In the present paper, we study a system of nonlinear differential equations with three-point boundary conditions. The given original problem is reduced to the equivalent integral equations using Green function.

Several theorems are proved concerning the existence and uniqueness of solutions to the boundary value problems for the first order nonlinear system of ordinary differential equations with three-point boundary conditions. Then, we describe different types of Ulam stability: Ulam-Hyers stability, generalized Ulam-Hyers stability. We discuss the stability results providing suitable example. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, vol.

Castro and A. Phys, AIP Conf. Nonlinear Sci. Hyers, On the stability of the linear functional equation, Proc. Agarwal, B.

Ahmad, D. Garout and A. Alsaedi, Existence results for coupled nonlinear fractional differential equations equipped with nonlocal coupled flux and multi-point boundary conditions, Chaos Solit.

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Images Donate icon An illustration of a heart shape Donate Ellipses icon An illustration of text ellipses. EMBED for wordpress. Want more? Advanced embedding details, examples, and help! In this work, we prove an existence theorem of the Hyers-Ulam stability for the nonlinear Volterra integral equations which improves and generalizes Castro-Ramos theorem by using some weak conditions.

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Finally, we give two examples to illustrate our main theorems. Fractional-order differential equations are important since their nonlocal property is suitable to characterize memory phenomena in economic, control, and materials sciences. Existence, stability, and control theory to fractional differential equations was investigated in [ 123456789101112131415161718192021 ].

In particular, the Ulam-type stability of delay differential equations was investigated in [ 222324252627282930 ]. In [ 22 ], results for a delay differential equation were obtained using the Picard operator method, and in [ 23 ] the authors adopted a similar approach to establish the existence and uniqueness results for a Caputo-type fractional-order delay differential equation.

In [ 3132 ], the authors gave stability and numerical schemes for two classes of fractional equations. We establish the existence and uniqueness of solutions for 1 using the Picard operator approach in a weight function space. We also introduce and present Ulam—Hyers—Mittag-Leffler stability of solutions to 1. From Theorems 2. Then 2 is equivalent to. Then y is a solution of the following integral inequality:. Then we have. In this section, we establish the existence, uniqueness, and Ulam—Hyers—Mittag-Leffler stability.

We impose the following conditions. Now apply the Banach contraction principle to establish i. Now we prove ii. As a result, we get. Podlubny, I. Mathematics in Science and Engineering, vol. Academic Press, San Diego Google Scholar. Kilbas, A. Elsevier, Amsterdam Abbas, S. Li, M.

Liu, S. Slovaca 68— Wang, J. Fixed Point Theory Appl.Castro and A. Ramos More by L. Castro Search this author in:. The paper is devoted to the study of Hyers, Ulam and Rassias types of stability for a class of nonlinear Volterra integral equations.

Both Hyers-Ulam-Rassias stability and Hyers-Ulam stability are obtained for such a class of Volterra integral equations when considered on a finite interval. In addition, for corresponding Volterra integral equations on infinite intervals the Hyers-Ulam-Rassias stability is also obtained.

Source Banach J. Zentralblatt MATH identifier Keywords Hyers-Ulam-Rassias stability Volterra integral equation fixed point. Castro, L. Stationary Hyers-Ulam-Rassias stability for a class of nonlinear Volterra integral equations. Banach J. More by L. More by A. Abstract Article info and citation First page References Abstract The paper is devoted to the study of Hyers, Ulam and Rassias types of stability for a class of nonlinear Volterra integral equations.

Article information Source Banach J. Export citation. Export Cancel. References T. Burton, Volterra Integral and Differential Equations 2nd ed. You have access to this content. You have partial access to this content. You do not have access to this content.Metrics details.

We will apply the successive approximation method for proving the Hyers-Ulam stability of a nonlinear integral equation. We say a functional equation is stable if, for every approximate solution, there exists an exact solution near it. InUlam posed the following problem concerning the stability of functional equations [ 1 ]: we are given a group and a metric group with metric Given does there exist a such that if satisfies.

Since then, the stability problems of functional equations have been extensively investigated by several mathematicians cf. Recently, Y. Li and L. Hua proved the stability of Banach's fixed point theorem [ 6 ]. Examples of some recent developments, discussions, and critiques of that idea of stability can be found, for example, in [ 8 — 12 ].

In this paper, we study the Hyers-Ulam stability for the nonlinear Volterra integral equation of second kind. Jung was the author who investigated the Hyers-Ulam stability of Volterra integral equation on any compact interval. Inhe proved the following [ 13 ]. Given andlet denote a closed interval and let be a continuous function which satisfies a Lipschitz condition for all andwhere is a constant with.

If a continuous function satisfies. The purpose of this paper is to discuss the Hyers-Ulam stability of the following nonhomogeneous nonlinear Volterra integral equation:.

We will use the successive approximation method, to prove that 1. The method of this paper is distinctive.

This new technique is simpler and clearer than methods which are used in some papers, cf. On the other hand, Hyers-Ulam stability constant obtained in our paper is different to the other works, [ 13 ].

Consider the nonhomogeneous nonlinear Volterra integral equation 1. We assume that is continuous on the interval and is continuous with respect to the three variables, and on the domain ; and is Lipschitz with respect to.

In this paper, we consider the complete metric space and assume that is a bounded linear transformation on. Note that, the linear mapping is called bounded, if there exists such thatfor all. In this case, we define. Thus is bounded if and only if[ 15 ].In this paper, we study Hyers—Ulam stability and Hyers—Ulam—Rassias stability of first order non-linear impulsive time varying delay dynamic system on time scales, via a fixed point approach.

We obtain some results of existence and uniqueness of solutions by using Picard operator. In order to overcome difficulties arises in our considered model, we pose some conditions along with Lipchitz condition. At the end, an example is given that shows the validity of our main results. This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Agarwal, R. Alsina, C.

Bainov, D. Ellis Horwood Limited, Chichester Google Scholar. Bohner, M. Dachunha, J. Hamza, A. Hilger, S. Result Math.

Hyers, D. USA 27— Jung, S. Springer, NewYork Li, Y. Li, T. Lupulescu, V. Theory Differ. Nenov, S. Nonlinear Anal. Theory Methods Appl. Rocznik Nauk. Prace Mat. Discrete Contin. Rassias, T.

## Ulam stability for a delay differential equation

Rus, I. Shah, R.We will apply the successive approximation method for proving the Hyers-Ulam stability of a nonlinear integral equation. We say a functional equation is stable if, for every approximate solution, there exists an exact solution near it. InUlam posed the following problem concerning the stability of functional equations [ 1 ]: we are given a group and a metric group with metric Given does there exist a such that if satisfies.

Since then, the stability problems of functional equations have been extensively investigated by several mathematicians cf. Recently, Y. Li and L. Hua proved the stability of Banach's fixed point theorem [ 6 ]. Examples of some recent developments, discussions, and critiques of that idea of stability can be found, for example, in [ 8 — 12 ]. In this paper, we study the Hyers-Ulam stability for the nonlinear Volterra integral equation of second kind. Jung was the author who investigated the Hyers-Ulam stability of Volterra integral equation on any compact interval.

Inhe proved the following [ 13 ]. Given andlet denote a closed interval and let be a continuous function which satisfies a Lipschitz condition for all andwhere is a constant with. If a continuous function satisfies. The purpose of this paper is to discuss the Hyers-Ulam stability of the following nonhomogeneous nonlinear Volterra integral equation:. We will use the successive approximation method, to prove that 1. The method of this paper is distinctive.

This new technique is simpler and clearer than methods which are used in some papers, cf. On the other hand, Hyers-Ulam stability constant obtained in our paper is different to the other works, [ 13 ].

Consider the nonhomogeneous nonlinear Volterra integral equation 1. We assume that is continuous on the interval and is continuous with respect to the three variables, and on the domain ; and is Lipschitz with respect to. In this paper, we consider the complete metric space and assume that is a bounded linear transformation on.

Note that, the linear mapping is called bounded, if there exists such thatfor all. In this case, we define. Thus is bounded if and only if[ 15 ]. Definition 2. One says that 1.

We call such a Hyers-Ulam stability constant for 1. Since is assumed Lipschitz, we can write. So, we can write.

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