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

Abstract

In this paper, by studying the solutions of the abstract operator equation A(x, x) + B(x, x) + e = x on ordered Banach spaces, where A, B are two mixed monotone operators and e ∈ P with θ ≤ e ≤ h, we prove a class of boundary value problems on elastic beam equation to have a unique solution. Furthermore, we also apply our abstract result to establish the existence and uniqueness theorem of nontrivial solutions for nonlinear fractional boundary value problems. The iterative sequences to approximate unique solutions for the above two classes of problems are also obtained.

Analysis of a Nonlinear ψ-Hilfer Fractional Integro-Differential Equation Describing Cantilever Beam Model with Nonlinear Boundary Conditions

“…This section contains several illustrated cases to highlight the relevance of our findings in this study. 7 10 ;sin( πt 2t+2 )…”

Solvability for p‐Laplacian generalized fractional coupled systems with two‐sided memory effects

“…Since Zhai and Wang 33 presented the − (h, e)−concave operator, various fixed-point theorems of relevant operators appear in the people's line of sight. Inspired by recent results in Sang and Ren, 34 the abstract operator Equation 31is studied on ordered Banach space E in this section. We give the following definitions and lemmas in partially ordered sets, which are fundamental to the proof of our main results.…”

“…B1) for ∈ (0, 1), there exists ( ) ∈ ( , 1) such that A( x + ( − 1)e, −1 + ( −1 − 1)e) ≥ ( )A(x, ) + ( ( ) − 1)e, ∀x, ∈ P h,e ; (B2) for ∈ (0, 1), there exists ( ) ∈ ( , 1) such that B( x + ( − 1)e) ≥ ( )Bx + ( ( ) − 1)e, x ∈ P h,e ; (B3) there exists a constant > 0 such that A(x, ) ≥ (Bx + e) − e.Then, operator Equation(31) has a unique solution x * in P h,e ;(2) for any x 0 , y 0 ∈ P h,e , making the sequencesx n = A(x n−1 , n−1 ) + Bx n−1 + e, n = A( n−1 , x n−1 ) + B n−1 + e, n = 1, 2, … , one has x n → x * , y n → x * as n → ∞.Proof. On account of conditions (B1) and (B2), for ∈ (0, 1), we haveA( −1 x + ( −1 − 1)e, + ( − 1)e) ≤ −1 ( )A(x, ) + ( −1 ( ) − 1)e, x, ∈ P h,e , B( −1 x + ( −1 − 1)e) ≤ −1 ( )Bx + ( −1 ( ) − 1)e, x ∈ P h,e .From theorem 3.1 in Sang and Ren,34 A ∶P h,e → P h,e is gotten; then, we prove that B ∶ P h,e → P h,e . Because Bh ∈ P h,e , there existr 1 , r 2 > 0 such that r 1 h + (r 1 − 1)e ≤ Bh ≤ r 2 h + (r 2 − 1)e.For x ∈ P h,e , we can choose c ∈ (0, 1) such that ch + (c − 1)e ≤ x ≤ c −1 h + (c −1 − 1)e; we thus obtain Bx ≤ B(c −1 h + (c −1 − 1)e) ≤ −1 (c)Bh + ( −1 (c) − 1)e ≤ −1 (c)(r 2 h + (r 2 − 1)e) + ( −1 (c) − 1)e = ( −1 (c) − 1)r 2 h + ( −1 (c)r 2 − 1)e, and Bx ≥ B(ch + (c − 1)e) ≥ (c)Bh + ( (c) − 1)e ≥ (c)(r 1 h + (r 1 − 1)e) + ( (c) − 1)e = ( (c) − 1)r 1 h + ( (c)r 1 − 1)e, that is, Bx ∈ P h,e , so B ∶ P h,e → P h,e .…”

Existence of positive solutions for a class of fractional differential equations with the derivative term via a new fixed point theorem

“…We should mention the main results obtained in [18][19][20][21][22], which motivated us to consider problem (1.1). In [18], Wang and Zhang studied the operator equation Ax + Bx + C(x, x) = x, where A is an increasing α-concave operator, B is a decreasing operator, and C is a mixed monotone operator.…”

“…The author proved problem (1.3) to have a unique positive solution based on a new mixed monotone fixed theorem. Very recently, Sang and Ren [21] investigated the following fractional boundary value problem:…”