Article

Mathematische Zeitschrift

, Volume 266, Issue 1, pp 43-63

Positive solutions for nonlinear operator equations and several classes of applications

  • Cheng-Bo ZhaiAffiliated withSchool of Mathematical Sciences, Shanxi University Email author 
  • , Chen YangAffiliated withDepartment of Mathematics and Electronic Science, Business College of Shanxi University
  • , Xiao-Qin ZhangAffiliated withSchool of Mathematical Sciences, Shanxi University

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Abstract

In this paper, we study a class of nonlinear operator equations x = Ax + x 0 on ordered Banach spaces, where A is a monotone generalized concave operator. Using the properties of cones and monotone iterative technique, we establish the existence and uniqueness of solutions for such equations. In particular, we do not demand the existence of upper-lower solutions and compactness and continuity conditions. As applications, we study first-order initial value problems and two-point boundary value problems with the nonlinear term is required to be monotone in its second argument. In the end, applications to nonlinear systems of equations and to nonlinear matrix equations are also considered.

Keywords

Positive solution Nonlinear operator equation Normal cone Initial value problem Boundary value problem Nonlinear algebra systems

Mathematics Subject Classification (2000)

47H10 47H07 34B18 35F25 15A30