Volume 21, Issue 6

In this paper, a sensitivity analysis of optimal control problem for a class of systems described by nonlinear fractional evolution inclusions (NFEIs, for short) on Banach spaces is investigated. Firstly, the nonemptiness as well as the compactness of the mild solutions set S ( ζ ) ( ζ being the initial condition) for the NFEIs are obtained, and we also present an extension Filippov’s theorem and whose proof differs from previous work only in some technical details. Finally, the optimal control problems described by NFEIs depending on the initial condition ζ and the parameter η are considered and the sensitivity properties of the optimal control problem are also established.

We prove the existence and finite-time attractivity of solutions to semilinear tempered fractional wave equations with sectorial operator and superlinear nonlinearity. Our analysis is based on the α -resolvent theory, the fixed point theory for condensing maps and the local estimates of solutions. An application to a class of partial differential equations will be given.

This paper discusses a construction of fractional differential geometry of curves (curvature of curve and Frenet-Serret formulas). A tangent vector of plane curve is defined by the Caputo fractional derivative. Under a simplification for the fractional derivative of composite function, a fractional expression of Frenet frame of curve is obtained. Then, the Frenet-Serret formulas and the curvature are derived for the fractional ordered frame. The different property from the ordinary theory of curve is given by the explicit expression of arclength in the fractional-order curvature. The arclength part of the curvature takes a large value around an initial time and converges to zero for a long period of time. This trend of curvature may reflect the memory effect of fractional derivative which is progressively weaken for a long period of time. Indeed, for a circle and a parabola, the curvature decreases over time. These results suggest that the basic property of fractional derivative is included in the fractional-order curvature appropriately.

This paper aims at obtaining a high precision numerical approximation for fractional partial differential equations. This is achieved through appropriate discretizations: firstly we consider the application of shifted Legendre or Chebyshev polynomials to get a spatial discretization, followed by a temporal discretization where we use the Implicit Euler method (although any other temporal integrator could be used). Finally, the use of an extrapolation technique is considered for improving the numerical results. In this way a very accurate solution is obtained. An algorithm is presented, and numerical results are shown to demonstrate the validity of the present technique.

Time optimal control problems governed by Riemann-Liouville fractional differential system are considered in this paper. Firstly, the existence results are obtained by using the theory of semigroup and Schauder’s fixed point. Secondly, the new approach of establishing time minimizing sequences twice is applied to acquire the time optimal pairs without the Lipschitz continuity of nonlinear function. Moreover, the reflexivity of state space is removed with the help of compact method. Finally, an example is given to illustrate the main conclusions. Our work essentially improves and generalizes the corresponding results in the existing literature.

This paper is devoted to the inverse generator problem in the setting of generators of integrated resolvent operator functions. It is shown that if the operator A is the generator of a tempered β -times integrated α -resolvent operator function (( α , β )-ROF) and it is injective, then the inverse operator A −1 is the generator of a tempered ( α , γ )-ROF for all γ > β + 1/2, by means of an explicit representation of the integrated resolvent operator function based in Bessel functions of first kind. Analytic resolvent operator functions are also considered, showing that A −1 is in addition the generator of a tempered ( δ , 0)-ROF for all δ < α . Moreover, the optimal decay rate of ( α , β )-ROFs as t → ∞ is given. These result are applied to fractional Cauchy problem unsolved in the fractional derivative.

By using an abstract critical point result for differentiable and parametric functionals due to B. Ricceri, we establish the existence of infinitely many classical solutions for fractional differential equations subject to boundary value conditions and impulses. More precisely, we determine some intervals of parameters such that the treated problems admit either an unbounded sequence of solutions, provided that the nonlinearity has a suitable oscillatory behaviour at infinity, or a pairwise distinct sequence of solutions that strongly converges to zero if a similar behaviour occurs at zero. No symmetric condition on the nonlinear term is assumed. Two examples are then given.

Variational methods and critical point theorems are used to discuss existence of infinitely many solutions to boundary value problem for fractional order differential equations where Riemann-Liouville fractional derivatives and Caputo fractional derivatives are used. An example is given to illustrate our result.

The paper presents the testing results of a semi-analytic collocation method, using five benchmark problems published in a paper by Xue and Bai in Fract . Calc . Appl . Anal ., Vol. 20 , No 5 (2017), pp. 1305–1312, DOI: 10.1515/fca-2017-0068.

In this paper, we study the blow-up and global existence of solutions to the following time fractional nonlinear diffusion equations 0CDtα u− △ u=0It1− γ (|u|p− 1u),x∈ RN,t> -->0,u(0,x)=u0(x),x∈ RN,$$\begin{array}{} \displaystyle \left\{\begin{array}{l}{_0^CD_t^\alpha u}-\triangle u={_0I_t^{1-\gamma}}(|u|^{p-1}u), \, \, x\in \mathbb{R}^N,\, \, t\gt 0,\\ u(0,x)=u_0(x),\, \, x\in \mathbb{R}^N, \end{array}\right. \end{array}$$ where 0 < α < γ < 1, p > 1, u 0 ∈ C 0 (ℝ N ), 0Itθ $\begin{array}{} {_0I_t^{\theta}} \end{array}$ denotes left Riemann-Liouville fractional integrals of order θ . 0CDtα u=∂ ∂ t0It1− α $\begin{array}{} {_0^CD_t^\alpha u}=\frac{\partial}{\partial t}{_0I_t^{1-\alpha}} \end{array}$ ( u ( t , x ) − u (0, x ))}. Let β = 1 − γ . We prove that if 1 < p < p ∗ = max{1+β α ,1+2(α +β )α N}$\begin{array}{} \max\big\{1+\frac{\beta}{\alpha},1+\frac{2(\alpha+\beta)}{\alpha N}\big\} \end{array}$ , the solutions of (1.1) blows up in a finite time. If N < 2(α +β )β $\begin{array}{} \frac{2(\alpha+\beta)}{\beta} \end{array}$ , p ≥ p ∗ or N ≥ 2(α +β )β $\begin{array}{} \frac{2(\alpha+\beta)}{\beta} \end{array}$ , p > p ∗ , and ∥ u 0 ∥ L q c (ℝ N ) is sufficiently small, where qc=Nα (p− 1)2(α +β )$\begin{array}{} q_c=\frac{N\alpha(p-1)}{2(\alpha+\beta)} \end{array}$ , the solutions of (1.1) exists globally.

The Blaschke-Petkantschin formula is a variant of the polar decomposition of the k -fold Lebesgue measure on ℝ n in terms of the corresponding measures on k -dimensional linear subspaces of ℝ n . We suggest a new elementary proof of this famous formula and discuss its connection with Riesz distributions associated with fractional powers of the Cayley-Laplace operator on matrix spaces. Another application of our proof is the celebrated Drury identity that plays a key role in the study of mapping properties of the Radon-John k -plane transforms. Our proof gives precise meaning to the constants in Drury’s identity and to the class of admissible functions.

The present paper investigates fractal dimension of fractional integral of continuous functions whose fractal dimension is 1 on [0, 1]. For any continuous functions whose Box dimension is 1 on [0, 1], Riemann-Liouville fractional integral of these functions of any positive order has been proved to still be 1-dimensional continuous functions on [0, 1].