Positive solutions for nonlinear operator equations and several classes of applications

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.

Positive solutions for a new class of Hadamard fractional differential equations on infinite intervals

“…For getting our results, it is essential to list some useful concepts in Banach spaces and the main tool, i.e., a fixed point theorem for generalized concave operators. For this knowledge, we can see [26,28,29] for details.…”

Existence and Uniqueness of Positive Solutions to Nonlinear Second Order Impulsive Differential Equations with Concave or Convex Nonlinearities

“…In this section, we state some definitions, notations, and known results. For convenience of readers, we suggest that one refers to [30] and references therein for details.…”

“…We now present a fixed point theorem of generalized concave operators which will be used in the latter proof. See [30] , and let be a normal cone. Assume that ( 1 ) : → is increasing and ℎ ∈ ℎ ; ( 2 ) for any ∈ and ∈ (0, 1), there exists ( ) ∈ ( , 1) with respect to such that ( ) ≥ ( ) .…”

Existence and uniqueness results to positive solutions of integral boundary value problem for fractional q-derivatives

“…These days the fractional q-difference equation have given fire to increasing scholars' imaginations. Some works considered the existence of positive solutions for nonlinear q-fractional boundary value problem [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. For example, Ferreira [13] where γ , β ≤ 0, and c D α q is the fractional q-derivative of Caputo type.…”