Mathematische Zeitschrift

, Volume 266, Issue 1, pp 43–63

Positive solutions for nonlinear operator equations and several classes of applications


DOI: 10.1007/s00209-009-0553-4

Cite this article as:
Zhai, CB., Yang, C. & Zhang, XQ. Math. Z. (2010) 266: 43. doi:10.1007/s00209-009-0553-4


In this paper, we study a class of nonlinear operator equations x = Ax + x0 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.


Positive solutionNonlinear operator equationNormal coneInitial value problemBoundary value problemNonlinear algebra systems

Mathematics Subject Classification (2000)


Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.School of Mathematical SciencesShanxi UniversityTaiyuanPeople’s Republic of China
  2. 2.Department of Mathematics and Electronic ScienceBusiness College of Shanxi UniversityTaiyuanPeople’s Republic of China