Nonlinear Differential Equations in Ordered Spaces

Positive and maximal positive solutions of singular mixed boundary value problem

“…Hence the sequence { } ≥ ρ is equicontinuous on [2ρ T − 2ρ]. In addition, inequalities (11) and (12) show that the sequence { } ≥ ρ is bounded in…”

“…In the literature extremal solutions (that is, maximal and minimal solutions) are studied for regular and discontinuous boundary value problems, and extremal solutions usually "lie" in a sector bounded by some functions α and β (as a rule α and β are lower and upper functions of the problems under discussion). The existence of extremal solutions is proved by monotone iterative techniques (see, e.g., [10]- [12], [14]), an approximation method ( [9,11]), and a technique which works with ordered normed spaces and assumes that the set of all solutions is upward and downward directed ( [15]). In this paper we prove the existence of the maximal positive solution of problem (1), (2) by a method which is a combination of the method of generalized lower and upper functions ( [19]) with regularization and sequential techniques.…”

On extending existence theory from scalar ordinary differential equations to infinite quasimonotone systems of functional equations

“…For positive g's Carl and Heikkilä improved Hassan and Rzymowski's result in the monograph [5] and Cid and Pouso gave in [7] an alternative result which, roughly speaking, interchanges the roles of t and x in the assumptions.…”

Trapping region for discontinuous quasilinear elliptic systems of mixed monotone type

“…It is well known that T η k : V → V are continuous and bounded, cf., e.g., [3,6]. By means of T η k we define a new vector field which is related to f = (…”