Three solutions for a three-point boundary value problem with instantaneous and non-instantaneous impulses

In this paper, we consider the multiplicity of solutions for the following three-point boundary value problem of second-order p-Laplacian differential equations with instantaneous and non-instantaneous impulses: ⎪ ⎪⎪⎪⎪⎪⎪⎧ ⎨ ⎪⎪ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ -(& rho;(t)& phi;p(u'(t)))' + g(t)& phi;p(u(t)) = & lambda;fj(t,u(t)), t E (sj, tj+1], j = 0, 1, ..., m, & UDelta;(& rho;(tj)& phi;p(u'(tj))) = & mu;Ij(u(tj)), j = 1, 2, ..., m, & rho;(t)& phi;p(u'(t)) = & rho;(t+ j )& phi;p(u'(t+ j )), t E (tj, sj], j = 1, 2, ..., m, & rho;(s+ j )& phi;p(u'(s+ j )) = & rho;(s-j)& phi;p(u'(s-j )), j = 1, 2, ..., m, u(0) = 0, u(1) = & zeta;u(& eta;), where & phi;p(u) := |u|p-2u, p > 1, 0 = s0 < t1 < s1 < t2 < ... < sm1 < tm1+1 = & eta; < ... < sm < tm+1 = 1, & zeta; > 0, 0 < & eta; < 1, & UDelta;(& rho;(tj)& phi;p(u'(tj))) = & rho;(t+j)& phi;p(u'(t+j))-& rho;(t-j)& phi;p(u'(t-j )) for u'(t & PLUSMN;j) = lim u'(t), j = 1, 2, ..., m, t & RARR;t & PLUSMN;j and f j E C((sj, tj+1] xR, R), Ij E C(R, R). & lambda; E (0, +& INFIN;), & mu; E R are two parameters. & rho;(t) & GE; 1, 1 & LE; g(t) & LE; c for t E (sj, tj+1], & rho;(t), g(t) E Lp[0,1], and c is a positive constant. By using variational methods and the critical points theorems of Bonanno-Marano and Ricceri, the existence of at least three classical solutions is obtained. In addition, several examples are presented to illustrate our main results.