Ulam-type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition

Zada, Akbar; Ali, Sartaj; Li, Yongjin

doi:10.1186/s13662-017-1376-y

Cited by 54 publications

(39 citation statements)

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“…Zada et al [38] studied existence and uniqueness of solutions by using Diaz-Margolis's fixed point theorem and presented Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability, and generalized Ulam-Hyers-Rassias stability for a class of nonlinear implicit fractional differential equation with noninstantaneous integral impulses and nonlinear integral boundary condition:…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of nonlinear implicit fractional Langevin equation with noninstantaneous impulses

2019

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Section: Introductionmentioning

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Stability analysis of nonlinear implicit fractional Langevin equation with noninstantaneous impulses

2019

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“…The implicit FDEs represent a very important class of fractional differential equations. For details see [57][58][59][60][61] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Analysis of coupled systems of implicit impulsive fractional differential equations involving Hadamard derivatives

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We present some results on the existence, uniqueness and Hyers-Ulam stability to the solution of an implicit coupled system of impulsive fractional differential equations having Hadamard type fractional derivative. Using a fixed point theorem of Kransnoselskii's type, the existence and uniqueness results are obtained. Along these lines, different kinds of Hyers-Ulam stability are discussed. An example is given to illustrate the main theorems.

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“…Wang et al studied generalized Ulam‐Hyers‐Rassias stability of the following fractional differential equation:

\{\begin{matrix} ^{c} D_{0, w}^{ν} z false(w false) = f false(w, z false(w false) false), w \in false(w_{k}, s_{k} false], k = 0, 1, ⋯, m, 0 < ν < 1, \\ z false(w false) = g_{k} false(w, z false(w false) false), w \in false(s_{k - 1}, w_{k} false], k = 1, 2, ⋯, m . \end{matrix}

Zada et al studied existence and uniqueness of solutions by using Diaz Margolis' fixed‐point theorem and presented different types of Ulam‐Hyers stability for a class of nonlinear implicit fractional differential equation with noninstantaneous integral impulses and nonlinear integral boundary conditions:

\{\begin{matrix} ^{c} D_{0, w}^{ν} z false(w false) = f false(w, z false(w false),^{c} D_{0, w}^{ν} z false(w false) false), w \in false(w_{k}, s_{k} false], k = 0, 1, ⋯, m, 0 < ν < 1, w \in false(0, 1 false], \\ z false(w false) = I_{s_{k - 1}, w_{k}}^{ν} false(ξ_{k} false(w, z false(w false) false) false), w \in false(s_{k - 1 ...} \end{matrix}

…”

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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Ulam-type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition

Cited by 54 publications

References 20 publications

Stability analysis of nonlinear implicit fractional Langevin equation with noninstantaneous impulses

Stability analysis of nonlinear implicit fractional Langevin equation with noninstantaneous impulses

Analysis of coupled systems of implicit impulsive fractional differential equations involving Hadamard derivatives

Nonlinear impulsive Langevin equation with mixed derivatives

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