Existence of solutions of nonlinear fractional pantograph equations

This article deals with the existence of solutions of nonlinear fractional pantograph equations. Such model can be considered suitable to be applied when the corresponding process occurs through strongly anomalous media. The results are obtained using fractional calculus and fixed point theorems. An example is provided to illustrate the main result obtained in this article.     

Existence of solutions of initial value problems for nonlinear fractional differential equations

In this paper, by using the Schauder fixed point theorem, we study the existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations and obtain some new results.     

Existence of positive solutions of nonlinear fractional differential equations

Existence of positive solutions for the nonlinear fractional differential equation D^su(x)=f(x,u(x)), 0<s<1, has been studied (S. Zhang, J. Math. Anal. Appl. 252 (2000) 804-812), where D^s denotes Riemann-Liouville fractional derivative. In the present work we study existence of positive solutions in case of the nonlinear fractional differential equation:

L(D)u=f(x,u),u(0)=0,0<x<1,     

具有逐项分数阶导数的微分方程边值问题解的存在性

研究了一类具有逐项分数阶导数的微分方程边值问题.对参数的各种取值情况进行了全面的分析,运用Banach压缩映射原理和Schauder不动点定理,得到并证明了边值问题解的存在性定理.最后,给出了两个例子来证明结论有效.     

非线性一阶常微分方程解的存在惟一性

讨论了一类非线性一阶常微分方程边值问题解的存在惟一性.得到了当参数在一定的范围取值时解存在惟一的充分条件,并包含了一些已知结果.主要结果基于Leray-Schauder非线性抉择理论和Banach不动点定理.     

Existence of solutions for fractional differential equation three-point boundary value problems

In this paper, by using some fixed point theorems, the existence of unique solution and the existence of at least one solution for a fractional differential equation three-point boundary value problems are established. Finally, some illustrative examples are presented to demonstrate the validity of the main results.     

Existence of pseudo-almost automorphic solutions for nonlinear differential equations

In this paper, we discuss the existence of pseudo-almost automorphic solutions to linear differential equation which has an exponential trichotomy~ and the results also hold for some nonlinear equations with the form x＇（t） = f（t,x（t）） ＋ λg（t,x（t））, where f,g are pseudo-almost automorphic functions. We prove our main result by the application of Leray-Schauder fixed point theorem.     

Existence of weak solutions for fractional porous medium equations with nonlinear term

We study the following fractional porous medium equations with nonlinear term $\{\begin{array}{cc}{u}_t+{(-\Delta )}^{\sigma /2}\left({\left|u\right|}^{m-1}u\right)+g\left(u\right)=h\text{,}& \text{in}\Omega \times {\mathbb{R}}^{+}\text{,}\\ u(x,t)=0\text{,}& \text{in}\partial \Omega \times {\mathbb{R}}^{+}\text{,}\\ u(x,0)=u_0\text{,}& \text{in}\Omega \text{.}\end{array}$ The authors in de Pablo et al. (2011) and de Pablo et al. (2012) established the existence of weak solutions for the case $g\left(u\right)\equiv 0$. Here, we consider the nonlinear term $g$ is without an upper growth restriction. The nonlinearity of $g$ leads to the invalidity of the Crandall–Liggett theorem, which is the critical method to establish the weak solutions in de Pablo et al. (2011) and de Pablo et al. (2012). In addition, because of $g$ does not have an upper growth restriction, we have to apply the weak compactness theorem in an Orlicz space to prove the existence of weak solutions by using the Implicit Time Discretization method.     

Existence results of positive solutions to boundary value problem for fractional differential equation

In this paper, we consider the existence, multiplicity, and nonexistence of positive solutions to some class of boundary vale problem for fractional differential equation of high order. Our analysis relies on the fixed point index.      

Approximate solutions of nonlinear fractional equations

Fractional exponential that are invariant under fractional derivatives, elementary and special fractional functions are introduced. Approximate solutions to fractional Burgers equation, by using the homotopy perturbation method, are obtained. Furthermore, real integral representations for some H-functions are found that may be very helpful in numerical computations.     

Existence-uniqueness result for a nonlinear n-term fractional equation

We prove the existence-uniqueness of the solution to the nonlinear n-term time-fractional differential equation with constant coefficients in the Banach space C([0,T]),

(1)     

Existence and uniqueness of solutions of sequential nonlinear fractional difference equations with three‐point fractional sum boundary conditions

In this paper, we consider a discrete fractional boundary value problem of the form: where 0 < α,β≤1, 1 < α + β≤2, λ and ρ are constants, γ > 0, , is a continuous function, and E_βx(t) = x(t + β ? 1). The existence and uniqueness of solutions are proved by using Banach's fixed point theorem. An illustrative example is also presented. Copyright © 2014 John Wiley & Sons, Ltd.     

Global attractivity for nonlinear fractional differential equations

We present some results for the global attractivity of solutions for fractional differential equations involving Riemann-Liouville fractional calculus. The results are obtained by employing Krasnoselskii’s fixed point theorem. Similar results for fractional differential equations involving Caputo fractional derivative are also obtained by using the classical Schauder’s fixed point theorem. Several examples are given to illustrate our main results.     

Multiple positive solutions to BVPS for higher order nonlinear differential equations in Banach spaces

1. IntroductionIn [1], F.H. Wong proved the ekistence of a nonnegative solution to a higher order scalarboUndary value problem (for short, BVP) by means of Schauder's fixed point theorem. Inthis article, we shajl investigate the ekistence of multiple positive solutions of such higherorder BVP in Banach space E by utilizing the conical expansion and compression foredpoied principle and the thed poied index theory, both completely dtherellt from that in [1].Consider the following higher ord…