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.
Highlights
It is generally accepted that the theory and applications of differential equations with impulsive effects are an important area of investigation, since it is far richer than the corresponding theory of differential equations without impulsive effects
By constructing a cone K1 × K2, 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 K1 × K2, we obtain conditions for the existence and multiplicity of positive solutions of a nonlocal indefinite impulsive differential system
1 Introduction It is generally accepted that the theory and applications of differential equations with impulsive effects are an important area of investigation, since it is far richer than the corresponding theory of differential equations without impulsive effects
Summary
It is generally accepted that the theory and applications of differential equations with impulsive effects are an important area of investigation, since it is far richer than the corresponding theory of differential equations without impulsive effects. By constructing a cone K1 × K2, which is the Cartesian product of two cones in the space PC[0, 1], and using the well-known fixed point theorem of cone expansion and compression, we obtain conditions for the existence and multiplicity of positive solutions of system (1.9). We remark that this is probably the first time that the existence and multiplicity of positive solutions of impulsive differential systems with indefinite weight and integral boundary conditions have been studied.
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