In this paper, we study Hyers-Ulam and Hyers-Ulam-Rassias stability of nonlinear Caputo-Fabrizio ... more In this paper, we study Hyers-Ulam and Hyers-Ulam-Rassias stability of nonlinear Caputo-Fabrizio fractional differential equations on a noncompact interval. We extend the corresponding uniqueness and stability results on a compact interval. Two examples are given to illustrate our main results.
Zeitschrift für Angewandte Mathematik und Physik, Jan 11, 2021
This paper proposes a nonlocal formulation regarding the modeling of Antarctic Circumpolar Curren... more This paper proposes a nonlocal formulation regarding the modeling of Antarctic Circumpolar Current by introducing flow functions to encode horizontal flow components without considering vertical motion. Using topological degree, zero exponent theory and fixed point technique, we show the existence of positive solutions to nonlocal boundary value problems with nonlinear vorticity.
In this paper, we study the approximate controllability of Sobolevtype fractional evolution syste... more In this paper, we study the approximate controllability of Sobolevtype fractional evolution systems with non-local conditions in Hilbert spaces. Sufficient conditions of approximate controllability of the desired problem are presented by supposing an approximate controllability of the corresponding linear system. By constructing a control function involving Gramian controllability operator, we transform our problem to a fixed point problem of nonlinear operator. Then the Schauder Fixed Point Theorem is applied to complete the proof. An example is given to illustrate our theoretical results. 2000 Mathematics Subject Classification. 47J35 and 93B05 and 93C25.
Qualitative Theory of Dynamical Systems, Nov 29, 2022
We investigate the Hyers-Ulam stability of the linear recurrence x n+p = a 1 x n+p−1 + ••• + a p−... more We investigate the Hyers-Ulam stability of the linear recurrence x n+p = a 1 x n+p−1 + ••• + a p−1 x n+1 + a p x n + b n , n ≥ 0, where x 0 ,x 1 ,...,x p−1 ∈ X.
International Journal of Bifurcation and Chaos, Mar 30, 2020
The dynamics of a discrete nonlinear prey–predator model is studied. The local dynamical results ... more The dynamics of a discrete nonlinear prey–predator model is studied. The local dynamical results are obtained on asymptotic properties of fixed points and Neimark–Sacker bifurcations. Then global dynamics is studied by finding invariant sets. Also some achievements on attraction are shown for certain trivial invariant sets including a shadowing type result, and estimates on boundedness of orbits. Some ergodic type results are also derived. Certain issues are extended to systems with impulses by showing the influence of impulses on dynamics. Moreover, backward dynamics is investigated as well. All these results are derived analytically and numerical computations are presented to support them.
Qualitative Theory of Dynamical Systems, Apr 10, 2021
In this paper, we transfer the approach of finding the explicit solutions to linear impulsive dif... more In this paper, we transfer the approach of finding the explicit solutions to linear impulsive differential equations (LIDEs) to quaternion-valued case. We develop the similar idea to find the solutions to LIDEs and give the details to derive the representations of the solutions to quaternion-valued LIDEs in the sense of complex numbers and quaternion. In addition, we present that two formulas are equivalent to each other. Finally, some examples are given to illustrate the theory approach.
In this research article, a novel Φ-fractional Bielecki-type norm introduced by Sousa and Oliveir... more In this research article, a novel Φ-fractional Bielecki-type norm introduced by Sousa and Oliveira [23] is used to obtain results on uniqueness and Ulam stability of solutions for a new class of multiterms fractional differential equations in the framework of generalized Caputo fractional derivative. The uniqueness results are obtained by employing Banach' and Perov's fixed point theorems. While the Φ-fractional Gronwall type inequality and the concept of the matrices converging to zero are implemented to examine different types of stabilities in the sense of Ulam-Hyers (UH) of the given problems. Finally, two illustrative examples are provided to demonstrate the validity of our theoretical findings.
Bulletin of National University of Uzbekistan Mathematics and Natural Sciences, Mar 14, 2023
In this paper, we show that the negative-order Korteweg-de Vries equation with a self-consistent ... more In this paper, we show that the negative-order Korteweg-de Vries equation with a self-consistent source can be solved by the inverse scattering method. The evolution of the spectral data of the Sturm-Liouville operator with the potential associated with the solution of the negative order Korteweg-de Vries equation with a self-consistent source is determined. The results obtained make it possible to apply the method of the inverse scattering problem to solve the problem under consideration.
Impulsive control is used to suppress the chaotic behavior in an one-dimensional discrete supply ... more Impulsive control is used to suppress the chaotic behavior in an one-dimensional discrete supply and demand dynamical system. By perturbing periodically the state variable with constant impulses, the chaos can be suppressed. It is proved analytically that the obtained orbits are bounded and periodic. Moreover, it is shown for the first time that the difference equations with impulses, used to control the chaos, can generate hidden chaotic attractors. To the best of the authors knowledge, this interesting feature has not yet been discussed. The impulsive algorithm can be used to stabilize chaos in other classes of discrete dynamical systems.
We prove the presence of chaos near a homoclinic orbit in the modified Li-Yorke sense [10] by imp... more We prove the presence of chaos near a homoclinic orbit in the modified Li-Yorke sense [10] by implementing chaotic perturbations. A Duffing oscillator is considered to show the effectiveness of our technique, and simulations that support the theoretical results are depicted. Ott-Grebogi-Yorke and Pyragas control methods are used to stabilize almost periodic motions.
In this paper the fractional-order Mandelbrot and Julia sets in the sense of q-th Caputolike disc... more In this paper the fractional-order Mandelbrot and Julia sets in the sense of q-th Caputolike discrete fractional differences, for q ∈ (0, 1), are introduced and several properties are analytically and numerically studied. Some intriguing properties of the fractional models are revealed. Thus, for q ↑ 1, contrary to expectations, it is not obtained the known shape of the Mandelbrot of integer order, but for q ↓ 0. Also, we conjecture that for q ↓ 0, the fractionalorder Mandelbrot set is similar to the integer-order Mandelbrot set, while for q ↓ 0 and c = 0, one of the underlying fractional-order Julia sets is similar to the integer-order Mandelbrot set. In support of our conjecture, several extensive numerical experiments were done. To draw the Mandelbrot and Julia sets of fractional order, the numerical integral of the underlying initial values problem of fractional order is used, while to draw the sets, the escape-time algorithm adapted for the fractional-order case is used. The algorithm is presented as pseudocode.
We study the existence of exact solutions either for linear systems of ODEs or for scalar second ... more We study the existence of exact solutions either for linear systems of ODEs or for scalar second order linear differential equations with Dirichlet boundary value conditions.
This paper focuses on the generalization of some multiplicative integral inequalities for twice d... more This paper focuses on the generalization of some multiplicative integral inequalities for twice differentiable functions. First, we derive multiplicative integral identity for multiplicatvely twice differentiable functions. Then, with the help of integral identity, we prove a family of integral inequalities such that Simpson's, Hermite-Hadamard, Midpoint, Trapezoid and Bullen's types inequalities for multiplicatively convex functions. Moreover, we provide some numerical examples and computational analysis of these newly established inequalities to show the validity of the results for multiplicatively convex functions. The generalized forms derive in this research offer valuable tools for researchers in various fields.
In this paper, we study Hyers-Ulam and Hyers-Ulam-Rassias stability of nonlinear Caputo-Fabrizio ... more In this paper, we study Hyers-Ulam and Hyers-Ulam-Rassias stability of nonlinear Caputo-Fabrizio fractional differential equations on a noncompact interval. We extend the corresponding uniqueness and stability results on a compact interval. Two examples are given to illustrate our main results.
Zeitschrift für Angewandte Mathematik und Physik, Jan 11, 2021
This paper proposes a nonlocal formulation regarding the modeling of Antarctic Circumpolar Curren... more This paper proposes a nonlocal formulation regarding the modeling of Antarctic Circumpolar Current by introducing flow functions to encode horizontal flow components without considering vertical motion. Using topological degree, zero exponent theory and fixed point technique, we show the existence of positive solutions to nonlocal boundary value problems with nonlinear vorticity.
In this paper, we study the approximate controllability of Sobolevtype fractional evolution syste... more In this paper, we study the approximate controllability of Sobolevtype fractional evolution systems with non-local conditions in Hilbert spaces. Sufficient conditions of approximate controllability of the desired problem are presented by supposing an approximate controllability of the corresponding linear system. By constructing a control function involving Gramian controllability operator, we transform our problem to a fixed point problem of nonlinear operator. Then the Schauder Fixed Point Theorem is applied to complete the proof. An example is given to illustrate our theoretical results. 2000 Mathematics Subject Classification. 47J35 and 93B05 and 93C25.
Qualitative Theory of Dynamical Systems, Nov 29, 2022
We investigate the Hyers-Ulam stability of the linear recurrence x n+p = a 1 x n+p−1 + ••• + a p−... more We investigate the Hyers-Ulam stability of the linear recurrence x n+p = a 1 x n+p−1 + ••• + a p−1 x n+1 + a p x n + b n , n ≥ 0, where x 0 ,x 1 ,...,x p−1 ∈ X.
International Journal of Bifurcation and Chaos, Mar 30, 2020
The dynamics of a discrete nonlinear prey–predator model is studied. The local dynamical results ... more The dynamics of a discrete nonlinear prey–predator model is studied. The local dynamical results are obtained on asymptotic properties of fixed points and Neimark–Sacker bifurcations. Then global dynamics is studied by finding invariant sets. Also some achievements on attraction are shown for certain trivial invariant sets including a shadowing type result, and estimates on boundedness of orbits. Some ergodic type results are also derived. Certain issues are extended to systems with impulses by showing the influence of impulses on dynamics. Moreover, backward dynamics is investigated as well. All these results are derived analytically and numerical computations are presented to support them.
Qualitative Theory of Dynamical Systems, Apr 10, 2021
In this paper, we transfer the approach of finding the explicit solutions to linear impulsive dif... more In this paper, we transfer the approach of finding the explicit solutions to linear impulsive differential equations (LIDEs) to quaternion-valued case. We develop the similar idea to find the solutions to LIDEs and give the details to derive the representations of the solutions to quaternion-valued LIDEs in the sense of complex numbers and quaternion. In addition, we present that two formulas are equivalent to each other. Finally, some examples are given to illustrate the theory approach.
In this research article, a novel Φ-fractional Bielecki-type norm introduced by Sousa and Oliveir... more In this research article, a novel Φ-fractional Bielecki-type norm introduced by Sousa and Oliveira [23] is used to obtain results on uniqueness and Ulam stability of solutions for a new class of multiterms fractional differential equations in the framework of generalized Caputo fractional derivative. The uniqueness results are obtained by employing Banach' and Perov's fixed point theorems. While the Φ-fractional Gronwall type inequality and the concept of the matrices converging to zero are implemented to examine different types of stabilities in the sense of Ulam-Hyers (UH) of the given problems. Finally, two illustrative examples are provided to demonstrate the validity of our theoretical findings.
Bulletin of National University of Uzbekistan Mathematics and Natural Sciences, Mar 14, 2023
In this paper, we show that the negative-order Korteweg-de Vries equation with a self-consistent ... more In this paper, we show that the negative-order Korteweg-de Vries equation with a self-consistent source can be solved by the inverse scattering method. The evolution of the spectral data of the Sturm-Liouville operator with the potential associated with the solution of the negative order Korteweg-de Vries equation with a self-consistent source is determined. The results obtained make it possible to apply the method of the inverse scattering problem to solve the problem under consideration.
Impulsive control is used to suppress the chaotic behavior in an one-dimensional discrete supply ... more Impulsive control is used to suppress the chaotic behavior in an one-dimensional discrete supply and demand dynamical system. By perturbing periodically the state variable with constant impulses, the chaos can be suppressed. It is proved analytically that the obtained orbits are bounded and periodic. Moreover, it is shown for the first time that the difference equations with impulses, used to control the chaos, can generate hidden chaotic attractors. To the best of the authors knowledge, this interesting feature has not yet been discussed. The impulsive algorithm can be used to stabilize chaos in other classes of discrete dynamical systems.
We prove the presence of chaos near a homoclinic orbit in the modified Li-Yorke sense [10] by imp... more We prove the presence of chaos near a homoclinic orbit in the modified Li-Yorke sense [10] by implementing chaotic perturbations. A Duffing oscillator is considered to show the effectiveness of our technique, and simulations that support the theoretical results are depicted. Ott-Grebogi-Yorke and Pyragas control methods are used to stabilize almost periodic motions.
In this paper the fractional-order Mandelbrot and Julia sets in the sense of q-th Caputolike disc... more In this paper the fractional-order Mandelbrot and Julia sets in the sense of q-th Caputolike discrete fractional differences, for q ∈ (0, 1), are introduced and several properties are analytically and numerically studied. Some intriguing properties of the fractional models are revealed. Thus, for q ↑ 1, contrary to expectations, it is not obtained the known shape of the Mandelbrot of integer order, but for q ↓ 0. Also, we conjecture that for q ↓ 0, the fractionalorder Mandelbrot set is similar to the integer-order Mandelbrot set, while for q ↓ 0 and c = 0, one of the underlying fractional-order Julia sets is similar to the integer-order Mandelbrot set. In support of our conjecture, several extensive numerical experiments were done. To draw the Mandelbrot and Julia sets of fractional order, the numerical integral of the underlying initial values problem of fractional order is used, while to draw the sets, the escape-time algorithm adapted for the fractional-order case is used. The algorithm is presented as pseudocode.
We study the existence of exact solutions either for linear systems of ODEs or for scalar second ... more We study the existence of exact solutions either for linear systems of ODEs or for scalar second order linear differential equations with Dirichlet boundary value conditions.
This paper focuses on the generalization of some multiplicative integral inequalities for twice d... more This paper focuses on the generalization of some multiplicative integral inequalities for twice differentiable functions. First, we derive multiplicative integral identity for multiplicatvely twice differentiable functions. Then, with the help of integral identity, we prove a family of integral inequalities such that Simpson's, Hermite-Hadamard, Midpoint, Trapezoid and Bullen's types inequalities for multiplicatively convex functions. Moreover, we provide some numerical examples and computational analysis of these newly established inequalities to show the validity of the results for multiplicatively convex functions. The generalized forms derive in this research offer valuable tools for researchers in various fields.
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