φ− (h,e)-concave operators and applications

Nonlinear sum operator equations and applications to elastic beam equation and fractional differential equation

“…Very recently, Wardowski [24] introduced the definition of (e, u)-concave-convex operator, and proved a fixed point theorem of such operator by analyzing some of its properties. By comparing with main result obtained in [25], we find that the above new operator is the same as ϕ -(h, e)-concave operator defined in Zhai and Wang [25].…”

Unique solutions for a new coupled system of fractional differential equations

“…Inspired by the works of coupled systems and recent papers [16,34], we study the coupled system (1.1) and give the existence and uniqueness of solutions. By using a fixed point theorem of increasing ϕ-(h, e)-concave operators, we establish the existence and uniqueness of solutions for the coupled system dependent on two constants.…”

“…Clearly, P h ⊂ P. Take another element e ∈ P with θ ≤ e ≤ h, we define P h,e = {x ∈ E|x + e ∈ P h }. Definition 2.1 (see [34]) Assume that A : P h,e → E is an operator which satisfies: for any x ∈ P h,e and λ ∈ (0, 1), there exists…”

“…Lemma 2.5 (see [34]) Suppose that P is normal and A is an increasing ϕ-(h, e)-concave operator satisfying Ah ∈ P h,e . Then A has a unique fixed point x * in P h,e .…”

Fixed point theorems for a class of nonlinear sum-type operators and application in a fractional differential equation

“…With a significant development and extensive applications in various differential and integral equations, nonlinear operators theory has been an active area of research in nonlinear functional analysis. Over the past several decades, much attention has been paid to various fixed point theorems for the single nonlinear operator, and a lot of important results have been obtained, see for example [1][2][3][4][5][6][7][8][9][10]. Thereinto, without requiring the operators to be continuous or compact or having the upper-lower solutions, the authors present some important and interesting fixed point theorems (see [1,[4][5][6][7][8][9][10]).…”