A class of second-order nonlocal indefinite impulsive differential systems

Abstract

We consider the second-order nonlocal impulsive differential system \t\t\t{−x″=a(t)xy+ω(t)f(x),0<t<1,t≠tk,−y″=b(t)x,0<t<1,t≠tk,Δx|t=tk=Ik(x(tk)),k=1,2,…,n,Δy|t=tk=Jk(y(tk)),k=1,2,…,n,x(0)=∫01h(t)x(t)dt,x′(1)=0,y(0)=∫01g(t)y(t)dt,y′(1)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} -x''=a(t)xy+\\omega (t)f(x), \\quad 0< t< 1, t\\neq t_{k}, \\\\ -y''=b(t)x, \\quad 0< t< 1, t\\neq t_{k}, \\\\ \\Delta x|_{t=t_{k}}=I_{k}(x(t_{k})), \\quad k=1,2,\\ldots,n, \\\\ \\Delta y|_{t=t_{k}}=J_{k}(y(t_{k})), \\quad k=1,2,\\ldots,n, \\\\ x(0)=\\int_{0}^{1}h(t)x(t)\\,dt, \\qquad x'(1)=0, \\\\ y(0)=\\int_{0}^{1}g(t)y(t)\\,dt, \\qquad y'(1)=0, \\end{cases} $$\\end{document} where the weight functions a(t), b(t), and omega (t) change sign on [0,1], and g(t)not equiv 0 and h(t)not equiv 0 on [0,1]. By constructing a cone K_{1}times K_{2}, which is the Cartesian product of two cones in space PC[0,1], and applying the well-known fixed point theorem of cone expansion and compression in K_{1}times K_{2}, we obtain conditions for the existence and multiplicity of positive solutions of a nonlocal indefinite impulsive differential system. An example is given to illustrate the main results.