Continuous Mild Solutions Research Articles

Approximate Controllability of Delayed Fractional Stochastic Differential Systems with Mixed Noise and Impulsive Effects

We herein report a new class of impulsive fractional stochastic differential systems driven by mixed fractional Brownian motions with infinite delay and Hurst parameter H^∈(1/2,1). Using fixed point techniques, a q-resolvent family, and fractional calculus, we discuss the existence of a piecewise continuous mild solution for the proposed system. Moreover, under appropriate conditions, we investigate the approximate controllability of the considered system. Finally, the main results are demonstrated with an illustrative example.

Optimal controls for fractional stochastic differential systems driven by Rosenblatt process with impulses

AbstractThe objective of this article is to consider a new class of fractional stochastic differential systems driven by the Rosenblatt process with impulses. We used fractional calculus, stochastic analysis, and Krasnoselskii's fixed point theorem to study the existence of piecewise continuous mild solutions for the proposed system. Further, we discussed the existence of optimal controls for the considered system. Our main results are well supported by an illustrative example.

Existence of mild solutions to semilinear fractional differential inclusion with deviated advanced nonlocal conditions

This paper is devoted to study the existence of at least one continuous mild solution to semilinear fractional differential inclusion with deviated-advanced nonlocal conditions. We develop the results obtained, by exchanging the deviated-advanced nonlocal condition to an integral form. Our results will be accomplished by using the nonlinear Leray-Schauder’s alternative fixed point theorem. The main results are well illustrated with the aid of an example.

Non-autonomous Evolution Equations of Parabolic Type with Non-instantaneous Impulses

In this paper, we study the Cauchy problem to a class of non-autonomous evolution equations of parabolic type with non-instantaneous impulses in Banach spaces, where the operators in linear part (possibly unbounded) depend on time t and generate an evolution family. New existence result of piecewise continuous mild solutions is established under more weaker conditions. At last, as a sample of application, the abstract result is applied to a class of non-autonomous partial differential equation of parabolic type with non-instantaneous impulses. The result obtained in this paper is a supplement to the existing literature and essentially extends some existing results in this area.

Non-autonomous parabolic evolution equations with non-instantaneous impulses governed by noncompact evolution families

In this paper, we study the initial value problem to a class of non-autonomous parabolic evolution equations with non-instantaneous impulses in Banach spaces, where the operators in linear part (possibly unbounded) depend on time t and generate an noncompact evolution family. Some new existence results of piecewise continuous mild solutions are established under more weaker conditions. At last, as a sample of application, these results are applied to a class of non-autonomous partial differential equation of parabolic type with non-instantaneous impulses. The results obtained in this paper are a supplement to the existing literature and essentially extend some existing results in this area.

Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions

We prove the immediate appearance of an exponential lower bound, uniform in time and space, for continuous mild solutions to the full Boltzmann equation in a $C^2$ convex bounded domain with the physical Maxwellian diffusion boundary conditions, under the sole assumption of regularity of the solution. We investigate a wide range of collision kernels, with and without Grad's angular cutoff assumption. In particular, the lower bound is proven to be Maxwellian in the case of cutoff collision kernels. Moreover, these results are entirely constructive if the initial distribution contains no vacuum, with explicit constants depending only on the \textit{a priori} bounds on the solution.

Periodic BVPs for fractional order impulsive evolution equations

In this paper, we study periodic BVPs for fractional order impulsive evolution equations. The existence and boundedness of piecewise continuous mild solutions and design parameter drift for periodic motion of linear problems are presented. Furthermore, existence results of piecewise continuous mild solutions for semilinear impulsive periodic problems are showed. Finally, an example is given to illustrate the results.

Existence of bounded uniformly continuous mild solutions on [formula omitted] of evolution equations and their asymptotic behaviour

We prove that u′=Au+ϕ has on R a mild solution uϕ∈BUC(R,X) (that is, bounded and uniformly continuous), where A is the generator of a C0-semigroup on the Banach space X with resolvent satisfying ‖R(it,A)‖=O(|t|−θ), |t|→∞, for some θ>12, ϕ∈L∞(R,X) and isp(ϕ)∩σ(A)=0̸ (sp=Beurlingspectrum). As a consequence it is shown that if F is the space of almost periodic, almost automorphic, bounded Levitan almost periodic or one of certain classes of recurrent functions and ϕ as above is such that only Mhϕ≔(1/h)∫0hϕ(⋅+s)ds∈F for each h>0, then uϕ∈F∩BUC(R,X). These results seem new and strengthen several recent theorems.

Relaxed Controls for Nonlinear Fractional Impulsive Evolution Equations

In this paper, we study optimal relaxed controls and relaxation of nonlinear fractional impulsive evolution equations. Firstly, existence of piecewise continuous mild solutions for the original fractional impulsive control system is presented. Secondly, fractional impulsive relaxed control system is constructed by using a regular countably additive measure and making the original control system convexified. Thirdly, optimal relaxed controls and relaxation theorems are obtained. Finally, application to initial-boundary value problem of fractional impulsive parabolic control system is considered.

Uniform $L^1$-stability of the relativistic Boltzmann equation near vacuum

We present the uniform $L^1$-stability estimate for the relativistic Boltzmann equation near vacuum. For this, we explicitly construct a relativistic counterpart of the nonlinear functional which is a linear combination of $L^1$-distance and a collision potential. This functional measures the $L^1$-distance between two continuous mild solutions. When the initial data is sufficiently small and decays exponentially fast, we show that the functional satisfies the uniform stability estimate leading to the uniform $L^1$-stability estimate with respect to initial data.

Stability estimates of the Boltzmann equation in a half space

In this paper, we study the large-time behavior and the stability of continuous mild solutions to the Boltzmann equation in a half space. For this, we introduce two nonlinear functionals measuring future binary collisions and L 1 L^1 -distance. Through the time-decay estimates of these functionals and the pointwise estimate of the gain part of the collision operator, we show that continuous mild solutions approach to collision free flows time-asymptotically in L 1 L^1 , and L 1 L^1 -distance at time t t is uniformly bounded by that of corresponding initial data, when initial datum is a small perturbation of the vacuum.

Stability estimates of the Boltzmann equation with quantum effects

We study uniform stability estimates to the Boltzmann equation for quantum particles such as Bose-Einstein particles and Fermi-Dirac particles. When the small amount of particles expands toward the vacuum, we show that continuous mild solutions are L1-stable and also satisfy BV-type estimates using a nonlinear functional approach.

Semilinear Stochastic Hyperbolic Systems in One Dimension

The article is concerned with two types of solutions to the Cauchy problems for some semilinear stochastic hyperbolic systems in one space dimension. In the symmetric hyperbolic case, by means of an energy inequality, the existence of unique local and global solutions in a Sobolev space is proved. In addition, for some random regular hyperbolic systems with additive noise, based on the method of characteristics, it is shown that the system has a unique continuous mild solution. Such a solution may exist locally or globally depending on the smoothness of the random coefficients.

A Massera‐type criterion for almost periodic solutions of higher‐order delay or advanceabstract functional differential equations

Let u be a given bounded uniformly continuous mild solution of a higher‐order abstract functional differential equation of delay or advance type. We give a so‐called Massera‐type criterion for the existence of a mild solution, which is a “spectral component” of u with spectrum similar to the one of the forcing term f. Various spectral criteria for the existence of almost periodic and quasiperiodic mild solutions are given.

A Decomposition Theorem for Bounded Solutions and the Existence of Periodic Solutions of Periodic Differential Equations

We prove a decomposition theorem for bounded uniformly continuous mild solutions to τ-periodic evolution equations of the form dx/dt=A(t)x+f(t) (*) with (in general, unbounded) τ-periodic A(·), τ-periodic f (·), and compact monodromy operator. By this theorem, every bounded uniformly continuous mild solution to (*) is a sum of a τ-periodic solution to (*) and a quasi periodic solution to its homogeneous equation. An analog of this for bounded solutions has been proved for abstract functional differential equations dx/dt=Ax+F(t)xt+f(t) with finite delay, where A generates a compact semigroup. As an immediate consequence, the existence of such a solution implies the existence of a τ-periodic solution to the inhomogeneous equation as well as a formula for its Fourier coefficients. This, even for the classical case of equations, improves considerably the previous results on the subject.

On a spectral criterion for almost periodicity of solutions of periodic evolution equations

This paper is concerned with equations of the form: u 0 = A(t)u + f(t) , where A(t) is (unbounded) periodic linear operator and f is almost periodic. We extend a central result on the spectral criteria for almost periodicity of solutions of evolution equations to some classes of periodic equations which says that if u is a bounded uniformly continuous mild solution and P is the monodromy operator, then their spectra satisfy e ispAP (u) (P ) S 1 , where S 1 is the unit circle. This result is then applied to nd almost periodic solutions to the above-mentioned equations. In particular, parabolic and functional dieren tial equations are considered. Existence conditions for almost periodic and quasi-periodic solutions are discussed.

Harmonic analysis and asymptotic behavior of solutions to the abstract cauchy problem

LetC ub ( $$\mathbb{J}$$ , X) denote the Banach space of all uniformly continuous bounded functions defined on $$\mathbb{J}$$ 2 e {ℝ+, ℝ} with values in a Banach spaceX. Let ℱ be a class fromC ub( $$\mathbb{J}$$ ,X). We introduce a spectrumspℱ(φ) of a functionφ eC ub (ℝ,X) with respect to ℱ. This notion of spectrum enables us to investigate all twice differentiable bounded uniformly continuous solutions on ℝ to the abstract Cauchy problem (*)ω′(t) =Aω(t) +φ(t),φ(0) =x,φ e ℱ, whereA is the generator of aC 0-semigroupT(t) of bounded operators. Ifφ = 0 andσ(A) ∩iℝ is countable, all bounded uniformly continuous mild solutions on ℝ+ to (*) are studied. We prove the bound-edness and uniform continuity of all mild solutions on ℝ+ in the cases (i)T(t) is a uniformly exponentially stableC 0-semigroup andφ eC ub(ℝ,X); (ii)T(t) is a uniformly bounded analyticC 0-semigroup,φ eC ub (ℝ,X) andσ(A) ∩i sp(φ) = O. Under the condition (i) if the restriction ofφ to ℝ+ belongs to ℱ = ℱ(ℝ+,X), then the solutions belong to ℱ. In case (ii) if the restriction ofφ to ℝ+ belongs to ℱ = ℱ(ℝ+,X), andT(t) is almost periodic, then the solutions belong to ℱ. The existence of mild solutions on ℝ to (*) is also discussed.