Hyers Ulam Rassias Research Articles

Investigation of infertility treatments on infertile couples through fractional-order modeling approach

In this paper, our aim is to construct a nonlinear mathematical model to study the impact of treatment on the infertility problem. We utilize a well-known Caputo derivative with fractional-order to decrease the issue of infertility treatments. Banach fixed-point theory is used to determine the existence and uniqueness of the suggested model. Equilibrium points have been calculated by generating reproduction numbers as a part of local stability analysis, and Ulam–Hyers–Rassias stability is discussed as a generalized form of system to prove global stability. We use a fractional-order Euler’s method to determine the numerical and graphical results. The result manifests the improvement of achieving a healthy pregnancy through hormonal and IVF treatment in women.

A Fixed Point Problem for a Hybrid Contraction and Ulam–Hyers–Rassias Stability Result with Respect to w-Distance

A Fixed Point Problem for a Hybrid Contraction and Ulam–Hyers–Rassias Stability Result with Respect to w-Distance

Existence and Stability of Neutral Stochastic Impulsive and Delayed Integro-Differential System via Resolvent Operator

In this paper, we present the existence of a mild solution for a class of a neutral stochastic integro-differential system over a Hilbert space. Such systems are influenced by both multiplicative and fractional noise, alongside non-instantaneous impulses, with a Hurst index H in the interval (12,1). Additionally, the systems under consideration feature state-dependent delays (SDDs). To address this, we develop an approach to reformulate the neutral stochastic integro-differential system, incorporating SDDs and non-instantaneous impulses, into an equivalent fixed-point (FP) problem via an appropriate integral operator. By integrating stochastic analysis with the theory of resolvent operators, we employ Banach’s FP theorem to establish both the existence and uniqueness of the solution. Furthermore, we explore the Ulam–Hyers–Rassias stability of the system. Lastly, we provide illustrative examples to demonstrate the practical applicability of our results.

Existence and k‐Mittag–Leffler–Ulam stabilities of a Volterra integro‐differential equation via (k,ϱ)‐Hilfer fractional derivative

In this paper, we investigate the existence, uniqueness, and analysis of two types of ‐Mittag–Leffler–Ulam stabilities in a Volterra integro‐differential fractional differential equation that involves the ‐Hilfer operator. We utilize the Banach fixed‐point theorem to establish the existence and uniqueness of solutions. We examine the stability properties, including the ‐Mittag–Leffler–Ulam–Hyers ‐ and k‐Mittag–Leffler–Ulam–Hyers–Rassias ‐ stabilities, by employing the Grönwall–Bellman inequality. Additionally, we provide an example to confirm our findings.

Existence and Stability for Fractional Differential Equations with a ψ–Hilfer Fractional Derivative in the Caputo Sense

This article aims to explore the existence and stability of solutions to differential equations involving a ψ-Hilfer fractional derivative in the Caputo sense, which, compared to classical ψ-Hilfer fractional derivatives (in the Riemann–Liouville sense), provide a clear physical interpretation when dealing with initial conditions. We discovered that the ψ-Hilfer fractional derivative in the Caputo sense can be represented as the inverse operation of the ψ-Riemann–Liouville fractional integral, and used this property to prove the existence of solutions for linear differential equations with a ψ-Hilfer fractional derivative in the Caputo sense. Additionally, we applied Mönch’s fixed-point theorem and knowledge of non-compactness measures to demonstrate the existence of solutions for nonlinear differential equations with a ψ-Hilfer fractional derivative in the Caputo sense, and further discussed the Ulam–Hyers–Rassias stability and semi-Ulam–Hyers–Rassias stability of these solutions. Finally, we illustrated our results through case studies.

A study on existence and stability analysis of implicit k,Ψ$$ \left(k,\Psi \right) $$‐Hilfer fractional differential equations and application to RC$$ \mathcal{RC} $$ circuit model

In the present study, we establish the existence and uniqueness of solutions for nonlinear initial value problems and nonlocal boundary value problems associated with implicit fractional differential equations involving ‐Hilfer derivative operator. Furthermore, we explore the stability properties of these solutions in the sense of the Ulam–Hyers, Ulam–Hyers–Rassias, and their generalized stability concepts by proving and applying generalized Gronwall inequality. Lastly, we present the fractional electric circuit model as an example to show the practical applicability of our main results.

Existence, Uniqueness, and Stability of Solutions for Nabla Fractional Difference Equations

In this paper, we study a class of nabla fractional difference equations with multipoint summation boundary conditions. We obtain the exact expression of the corresponding Green’s function and deduce some of its properties. Then, we impose some sufficient conditions in order to ensure existence and uniqueness results. Also, we establish some conditions under which the solution to the considered problem is generalized Ulam–Hyers–Rassias stable. In the end, some examples are included in order to illustrate our main results.

Analysis of Caputo fractional variable order multi-point initial value problems: existence, uniqueness, and stability

In this paper, we examine the existence, uniqueness, and stability of solutions for a Caputo variable order ϑ-initial value problem (ϑ-IVP) with multi-point initial conditions. The proofs for uniqueness and existence leverage Sadovski’s and Banach’s fixed point theorems, along with the Kuratowski measure of noncompactness. Furthermore, we explore the Ulam–Hyers–Rassias (UHR) stability of the solution. To validate our findings, we present a numerical example.

Existence and Uniqueness Solutions of Multi-Term Delay Caputo Fractional Differential Equations

This study investigates a novel type of nonlocal boundary value problem with multipoint-integral boundaries and multi-term delay Caputo fractional differential equations (FDE). The provided problem is turned into an analogous fixed-point problem using fixed-point (F P) theory tools. Additionally, discussing about stability, in Ulam-Hyers-Rassias (UHR), Ulam-Hyers (UH), generalized Ulam-Hyers-Rassias (GUHR) and generalized Ulam-Hyers (GUH) stability, for finding the problem. Based on our obtained results we given some examples. As of our obtained results are very useful to multi-term caputo FDE related to hydrodynamics.

Computational techniques to monitoring fractional order type-1 diabetes mellitus model for feedback design of artificial pancreas

In this paper, we developed a significant class of control issues regulated by nonlinear fractal order systems with input and output signals, our goal is to design a direct transcription method with impulsive instant order. Recent advances in the artificial pancreas system provide an emerging treatment option for type 1 diabetes. The performance of the blood glucose regulation directly relies on the accuracy of the glucose-insulin modeling. This work leads to the monitoring and assessment of comprehensive type-1 diabetes mellitus for controller design of artificial panaceas for the precision of the glucose-insulin glucagon in finite time with Caputo fractional approach for three primary subsystems. For the proposed model, we admire the qualitative analysis with equilibrium points lying in the feasible region. Model satisfied the biological feasibility with the Lipschitz criteria and linear growth is examined, considering positive solutions, boundedness and uniqueness at equilibrium points with Leray-Schauder results under time scale ideas. Within each subsystem, the virtual control input laws are derived by the application of input to state theorems and Ulam Hyers Rassias. Chaotic Relation of Glucose insulin glucagon compartmental in the feasible region and stable in finite time interval monitoring is derived through simulations that are stable and bounded in the feasible regions. Additionally, as blood glucose is the only measurable state variable, the unscented power-law kernel estimator appropriately takes into account the significant problem of estimating inaccessible state variables that are bound to significant values for the glucose-insulin system. The comparative results on the simulated patients suggest that the suggested controller strategy performs remarkably better than the compared methods. In the model under investigation, parametric uncertainties are identified since the glucose, insulin, and glucagon system's parameters are accurately measured numerically at different fractional order values. In terms of algorithm resilience and Caputo tracking in the presence of glucagon and insulin intake disturbance to maintain the glucose level. A comprehensive analysis of numerous difficult test issues is conducted in order to offer a thorough justification of the planned strategy to control the type 1 diabetes mellitus with designed the artificial pancreas.

Ulam–Hyers–Rassias Mittag-Leffler stability of ϖ–fractional partial differential equations

This paper offers a comprehensive analysis of solution representations for ϖ-fractional partial differential equations, specifically focusing on the linear case of the Darboux problem. We exhibit a representation of the solutions for the Darboux problem of ϖ-fractional partial differential equations in the linear case in the space of continuous functions. Through the application of the generalized Gronwall inequality, we establish the Ulam–Hyers–Rassias Mittag–Leffler stability in the space of continuous functions. Three numerical examples are presented to show the effectiveness and the applicability of our results.

On the qualitative behaviors of Volterra-Fredholm integro differential equations with multiple time-varying delays

This article considers a Volterra-Fredholm integro-differential equation including multiple time-varying delays. The aim of this article is to study the uniqueness of solution, the Ulam–Hyers–Rassias stability and the Ulam–Hyers stability of the Volterra-Fredholm integro-differential equation including multiple time-varying delays. We prove four new results in connection with the uniqueness of solution, the Ulam–Hyers–Rassias stability and the Ulam–Hyers stability of the considered Volterra-Fredholm integro-differential equation, respectively. The new results of this article involve sufficient conditions. The techniques of the proofs depend on the fixed point method according to the definitions of a suitable metric, operators and the related calculations. In particular case of the considered Volterra-Fredholm integro-differential equation, two illustrative examples are presented to verify the applications of the results. This article also involves some new complementary outcomes in connection with qualitative theory of Volterra-Fredholm integro-differential equations with delays.

Impulsive integro‐differential inclusions with nonlocal conditions: Existence and Ulam's type stability

This article focuses on the existence and Ulam–Hyers–Rassias stability outcomes pertaining to a specific category of impulsive integro‐differential inclusions (with instantaneous and non‐instantaneous impulses). These problems are examined using resolvent operators, drawing from the Grimmer perspective. Our analysis is based on Bohnenblust–Karlin's and Darbo's fixed point theorems for multivalued mappings in Banach spaces. Additionally, we provide an illustrative example to reinforce and demonstrate the validity of our findings.

Quantum Laplace Transforms for the Ulam–Hyers Stability of Certain q-Difference Equations of the Caputo-like Type

We aim to investigate the stability property for the certain linear and nonlinear fractional q-difference equations in the Ulam–Hyers and Ulam–Hyers–Rassias sense. To achieve this goal, we prove that three types of the linear q-difference equations of the q-Caputo-like type are Ulam–Hyers stable by using the quantum Laplace transform and quantum Mittag–Leffler function. Moreover, after proving the existence property for a nonlinear Cauchy q-difference initial value problem, we use the same quantum Laplace transform and the q-Gronwall inequality to show that it is generalized Ulam–Hyers–Rassias stable.

Ψ-Bielecki-type norm inequalities for a generalized Sturm–Liouville–Langevin differential equation involving Ψ-Caputo fractional derivative

The present research work investigates some new results for a fractional generalized Sturm–Liouville–Langevin (FGSLL) equation involving the Ψ-Caputo fractional derivative with a modified argument. We prove the uniqueness of the solution using the Banach contraction principle endowed with a norm of the Ψ-Bielecki-type. Meanwhile, the fixed-point theorems of the Leray–Schauder and Krasnoselskii type associated with the Ψ-Bielecki-type norm are used to derive the existence properties by removing some strong conditions. We use the generalized Gronwall-type inequality to discuss Ulam–Hyers (UH), generalized Ulam–Hyers (GUH), Ulam–Hyers–Rassias (UHR), and generalized Ulam–Hyers–Rassias (GUHR) stability of these solutions. Lastly, three examples are provided to show the effectiveness of our main results for different cases of (FGSLL)-problem such as Caputo-type Sturm–Liouville, Caputo-type Langevin, Caputo–Erdélyi–Kober-type Langevin problems.

Modelling the transmission behavior of measles disease considering contaminated environment through a fractal-fractional Mittag-Leffler kernel

This work utilises a fractal-fractional operator to examine the dynamics of transmission of measles disease. The existence and uniqueness of the measles model have been thoroughly examined in the context of the fixed point theorem, specifically utilising the Atangana-Baleanu fractal and fractional operators. The model has been demonstrated to possess both Hyers-Ulam stability and Hyers-Ulam Rassias stability. Furthermore, a qualitative analysis of the model was performed, including examination of key parameters such as the fundamental reproduction number, the measles-free and measles-present equilibria, and assessment of global stability. This research has shown that the transmission of measles disease is affected by natural phenomena, as changes in the fractal-fractional order lead to changes in the disease dynamics. Furthermore, environmental contamination has been shown to play a significant role in the transmission of the measles disease.

The Existence and Ulam Stability Analysis of a Multi-Term Implicit Fractional Differential Equation with Boundary Conditions

In this paper, we investigate a class of multi-term implicit fractional differential equation with boundary conditions. The application of the Schauder fixed point theorem and the Banach fixed point theorem allows us to establish the criterion for a solution that exists for the given equation, and the solution is unique. Afterwards, we give the criteria of Ulam–Hyers stability and Ulam–Hyers–Rassias stability. Additionally, we present an example to illustrate the practical application and effectiveness of the results.

Hyers–Ulam–Rassias stability of fractional delay differential equations with Caputo derivative

This paper is devoted to the study of Hyers–Ulam–Rassias (HUR) stability of a nonlinear Caputo fractional delay differential equation (CFrDDE) with multiple variable time delays. We obtain two new theorems with regard to HUR stability of the CFrDDE on bounded and unbounded intervals. The method of the proofs is based on the fixed point approach. The HUR stability results of this paper have indispensable contributions to theory of Ulam stabilities of CFrDDEs and some earlier results in the literature.

An investigation into the characteristics of VFIDEs with delay: solvability criteria, Ulam–Hyers–Rassias and Ulam–Hyers stability

An investigation into the characteristics of VFIDEs with delay: solvability criteria, Ulam–Hyers–Rassias and Ulam–Hyers stability

Fundamental Matrix, Integral Representation and Stability Analysis of the Solutions of Neutral Fractional Systems with Derivatives in the Riemann—Liouville Sense

The paper studies a class of nonlinear disturbed neutral linear fractional systems with derivatives in the the Riemann–Liouville sense and distributed delays. First, it is proved that the initial problem for these systems with discontinuous initial functions under some natural assumptions possesses a unique solution. The assumptions used for the proof are similar to those used in the case of systems with first-order derivatives. Then, with the obtained result, we derive the existence and uniqueness of a fundamental matrix and a generalized fundamental matrix for the homogeneous system. In the linear case, via these fundamental matrices we obtain integral representations of the solutions of the homogeneous system and the corresponding inhomogeneous system. Furthermore, for the fractional systems with Riemann–Liouville derivatives we introduce a new concept for weighted stabilities in the Lyapunov, Ulam–Hyers, and Ulam–Hyers–Rassias senses, which coincides with the classical stability concepts for the cases of integer-order or Caputo-type derivatives. It is proved that the zero solution of the homogeneous system is weighted stable if and only if all its solutions are weighted bounded. In addition, for the homogeneous system it is established that the weighted stability in the Lyapunov and Ulam–Hyers senses are equivalent if and only if the inequality appearing in the Ulam–Hyers definition possess only bounded solutions. Finally, we derive natural sufficient conditions under which the property of weighted global asymptotic stability of the zero solution of the homogeneous system is preserved under nonlinear disturbances.