Multiple solutions of nonlocal boundary value problems for fractional differential equations on half-line

In this paper, we study the existence of multiple solutions of nonlocal boundary value problems for fractional differential equations with integral boundary conditions on the half-line. Applying the fixed point theory and the upper and lower solutions method, some new results on the existence of at least three nonnegative solutions are obtained. An example is presented to illustrate the application of our main results

A Computational Cognitive Model of User Applying Creativity Technique in Creativity Support Systems

AbstractNumerous creativity techniques have been purposed and applied in creativity support system. Because most creativity techniques are used informally and hardly represented formally in computer, it becomes very difficult to build the computational cognitive model of user applying those techniques. However the model is necessary for creativity support systems to detect or predict the change of user's cognitive state in time and make some adaption to avoid inhibiting creativity of user. In this paper we introduce extension creative idea generation method which has the characteristics of formalization and systematization. The method can be represented by extension rules which provide the precondition to build computational cognitive model of user in creativity support systems. The computational cognitive model of user learning in applying extension creative idea generation method is presented through experiments. The experimental results show how and when the user will develop the application skill of creativity technique and inhibit his creativity

Existence and uniqueness of solution for fractional differential equations with integral boundary conditions

This paper is devoted to the existence and uniqueness results of solutions for fractional differential equations with integral boundary conditions. $$\left\{ \begin{array}{l} ^C\hspace{-0.2em}D^\alpha x(t)+f(t,x(t),x'(t))=0,\quad t\in(0,1),\\ x(0)=\int^1_0 g_0(s,x(s))\mathrm{d}s ,\\ x(1)=\int^1_0 g_1(s,x(s))\mathrm{d}s ,\\ x^{(k)}(0)=0,\,\ k=2,3,\cdots, [\alpha]-1. \end{array} \right.$$ By means of the Banach contraction mapping principle, some new results on the existence and uniqueness are obtained. It is interesting to note that the sufficient conditions for the existence and uniqueness of solutions are dependent on the order $\alpha$

On the boundary value problems of piecewise differential equations with left-right fractional derivatives and delay

In this paper, we study the multi-point boundary value problems for a new kind of piecewise differential equations with left and right fractional derivatives and delay. In this system, the state variables satisfy the different equations in different time intervals, and they interact with each other through positive and negative delay. Some new results on the existence, no-existence and multiplicity for the positive solutions of the boundary value problems are obtained by using Guo–Krasnoselskii’s fixed point theorem and Leggett–Williams fixed point theorem. The results for existence highlight the influence of perturbation parameters. Finally, an example is given out to illustrate our main results

Image Super-Resolution Based on Sparse Coding with Multi-Class Dictionaries

Sparse coding-based single image super-resolution has attracted much interest. In this paper, a super-resolution reconstruction algorithm based on sparse coding with multi-class dictionaries is put forward. We propose a novel method for image patch classification, using the phase congruency information. A sub-dictionary is learned from patches in each category. For a given image patch, the sub-dictionary that belongs to the same category is selected adaptively. Since the given patch has similar pattern with the selected sub-dictionary, it can be better represented. Finally, iterative back-projection is used to enforce global reconstruction constraint. Experiments demonstrate that our approach can produce comparable or even better super-resolution reconstruction results with some existing algorithms, in both subjective visual quality and numerical measures

The existence of positive solutions for high order fractional differential equations with sign changing nonlinearity and parameters

By constructing an auxiliary boundary value problem, the difficulty caused by sign changing nonlinearity terms is overcome by means of the linear superposition principle. Using the Guo-Krasnosel'skii fixed point theorem, the results of the existence of positive solutions for boundary value problems of high order fractional differential equation are obtained in different parameter intervals under a more relaxed condition compared with the existing literature. As an application, we give two examples to illustrate our results

Existence and Uniqueness of the Solutions for Fractional Differential Equations with Nonlinear Boundary Conditions

We study the existence and uniqueness of the solutions for the boundary value problem of fractional differential equations with nonlinear boundary conditions. By using the upper and lower solutions method in reverse order and monotone iterative techniques, we obtain the sufficient conditions of both the existence of the maximal and minimal solutions between an upper solution and a lower solution and the uniqueness of the solutions for the boundary value problem and present the iterative sequence for calculating the approximate analytical solutions of the boundary value problem and the error estimate. An example is also given to illustrate the main results

Accelerating Wireless Federated Learning via Nesterov's Momentum and Distributed Principle Component Analysis

A wireless federated learning system is investigated by allowing a server and workers to exchange uncoded information via orthogonal wireless channels. Since the workers frequently upload local gradients to the server via bandwidth-limited channels, the uplink transmission from the workers to the server becomes a communication bottleneck. Therefore, a one-shot distributed principle component analysis (PCA) is leveraged to reduce the dimension of uploaded gradients such that the communication bottleneck is relieved. A PCA-based wireless federated learning (PCA-WFL) algorithm and its accelerated version (i.e., PCA-AWFL) are proposed based on the low-dimensional gradients and the Nesterov's momentum. For the non-convex loss functions, a finite-time analysis is performed to quantify the impacts of system hyper-parameters on the convergence of the PCA-WFL and PCA-AWFL algorithms. The PCA-AWFL algorithm is theoretically certified to converge faster than the PCA-WFL algorithm. Besides, the convergence rates of PCA-WFL and PCA-AWFL algorithms quantitatively reveal the linear speedup with respect to the number of workers over the vanilla gradient descent algorithm. Numerical results are used to demonstrate the improved convergence rates of the proposed PCA-WFL and PCA-AWFL algorithms over the benchmarks