Existence, Uniqueness and UHR Stability of Solutions to Nonlinear Ordinary Differential Equations with Noninstantaneous Impulses

2020 ◽  
Vol 21 (2) ◽  
pp. 195-203
Author(s):  
Xuping Zhang ◽  
Zhen Xin
Keyword(s):  
Measure Of Noncompactness ◽  
Perturbation Technique ◽  
Nonlinear Ordinary Differential Equation ◽  
Monotone Iterative Method ◽  
Stability Of Solutions ◽  
Uniqueness Of Solutions ◽  
Monotone Iterative ◽  
Noninstantaneous Impulses ◽  
Rassias Stability ◽  
Ordered Banach Spaces

AbstractWe consider the existence, uniqueness and Ulam–Hyers–Rassias stability of solutions to the initial value problem with noninstantaneous impulses on ordered Banach spaces. The existence and uniqueness of solutions for nonlinear ordinary differential equation with noninstantaneous impulses is obtained by using perturbation technique, monotone iterative method and a new estimation technique of the measure of noncompactness under the situation that the corresponding noninstantaneous impulsive functions gi are compact and not compact, respectively. Furthermore, the UHR stability of solutions is also obtained, which provides an approach to find approximate solution to noninstantaneous impulsive equations in the sense of UHR stability.

scholarly journals Existence of positive S-asymptotically periodic solutions of the fractional evolution equations in ordered Banach spaces

2021 ◽  
Vol 26 (5) ◽  
pp. 928-946
Author(s):  
Qiang Li ◽  
Lishan Liu ◽  
Mei Wei
Keyword(s):  
Evolution Equations ◽  
Mild Solutions ◽  
Upper And Lower Solutions ◽  
Periodic Problem ◽  
Growth Conditions ◽  
Monotone Iterative Method ◽  
Monotone Iterative ◽  
Asymptotically Periodic ◽  
Asymptotically Periodic Solutions ◽  
Ordered Banach Spaces

In this paper, we discuss the asymptotically periodic problem for the abstract fractional evolution equation under order conditions and growth conditions. Without assuming the existence of upper and lower solutions, some new results on the existence of the positive S-asymptotically ω-periodic mild solutions are obtained by using monotone iterative method and fixed point theorem. It is worth noting that Lipschitz condition is no longer needed, which makes our results more widely applicable.

scholarly journals Monotone iterative technique for non-autonomous semilinear differential equations with nonlocal condition

2019 ◽  
Vol 52 (1) ◽  
pp. 29-39 ◽  
Author(s):  
Arshi Meraj ◽  
Dwijendra Narain Pandey
Keyword(s):  
Differential Equations ◽  
Measure Of Noncompactness ◽  
Nonlocal Condition ◽  
Mild Solutions ◽  
Upper And Lower Solutions ◽  
Monotone Iterative Technique ◽  
Monotone Iterative Method ◽  
Evolution System ◽  
Monotone Iterative ◽  
Norm Space

AbstractThe objective of this article is to discuss the existence and uniqueness of mild solutions for a class of non-autonomous semilinear differential equations with nonlocal condition via monotone iterative method with upper and lower solutions in an ordered complete norm space X, using evolution system and measure of noncompactness.

scholarly journals Existence and uniqueness of extremal mild solutions for non-autonomous nonlocal integro-differential equations via monotone iterative technique

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 2985-2993 ◽  
Author(s):  
Arshi Meraj ◽  
Dwijendra Pandey
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Measure Of Noncompactness ◽  
Nonlocal Condition ◽  
Mild Solutions ◽  
Upper And Lower Solutions ◽  
Monotone Iterative Method ◽  
Evolution System ◽  
Ordered Banach Space ◽  
Monotone Iterative

In this work, we will discuss the existence and uniqueness of extremal mild solutions for non-autonomous integro-differential equations having nonlocal condition via monotone iterative method with upper and lower solutions in an ordered Banach space X, using evolution system and measure of noncompactness.

scholarly journals Existence and Ulam stability for implicit fractional q-difference equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Saïd Abbas ◽  
Mouffak Benchohra ◽  
Nadjet Laledj ◽  
Yong Zhou
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Difference Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Ulam Stability ◽  
Uniqueness Of Solutions ◽  
Stability Results ◽  
Rassias Stability

AbstractThis paper deals with some existence, uniqueness and Ulam–Hyers–Rassias stability results for a class of implicit fractional q-difference equations. Some applications are made of some fixed point theorems in Banach spaces for the existence and uniqueness of solutions, next we prove that our problem is generalized Ulam–Hyers–Rassias stable. Two illustrative examples are given in the last section.

scholarly journals Discrete fractional order two-point boundary value problem with some relevant physical applications

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
A. George Maria Selvam ◽  
Jehad Alzabut ◽  
R. Dhineshbabu ◽  
S. Rashid ◽  
M. Rehman
Keyword(s):  
Boundary Value Problem ◽  
Fractional Order ◽  
Contraction Mapping ◽  
Boundary Value ◽  
Contraction Mapping Principle ◽  
Difference Operators ◽  
Point Boundary ◽  
Uniqueness Of Solutions ◽  
Fractional Difference ◽  
Rassias Stability

Abstract The results reported in this paper are concerned with the existence and uniqueness of solutions of discrete fractional order two-point boundary value problem. The results are developed by employing the properties of Caputo and Riemann–Liouville fractional difference operators, the contraction mapping principle and the Brouwer fixed point theorem. Furthermore, the conditions for Hyers–Ulam stability and Hyers–Ulam–Rassias stability of the proposed discrete fractional boundary value problem are established. The applicability of the theoretical findings has been demonstrated with relevant practical examples. The analysis of the considered mathematical models is illustrated by figures and presented in tabular forms. The results are compared and the occurrence of overlapping/non-overlapping has been discussed.

scholarly journals Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Abdelkrim Salim ◽  
Mouffak Benchohra ◽  
Erdal Karapınar ◽  
Jamal Eddine Lazreg
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Banach Spaces ◽  
Fractional Differential Equations ◽  
Fixed Point Theorems ◽  
Measure Of Noncompactness ◽  
Ulam Stability ◽  
Stability Of Solutions ◽  
Fractional Differential ◽  
Implicit Fractional Differential Equations

Abstract In this manuscript, we examine the existence and the Ulam stability of solutions for a class of boundary value problems for nonlinear implicit fractional differential equations with instantaneous impulses in Banach spaces. The results are based on fixed point theorems of Darbo and Mönch associated with the technique of measure of noncompactness. We provide some examples to indicate the applicability of our results.

Sign in / Sign up

Export Citation Format

Share Document