On Concave Operators

Positive solutions for nonlinear operator equations and several classes of applications

“…And then, as two special cases of (1.1), we consider the existence and uniqueness of positive solutions for (1.1) with an h-concave operator and a generalized β-concave operator. The corresponding results in [7,10,11,13,21,[26][27][28]32,33] turn out to be special cases of our main results. As applications, we first utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for first-order initial value problems and two-point boundary value problems.…”

“…Remark 2.1 In many papers, the condition that operator A : P h → P h or A :P →P was always required (for example, see [7,10,21,28,[31][32][33]). It is clear that Ah ∈ P h is automatically satisfied.…”

“…(1.1) with a monotone generalized concave operator A and not suppose the continuity and compactness of the operator A. As we know, a series of significant results have been obtained in [3,7,11,13,21,[26][27][28] for an operator equation x = Ax with a nonlinear monotone concave positive operator that acts in a ordered Banach space. Some of these results in [11,13,15,21,28] are valid for h-concave operator with h is a nonzero element.…”

“…As we know, a series of significant results have been obtained in [3,7,11,13,21,[26][27][28] for an operator equation x = Ax with a nonlinear monotone concave positive operator that acts in a ordered Banach space. Some of these results in [11,13,15,21,28] are valid for h-concave operator with h is a nonzero element. Others in [7,10,21,26,27,33] are also valid for β-concave operator or β-sublinear operator with β ∈ (0, 1).…”

“…Some of these results in [11,13,15,21,28] are valid for h-concave operator with h is a nonzero element. Others in [7,10,21,26,27,33] are also valid for β-concave operator or β-sublinear operator with β ∈ (0, 1). However, to the best of our knowledge, the results on the operator Eq.…”