In this paper, we investigate the non-confluence property of a class of stochastic differential equations with Markovian switching driven by fractional Brownian motion with Hurst parameter \(H\in (1/2,1)\). By using the generalized Itô formula and stopping time techniques, we obtain some sufficient conditions ensuring the non-confluence property for the considered equations. Additionally, we present

In this paper, we introduce the concept of Stepanov-like (weighted) pseudo S-asymptotically Bloch type periodic processes in the square mean sense, and establish some basic results on the function space of such processes like completeness, convolution and composition theorems. Under the situation that the functions forcing are Stepanov-like (weighted) pseudo S-asymptotically Bloch type periodic and

This article aims to investigate the semi-classical analog of the general Caputo-type diffusion equation with time-dependent diffusion coefficient associated with the discrete Schrödinger operator, \(\mathcal {H}_{\hbar ,V}:=-\hbar ^{-2}\mathcal {L}_{\hbar }+V\) on the lattice \(\hbar \mathbb {Z}^{n},\) where V is a positive multiplication operator and \(\mathcal {L}_{\hbar }\) is the discrete Laplacian

This paper study a class of time-space fractional reaction-diffusion equations with nonlocal initial conditions and construct an abstract theory in fractional power spaces to discuss the results related S-asymptotically \(\omega \)-periodic mild solutions. When the coefficients are sufficiently small, under the condition that the nonlinear term can grow any number of orders, we discuss the existence

In this paper, the averaging principle for stochastic Caputo fractional differential equations with the nonlinear terms satisfying the non-Lipschitz condition is considered. The work in the article is roughly divided into three parts. Firstly, we establish a generalized Gronwall inequality with singular integral kernel which is a key part in our analysis. Secondly, we discuss the existence and uniqueness

This paper has two primary objectives. The first one is to demonstrate that the solutions of master equation $$\begin{aligned} (\partial _t-\Delta )^s u(x,t) =f(u(x, t)), \,\,(x, t)\in B_1(0)\times \mathbb {R}, \end{aligned}$$ subject to the vanishing exterior condition, are radially symmetric and strictly decreasing with respect to the origin in \(B_1(0)\) for any \(t\in \mathbb {R}\). Another one

The paper investigates fractional differential equations involving the Liouville derivative. Solution to these equations under a boundary condition inside the time interval are derived in explicit form, their uniqueness is established using integral transforms technique.

Bearing in mind an increasing popularity of the fractional calculus the main aim of this paper is to derive several new representation formulae for the cumulative distribution function (cdf) of the McKay \(I_\nu \) Bessel distribution including the Grünwald-Letnikov fractional derivative; also, two connection formulae between cdf of the McKay \(I_\nu \) random variable and the so–called Neumann series

We investigate the fractional stochastic heat equation, driven by a random noise which admits a covariance measure structure with respect to the time variable and has a spatial covariance given by the Riesz kernel. This class of process includes White-colored noise, fractional colored noise and other related processes. We give a sufficient condition for the existence of the mild solution and we establish

The causal shift-invariant convolution is studied from the point of view of inversion. Abel’s algorithm, used in the tautochrone problem, is considered and Sonin’s existence condition is deduced. To generate pairs of functions verifying Sonin’s condition, the class of Mittag-Leffler type functions is used. In particular, functions that are impulse responses of ARMA(N,N) systems serve as a basis. The

The Volterra integral equation associated with autonomous Caputo fractional differential equation (FDE) of order \(\alpha \in (0,1)\) in \({\mathbb {R}}^d\) was shown by the authors [4] to generate a semi-group on the space \({\mathfrak {C}}\) of continuous functions \(f:{\mathbb {R}}^+\rightarrow {\mathbb {R}}^d\) with the topology uniform convergence on compact subsets. It serves as a semi-dynamical

The paper studies the fractional order version of the reinforced membrane problem introduced in [A. Henrot and H. Maillot, 2001]. Existence and uniqueness of the solutions of the corresponding non-local equations has been proven for the relaxed problem. In addition, for the radial symmetric case the existence of the optimal domain has been shown.

Let \(p,q\in [1,\infty )\), s be a nonnegative integer, \(\alpha \in \mathbb {R}\), and \(\mathcal {X}\) be \(\mathbb {R}^n\) or a cube \(Q_0\subsetneqq \mathbb {R}^n\). In this article, the authors introduce the localized special John–Nirenberg–Campanato spaces via congruent cubes, \(jn_{(p,q,s)_{\alpha }}^{\textrm{con}}(\mathcal {X})\), and show that, when \(p\in (1,\infty )\), the predual of \(jn_{(p

We study the fractional Fokker-Planck-Kolmogorov equation with the fractional index \(\alpha \in [1,2)\) and use a vector-valued Calderón-Zygmund theorem to obtain the existence and uniqueness of \(L^p([0,T];{{\mathcal {C}}}_b^{\alpha +\beta }({{\mathbb {R}}}^d))\cap W^{1,p}([0,T];{{\mathcal {C}}}_b^\beta ({{\mathbb {R}}}^d))\) solution under the assumptions that the drift coefficient and nonhomogeneous

A necessary and sufficient condition for the existence or nonexistence of global solutions to the following fractional reaction-diffusion equations $$\begin{aligned} {\left\{ \begin{array}{ll} u_{t}=\Delta _{\alpha } u + \psi (t)f(u),\,\,&{} \text{ in } \mathbb {R}^{N}\times (0,\infty ),\\ u(\cdot ,0)=u_{0}\ge 0,\,\,&{} \text{ in } \mathbb {R}^{N}, \end{array}\right. } \end{aligned}$$ has not been

The maximal \(B_{p,q}^{s}\)-regularity properties of a fractional convolution elliptic equation is studied. Particularly, it is proven that the operator generated by this nonlocal elliptic equation is sectorial in \( B_{p,q}^{s}\) and also is a generator of an analytic semigroup. Moreover, well-posedeness of nonlocal fractional parabolic equation in Besov spaces is obtained. Then by using the \(B_{p

The theory of fractal surfaces, in its basic setting, asserts the existence of a bivariate continuous function defined on a triangular domain. The extant literature on the construction of fractal surfaces over triangular domains use certain assumptions for the construction and deal primarily with the interpolation aspects. Working in the framework of fractal surfaces over triangular domains, this note

This article proposes a predictor-corrector scheme for solving the fractional differential equations \({}_0^C D_t^\alpha y(t) = f(t,y(t)), \alpha >0\) with non-uniform meshes. We reduce the fractional differential equation into the Volterra integral equation. Detailed error analysis and stability analysis are investigated. The convergent order of this method on non-uniform meshes is still 3 though

Let \(({{\mathcal {X}}},d,\mu )\) be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, we first establish several weighted norm estimates for various maximal functions. Then we show the weighted boundedness of the fractional integral \(I_\beta \) associated with admissible functions and its commutators. Similarly to \(I_\beta \), corresponding results for Calderón–Zygmund

In this paper, we establish the existence of standing wave solutions for a class of nonlinear fractional Schrödinger-Poisson system involving nonlinearity with subcritical and critical growth. We suppose that the potential V satisfies either Palais-Smale type condition or there exists a bounded domain \(\varOmega \) such that V has no critical point in \(\partial \varOmega \). To overcome the “lack

Nonlinear fractional equations for Caputo differential operators with two fractional orders are studied. One case is a generalization of the Bagley-Torvik equation, another is of Langevin type. These can be confused as being the same but because fractional derivatives do not commute these are different problems. However it is possible to use some common methodology. Some new regularity results for

We consider a problem which describes the heat diffusion in a complex media with fading memory. The model involves a fractional time derivative of order between zero and one instead of the classical first order derivative. The model takes into account also the effect of a neutral delay. We discuss the existence and uniqueness of a mild solution as well as a classical solution. Then, we prove a Mittag-Leffler

In this paper, we study a class of backward problems for nonlinear fractional super-diffusion equations in Banach spaces. We consider the time fractional derivative in the sense of Caputo type. First, we establish some results for the existence of the mild solutions. Moreover, we obtain regularity results of the first order and fractional derivatives of mild solutions. These conclusions are mainly

In this paper, we prove Hopf’s lemma for the Logarithmic Laplacian. At first, we introduce the strong minimum principle. Then Hopf’s lemma for the Logarithmic Laplacian in the ball is proved. On this basis, Hopf’s lemma of the Logarithmic Laplacian is extended to any open set with the property of the interior ball. Finally, an example is given to explain Hopf’s lemma can be applied to the study of

An initial-boundary value problem for the multi-term time-fractional wave equation on a bounded domain is considered. For the largest and smallest orders of the involved Caputo fractional time-derivatives, \(\alpha \) and \(\alpha _m\), it is assumed \(1<\alpha <2\) and \(\alpha -\alpha _m\le 1\). Subordination principle with respect to the corresponding single-term time-fractional wave equation of

In this paper, we investigate the existence and uniqueness of the S-asymptotically \(\omega \)-periodic mild solutions to a class of multi-term time-fractional measure differential equations with nonlocal conditions in an ordered Banach spaces. Firstly, we look for suitable concept of S-asymptotically \(\omega \)-periodic mild solution to our concern problem, by means of Laplace transform and \((\beta

This paper studies the following coupled k-Hessian system with different order fractional Laplacian operators: $$\begin{aligned} {\left\{ \begin{array}{ll} {S_k}({D^2}w(x))-A(x)(-\varDelta )^{\alpha /2}w(x)=f(z(x)),\\ {S_k}({D^2}z(x))-B(x)(-\varDelta )^{\beta /2}z(x)=g(w(x)). \end{array}\right. } \end{aligned}$$ Firstly, we discuss decay at infinity principle and narrow region principle for the k-Hessian

In this paper, we define the \(p\)-adic mixed Morrey type spaces and study the boundedness of \(p\)-adic fractional integral operators and their commutators on these spaces. More precisely, we first obtain the boundedness of \(p\)-adic fractional integral operators and their commutators on \(p\)-adic mixed central Morrey spaces. Moreover, we further extend these results on \(p\)-adic generalized mixed

Jeffreys equation and its fractional generalizations provide extensions of the classical diffusive laws of Fourier and Fick for heat and particle transport. In this work, a class of multi-term time-fractional generalizations of the classical Jeffreys equation is studied. Restrictions on the parameters are derived, which ensure that the fundamental solution to the one-dimensional Cauchy problem is a

The paper is concerned with the three-dimensional Boussinesq-Coriolis equations with Caputo time-fractional derivatives. Specifically, by striking new balances between the dispersion effects of the Coriolis force and the smoothing effects of the Laplacian dissipation involving with a time-fractional evolution mechanism, we obtain the global existence of mild solutions to Cauchy problem of three-dimensional