∇F(σ(t),u(σ(t)))=0,Δ-a.e. t∈[0,T]Tκ;u(0)−u(T)=uΔ(0)−uΔ(T)=0,(ui)Δ(tj+)−(ui)Δ(tj−)=Iij(ui(tj)),i=1,2,…,N,j=1,2,…,p,wheret0=0<t1<t2<⋯<tp<tp+1=T,tj∈[0,T]T(j=0,1,2,…,p+1),u(t)=(u1(t),u2(t),…,uN(t))∈RN,A(t)=[dlm(t)]is a symmetricN×Nmatrix-valued function defined on[0,T]Twithdlm∈L∞([0,T]T,R)for alll,m=1,2,…,N,Iij:R→R(i=1,2,…,N,j=1,2,…,p) are continuous andF:[0,T]T×RN→R. Finally, two examples are presented to illustrate the feasibility and effectiveness of our results.MSC:34B37, 34N05.