In this paper we consider the stability problem of a general class of differential equations in t... more In this paper we consider the stability problem of a general class of differential equations in the sense of Hyers-Ulam and Hyers-Ulam-Rassias with the aid of a fixed point technique. We extend and improve the literature by dropping some assumptions of some well known and commonly cited results in this topic. Some illustrative examples are also given to visualize the improvement.
In this paper, we study the stability in the sense of Hyers-Ulam for the following fractional dif... more In this paper, we study the stability in the sense of Hyers-Ulam for the following fractional differential equations including the new Caputo-Fabrizio fractional derivative: CF D α y (x) = f (x) and CF D α y (x) − λy (x) = f (x). Finally, two examples are given to illustrate our results.
Highlights • The paper focused on the stability of Hyers-Ulam, Hyers-Ulam-Rassias and Hyers-Ulam-... more Highlights • The paper focused on the stability of Hyers-Ulam, Hyers-Ulam-Rassias and Hyers-Ulam-Rassias-Gavruta. • This is the generalization of many previous studies. • The equation includes linear, Bernoulli, Riccati and Abel equations. • It is also the first work related to the stability of Abel equations in the literature.
In this paper, we examine the Hyers-Ulam and Hyers-Ulam-Rassias stability of solutions of a gener... more In this paper, we examine the Hyers-Ulam and Hyers-Ulam-Rassias stability of solutions of a general class of nonlinear Volterra integral equations. By using a fixed point alternative and improving a technique commonly used in similar problems, we extend and improve some well-known results on this problem. We also provide some examples visualizing the improvement of the results mentioned.
Hacettepe Journal of Mathematics and Statistics, Sep 21, 2017
By using the positive linear functional, including the generalized averaging technique, some new ... more By using the positive linear functional, including the generalized averaging technique, some new Kamenev-type oscillation criteria are established for the second order matrix differential system (r(t)P (t)ψ(X(t))K(X (t))) + p(t)R(t)ψ(X(t))K(X (t)) +Q(t)F (X (t))G(X(t)) = 0. The results improve and generalize those given in some previous papers.
Journal of Inequalities and Applications, Nov 8, 2018
In this manuscript, we developed the Hardy-type inequality within the Caputo-Fabrizio fractional ... more In this manuscript, we developed the Hardy-type inequality within the Caputo-Fabrizio fractional derivative. We presented some illustrative examples to confirm our work.
In this manuscript, we prove new aspects for several Opial-type integral inequalities for the lef... more In this manuscript, we prove new aspects for several Opial-type integral inequalities for the left and right Caputo-Fabrizio operators with nonsingular kernel. For this purpose we use the inequalities obtained by Andrić et al. (Integral Transforms Spec. Funct. 25(4):324-335, 2014), which is the generalization of an inequality of Agarwal and Pang (Opial Inequalities with Applications in Differential and Difference Equations, 1995). Besides, examples are presented to validate the reported results.
Mathematical Methods in The Applied Sciences, Dec 23, 2022
In this paper, we investigate the Hyers–Ulam and Hyers–Ulam–Rassias stability of solutions of a g... more In this paper, we investigate the Hyers–Ulam and Hyers–Ulam–Rassias stability of solutions of a general class of nonlinear Volterra integral equations. By applying a fixed point theorem and modifying a technique widely used in similar problems, we improve some well‐known results on this problem. We also provide some examples illustrating the improvement of the results mentioned.
Sakarya University Journal of Science, Dec 1, 2017
In this study, we prove continuous dependence of solutions on coefficients of a coupled system of... more In this study, we prove continuous dependence of solutions on coefficients of a coupled system of waveplate type.
In this paper, we deal with the Hyers-Ulam-Rassias (HUR) and Hyers-Ulam (HU) stability of Hadamar... more In this paper, we deal with the Hyers-Ulam-Rassias (HUR) and Hyers-Ulam (HU) stability of Hadamard type fractional integral equations on compact intervals. The stability conditions are developed using a new generalized metric (GM) definition and the fixed point technique by motivating Wang and Lin Ulam's type stability of Hadamard type fractional integral equations. Filomat 2014; 28(7): 1323-1331. Moreover, our approach is efficient and ease in use than to the previously studied approaches. Finally, we give two examples to explain our main results.
Rendiconti Del Circolo Matematico Di Palermo, Jul 3, 2020
In this paper, we present some new stability criteria in the sense of Ulam for the solutions of f... more In this paper, we present some new stability criteria in the sense of Ulam for the solutions of fractional differential equations involving the conformable fractional derivative. Our results are based on a fixed point alternative which is developed for generalized metric spaces. This study improves and extends the literature in this topic since there is no previous progress on the problem we consider. We also provide examples to illustrate our results in a separate section.
In this study we introduce several new Ostrowski-type inequalities for both left and right sided ... more In this study we introduce several new Ostrowski-type inequalities for both left and right sided fractional integrals of a function g with respect to another function ψ. Our results generalized the ones presented previously by Farid. Furthermore, two illustrative examples are presented to support our results.
The main aim of this paper is to investigate various types of Ulam stability and Mittag-Leffler s... more The main aim of this paper is to investigate various types of Ulam stability and Mittag-Leffler stability of linear differential equations of second order with constant coefficients having damping term using the Aboodh transform method. We also obtain the Hyers-Ulam stability constants of these differential equations using the Aboodh transform and some examples to illustrate our main results are given.
In this paper, we deal with the Hyers-Ulam-Rassias (HUR) and Hyers-Ulam (HU) stability of Hadamar... more In this paper, we deal with the Hyers-Ulam-Rassias (HUR) and Hyers-Ulam (HU) stability of Hadamard type fractional integral equations on compact intervals. The stability conditions are developed using a new generalized metric (GM) definition and the fixed point technique by motivating Wang and Lin Ulam's type stability of Hadamard type fractional integral equations. Filomat 2014; 28(7): 1323-1331. Moreover, our approach is efficient and ease in use than to the previously studied approaches. Finally, we give two examples to explain our main results.
In this paper we consider the stability problem of a general class of differential equations in t... more In this paper we consider the stability problem of a general class of differential equations in the sense of Hyers-Ulam and Hyers-Ulam-Rassias with the aid of a fixed point technique. We extend and improve the literature by dropping some assumptions of some well known and commonly cited results in this topic. Some illustrative examples are also given to visualize the improvement.
By using the positive linear functional, including the general means and Riccati technique, some ... more By using the positive linear functional, including the general means and Riccati technique, some new oscillation criteria are established for the second order matrix differential equations (r(t)P(t)?(X(t))K(X'(t)))' + p(t)R(t)?(X(t))K(X'(t)) + Q(t)F(X'(t))G(X(t)) = 0,t ? t0 > 0. The results improve and generalize those given in some previous papers.
This paper examines Hyers-Ulam (HU), Hyers-Ulam-Rassias (HUR) and Hyers-Ulam-Rassias-Gavruta (HUR... more This paper examines Hyers-Ulam (HU), Hyers-Ulam-Rassias (HUR) and Hyers-Ulam-Rassias-Gavruta (HURG) stability of the first-order differential equation including Bernoulli’s, Riccati and Abel with given initial condition.
In this paper we consider the stability problem of a general class of differential equations in t... more In this paper we consider the stability problem of a general class of differential equations in the sense of Hyers-Ulam and Hyers-Ulam-Rassias with the aid of a fixed point technique. We extend and improve the literature by dropping some assumptions of some well known and commonly cited results in this topic. Some illustrative examples are also given to visualize the improvement.
In this paper, we study the stability in the sense of Hyers-Ulam for the following fractional dif... more In this paper, we study the stability in the sense of Hyers-Ulam for the following fractional differential equations including the new Caputo-Fabrizio fractional derivative: CF D α y (x) = f (x) and CF D α y (x) − λy (x) = f (x). Finally, two examples are given to illustrate our results.
Highlights • The paper focused on the stability of Hyers-Ulam, Hyers-Ulam-Rassias and Hyers-Ulam-... more Highlights • The paper focused on the stability of Hyers-Ulam, Hyers-Ulam-Rassias and Hyers-Ulam-Rassias-Gavruta. • This is the generalization of many previous studies. • The equation includes linear, Bernoulli, Riccati and Abel equations. • It is also the first work related to the stability of Abel equations in the literature.
In this paper, we examine the Hyers-Ulam and Hyers-Ulam-Rassias stability of solutions of a gener... more In this paper, we examine the Hyers-Ulam and Hyers-Ulam-Rassias stability of solutions of a general class of nonlinear Volterra integral equations. By using a fixed point alternative and improving a technique commonly used in similar problems, we extend and improve some well-known results on this problem. We also provide some examples visualizing the improvement of the results mentioned.
Hacettepe Journal of Mathematics and Statistics, Sep 21, 2017
By using the positive linear functional, including the generalized averaging technique, some new ... more By using the positive linear functional, including the generalized averaging technique, some new Kamenev-type oscillation criteria are established for the second order matrix differential system (r(t)P (t)ψ(X(t))K(X (t))) + p(t)R(t)ψ(X(t))K(X (t)) +Q(t)F (X (t))G(X(t)) = 0. The results improve and generalize those given in some previous papers.
Journal of Inequalities and Applications, Nov 8, 2018
In this manuscript, we developed the Hardy-type inequality within the Caputo-Fabrizio fractional ... more In this manuscript, we developed the Hardy-type inequality within the Caputo-Fabrizio fractional derivative. We presented some illustrative examples to confirm our work.
In this manuscript, we prove new aspects for several Opial-type integral inequalities for the lef... more In this manuscript, we prove new aspects for several Opial-type integral inequalities for the left and right Caputo-Fabrizio operators with nonsingular kernel. For this purpose we use the inequalities obtained by Andrić et al. (Integral Transforms Spec. Funct. 25(4):324-335, 2014), which is the generalization of an inequality of Agarwal and Pang (Opial Inequalities with Applications in Differential and Difference Equations, 1995). Besides, examples are presented to validate the reported results.
Mathematical Methods in The Applied Sciences, Dec 23, 2022
In this paper, we investigate the Hyers–Ulam and Hyers–Ulam–Rassias stability of solutions of a g... more In this paper, we investigate the Hyers–Ulam and Hyers–Ulam–Rassias stability of solutions of a general class of nonlinear Volterra integral equations. By applying a fixed point theorem and modifying a technique widely used in similar problems, we improve some well‐known results on this problem. We also provide some examples illustrating the improvement of the results mentioned.
Sakarya University Journal of Science, Dec 1, 2017
In this study, we prove continuous dependence of solutions on coefficients of a coupled system of... more In this study, we prove continuous dependence of solutions on coefficients of a coupled system of waveplate type.
In this paper, we deal with the Hyers-Ulam-Rassias (HUR) and Hyers-Ulam (HU) stability of Hadamar... more In this paper, we deal with the Hyers-Ulam-Rassias (HUR) and Hyers-Ulam (HU) stability of Hadamard type fractional integral equations on compact intervals. The stability conditions are developed using a new generalized metric (GM) definition and the fixed point technique by motivating Wang and Lin Ulam's type stability of Hadamard type fractional integral equations. Filomat 2014; 28(7): 1323-1331. Moreover, our approach is efficient and ease in use than to the previously studied approaches. Finally, we give two examples to explain our main results.
Rendiconti Del Circolo Matematico Di Palermo, Jul 3, 2020
In this paper, we present some new stability criteria in the sense of Ulam for the solutions of f... more In this paper, we present some new stability criteria in the sense of Ulam for the solutions of fractional differential equations involving the conformable fractional derivative. Our results are based on a fixed point alternative which is developed for generalized metric spaces. This study improves and extends the literature in this topic since there is no previous progress on the problem we consider. We also provide examples to illustrate our results in a separate section.
In this study we introduce several new Ostrowski-type inequalities for both left and right sided ... more In this study we introduce several new Ostrowski-type inequalities for both left and right sided fractional integrals of a function g with respect to another function ψ. Our results generalized the ones presented previously by Farid. Furthermore, two illustrative examples are presented to support our results.
The main aim of this paper is to investigate various types of Ulam stability and Mittag-Leffler s... more The main aim of this paper is to investigate various types of Ulam stability and Mittag-Leffler stability of linear differential equations of second order with constant coefficients having damping term using the Aboodh transform method. We also obtain the Hyers-Ulam stability constants of these differential equations using the Aboodh transform and some examples to illustrate our main results are given.
In this paper, we deal with the Hyers-Ulam-Rassias (HUR) and Hyers-Ulam (HU) stability of Hadamar... more In this paper, we deal with the Hyers-Ulam-Rassias (HUR) and Hyers-Ulam (HU) stability of Hadamard type fractional integral equations on compact intervals. The stability conditions are developed using a new generalized metric (GM) definition and the fixed point technique by motivating Wang and Lin Ulam's type stability of Hadamard type fractional integral equations. Filomat 2014; 28(7): 1323-1331. Moreover, our approach is efficient and ease in use than to the previously studied approaches. Finally, we give two examples to explain our main results.
In this paper we consider the stability problem of a general class of differential equations in t... more In this paper we consider the stability problem of a general class of differential equations in the sense of Hyers-Ulam and Hyers-Ulam-Rassias with the aid of a fixed point technique. We extend and improve the literature by dropping some assumptions of some well known and commonly cited results in this topic. Some illustrative examples are also given to visualize the improvement.
By using the positive linear functional, including the general means and Riccati technique, some ... more By using the positive linear functional, including the general means and Riccati technique, some new oscillation criteria are established for the second order matrix differential equations (r(t)P(t)?(X(t))K(X'(t)))' + p(t)R(t)?(X(t))K(X'(t)) + Q(t)F(X'(t))G(X(t)) = 0,t ? t0 > 0. The results improve and generalize those given in some previous papers.
This paper examines Hyers-Ulam (HU), Hyers-Ulam-Rassias (HUR) and Hyers-Ulam-Rassias-Gavruta (HUR... more This paper examines Hyers-Ulam (HU), Hyers-Ulam-Rassias (HUR) and Hyers-Ulam-Rassias-Gavruta (HURG) stability of the first-order differential equation including Bernoulli’s, Riccati and Abel with given initial condition.
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