Hyers Ulam Research Articles

HYERS - ULAM STABILITY OF FIRST AND SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we prove the Hyers-Ulam (HU) stability of the first and second order partial differential equations: ux(x,t)+K(x, u(x,t))=0 and uxx(x,t)+F(x,u)ux(x,t)+H(x,u)=0 respectively.

Hyers-Ulam-Rassias Stability for Abel-Riccati Type First-Order Differential Equations

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.

Existence and Stability of Solutions for Linear and Nonlinear Stieltjes Differential Equations

This paper deals with Cauchy problems and nonlocal problems for non-linear Stieltjes differential equations corresponding to a certain function g. We establish existence and uniqueness results for nonlinear equations with initial value or nonlocal conditions in the space ℬ𝒞 g ([0, H], ℝ) using fixed point methods and g-topology theory. We introduce the concepts of Ulam-Hyers (UH) and generalized Ulam-Hyers-Rassias (UHR) stability and present Ulam type stability results for linear and nonlinear equations in the spaces 𝒜𝒞 g ([0, H], ℝ) ⊂ ℬ𝒞 g ([0, H], ℝ) and ℬ𝒞 g ([0, H], ℝ). Finally, numerical examples are given to illustrate our results.

Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel

In this paper we are established the existence of positive solutions (EPS) and the Hyers-Ulam (HU) stability of a general class of nonlinear Atangana-Baleanu-Caputo (ABC) fractional differential equations (FDEs) with singularity and nonlinear p-Laplacian operator in Banach’s space. To find the solution for the EPS, we use the Guo-Krasnoselskii theorem. The fractional differential equation is converted into an alternative integral structure using the Atangana-Baleanu fractional integral operator. Also, HU-stability is analyzed. We include an example with specific parameters and assumptions to show the results of the proposal.

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

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.

Existence of positive solution and Hyers\u2013Ulam stability for a nonlinear singular-delay-fractional differential equation

In this article, we consider a study of a general class of nonlinear singular fractional DEs with p-Laplacian for the existence and uniqueness (EU) of a positive solution and the Hyers–Ulam (HU) stability. To proceed, we use classical fixed point theorem and properties of a p-Laplacian operator. The fractional DE is converted into an integral alternative form with the help of the Green’s function. The Green’s function is analyzed as regards its nature and then, with the help of a fixed point approach, the existence of a positive solution and uniqueness are studied. After the EU of a positive solution, the HU-stability and an application are considered. The suggested singular fractional DE with phi _{p} is more general than the one considered in (Khan et al. in Eur. Phys. J. Plus 133:26, 2018)

On Ulam Stability and Multiplicity Results to a Nonlinear Coupled System with Integral Boundary Conditions

This manuscript is devoted to establishing existence theory of solutions to a nonlinear coupled system of fractional order differential equations (FODEs) under integral boundary conditions (IBCs). For uniqueness and existence we use the Perov-type fixed point theorem. Further, to investigate multiplicity results of the concerned problem, we utilize Krasnoselskii’s fixed-point theorems of cone type and its various forms. Stability analysis is an important aspect of existence theory as well as required during numerical simulations and optimization of FODEs. Therefore by using techniques of functional analysis, we establish conditions for Hyers-Ulam (HU) stability results for the solution of the proposed problem. The whole analysis is justified by providing suitable examples to illustrate our established results.

Study of implicit type coupled system of non‐integer order differential equations with antiperiodic boundary conditions

In this paper, the first purpose is to study existence and uniqueness of solutions to a system of implicit fractional differential equations (IFDEs) equipped with antiperiodic boundary conditions (BCs). To obtain the mentioned results, we use Schauder's and Banach fixed point theorem. The second purpose is discussing the Ulam‐Hyers (UH) and generalized Ulam‐Hyers (GUH) stabilities for the respective solutions. An example is provided to illustrate the established results.

Existence and Stability Analysis of Three Point Boundary Value Problem

In this research article, we study the existence and uniqueness of solution for a three points boundary value problem of fractional order differential equations. Further, we also investigate different kinds of Ulam stability such as Ulam-Hyers (UH) stability, generalized Ulam-Hyers (GUH) stability, Ulam-Hyers-Rassias (UHR) stability and generalized Ulam-Hyers-Rassias (GUHR) stability for the proposed problem. The concerned analysis is carried out via using classical technique of nonlinear functional analysis. The main results are demonstrated by providing couple of examples.

The Fixed Point Approach to the Stability of Fractional Differential Equations with Causal Operators

In this paper, the Hyers–Ulam (HU) stability and Hyers–Ulam–Rassias (HUR) stability of fractional differential equations with causal operators (FDEwCO) are investigated. The techniques rely on a fixed point theorem which is employed to study the HUR stability for FDEwCO on both bounded and unbounded time intervals as well as HU stability on bounded time interval. Finally, two typical examples are given to demonstrate the applications of theoretical results proposed.

On Tabor groupoids and stability of some functional equations

Two results are given, which use Tabor groupoids for questions of stability in the sense of Pólya–Szegő–Hyers–Ulam. We also start to study Tabor groupoids in their own right.