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In this paper, we analyses the existence and Hyers–Ulam stability of a coupled system of three sequential fractional differential equations with coupled integral boundary conditions. This manuscript can be categorized into three parts: The Leray–Schauder alternative is used to prove the existence of a solution in the first section. The second section emphasizes the analysis of uniqueness, which is based on the Banach fixed point theorem's concept of contraction mapping, and the third section establishes the Hyers–Ulam stability results. In addition, we provide examples to demonstrate our findings. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and uniqueness of a fixed point in a dislocated quasi-metric space. Several fixed point theorems for distinct type of contractive conditions are presented that generalize, extend, and unify a number of related results reported in the literature. Illustrative examples are provided. [ABSTRACT FROM AUTHOR]

We study the existence of nonoscillatory solutions to a class of third-order neutral functional dynamic equations on time scales. The integral convergence and divergence of the reciprocal of the coefficients in the equations are different. Two examples are given to demonstrate the results. [ABSTRACT FROM AUTHOR]

We prove that the system $\dot{\theta}(t) =\Lambda(t)\theta(t)$ , $t\in\mathbb{R}_{+}$ , is Hyers-Ulam stable if and only if it is uniformly exponentially stable under certain conditions; we take the exact solutions of the Cauchy problem $\dot{\phi}(t)=\Lambda(t)\phi(t)+e^{i\gamma t}\xi(t)$ , $t\in\mathbb{R}_{+}$ , $\phi(0)=\theta_{0}$ as the approximate solutions of $\dot{\theta}(t)=\Lambda(t)\theta(t)$ , where γ is any real number, ξ is a 2-periodic, continuous, and bounded vectorial function with $\xi(0)=0$ , and $\Lambda(t)$ is a 2-periodic square matrix of order l. [ABSTRACT FROM AUTHOR]

We discuss the notion of Volterra, ℓ-Volterra and separable quadratic stochastic operators defined on (m-1)-dimensional simplex, where ℓ ∈ {0, 1, ...,m}. The ℓ-Volterra operator is a Volterra operator if and only if ℓ = m. We study the structure of the set of all Volterra and ℓ-Volterra operators and describe their several fixed and periodic points. For m = 2 and m = 3 we describe behavior of trajectories of (m-1)-Volterra operators. We also mention many remarks with comparisons of ℓ-Volterra operators and Volterra ones. Also we discuss the dynamics of separable quadratic stochastic operators. [ABSTRACT FROM AUTHOR]

Let $$v_{n}$$ be the solution of the nonhomogeneous evolution difference equation $$v_{n+1} = \mathcal {A}_nv_{n}+ \alpha _{n+1}$$ , $$v_0= 0$$ for all $$\ n\in \mathbb {Z}_+$$ , where $$\mathcal {A}_n$$ is a sequence of almost periodic (possibly unbounded) linear operators on a Banach space $$\mathcal {W}$$ . Let $$\mathcal {C}_{00}(\mathbb {Z}_+,\mathcal {W})$$ is the space of all $$\mathcal {W}$$ -valued bounded sequences which decays at zero and at infinity and $$\mathcal {AP}_0(\mathbb {Z}_+,\mathcal {W})$$ is the space of all $$\mathcal {W}$$ -valued almost periodic sequences decaying at zero. We consider the space $$\mathcal {AAP}_0^r(\mathbb {Z}_+,\mathcal {W})$$ consisting of all sequences $$\alpha (n)$$ with relatively compact ranges for which there exists $$\beta (n)\in \mathcal {AP}_0(\mathbb {Z}_+,\mathcal {W})$$ and $$\gamma (n) \in \mathcal {C}_{00}(\mathbb {Z}_+,\mathcal {W})$$ such that $$\alpha (n) = \beta (n) + \gamma (n)$$ . We prove that $$v_{n}$$ belongs to $$\mathcal {AAP}_0^r(\mathbb {Z}_+,\mathcal {W})$$ if and only if for each $$x\in \mathcal {W}$$ the discrete evolution family of operators $$E = \{\mathcal {E}(n, m): n,m \in \mathbb {Z}_+, \ n\ge m \}$$ is uniformly exponentially stable. Our results are based on evolution semigroup approach. [ABSTRACT FROM AUTHOR]

In this article, we prove that the ω-periodic discrete evolution family $\Gamma:= \{\rho(n,k): n, k \in\mathbb{Z}_{+}, n\geq k\}$ of bounded linear operators is Hyers-Ulam stable if and only if it is uniformly exponentially stable under certain conditions. More precisely, we prove that if for each real number γ and each sequence $(\xi(n))$ taken from some Banach space, the approximate solution of the nonautonomous ω-periodic discrete system $\theta _{n+1} = \Lambda_{n}\theta_{n}$, $n\in\mathbb{Z}_{+}$ is represented by $\phi _{n+1}=\Lambda_{n}\phi_{n}+e^{i\gamma(n+1)}\xi(n+1)$, $n\in\mathbb{Z}_{+}$; $\phi_{0}=\theta_{0}$, then the Hyers-Ulam stability of the nonautonomous ω-periodic discrete system $\theta_{n+1} = \Lambda_{n}\theta_{n}$, $n\in\mathbb{Z}_{+}$ is equivalent to its uniform exponential stability. [ABSTRACT FROM AUTHOR]

For a higher-order nonlinear impulsive ordinary differential equation, we present the concepts of Hyers-Ulam stability, generalized Hyers-Ulam stability, Hyers-Ulam-Rassias stability, and generalized Hyers-Ulam-Rassias stability. Furthermore, we prove the generalized Hyers-Ulam-Rassias stability by using integral inequality of Grönwall type for piecewise continuous functions. These results extend related contributions to the corresponding first-order impulsive ordinary differential equation. Hyers-Ulam stability, generalized Hyers-Ulam stability, and Hyers-Ulam-Rassias stability can be discussed by the same methods. [ABSTRACT FROM AUTHOR]

For nth order linear homogeneous and nonhomogeneous differential equations with nonconstant coef- ficients, we prove Hyers-Ulam stability by using open mapping theorem. The generalized Hyers-Ulam stability is also investigated. [ABSTRACT FROM AUTHOR]

In the current work, some fixed point theorems are proved on ordered cone b-metric space motivated by [2], [14] and [15]. The proposed theorems extend and unify several well-known compa- rable results in the literature to ordered cone b-metric spaces. Some supporting examples are given. [ABSTRACT FROM AUTHOR]

Nowadays, the rapid spread of various tick-borne viruses has caused various diseases in the animal population of livestock and poultry, in which the human population is not safe. Crimean-Congo hemorrhagic fever is one of the common diseases between animals and humans that causes many deaths of both populations in the large areas of the world every year. Identification and control methods of this epidemic have led virologists to study the dynamics and behavior of these viruses in different transmission cycles in recent years based on mathematical models. In this paper, we present an advanced mathematical model of transmission cycle of viruses of the Crimean-Congo hemorrhagic fever between livestock, ticks and humans in a fractal–fractional system of six initial value problems. In fact, we extend the standard integer-order model to a two-parametric six-compartmental fractal–fractional hybrid model with power-law type kernels. To study the existence of solution for such a system, we first use a special family of contractions titled ϕ − ψ -contractions and also in the next step, we use the Leray–Schauder fixed point theorem. The Banach contraction principle helps us to prove the uniqueness of solutions. We try to investigate the stability behaviors of the solutions in the context of the Ulam's criterion, and then use Lagrange polynomials to obtain a numerical algorithm to find the approximate solutions of the mathematical model of Crimean-Congo hemorrhagic fever. Finally, by changing the values of the fractal dimension and fractional order in a closed interval, we analyze the convergence and stability of the solutions graphically. We see that all solutions have stable behaviors and at smaller fractal dimensions, decay and growth rates in susceptible and infected groups are slower, and vice versa. The accurate results of fractal–fractional operators in mathematical modeling motivate us to use them in different models. • A computational model of the Crimean-Congo hemorrhagic fever disease is studied. • We used the fractal–fractional derivatives of the power-law type kernels. • We analyze the convergence and stability of the solutions graphically. • Some concluding remarks for future research is presented. [ABSTRACT FROM AUTHOR]

This primer article focuses on the representation of solutions and finite-time stability of impulsive first-order delay differential systems. We define delayed matrix function with impulses and use variation of parameters to obtain a representation of solutions of linear systems with impulse effects. The famous classical Grownwall inequalities and properties of delayed matrix exponential with impulses are used to develop sufficient conditions for finite-time stability. In the end, we provide some examples to support the results. [ABSTRACT FROM AUTHOR]

The major goal of this manuscript is to investigate the existence, uniqueness, and stability of a q-fractional Langevin differential equation with q-fractional integral conditions. We demonstrate the existence and uniqueness of the solution to the proposed q-fractional Langevin differential equation using the Banach contraction principle and Schaefer's fixed-point theorem. We also elaborate on different kinds of Ulam stability. The theoretical outcomes are verified by examples. [ABSTRACT FROM AUTHOR]

This manuscript is committed to deal with the existence and uniqueness of positive solutions for fractional relaxation equation involving ψ-Caputo fractional derivative. The existence of solution is carried out with the help of Schauder's fixed point theorem, while the uniqueness of the solution is obtained by applying the Banach contraction principle, along with Bielecki type norm. Moreover, two explicit monotone iterative sequences are constructed for the approximation of the extreme positive solutions to the proposed problem. Lastly, two examples are presented to support the obtained results. [ABSTRACT FROM AUTHOR]

We prove that a quadratic matrix of order n having complex entries is dichotomic (i.e. its spectrum does not intersect the imaginary axis) if and only if there exists a projection P on ℁n such that PetA = etAP for all t ≥ 0 and for each real number μ and each vector b ∈ ℁n the solutions of the following two Cauchy problems are bounded: ẋ(t) = Ax(t) + eiµtPb, t ≥ 0, x(0) = 0 and ẏ(t) = -Ay(t) + eiµt(I - P)b, t ≥ 0, y(0) = 0 . [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlocal boundary value problem of nonlinear implicit impulsive Langevin equation involving mixed order derivatives. Sufficient conditions are constructed to discuss the qualitative properties like existence and Ulam's stability of the proposed problem. The main result is verified by an example. [ABSTRACT FROM AUTHOR]

This paper discusses the synchronization problem of N-coupled fractional-order chaotic systems with ring connection via bidirectional coupling. On the basis of the direct design method, we design the appropriate controllers to transform the fractional-order error dynamical system into a nonlinear system with antisymmetric structure. By choosing appropriate fractional-order Lyapunov functions and employing the fractional-order Lyapunov-based stability theory, several sufficient conditions are obtained to ensure the asymptotical stabilization of the fractional-order error system at the origin. The proposed method is universal, simple, and theoretically rigorous. Finally, some numerical examples are presented to illustrate the validity of theoretical results. [ABSTRACT FROM AUTHOR]

This paper deals with existence, uniqueness, and Hyers–Ulam stability of solutions to a nonlinear coupled implicit switched singular fractional differential system involving Laplace operator ϕ p . The proposed problem consists of two kinds of fractional derivatives, that is, Riemann–Liouville fractional derivative of order β and Caputo fractional derivative of order σ, where m − 1 < β , σ < m , m ∈ { 2 , 3 , ... } . Prior to proceeding to the main results, the system is converted into an equivalent integral form by the help of Green's function. Using Schauder's fixed point theorem and Banach's contraction principle, the existence and uniqueness of solutions are proved. The main results are demonstrated by an example. [ABSTRACT FROM AUTHOR]

We present some results on the existence, uniqueness and Hyers–Ulam stability to the solution of an implicit coupled system of impulsive fractional differential equations having Hadamard type fractional derivative. Using a fixed point theorem of Kransnoselskii's type, the existence and uniqueness results are obtained. Along these lines, different kinds of Hyers–Ulam stability are discussed. An example is given to illustrate the main theorems. [ABSTRACT FROM AUTHOR]

We study almost periodic solutions for a class of nonlinear second-order differential equations involving reflection of the argument. We establish existence results of almost periodic solutions as critical points by a variational approach. We also prove structure results on the set of strong almost periodic solutions, existence results of weak almost periodic solutions, and a density result on the almost periodic forcing term for which the equation possesses usual almost periodic solutions. [ABSTRACT FROM AUTHOR]

In this manuscript, we give some sufficient conditions for existence, uniqueness and various kinds of Ulam stability for a toppled system of fractional order boundary value problems involving the Riemann-Liouville fractional derivative. Applying the Banach contraction principle and the Leray-Schauder result of cone type, uniqueness and existence results are proved for the proposed toppled system. Stability is investigated by using the classical technique of nonlinear functional analysis. The results obtained are well illustrated with the aid of an example. [ABSTRACT FROM AUTHOR]

In this research study, we formulate two generalized nonlinear multi-order fractional boundary value problems with the help of quantum diffeerence operators. To investigate the exis- tence property for possible solutions of these q-di_erence FBVPs, we apply two separate methods motivated by some notions in relation to the measure of noncompactness and end-point technique. The condensing functions and multifunctions having the (AE)-property play an important role in our study. As well as, two examples corresponding to both techniques are provided to ensure the compatibility of the findings numerically. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the system of switched coupled implicit fractional Langevin equations of nonlinear form with anti‐periodic boundary conditions. In the first step, problem's equivalence and the corresponding integral equation by applying fractional calculus tools are established, and a fixed point problem is defined. For mixed monotone mappings, we have used coupled fixed point theorems to achieve the existence and uniqueness of solutions of these equations. In the next step, by using Banach's fixed point theorem, Ulam–Hyers, Ulam–Hyers–Rassias, generalized Ulam–Hyers and generalized Ulam–Hyers–Rassias stabilities of our considered model are discussed. An example is given at the end for the verification of our results. [ABSTRACT FROM AUTHOR]

Waterborne diseases are illnesses caused by pathogenic bacteria that spread through water and have a negative influence on human health. Due to the involvement of most countries in this vital issue, accurate analysis of mathematical models of such diseases is one of the first priorities of researchers. In this regard, in this paper, we turn to a waterborne disease model for solution's existence, HU-stability, and computational analysis. We transform the model to an analogous fractal-fractional integral form and study its qualitative analysis using an iterative convergent sequence and fixed-point technique to see whether there is a solution. We use Lagrange's interpolation to construct numerical algorithms for the fractal-fractional waterborne disease model in terms of computations. The approach is then put to the test in a case study, yielding some interesting outcomes. [ABSTRACT FROM AUTHOR]

In the current paper, we analyzed the existence and uniqueness of a solution for a coupled system of impulsive hybrid fractional differential equations involving ψ -Caputo fractional derivatives with generalized slit-strips-type integral boundary conditions. We also study the Ulam–Hyers stability for the considered system. For the existence and uniqueness of the solution, we use the Banach contraction principle. With the help of Schaefer's fixed-point theorem and some assumptions, we also obtain at least one solution of the mentioned system. Finally, the main results are verified with an appropriate example. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to investigate the existence, uniqueness, and Ulam–Hyers stability of positive solutions for fractional integro-differential boundary values problem. Uniqueness result is obtained by using the Banach principle. For obtaining two positive solutions, we apply another fixed point criterion due to Avery–Anderson–Henderson on cones by establishing some inequalities. An illustrative example is presented to indicate the validity of the obtained results. The results are new and provide a generalization to some known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we examine the consequences of existence, uniqueness and stability of a multi-point boundary value problem defined by a system of coupled fractional differential equations involving Hadamard derivatives. To prove the existence and uniqueness, we use the techniques of fixed point theory. Stability of Hyers-Ulam type is also discussed. Furthermore, we investigate variations of the problem in the context of different boundary conditions. The current results are verified by illustrative examples. [ABSTRACT FROM AUTHOR]

Our basic purpose is to derive several existence aspects of solutions for a novel class of the fractional inclusion problem in terms of the well-defined generalized φ-Caputo and φ-Riemann–Liouville operators. The existing boundary conditions in such an inclusion problem are endowed with mixed generalized φ-Riemann–Liouville conditions. To reach this goal, we utilize the analytical methods on α-ψ-contractive maps and multifunctions involving approximate endpoint specification to derive the required results. In the final part, we formulate an illustrative simulation example to examine obtained theoretical outcomes by computationally and numerically. [ABSTRACT FROM AUTHOR]