Analysis of positive solution and Hyers‐Ulam stability for a class of singular fractional differential equations with <i>p</i>‐Laplacian in Banach space

Khan, Hasib; Chen, Wen; Sun, HongGuang

doi:10.1002/mma.4835

Cited by 49 publications

(30 citation statements)

References 39 publications

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“…With the help of conditions (37) to (39) and our results in (44), (46), and (48), we have ||u 1 − u * 1 || = 0, ||u 2 − u * 2 || = 0, ||u 3 − u * 3 || = 0, which implies u * 1 = u 1 , u * 2 = u 2 , u * 3 = u 3 , respectively. Thus, the model (6) has a unique solution.…”

Section: Uniqueness Analysismentioning

confidence: 65%

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Existence of solution for a fractional‐order Lotka‐Volterra reaction‐diffusion model with Mittag‐Leffler kernel

Khan

et al. 2019

Math Methods in App Sciences

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In the literature, many researchers have studied Lotka‐Volterra (L‐V) models for different types of studies. In order to continue the study, we consider a fractional‐order L‐V model involving three different species in the Atangana‐Baleanu‐Caputo (ABC) sense of fractional derivative. This new model has potentials for a large number of research‐oriented studies. The first point that arises is whether the new model has a solution or not. Therefore, to answer this question, we consider the existence and uniqueness (EU) of the solutions and then Hyers‐Ulam (HU) stability for the proposed L‐V model.

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Section: Uniqueness Analysismentioning

confidence: 65%

“…With the help of recent development on the HUS, [37][38][39][40] we analyze the suggested ABC fractional-order L-V reaction-diffusion model (6) and initiate the following definition of HUS:…”

Section: Hyers-ulam Stabilitymentioning

confidence: 99%

Existence of solution for a fractional‐order Lotka‐Volterra reaction‐diffusion model with Mittag‐Leffler kernel

Khan

et al. 2019

Math Methods in App Sciences

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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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“…Recently, fractional differential equations with -Laplacian operator have gained its popularity and importance due to its distinguished applications in numerous diverse fields of science and engineering, such as viscoelasticity mechanics, non-Newtonian mechanics, electrochemistry, fluid mechanics, combustion theory, and material science. There have appeared some results for the existence of solutions or positive solutions of boundary value problems for fractional differential equations with -Laplacian operator; see [15,[21][22][23][24][25][26] and the references therein. For example, under different conditions K. Hasib, W. Chen and H. Sun [21] apply some classical fixed-point theorems to study the existence of positive solution for a class of singular fractional differential equations with nonlinear -Laplacian operator in Caputo sense…”

Section: Introductionmentioning

confidence: 99%

Existence of Nontrivial Solutions for Fractional Differential Equations with p-Laplacian

Zhang

Wang

2019

Journal of Function Spaces

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Analysis of positive solution and Hyers‐Ulam stability for a class of singular fractional differential equations with p‐Laplacian in Banach space

Cited by 49 publications

References 39 publications

Existence of solution for a fractional‐order Lotka‐Volterra reaction‐diffusion model with Mittag‐Leffler kernel

Existence of solution for a fractional‐order Lotka‐Volterra reaction‐diffusion model with Mittag‐Leffler kernel

Nonlinear impulsive Langevin equation with mixed derivatives

Existence of Nontrivial Solutions for Fractional Differential Equations with p-Laplacian

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