Stability of higher-order nonlinear impulsive differential equations

Tang, Shuhong; Zada, Akbar; Faisal, Shah; El-Sheikh, M. M. A.; Li, Tongxing

doi:10.22436/jnsa.009.06.110

Cited by 43 publications

(22 citation statements)

References 15 publications

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“…Later on, Hyers results are extended by many mathematicians; for details, reader may see [32][33][34][35][36][37][38][39] and the reference therein. The mentioned stability analysis is extremely helpful in numerous applications, for example, numerical analysis and optimization, where it is very tough to find the exact solution of a nonlinear problem.…”

Section: Introductionmentioning

confidence: 99%

Analysis of Implicit Type Nonlinear Dynamical Problem of Impulsive Fractional Differential Equations

Ahmad¹,

Ali

Shah

et al. 2018

Complexity

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We study the existence, uniqueness, and various kinds of Ulam-Hyers stability of the solutions to a nonlinear implicit type dynamical problem of impulsive fractional differential equations with nonlocal boundary conditions involving Caputo derivative. We develop conditions for uniqueness and existence by using the classical fixed point theorems such as Banach fixed point theorem and Krasnoselskii's fixed point theorem. For stability, we utilized classical functional analysis. Also, an example is given to demonstrate our main theoretical results.

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Section: Introductionmentioning

confidence: 99%

Analysis of Implicit Type Nonlinear Dynamical Problem of Impulsive Fractional Differential Equations

Ahmad¹,

Ali

Shah

et al. 2018

Complexity

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“…Progressively, fractional differential equations play a very important role in fields such as thermodynamics, statistical physics viscoelasticity, nonlinear oscillation of earthquakes, defence, optics, control, electrical circuits, signal processing, and astronomy. There are some outstanding articles that provide the main theoretical tools for the qualitative analysis of this research field and, at the same time, shows the interconnection as well as the distinction between integral models of classical and fractional differential equations, see previous studies …”

Section: Introductionmentioning

confidence: 99%

Nonlinear impulsive Langevin equation with mixed derivatives

Rizwan

Zada

2019

Math Methods in App Sciences

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In this paper, we consider a nonlocal boundary value problem of nonlinear impulsive Langevin equation with mixed derivatives. Some sufficient conditions are constructed to observe the existence, uniqueness, and generalized Ulam-Hyers-Rassias stability of our proposed model, with the help of Diaz-Margolis' fixed-point approach over generalized complete metric space. We give an example that supports our main result. KEYWORDS Caputo derivative, Langevin equation, noninstantaneous impulses, Ulam-Hyers-Rassias stability MSC CLASSIFICATION26A33; 34A08; 34B27 INTRODUCTIONAt Wisconsin University, Ulam raised a question about the stability of functional equations in the year 1940. The question of Ulam was under what conditions does there exist an additive mapping near an approximately additive mapping? 1 In 1941, Hyers was the first mathematician who gave partial answer to Ulam's question, 2 over Banach space. Afterwards, stability of such form is known as Ulam-Hyers stability. In 1978, Rassias 3 provided a remarkable generalization of the Ulam-Hyers stability of mappings by considering variables. For more information and different approaches about the topic, we refer the reader to the previous studies. [4][5][6][7][8][9][10][11][12][13][14][15][16] An equation of the form m d 2 x dt 2 = dx dt + (t) is called Langevin equation, introduced by Paul Langevin in 1908. Langevin equations are broadly used to describe stochastic problems in image processing, physics, astronomy, chemistry, defence system, electrical, and mechanical engineering. Brownian motion is well described by the Langevin equations when the random oscillation force is supposed to be Gaussian noise. For the removal of noise, mathematicians used fractional order differential equations, also it performs well in reducing the staircase effects as compared with ordinary differential equations. Thus, it is very important to study fractional Langevin equations, for more details, see previous studies. [17][18][19][20] Fractional order differential equations are the generalizations of the classical integer order differential equations. Fractional calculus has become a speedily developing area, and its applications can be found in diverse fields ranging from physical sciences, porous media, electrochemistry, economics, electromagnetics, medicine, and engineering to biological sciences. Progressively, fractional differential equations play a very important role in fields such as thermodynamics, statistical physics viscoelasticity, nonlinear oscillation of earthquakes, defence, optics, control, electrical circuits, signal processing, and astronomy. There are some outstanding articles that provide the main theoretical tools for the qualitative analysis of this research field and, at the same time, shows the interconnection as well as the distinction between integral models of classical and fractional differential equations, see previous studies. [21][22][23][24][25][26][27][28][29][30][31][32][33] Impulsive fractional differential equations are used to describe both physi...

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“…To the best of our knowledge, the mathematicians who first investigated Ulam's type stability of impulsive ordinary differential equations were Wang et al Following their own work, in 2014, they proved the Hyers–Ulam–Rassias stability and generalized Hyers–Ulam–Rassias stability for impulsive evolution equations on a compact interval, which then they extended for infinite impulses in the same paper. For more details, the reader may see the previous studies . The problems with instantaneous impulses can not characterize processes in which the problem solutions has an interval discontinuity, for example, the introduction of drugs in the bloodstream and the consequent absorption for the body is gradual and a continuous process.…”

Section: Introductionmentioning

confidence: 99%

“…For more details, the reader may see the previous studies. [24][25][26][27][28][29] The problems with instantaneous impulses can not characterize processes in which the problem solutions has an interval discontinuity, for example, the introduction of drugs in the bloodstream and the consequent absorption for the body is gradual and a continuous process. These type of processes are characterized by noninstantaneous impulses, which starts from an arbitrary fixed point and stays alive on a finite interval.…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of higher order nonlinear differential equations in β–normed spaces

Zada

Shaleena

2018

Math Methods in App Sciences

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In this paper, we interrogate different Ulam type stabilities, ie, β–Ulam–Hyers stability, generalized β–Ulam–Hyers stability, β–Ulam–Hyers–Rassias stability, and generalized β–Ulam–Hyers–Rassias stability, for nth order nonlinear differential equations with integrable impulses of fractional type. The existence and uniqueness of solutions are investigated by using the Banach contraction principle. In the end, we give an example to support our main result.

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Stability of higher-order nonlinear impulsive differential equations

Cited by 43 publications

References 15 publications

Analysis of Implicit Type Nonlinear Dynamical Problem of Impulsive Fractional Differential Equations

Analysis of Implicit Type Nonlinear Dynamical Problem of Impulsive Fractional Differential Equations

Nonlinear impulsive Langevin equation with mixed derivatives

Stability analysis of higher order nonlinear differential equations in β–normed spaces

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