Volume 20, Issue 6

We study the Cauchy problem for the semilinear fractional heat equation u t = △ α /2 u + f ( u ) with non-negative initial value u 0 ∈ L q (ℝ n ) and locally Lipschitz, non-decreasing, non-negative source term f . For f satisfying the Osgood-type condition ∫ 1∞ dsf(s)=∞ $\begin {array}{} \displaystyle\int_{1}^{\infty}\frac{ds}{f(s)}=\infty \end {array}$ , we show that there exist initial conditions such that the equation has no local solution in Lloc1(Rn) $\begin {array}{} L^{1}_{loc}(\mathbb{R}^{n}) \end {array}$ .

In this paper, we study a class of fractional semilinear integro-differential equations of order β ∈ (1,2] with nonlocal conditions. By using the solution operator, measure of noncompactness and some fixed point theorems, we obtain the existence of local and global mild solutions for the problem. The results presented in this paper improve and generalize many classical results. An example about fractional partial differential equations is given to show the application of our theory.

Fractional differential equation (FDE) provides an accurate description of transport processes that exhibit anomalous diffusion but introduces new mathematical difficulties that have not been encountered in the context of integer-order differential equation. For example, the wellposedness of the Dirichlet boundary-value problem of one-dimensional variable-coefficient FDE is not fully resolved yet. In addition, Neumann boundary-value problem of FDE poses significant challenges, partly due to the fact that different forms of FDE and different types of Neumann boundary condition have been proposed in the literature depending on different applications. We conduct preliminary mathematical analysis of the wellposedness of different Neumann boundary-value problems of the FDEs. We prove that five out of the nine combinations of three different forms of FDEs that are closed by three types of Neumann boundary conditions are well posed and the remaining four do not admit a solution. In particular, for each form of the FDE there is at least one type of Neumann boundary condition such that the corresponding boundary-value problem is well posed, but there is also at least one type of Neumann boundary condition such that the corresponding boundary-value problem is ill posed. This fully demonstrates the subtlety of the study of FDE, and, in particular, the crucial mathematical modeling question: which combination of FDE and fractional Neumann boundary condition, rather than which form of FDE or fractional Neumann boundary condition, should be used and studied in applications.

In this paper, from the classical short memory principle under Grünwald-Letnikov definition, several novel short memory principles are presented and investigated. On one hand, the classical principle is extended to Riemann-Liouville and Caputo cases. On the other hand, a special kind of principles are formulated by introducing a discrete argument instead of the continuous time, resulting in principles with fixed memory length or fixed memory step. Apart from these, several interesting properties of the proposed principles are revealed profoundly.

The main goal of this article is to show a new method to solve some Fractional Order Integral Equations (FOIE), more precisely the ones which are linear, have constant coefficients and all the integration orders involved are rational. The method essentially turns a FOIE into an Ordinary Integral Equation (OIE) by applying a suitable fractional integral operator. After discussing the state of the art, we present the idea of our construction in a particular case (Abel integral equation). After that, we propose our method in a general case, showing that it does work when dealing with a family of “additive” operators over a vector space. Later, we show that our construction is always possible when dealing with any FOIE under the above-mentioned hypotheses. Furthermore, it is shown that our construction is “optimal” in the sense that the OIE that we obtain has the least possible order.

We prove existence of mild solutions to a class of semilinear fractional differential inclusions with non local conditions in a reflexive Banach space. We are able to avoid any kind of compactness assumptions both on the nonlinear term and on the semigroup generated by the linear part. We apply the obtained theoretical results to two diffusion models described by parabolic partial integro-differential inclusions.

In this paper, we apply the classical control theory to a variable order fractional differential system in a bounded domain. The Fractional Optimal Control Problem (FOCP) for variable order differential system is considered. The fractional time derivative is considered in a Caputo sense. We first study by using the Lax-Milgram Theorem, the existence and the uniqueness of the solution of the variable order fractional differential system in a Hilbert space. Then we show that the considered optimal control problem has a unique solution. The performance index of a (FOCP) is considered as a function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE) with variable order. The time horizon is fixed. Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of right fractional Caputo derivative, we obtain an optimality system for the optimal control. Some examples are analyzed in details.

In this article, we prove the existence and uniqueness of solution for some higher-order fractional differential equations with conjugate type integral conditions. The interesting point lies in that the Lipschitz constant is closely associated with the first eigenvalues corresponding to the relevant linear operator. The discussion is based on the Banach contraction map principle and the theory of u 0 -positive linear operator.

In this paper, we establish Lyapunov-type inequalities for a fractional p -Laplacian system in an open bounded subset Ω⊂ℝ N RN under Dirichlet boundary conditions. As an application of the obtained inequalities, we establish some geometric properties of the generalized spectrum associated to the considered system. MSC 2010 : Primary 35R11; Secondary , 35P15,

In this work we study the following singular, fractional critical problem (Pλ)(−Δ)su=λuγ+uqinΩ;u<0inΩ,u=0,inRN∖Ω,$$\begin{array}{*{20}{c}} {({{\text{P}}_\lambda })\left\{ {\begin{array}{*{20}{l}} {{{( - {\Delta})}^s}u = \frac{\lambda }{{{u^\gamma }}} + {u^q}\quad{\text{in}}\; {\Omega};} \\ {u \lt 0\quad {\text{in}} \;\Omega,} \\ {u = 0,\quad{\text{in}} \;{\mathbb{R}^N} \setminus {\Omega},} \end{array}} \right.} \end{array}$$ where Ω ⊂ ℝ N ( N ≥ 3) is a bounded domain with smooth boundary ∂ Ω, N < 2 s , 0 < 1, λ < 0, 0<γ<1<q≤2s∗−1=N+2sN−2s.$ 0 \lt \gamma \lt 1 \lt q \le 2_s^ * - 1 = \frac{{N + 2s}}{{N - 2s}}.$ Here (–Δ) s is the fractional Laplace operator defined as (−Δ)su(x)=−12∫RNux+y+ux−y−2u(x)|y|N+2sdy,for allx∈RN.$$\begin{array}{*{20}{c}} {{{( - {\Delta})}^s}u(x) = - \frac{1}{2}\int_{{\mathbb R}^{N}} {\frac{{u\left( {x + y} \right) + u\left( {x - y} \right) - 2u(x)}}{{|y{|^{N + 2s}}}} dy} ,\:\:{\text{for all}}\;x \in {\mathbb{R}^N}.} \end{array}$$ We use variational methods, in order to show the existence of multiple positive solutions to the problem ( P λ ) for different values of λ.

The paper gives the construction of the half-plane analog of the part of the factorization theory of M.M. Djrbashian – V.S. Zakaryan, where Djrbashian’s generalized fractional integral was used to establish the descriptive representations and boundary properties of meromorphic in the unit disc functions of the classes N { ω } contained in the Nevanlinna class N of functions of bounded type. Some results of nearly the same type are obtained for several weighted classes of meromorphic in the upper half-plane functions with bounded Tsuji characteristics by application of the Laplace transform along with an Hadamard–Liouville type generalized integro-differential operator with an unbounded integration contour, which becomes the Liouville integro-differentiation in a particular case.

We study weighted generalized Hardy and fractional operators acting from generalized Morrey spaces L p , φ ,(ℝ n ) into Orlicz-Morrey spaces L Φ, φ ,(ℝ n ). We deal with radial quasi-monotone weights and assumptions imposed on weights are given in terms of Zigmund-type integral conditions. We find conditions on φ ,Φ, the weight w and the kernel of the fractional operator, which insures such a boundedness. We prove some pointwise estimates for weighted generalized fractional operators via generalized Hardy operators, which allow to obtain the weighted boundedness for fractional operators from those for Hardy operators. We provide also some easy to check numerical inequalities to verify the obtained conditions.

In some of the statements of the author’s paper “The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus” ( Fract. Calc. Appl. Anal . 15 , No 2 (2012), pp. 304–313), a factor α was missing. This note provides the correct formulations.