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In this paper we consider a concept of uniform polynomial splitting for a discrete cocycle over a discrete semiflow in Banach spaces. We obtain some characterizations of Datko type and also in terms of Lyapunov functions. The study is made from the point of view of the invariant projectors and also strongly invariant projectors.

In this dissertation, we study the coupled system of q$$ q $$‐fractional Langevin differential equations involving q$$ q $$‐Caputo derivative having q$$ q $$‐fractional integral conditions. With the help of some adequate conditions, we investigate the uniqueness and existence of mild solution of the aforementioned system. We also analyze various kinds of Ulam's stability. Banach fixed point theorem and Leray–Schauder of cone type are used to illustrate the existence and uniqueness results. We also used non‐linear functional analysis methods to explore variety of stability types. An example is provided to clearly demonstrate our theoretical outcomes. [ABSTRACT FROM AUTHOR]

We establish sufficient conditions for the exponential stability of nonsingular impulsive delayed integro‐differential systems. Our approach to addressing nonsingular differential problems involves the application of permutable matrices and their associated delayed exponential. Furthermore, we investigate the controllability of a nonlinear impulsive and delayed problem by employing the corresponding Gramian matrix. Finally, to illustrate the theoretical outcomes, we provide examples and graphical representations for each situation. [ABSTRACT FROM AUTHOR]

In this article, with the help of Laplace transform, the existence of solution was established in a finite dimensional setting for nonlinear ψ-Hilfer fractional stochastic equation with both retarded and advanced arguments driven by multiplicative and fractional noises, with Hurst index H∈(1/2,1). At first, we obtained the existence and uniqueness results by using the Banach fixed point theorem (FPT). Second, the existence result was also obtained by applying Schaefer's fixed point theorem with less conservative conditions. Furthermore, we investigated the Hyers Ulam Rasisas stability for the aforementioned system. At the end, an example was illustrated to validate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the problem of indefinite stochastic linear quadratic (LQ) optimal control by piecewise constant controls in an infinite horizon case. By restricting the set of admissible controls to the class of piecewise constant stochastic processes, we reformulated the above control problem under the setting of systems modeled by Itô differential equations controlled by impulses. We show that the solution in a state feedback form of the indefinite stochastic LQ control problem is equivalent to the existence of a global stabilizing solution associated to a class of backward matrix linear differential equations with a Riccati type jumping operators. [ABSTRACT FROM AUTHOR]

In this work, we analyze a q-fractional jerk problem having anti-periodic boundary conditions. The focus is on investigating whether a unique solution exists and remains stable under specific conditions. To prove the uniqueness of the solution, we employ a Banach fixed point theorem and a mathematical tool for establishing the presence of distinct fixed points. To demonstrate the availability of a solution, we utilize Leray–Schauder's alternative, a method commonly employed in mathematical analysis. Furthermore, we examine and introduce different kinds of stability concepts for the given problem. In conclusion, we present several examples to illustrate and validate the outcomes of our study. [ABSTRACT FROM AUTHOR]

The concept of generalized exponential trichotomy for linear time-varying systems is investigated in relationship with the classical notion of uniform exponential trichotomy. Some key properties of generalized exponential trichotomy are explored through supplementary projections. These results are also extended to the case of projection sequences, while certain applications for adjoint systems are suggested.

The aim of this paper is to study the concept of uniform exponential trisplitting for skew-product semiflow in Banach spaces. This concept is a generalisation of the well-known concept of uniform exponential trichotomy. We obtain necessary and sufficient conditions for this concept of Datko’s type. a character-isation in terms of Lyapunov functions is provided. The results are obtained from the point of view of the projector families, i.e. invariant and strongly invariant.

In this article, we analyze the existence and uniqueness of mild solution to the Stieltjes integral boundary value problem involving a nonlinear multi-term, Caputo-type sequential fractional integro-differential equation. Krasnoselskii's fixed-point theorem and the Banach contraction principle are utilized to obtain the existence and uniqueness of the mild solution of the aforementioned problem. Furthermore, the Hyers-Ulam stability is obtained with the help of established methods. Our proposed model contains both the integer order and fractional order derivatives. As a result, the exponential function appears in the solution of the model, which is a fundamental and naturally important function for integer order differential equations and its many properties. Finally, two examples are provided to illustrate the key findings. [ABSTRACT FROM AUTHOR]

In this paper, the existence of a unique solution is established for a coupled system of Langevin fractional problems of ψ -Caputo fractional derivatives with generalized slit-strip-type integral boundary conditions and impulses using the Banach contraction principle. We also find at least one solution to the aforementioned system using some assumptions and Schaefer's fixed point theorem. After that, Ulam–Hyers stability is discussed. Finally, to provide additional support for the main results, pertinent examples are presented. [ABSTRACT FROM AUTHOR]

In this article, we examine a coupled system of fractional integrodifferential equations of Liouville–Caputo form with instantaneous impulsive conditions in a Banach space. We obtain the existence and uniqueness results by applying the theory of fixed point theorems. In a similar manner, we discuss Hyers–Ulam stability and controllability. We also present an example to show the validity of the obtained results. [ABSTRACT FROM AUTHOR]

In this article, we study the existence and uniqueness along with Ulam‐type stability of fractional integro causal evolution impulsive systems on time scales. To establish our main results, we use Picard operator along with abstract Grönwall inequality and Gröwall lemma. We constitute some assertions to remove difficulties. At the end, the results are demonstrated via the support of an example. [ABSTRACT FROM AUTHOR]

This article investigates the exact and approximate controllability, finite time stability, and B$$ \mathcal{B} $$–Hyers–Ulam–Rassias stability of a neutral impulsive system of differential equations with two delays. Fixed point theory is associated in the analysis of controllability of the system. The stability analysis is carried out with the help of Grönwall type inequality. [ABSTRACT FROM AUTHOR]

Horadam sequence is a general recurrence of second order in the complex plane, depending on four complex parameters (two initial values and two recurrence coefficients). These sequences have been investigated over more than 60 years, but new properties and applications are still being discovered. Small parameter variations may dramatically impact the sequence orbits, generating numerous patterns: periodic, convergent, divergent, or dense within one dimensional curves. Here we explore Horadam sequences whose orbit is dense within a 2D region of the complex plane, while the complex argument is uniformly distributed in an interval. This enables the design of a pseudo-random number generator (PRNG) for the uniform distribution, for which we test periodicity, correlation, Monte Carlo estimation of π , and the NIST battery of tests. We then calculate the probability distribution of the radii of the sequence terms of Horadam sequences. Finally, we propose extensions of these results for generalized Horadam sequences of third order. [ABSTRACT FROM AUTHOR]

In this paper, we solve a stochastic linear quadratic tracking problem. The controlled dynamical system is modeled by a system of linear Itô differential equations subject to jump Markov perturbations. We consider the case when there are two decision-makers and each of them wants to minimize the deviation of a preferential output of the controlled dynamical system from a given reference signal. We assume that the two decision-makers do not cooperate. Under these conditions, we state the considered tracking problem as a problem of finding a Nash equilibrium strategy for a stochastic differential game. Explicit formulae of a Nash equilibrium strategy are provided. To this end, we use the solutions of two given terminal value problems (TVPs). The first TVP is associated with a hybrid system formed by two backward nonlinear differential equations coupled by two algebraic nonlinear equations. The second TVP is associated with a hybrid system formed by two backward linear differential equations coupled by two algebraic linear equations. [ABSTRACT FROM AUTHOR]

We investigate a linear quadratic tracking problem for impulsive controlled stochastic systems. The main tool in the derivation of the optimal control is the stabilizing solution for a class of backward jump matrix linear differential equation. We introduce the concept of mean square stability by impulses and consider the properties of stabilizing solution of the associated backward jump matrix linear differential equation with Riccati‐type operators. Necessary and sufficient conditions that guarantee the existence of the unique optimal control for the linear quadratic tracking problem for impulsive controlled stochastic systems are proved. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of a stabilizing solution to a periodic backward nonlinear differential equation. This class of nonlinear equations includes as special cases many of the continuous-time Riccati equations arising both in deterministic and stochastic linear quadratic (LQ) type control problems. [ABSTRACT FROM AUTHOR]

This paper presents a new approach to formulating exponential behaviors like stability/instability for the linear time-varying systems and for the adjoint one. The classical concept of uniform exponential stability is generalized. Using this generalized concepts, some results extending existing uniform exponential stability conditions for linear time-varying systems are derived. As special cases for these results, some conditions are derived for the adjoint system. A characterization of the generalized concepts in terms of Lyapunov sequences is also given. Also, an example is included to further illustrate the connection with the classical concept of uniform exponential stability. [ABSTRACT FROM AUTHOR]

Abstract This note focuses on the problem of generalized exponential stability for linear time-varying discrete-time systems in Banach spaces. Characterizations for this concept are presented and connections with the classical concept of exponential stability existent in the literature is pointed out. An illustrative example clarifies the implications between these concepts. [ABSTRACT FROM AUTHOR]

An optimal control problem for a system of linear Itô differential equations with two fast time scales and Markovian jumping is considered. The asymptotic structure for the stabilizing solution satisfying a sign condition for the coupled stochastic algebraic Riccati equations is derived. Furthermore, a near optimal control whose gain matrices do not depend upon small parameters is discussed. [ABSTRACT FROM AUTHOR]

Horadam sequences are second-order linear recurrences in the complex plane which depend on two initial conditions and two recurrence coefficients which are complex numbers. Recently, numerous papers have been devoted to the periodicity of these sequences, as well as to generalizations and applications. In this paper we investigate aspects related to the sequence of rations of consecutive terms and geometric bounds of Horadam sequences. We also propose some directions for further study. [ABSTRACT FROM AUTHOR]

In the present paper the concept of uniform exponential trisplitting for skew-product flows in Banach space is considered. This concept is a generalisation of the well-known concept of uniform exponential trichotomy. Connections between these concepts are presented and some illustrating examples prove that these are distinct. Also, we present necessary and sufficient conditions for the uniform exponential trisplitting concept with invariant and strongly invariant projectors. Finally, a characterisation in terms of Lyapunov sequences is given. [ABSTRACT FROM AUTHOR]

The generalized exponential trichotomy (g.e.t) concept is explored for linear time-varying systems. Characterizations and relations with the notion of uniform exponential trichotomy (u.e.t) in the sense of Elaydi-Janglajew are also provided. [ABSTRACT FROM AUTHOR]

Horadam sequences are second-order recurrences depending on a family of four complex parameters: two initial conditions and two recurrence coefficients. The periodicity conditions, as well as the number and geometric structure of self-repeating Horadam sequences of fixed length have been recently investigated. In this paper various geometric properties of non-homogeneous Horadam sequences are explored, including periodicity conditions and the structure of certain periodic orbits. [ABSTRACT FROM AUTHOR]