Positive solutions for nonlinear operator equations and several classes of applications

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...

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Bibliographic Details
Published inMathematische Zeitschrift Vol. 266; no. 1; pp. 43 - 63
Main Authors Zhai, Cheng-Bo, Yang, Chen, Zhang, Xiao-Qin
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer-Verlag 01.09.2010
Subjects
Mathematics
Mathematics and Statistics
Nonlinear algebra systems
35F25
15A30
47H10
Boundary value problem
Normal cone
Initial value problem
Positive solution
47H07
Nonlinear operator equation
34B18
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Summary: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.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-009-0553-4