non-Lipschitz Conditions Research Articles

An Improved Three-Term Conjugate Gradient Algorithm for Constrained Nonlinear Equations under Non-Lipschitz Conditions and Its Applications

This paper proposes an improved three-term conjugate gradient algorithm designed to solve nonlinear equations with convex constraints. The key features of the proposed algorithm are as follows: (i) It only requires that nonlinear equations have continuous and monotone properties; (ii) The designed search direction inherently ensures sufficient descent and trust-region properties, eliminating the need for line search formulas; (iii) Global convergence is established without the necessity of the Lipschitz continuity condition. Benchmark problem numerical results illustrate the proposed algorithm’s effectiveness and competitiveness relative to other three-term algorithms. Additionally, the algorithm is extended to effectively address the image denoising problem.

Stability analysis of damped fractional stochastic differential systems with Poisson jumps: an successive approximation approach

This paper aims to investigate a new class of fractional stochastic differential systems under the influence of damping and Poisson jumps. First, the existence and uniqueness of mild solutions for the proposed system are investigated under the non-Lipschitz conditions. The results are formulated and proved by using the ( β , δ ) -regularised family, Grönwall's inequality, and successive approximation technique, which is different from the fixed point approach. Moreover, novel stability criteria for the considered system are obtained by utilising the corollary of the Bihari inequality. Finally, the correctness of the obtained results is verified by example.

The averaging principle of Hilfer fractional stochastic pantograph equations with non-Lipschitz conditions

This paper is devoted to presenting an averaging principle for Hilfer fractional stochastic differential pantograph equations (HFSDPEs). The probability of the solutions to averaged stochastic systems in the means square sence can be used to approximate the solutions to HFSDPEs under appropriate non-Lipschitz conditions. Furthermore, certain previous results have been significantly generalised by our results. Finally, an example is given to demonstrate the feasibility of the results.

Exponential Stability of Impulsive Stochastic Neutral Neural Networks with Lévy Noise Under Non-Lipschitz Conditions

In this paper, the exponential stability issue of stochastic impulsive neutral neural networks driven by Lévy noise is explored. By resorting to the Lyapunov-Krasovskii function that involves neutral time-delay components, the properties of the Lévy process, as well as various inequality approaches, some sufficient exponential stability criteria in non-Lipschitz cases are obtained. Besides, the achieved results depend on the time-delay, noise intensity, and impulse factor. At the end of the paper, two numerical examples with simulations are presented to demonstrate the effectiveness and feasibility of the addressed results

Approximations of the Euler–Maruyama Method of Stochastic Differential Equations with Regime Switching

This work focuses on a class of regime-switching diffusion processes with both continuous components and discrete components. Under suitable conditions, we adopt the Euler–Maruyama method to deal with the convergence of numerical solutions of the corresponding stochastic differential equations. More precisely, we first show the convergence rates in the Lp-norm of the stochastic differential equations with regime switching under Lipschitz conditions. Then, we also discuss L1 and L2 convergence under non-Lipschitz conditions.

Finite-time stability and uniqueness theorem of solutions of nabla fractional $ (q, h) $-difference equations with non-Lipschitz and nonlinear conditions

<abstract><p>In this paper, the discrete $ (q, h) $-fractional Bihari inequality is generalized. On the grounds of inequality, the finite-time stability and uniqueness theorem of solutions of $ (q, h) $-fractional difference equations with non-Lipschitz and nonlinear conditions is concluded. In addition, the validity of our conclusion is illustrated by a nonlinear example with a non-Lipschitz condition.</p></abstract>

Adaptive exponential synchronization of impulsive coupled neutral stochastic neural networks with Lévy noise and probabilistic delays under non-Lipschitz conditions

<p>In this paper, we investigated the adaptive exponential synchronization problem of impulsive coupled neutral stochastic neural networks with Lévy noise and probabilistic delays under non-Lipschitz conditions. A stochastic variable with a Bernoulli distribution was utilized to transform the information regarding probabilistic delays into a model featuring deterministic time delays and stochastic parameters. In the context of adaptive controllers, exponential synchronization conditions depending on the delay, noise intensity, and impulse factor were derived using Lyapunov-Krasovskii functions, the nature of Lévy noise, and some inequality methods. To provide further support for the proposed approach, two numerical illustrations were presented.</p>

Well-posedness and order preservation for neutral type stochastic differential equations of infinite delay with jumps

<abstract><p>In this work, we are concerned with the order preservation problem for multidimensional neutral type stochastic differential equations of infinite delay with jumps under non-Lipschitz conditions. By using a truncated Euler-Maruyama scheme and adopting an approximation argument, we have developed the well-posedness of solutions for a class of stochastic functional differential equations which allow the length of memory to be infinite, and the coefficients to be non-Lipschitz and even unbounded. Moreover, we have extended some existing conclusions on order preservation for stochastic systems to a more general case. A pair of examples have been constructed to demonstrate that the order preservation need not hold whenever the diffusion term contains a delay term, although the jump-diffusion coefficient could contain a delay term.</p></abstract>

EULER MARUYAMA APPROXIMATIONS FOR A GENERAL CLASS OF STOCHASTIC FRACTIONAL INTEGRODIFFERENTIAL EQUATIONS

In this paper, we study a class of stochastic fractional integro differential equations under the non-Lipschitz conditions. Thanks to Euler Maruy- ama's numerical scheme, we prove existence and uniqueness of a solution. We obtain also the strong convergence.

Stepanov-like doubly weighted pseudo almost automorphic mild solutions for fractional stochastic neutral functional differential equations

This paper first investigates the equivalence of the space and translation invariance of Stepanov-like doubly weighted pseudo almost automorphic stochastic processes for nonequivalent weight functions; secondly, based on semigroup theory, fractional calculations, and the Krasnoselskii fixed-point theorem, we obtain the existence and uniqueness of Stepanov-like doubly weighted pseudo almost automorphic mild solutions for a class of nonlinear fractional stochastic neutral functional differential equations under non-Lipschitz conditions. These results enrich the complex dynamics of Stepanov-like doubly weighted pseudo almost automorphic stochastic processes.

A moderate deviation principle for stochastic Hamiltonian systems

We prove a moderate deviation principle for stochastic differential equations (SDEs) with non-Lipschitz conditions. As an application of our result, we also study the stochastic Hamiltonian systems.

Existence and Uniqueness of the Solutions to High-Dimensional McKean-Vlasov SDEs Under Non-Lipschitz Conditions

Existence and Uniqueness of the Solutions to High-Dimensional McKean-Vlasov SDEs Under Non-Lipschitz Conditions

Controllability results of neutral Caputo fractional functional differential equations

<abstract><p>In this paper, using the properties of the phase space on infinite delay, generalized Gronwall inequality and fixed point theorems, the existence and controllability results of neutral fractional functional differential equations with multi-term Caputo fractional derivatives were obtained under Lipschitz and non-Lipschitz conditions.</p></abstract>

Sensitivity analysis for a fractional stochastic differential equation with $$S^{p}$$-weighted pseudo almost periodic coefficients and infinite delay

In this paper, we study the parameter sensitivity analysis a class of fractional impulsive stochastic differential equation with \(S^{p}\)-weighted pseudo almost periodic coefficients and infinite delay in Hilbert spaces. Firstly, a more appropriate concept of piecewise \(S^{p}\)-weighted pseudo almost periodic in distribution for stochastic processes of class \(\infty \) is introduced. The existence of piecewise weighted pseudo almost periodic mild solutions in distribution is presented by using \(\alpha \)-order fractional resolvent operator theory, the stochastic analysis techniques and a fixed-point theorem. Secondly, the sensitivity properties of these infinite delay systems under non-Lipschitz conditions is also obtained. Finally, two examples are given for demonstration.

Hilfer fractional stochastic evolution equations on infinite interval

Abstract This paper concerns the global existence of mild solutions for a class of Hilfer fractional stochastic evolution equations on infinite interval (0, +∞), while the existing work were considered on finite interval. The main difficulties here are how to construct suitable Banach spaces, proper operator relations, and then how to formulate the new criteria to guarantee the global existence of mild solutions on the previous constructed spaces under non-Lipschitz conditions. We mainly rely on the generalized Ascoli–Arzela theorem we established, Wright function, Schauder’s fixed point principle, and Kuratowski’s measure of noncompactness to handle with the infinite interval problems. Moreover, we give two examples to demonstrate the feasibility and utility of our results.

Continuity and approximation properties of solutions to fractional neutral stochastic functional differential equations with non-Lipschitz coefficients

ABSTRACT This paper aims to investigate a fractional neutral stochastic functional differential equation (FNSFDE) with non-Lipschitz coefficients. Under the assumptions, we first establish the continuity of the solution in the fractional order of the equation. Furthermore, an Euler-Maruyama (EM) approximation is constructed and then we obtain the strong convergence of the numerical scheme. Specially, if the non-Lipschitz conditions are replaced with the Lipschitz conditions, we shall get a definite convergence rate, which is related to the fractional order of the equation. Finally, we consider the averaging principle for the fractional neutral stochastic equation, which provides us with an easy way to study the properties of the equation.

Averaging principle for stochastic variational inequalities with application to PDEs with nonlinear Neumann conditions

Stochastic variational inequalities have been widely used in various areas. In this paper we establish averaging principles for a separated time-scale system of fully coupled stochastic system characterized by stochastic variational inequalities. Under non-Lipschitz continuous conditions, we show that the classical weak convergence result holds for this type of stochastic systems. Strong convergence is also studied for the cases when the diffusion coefficients of the slow motions do not depend on the fast motion components. As an application, we study the homogenization of generalized backward SDEs and semilinear parabolic variational inequalities with nonlinear Neumann boundary conditions.

Parameter Estimation of Path-Dependent McKean-Vlasov Stochastic Differential Equations

This work concerns a class of path-dependent McKean-Vlasov stochastic differential equations with unknown parameters. First, we prove the existence and uniqueness of these equations under non-Lipschitz conditions. Second, we construct maximum likelihood estimators of these parameters and then discuss their strong consistency. Third, a numerical simulation method for the class of path-dependent McKean-Vlasov stochastic differential equations is offered. Finally, we estimate the errors between solutions of these equations and that of their numerical equations.

Viability for mixed stochastic differential equations driven by fractional Brownian motion and its application

ABSTRACT In this paper, we focus on a class of stochastic differential equations driven by the standard Brownian motion and fractional Brownian motion with Hurst parameter under a weaker conditions than Lipschitz one. In the sense of the pathwise Riemann–Stieltjes integral, we prove the convergence of solutions for the considered equations. By making use of some transformation techniques and approximation means, we obtain some sufficient conditions on the viability for the stochastic systems under investigation. Subsequently, by using some distance functions, we establish some necessary conditions and some sufficient conditions for the viability property with respect to a given non-empty closed set K, but K can be not smooth. As applications of the results, we explore the comparison theorems about the mixed stochastic differential equations driven by fractional Brownian motion under some non-Lipschitz conditions.

Stochastic functional differential equations with infinite delay under non-Lipschitz coefficients: Existence and uniqueness, Markov property, ergodicity, and asymptotic log-Harnack inequality

This paper focuses on a class of stochastic functional differential equations with infinite delay and non-Lipschitz coefficients. Under one-sided super-linear growth and non-Lipschitz conditions, this paper establishes the existence and uniqueness of strong solutions and strong Markov properties of the segment processes. Under additional assumption on non-degeneracy of the diffusion coefficient, exponential ergodicity for the segment process is derived by using asymptotic coupling method. In addition, the asymptotic log-Harnack inequality is established for the associated Markovian semigroup by using coupling and change of measures, which implies the asymptotically strong Feller property. Finally, an example is given to demonstrate these results.