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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.

The primary objective of this manuscript is to investigate the existence and uniqueness of solutions for the Langevin (k , φ) -Hilfer fractional differential equation of different orders with multipoint nonlocal fractional integral boundary conditions. We consider the generalized version of the Hilfer fractional diferential equation called as (k , φ) -Hilfer fractional differential equation. We provide some significant outcomes about (k , φ) -Hilfer fractional Langevin differential equation that requires deriving equivalent fractional integral equation to (k , φ) -Hilfer Langevin fractional differential equation. The existence result is established using the Krasnoselskii's fixed-point theorem, while the uniqueness is addressed with the help of Banach contraction principle. Additionally, we investigate the different forms of Ulam stability for the solution of the mentioned problem, under specific conditions. To validate our main outcomes, we present a detailed example at the end of the manuscript. [ABSTRACT FROM AUTHOR]

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]

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 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]