Semilinear Differential Equations Research Articles

On the periodic boundary value problems for fractional nonautonomous differential equations with non-instantaneous impulses

In this paper, we investigate periodic boundary value problems for Caputo type fractional semilinear nonautonomous differential equations with non-instantaneous impulses. By using semigroup theory combined with the measure of noncompactness and some fixed point theorems, the existence of PC-mild solutions for the equations is established. At the end, an example is presented to illustrate the application of our main results.

A generalized change of variable formula for the Young integral

This article shows an Itô-Wentzell type formula adapted to Young integral. We apply this formula to study differential equations driven by α-Hölder paths with α∈121.We obtain a formula for the composition and the inverse of solution maps associated with a Young differential equation. Also, we prove a Leibniz rule for interchanging derivative and the Young integral, and a substitution formula for Young integrals.We also study the Cauchy problem for first-order semilinear differential equations driven by a Hölder path via the Method of Characteristics. We also prove a version of the structure theorem of characteristics in the Young context. We conclude by showing the explicit solution of a transport equation perturbed by the H-fractional Brownian motion with Hurst parameter 1/2 <H ≤ 1.

Exponential stability and synchronisation of fuzzy Mittag–Leffler discrete-time Cohen–Grossberg neural networks with time delays

Exponential Euler differences for semi-linear differential equations of first order have got rapid development in the past few years and a variety of exponential Euler difference methods have become very significant researching topics. Therefore, exponential Euler differences for fractional-order differential models are novel and significant. In allusion to fuzzy Cohen–Grossberg neural networks of fractional order, this paper establishes a Mittag–Leffler Euler difference model on the basis of the constant variation formula in fractional calculus. In the second place, the existence of a unique global bounded solution and equilibrium point is studied by using the contractive mapping principle. Thirdly, global exponential stability and synchronisation of the derived difference models are investigated by employing some inequality techniques and reductio. At last, some illustrative examples and numerical simulations are employed to demonstrate the main results of this paper.

Generalized Bloch type periodicity and applications to semi-linear differential equations in banach spaces

AbstractIn this paper, we mainly introduce some new notions of generalized Bloch type periodic functions namely pseudo Bloch type periodic functions and weighted pseudo Bloch type periodic functions. A Bloch type periodic function may not be Bloch type periodic under certain small perturbations while it can be quasi Bloch type periodic in sense of generalized Bloch type periodic functions. We firstly show the completeness of spaces of generalized Bloch type periodic functions and establish some further properties such as composition and convolution theorems of such functions. We then apply these results to investigate existence results for generalized Bloch type periodic mild solutions to some semi-linear differential equations in Banach spaces. The obtained results show that for each generalized Bloch type periodic input forcing disturbance, the output mild solutions to reference evolution equations remain generalized Bloch type periodic.

Modelling of the rotational motion of 6-DOF rigid body according to the Bobylev-Steklov conditions

This work demonstrates the 3D motion of 6-DOF rigid body around the main inertial axes relative to a fixed point. The novelty of this work is to examine the influence of the gyrostatic moment vector and the magnetic field on the gyrostatic motion besides the action of the Newtonian field in accordance with the Bobylev-Steklov conditions. This motion is characterized by the possibility of placing an acceptable constraint on the governing system of motion in the form of imposing a large value of the projection of the angular velocity around the axial axis of the body. The importance of this restriction is that the governing system in addition to the first-integrals can be reduced to an appropriate system of two semi-linear differential equations from second-order and one first integral. The new analytic approximate solutions of the governing equations are achieved using the small parameter technique of Poincaré. Euler’s angles are determined to evaluate the motion at any instance. The realized solutions are represented graphically for different values ​​of the gyrostatic moment and the point charge causing the magnetic field to estimate the impact of these values ​​on the rotational motion. Moreover, the phase plane plots which illustrate the stability behavior of the body are portrayed. The significance of this work goes back to its several applications in practical life and in engineering application like satellites, submarines, and aircraft.

Besicovitch Almost Periodic Solutions of Abstract Semi-Linear Differential Equations with Delay

In this paper, first, we give a definition of Besicovitch almost periodic functions by using the Bohr property and the Bochner property, respectively; study some basic properties of Besicovitch almost periodic functions, including composition theorem; and prove the equivalence of the Bohr definition and the Bochner definition. Then, using the contraction fixed point theorem, we study the existence and uniqueness of Besicovitch almost periodic solutions for a class of abstract semi-linear delay differential equations. Even if the equation we consider degenerates into ordinary differential equations, our result is new.

Positive solutions for a fractional configuration of the Riemann‐Liouville semilinear differential equation

The objective of this work is to analyze some criteria of the existence of positive solutions for a fractional configuration of the semilinear differential equation equipped with the Riemann‐Liouville operator. To achieve our aim, we first introduce an operator and transform our main problems into equivalent fixed point problems. After that, based on the fixed point theorem attributed to Krasnoselskii and the nonlinear alternative of Leray‐Schauder in a cone, we prove our main results of existence of solutions to our problems in a well‐defined Banach spaces. With the help of illustrative examples at the end of the study, we validate our theoretical outcomes according to the method implemented in theorems.

Random neutral semilinear differential equations with delay

Random neutral semilinear differential equations with delay

Two combined methods for the global solution of implicit semilinear differential equations with the use of spectral projectors and Taylor expansions

Two combined numerical methods for solving semilinear differential-algebraic equations (DAEs) are obtained and their convergence is proved. The comparative analysis of these methods is carried out and conclusions about the effectiveness of their application in various situations are made. In comparison with other known methods, the obtained methods require weaker restrictions for the nonlinear part of the DAE. Also, the obtained methods enable to compute approximate solutions of the DAEs on any given time interval and, therefore, enable to carry out the numerical analysis of global dynamics of mathematical models described by the DAEs. The examples demonstrating the capabilities of the developed methods are provided. To construct the methods we use the spectral projectors, Taylor expansions and finite differences. Since the used spectral projectors can be easily computed, to apply the methods it is not necessary to carry out additional analytical transformations.

On a study of the representation of solutions of a $ \Psi $-Caputo fractional differential equations with a single delay

&lt;abstract&gt;&lt;p&gt;We give a representation of solutions to linear nonhomogeneous $ \Psi $-fractional delayed differential equations with noncommutative matrices. We newly define $ \Psi $-delay perturbation of Mittag-Leffler type matrix function with two parameters and apply the method of variation of constants to obtain the representation of the solutions. We investigate the existence and uniqueness of solutions for a class of $ \Psi $-fractional delayed semilinear differential equations by using Banach Fixed Point Theorem. Further, we establish the Ulam-Hyers stability result for the analyzed problem. Finally, we provide some examples to illustrate the applicability of our results.&lt;/p&gt;&lt;/abstract&gt;

Weighted pseudo asymptotically antiperiodic sequential solutions to semilinear difference equations

In this paper, we firstly introduce the notion of weighted pseudo S-asymptotically ω-antiperiodic sequence and prove some fundamental properties of such sequences. Then we investigate some existence results for weighted pseudo S-asymptotically ω-antiperiodic solutions to a semilinear difference equation of convolution type. We establish some existence and uniqueness of weighted pseudo S-asymptotically ω-antiperiodic sequential solutions under the global Lipschitz growth condition on the second variable of the force term with different Lipschitz coefficients such as a constant, a summable function and a qth summable function, respectively. Particularly with an appropriate selection of a weight, one of our results shows that the strict contraction on the norm of Lipschitz coefficients is not necessarily required for the existence and uniqueness of weighted pseudo S-asymptotically ω-antiperiodic sequential solutions. We also prove some existence results for weighted pseudo S-asymptotically ω-antiperiodic sequential solutions under a local Lipschitz or a non-Lipschitz growth condition on the second variable of the force term, respectively.

Pseudo Almost Automorphic Solutions for Stochastic Differential Equations Driven by Lévy Noise and Its Optimal Control

As we know, the periodic functions are symmetric within a cycle time, and it is meaningful to generalize the periodicity into more general cases, such as almost periodicity or almost automorphy. In this work, we introduce the concept of Poisson Sγ2-pseudo almost automorphy (or Poisson generalized Stepanov-like pseudo almost automorphy) for stochastic processes, which are almost-symmetric within a suitable period, and establish some useful properties of such stochastic processes, including the composition theorems. In addition, we apply a Krasnoselskii–Schaefer type fixed point theorem to obtain the existence of pseudo almost automorphic solutions in distribution for some semilinear stochastic differential equations driven by Lévy noise under Sγ2-pseudo almost automorphic coefficients. In addition, then we establish optimal control results on the bounded interval. Finally, an example is provided to illustrate the theoretical results obtained in this paper.

Time-Optimal Control for Semilinear Stochastic Functional Differential Equations with Delays

The purpose of this paper is to find the time-optimal control to a target set for semilinear stochastic functional differential equations involving time delays or memories under general conditions on a target set and nonlinear terms even though the equations contain unbounded principal operators. Our research approach is to construct a fundamental solution for corresponding linear systems and establish variations of a constant formula of solutions for given stochastic equations. The existence result of time-optimal controls for one point target set governed by the given semilinear stochastic equation is also established.

Stepanov pseudo‐almost periodic functions and applications

In this work, we present basic results and applications of Stepanov pseudo‐almost periodic functions with measure. Using only the continuity assumption, we prove a new composition result of μ‐pseudo‐almost periodic functions in Stepanov sense. Moreover, we present different applications to semilinear differential equations and inclusions with weak regular forcing terms in Banach spaces. We prove the existence and uniqueness of μ‐pseudo‐almost periodic solutions (in the strong sense) to a class of semilinear fractional inclusions and semilinear evolution equations respectively, provided that the nonlinear forcing terms are only Stepanov μ‐pseudo‐almost periodic in the first variable and not a uniformly strict contraction with respect to the second argument. Our results are obtained using the Meir–Keeler principle and the Banach fixed point principle respectively. Some examples of fractional and nonautonomous partial differential equations illustrating our theoretical results are also presented.

An Analysis on the Positive Solutions for a Fractional Configuration of the Caputo Multiterm Semilinear Differential Equation

In this paper, we consider a multiterm semilinear fractional boundary value problem involving Caputo fractional derivatives and investigate the existence of positive solutions by terms of different given conditions. To do this, we first study the properties of Green’s function, and then by defining two lower and upper control functions and using the wellknown Schauder’s fixed-point theorem, we obtain the desired existence criteria. At the end of the paper, we provide a numerical example based on the given boundary value problem and obtain its upper and lower solutions, and finally, we compare these positive solutions with exact solution graphically.

A new numerical method for solving semilinear fractional differential equation

A new numerical method for solving semilinear fractional differential equation

Adaptive Euler methods for stochastic systems with non-globally Lipschitz coefficients

We present strongly convergent explicit and semi-implicit adaptive numerical schemes for systems of semi-linear stochastic differential equations (SDEs) where both the drift and diffusion are not globally Lipschitz continuous. Numerical instability may arise either from the stiffness of the linear operator or from the perturbation of the nonlinear drift under discretization, or both. Typical applications arise from the space discretization of an SPDE, stochastic volatility models in finance, or certain ecological models. Under conditions that include montonicity, we prove that a timestepping strategy which adapts the stepsize based on the drift alone is sufficient to control growth and to obtain strong convergence with polynomial order. The order of strong convergence of our scheme is (1 − ε)/2, for ε ∈ (0,1), where ε becomes arbitrarily small as the number of finite moments available for solutions of the SDE increases. Numerically, we compare the adaptive semi-implicit method to a fully drift-implicit method and to three other explicit methods. Our numerical results show that overall the adaptive semi-implicit method is robust, efficient, and well suited as a general purpose solver.

Mild Solution and Approximate Controllability of Retarded Semilinear Systems with Control Delays and Nonlocal Conditions

This work studies the approximate controllability of a class of first-order retarded semilinear differential equations with delays in control and nonlocal conditions. First we deduce the existence of mild solutions using fixed point approach. For this, the nonlinear function is supposed to be locally Lipschitz, which is a weaker condition than Lipschitz continuity. Controllability results of the system are shown by using the approximate and iterative technique. The results are illustrated by providing an example.

Asymptotic stability of solutions for some classes of impulsive differential equations with distributed delay

In this paper we show the asymptotic stability of the solutions of some differential equations with delay and subject to impulses. After proving the existence of mild solutions on the half-line, we give a Gronwall–Bellman-type theorem. These results are prodromes of the theorem on the asymptotic stability of the mild solutions to a semilinear differential equation with functional delay and impulses in Banach spaces and of its application to a parametric differential equation driving a population dynamics model.

Controllability of Semilinear Multi-Valued Differential Inclusions with Non-Instantaneous Impulses of Order α ∈ (1,2) without Compactness

Herein, we investigated the controllability of a semilinear multi-valued differential equation with non-instantaneous impulses of order α∈(1,2), where the linear part is a strongly continuous cosine family without compactness. We did not assume any compactness conditions on either the semi-group, the multi-valued function, or the inverse of the controllability operator, which is different from the previous literature.