Continuous Dependence Of Solutions Research Articles

On stability of nonlocal neutral stochastic integro differential equations with random impulses and Poisson jumps

This article aims to examine the existence and Hyers-Ulam stability of non-local random impulsive neutral stochastic integrodifferential delayed equations with Poisson jumps. Initially, we prove the existence of mild solutions to the equations by using the Banach fixed point theorem. Then, we investigate stability via the continuous dependence of solutions on the initial value. Next, we study the Hyers-Ulam stability results under the Lipschitz condition on a bounded and closed interval. Finally, we give an illustrative example of our main result.

Well-posedness of the discrete collision-induced breakage equation with unbounded fragment distribution

A discrete version of the nonlinear collision-induced breakage equation is studied. Existence of solutions is investigated for a broad class of unbounded collision kernels and daughter distribution functions, the collision kernel ai,j satisfying ai,j≤Aij for some A>0. More precisely, it is proved that, given suitable initial conditions, there exists at least one mass-conserving solution for all times. A result on the uniqueness of solutions is also demonstrated under reasonably general conditions. Furthermore, the propagation of moments, differentiability, and the continuous dependence of solutions are established, along with some invariance properties and the large time behaviour of solutions.

On the stochastic Euler-Poincaré equations driven by pseudo-differential/multiplicative noise

The stochastic Euler-Poincaré equations with pseudo-differential/multiplicative noise are considered in this work. We first establish two new cancellation properties on pseudo-differential operators, which considerably extend the previous results for transport type noise only involving gradient operator. Then, we obtain results on local solution, blow-up criterion, and global existence. The interplay between stability on exiting times and continuous dependence of solution on initial data is also studied for the multiplicative noise case.

On the single partial Caputo derivatives for functions of two variables

In this work we propose definitions (distinguishable from the standard ones) of single partial derivatives in a Caputo sense of functions of two variables on the rectangle P=[0,a]times [0,b]. Next, we give an integral representation of functions possessing such derivatives. Finally, we apply these derivatives to the study of the existence and uniqueness of solutions and the continuous dependence of solutions on controls to a fractional counterpart of a nonlinear continuous Roesser type model.

The two‐dimensional stationary Navier–Stokes equations in toroidal Besov spaces

AbstractWe consider the stationary Navier–Stokes equations in the two‐dimensional torus . For any , we show the existence, uniqueness, and continuous dependence of solutions in homogeneous toroidal Besov spaces for given small external forces in when . These spaces become closer to the scaling invariant ones if the difference ε becomes smaller. This well‐posedness is proved by using the embedding property and the para‐product estimate in homogeneous Besov spaces. In addition, for the case , we can show the ill‐posedness, even in the scaling invariant spaces. Actually in such cases of p and q, we can prove that ill‐posedness by showing the discontinuity of a certain solution map from to .

Qualitative results for a relativistic wave equation with multiplicative noise and damping terms

<abstract><p>Wave equations describing a wide variety of wave phenomena are commonly seen in mathematical physics. The inclusion of a noise term in a deterministic wave equation allows neglected degrees of freedom or fluctuations of external fields describing the environment to be considered in the equation. Moreover, adding a noise term to the deterministic equation reveals remarkable new features in the qualitative behavior of the solution. For example, noise can lead to singularities in some equations and prevent singularities in others. Taking into account the effects of the fluctuations along with a space-time white noise, we consider a relativistic wave equation with weak and strong damping terms and investigate the effect of multiplicative noise on the behavior of solutions. The existence of local and global solutions is provided, and some qualitative properties of solutions, such as continuous dependence of solutions on initial data, and blow up of solutions, are given. Moreover, an upper bound is provided for the blow up time.</p></abstract>

On mixed problem in thermoelasticity of type III for Cosserat media

In our present study, we approach a linear theory for the thermoelasticity of type III for Cosserat media. At the beginning we introduce the equations and conditions, specific for a mixed problem in this context, namely the motion and energy equations, the initial condition and boundary relations. After that, we establish two results regarding the solution unique to the problem and two results on the continuous dependence of solutions, for the same mixed problem. Both problems that we address in our paper (uniqueness and continuous dependence) are not based on the material symmetries of the medium. Moreover, our results concern the theory of type III thermoelasticity for Cosserat media in their most general form, namely anisotropic.

A two-dimensional stochastic fractional non-local diffusion lattice model with delays

The well-posedness, regularity and general stability of solutions to a two-dimensional stochastic non-local delay diffusion lattice system with a time Caputo fractional operator of order [Formula: see text] are investigated in [Formula: see text] spaces for [Formula: see text]. First, the global existence and uniqueness of solutions are established by using a temporally weighted norm, the Burkholder–Davis–Gundy inequality and the Banach fixed point theorem. Then the continuous dependence of solutions on initial values is established in the sense of [Formula: see text]th moment. In particular, the [Formula: see text]th moment Hölder regularities in time and [Formula: see text]th moment general stability, including polynomial and logarithmic stability of solutions, are obtained.

Existence and uniqueness of the mild solution of an abstract nonlocal semilinear fractional integrodifferential equation

The purpose of this paper is to study the existence and uniqueness of mild solutions to a semilinear Cauchy problem for an abstract nonlocal fractional integrodifferential equation, which has a distinctive nonlinear term. Two sufficient conditions on the existence of mild solutions will be displayed. Continuous dependence of solutions on initial conditions and local [Formula: see text]-approximate mild solutions will be discussed. An example will be given to elucidate the main results. The results obtained are based upon the method of semigroups, contraction mapping principle, and Krasnoselskii’s fixed point theorem.

On the Bounds of Lyapunov Exponents for Fractional Differential Systems with an Exponential Kernel

This paper focuses on the bounds of the Lyapunov exponents for fractional differential systems, where the fractional derivatives are Riemann–Liouville and Caputo fractional derivatives with the exponential kernel. First, the essential properties of fractional integral and derivatives with the exponential kernel are given. Then the continuous dependence of solutions on the initial value problems of some particular parameters is studied. On these bases, the bounds of Lyapunov exponents are estimated. Finally, the theoretical results are illustrated by numerical simulations.

Permanence of equilibrium points in the basin of attraction and existence of periodic solutions for autonomous measure differential equations and dynamic equations on time scales via generalized ODEs

It is well-known that generalized ODEs encompass several types of differential equations as, for instance, functional differential equations, measure differential equations, dynamic equations on time scales, impulsive differential equations and any combinations among them, not to mention integrals equations, among others. The aim of this paper is to establish a theory of autonomous equations in the setting of generalized ODEs. Thus, we introduce the notion of autonomous generalized ODEs as well as new classes of right-hand sides for nonautonomous generalized ODEs. Amongst the main results, we prove that one of these new classes coincide with the original class of right-hand sides introduced by Kurzweil in 1957. We also prove that autonomous generalized ODEs do not enlarge the class of autonomous ODEs with uniformly continuous right-hand sides. Motivated by this fact, we then enlarge the class of autonomous generalized ODEs so that discontinuities can be taken into account. We then introduce a more general class of autonomous generalized ODEs, in whose integral form, a Stieltjes-type integral appears. A correspondence between these equations and autonomous measure differential equations is established and several results are obtained. We mention local existence and uniqueness of solutions, continuous dependence of solutions on initial values, existence of periodic solutions and permanence of asymptotically stable equilibrium point in the basin of attraction. All these results are, then, specified not only for autonomous generalized ODEs, but also for autonomous measure differential equations and dynamic equations on time scales.

On the Cauchy Problem for the Biharmonic Equation

The work is devoted to the study of continuation and stability estimation of the solution of the Cauchy problem for the biharmonic equation in the domain G from its known values on the smooth part of the boundary @G. The problem under consideration belongs to the problems of mathematical physics in which there is no continuous dependence of solutions on the initial data. In this work, using the Carleman function, not only the biharmonic function itself, but also its derivatives are restored from the Cauchy data on a part of the boundary of the region. The stability estimates for the solution of the Cauchy problem in the classical sense are obtained

Solvability and stability of stochastic singular difference equations with constant coefficient matrices of index-ν

In this paper, we shall deal with stochastic singular difference equations (SSDEs) with constant coefficient matrices and nonlinear stochastic perturbations. The solvability and stability of SSDEs are difficult to study because of the singularity of the leading coefficient matrix. An index-ν concept is derived and formulas of solution are established for these equations. The continuous dependence of solution on initial condition is also considered. Finally, the stability of SSDEs is studied by using the method of Lyapunov functions. Some examples are given to illustrate the results.

Optimal harvesting in a unidirectional consumer–resource mutualisms system with size structure in the consumer

This paper considers the optimal harvesting problem for a size-structured model of unidirectional consumer–resource mutualisms in which the consumer species has both positive and negative effects on the resource species, while the resource has only a positive effect on the consumer. First, we show the existence of a unique nonnegative solution of the system and give the continuous dependence of solutions on the control variable. Next, the adjoint system is derived, which is necessary for optimality and the existence of a unique optimal policy. Then necessary conditions for optimality are established via the normal cone and adjoint system. Moreover, the existence of a unique optimal strategy is proved via Ekeland’s variational principle and fixed-point reasoning in convex analysis. Finally, we use numerical simulations to verify the main results and find other dynamic properties of the system.

Existence and uniqueness of mild solutions for mixed Caputo and Riemann-Liouville semilinear fractional integrodifferential equations with nonlocal conditions

The purpose of this paper is to investigate the existence and uniqueness of the mild solution to a class of semilinear fractional integrodifferential equations with state-dependent nonlocal fractional conditions. Our problem includes both Caputo and Riemann-Liouville fractional derivatives. Continuous dependence of solutions on initial conditions and $\epsilon$-approximate mild solutions of the considered problem will be discussed.

On a boundary value problem of arbitrary orders differential inclusion with nonlocal, integral and infinite points boundary conditions

<abstract><p>In this work, we are concerned with a boundary value problem of fractional orders differential inclusion with nonlocal, integral and infinite points boundary conditions. We prove some existence results for that nonlocal boundary value problem. Next, the existence of maximal and minimal solutions is proved. Finally, the sufficient condition for the uniqueness and continuous dependence of solution are studied.</p></abstract>

Existence and Uniqueness of the Mild Solution of an Abstract Semilinear Fractional Differential Equation with State Dependent Nonlocal Condition

The purpose of this paper is to investigate the existence and uniqueness of mild solutions to a semilinear Cauchy problem for an abstract fractional differential equation with state dependent nonlocal condition. Continuous dependence of solutions on initial conditions and local ????-approximate mild solution of the considered problem will be discussed.

On a nonlocal implicit problem under Atangana\u2013Baleanu\u2013Caputo fractional derivative

In this paper, we study a class of initial value problems for a nonlinear implicit fractional differential equation with nonlocal conditions involving the Atangana–Baleanu–Caputo fractional derivative. The applied fractional operator is based on a nonsingular and nonlocal kernel. Then we derive a formula for the solution through the equivalent fractional functional integral equations to the proposed problem. The existence and uniqueness are obtained by means of Schauder’s and Banach’s fixed point theorems. Moreover, two types of the continuous dependence of solutions to such equations are discussed. Finally, the paper includes two examples to substantiate the validity of the main results.

On global existence and blowup of solutions of Stochastic Keller–Segel type equation

In this paper we consider a stochastic Keller-Segel type equation, perturbed with random noise. We establish that for special types of random pertubations (i.e. in a divergence form), the equation has a global weak solution for small initial data. Furthermore, if the noise is not in a divergence form, we show that the solution has a finite time blowup (with nonzero probability) for any nonzero initial data. The results on the continuous dependence of solutions on the small random perturbations, alongside with the existence of local strong solutions, are also derived in this work.

Solvability of nonlinear fractional integro-differential equation with nonlocal condition

PurposeThis study describes the applicability of the a priori estimate method on a nonlocal nonlinear fractional differential equation for which the weak solution's existence and uniqueness are proved. The authors divide the proof into two sections for the linear associated problem; the authors derive the a priori bound and demonstrate the operator range density that is generated. The authors solve the nonlinear problem by introducing an iterative process depending on the preceding results.Design/methodology/approachThe functional analysis method is the a priori estimate method or energy inequality method.FindingsThe results show the efficiency of a priori estimate method in the case of time-fractional order differential equations with nonlocal conditions. Our results also illustrate the existence and uniqueness of the continuous dependence of solutions on fractional order differential equations with nonlocal conditions.Research limitations/implicationsThe authors’ work can be considered a contribution to the development of the functional analysis method that is used to prove well-positioned problems with fractional order.Originality/valueThe authors confirm that this work is original and has not been published elsewhere, nor is it currently under consideration for publication elsewhere.