The Defect of a Cauchy Type Problem for Linear Equations with Several Riemann–Liouville Derivatives

We consider the existence and uniqueness of solutions to initial value problems for general linear nonhomogeneous equations with several Riemann–Liouville fractional derivatives in Banach spaces. Considering the equation solved for the highest fractional derivative \( D^{\alpha}_{t} \), we introduce the concept of the defect \( m^{*} \) of a Cauchy type problem which determines the number of the zero initial conditions \( D^{\alpha-m+k}_{t}z(0)=0 \), \( k=0,1,\dots,m^{*}-1 \), necessary for the existence of the finite limits \( D^{\alpha-m+k}_{t}z(t) \) as \( t\to 0+ \) for all \( k=0,1,\dots,m-1 \). We show that the defect \( m^{*} \) is uniquely determined by the set of orders of the Riemann–Liouville fractional derivatives in the equation. Also we prove the unique solvability of the incomplete Cauchy problem \( D^{\alpha-m+k}_{t}z(0)=z_{k} \), \( k=m^{*},m^{*}+1,\dots,m-1 \), for the equation with bounded operator coefficients solved for the highest Riemann–Liouville derivative. The obtained result allowed us to investigate initial problems for a linear nonhomogeneous equation with a degenerate operator at the highest fractional derivative, provided that the operator at the second highest order derivative is 0-bounded with respect to this operator, while the cases are distinguished that the fractional part of the order of the second derivative coincides or does not coincide with the fractional part of the order of the highest derivative. The results for equations in Banach spaces are used for the study of initial boundary value problems for a class of equations with several Riemann–Liouville time derivatives and polynomials in a selfadjoint elliptic differential operator of spatial variables.

     

Solvability and Stability for Singular Fractional (p,q)-difference Equation

In this paper, we initiate the solvability and stability for a class of singular fractional $(p,q)$-difference equations. First, we obtain an existence theorem of solution for the fractional $(p,q)$-difference equation. Then, by using a fractional $(p,q)$-Gronwall inequality, some stability criteria of solution are established, which also implies the uniqueness of solution.     

Initial value problem for a class of semi-linear fractional iterative differential equations

An initial value problem of a class of semi-linear fractional order iterative differential equations is studied in this paper. The existence of solution for fractional order iterative differential equations is obtained in Banach space $C(J,J)$ and $C_{K,\alpha}(J,J)$ respectively. However, uniqueness results can not be acquired since the operator only is H\"{o}lder continuous but not Lipschitz continuous. Furthermore, a small perturbation of the initial value will cause a change in the solution on $[k,b]$ for the $k\in J$. Our analysis is based on Schauder"s fixed point theorem and the properties of Mittag-Leffler function. Finally, an example is given to illustrate our results.     

Well-Posedness for Fractional $(p,q)$-Difference Equations Initial Value Problem

In this paper, we investigate a class of the fractional $(p,q)$-difference initial value problem with the fractional $(p,q)$-integral boundary conditions with the aid of the method of successive approximations(Picard method) and fractional $(p,q)$-Gronwall inequality, obtaining sufficient conditions for the existence, uniqueness and continuous dependence results of solutions.     

The Initial Value Problems and Nonlinear Boundary Value Problems for a Class of the Second order Quasilinear Parabolio Systems

In this paper initial value problems and nonlinear mixed boundary value problems for the quasilinear parabolic systems below $\[\frac{{\partial {u_k}}}{{\partial t}} - \sum\limits_{i,j = 1}^n {a_{ij}^{(k)}} (x,t)\frac{{{\partial ^2}{u_k}}}{{\partial {x_i}\partial {x_j}}} = {f_k}(x,t,u,{u_x}),k = 1, \cdots ,N\]$ are discussed.The boundary value conditions are $\[{u_k}{|_{\partial \Omega }} = {g_k}(x,t),k = 1, \cdots ,s,\]$ $\[\sum\limits_{i = 1}^n {b_i^{(k)}} (x,t)\frac{{\partial {u_k}}}{{\partial {x_i}}}{|_{\partial \Omega }} = {h_k}(x,t,u),k = s + 1, \cdots N.\]$ Under some "basically natural" assumptions it is shown by means of the Schauder type estimates of the linear parabolic equations and the embedding inequalities in Nikol'skii spaces,these problems have solutions in the spaces $\[{H^{2 + \alpha ,1 + \frac{\alpha }{2}}}(0 < \alpha < 1)\]$.For the boundary value problem with $\[b_i^{(k)}(x,t) = \sum\limits_{j = 1}^n {a_{ij}^{(k)}} (x,t)\cos (n,{x_j})\]$ uniqueness theorem is proved.     

A Class of Nonlocal Impulsive Problems for Integrodifferential Equations in Banach Spaces

In this paper, we study the existence and uniqueness of the PC-mild solution for a class of nonlinear integrodifferential impulsive differential equations with nonlocal conditions $$\left\{\begin{array}{l} x'(t)=Ax(t)+f\left(t,x(t), \int_{0}^{t}k(t,s,x(s))ds\right), \quad t\in J=[0,b], \,\, t\neq t_{i},\\ x(0)=g(x)+x_{0},\\ \Delta x(t_{i})=I_{i}(x(t_{i})), \quad i=1,2,\ldots,p, \,\, 0=t_{0} < t_{1} < \cdots < t_{p} < t_{p+1}=b.\end{array} \right.$$ Using the generalized Ascoli-Arzela theorem given by us, some fixed point technique including Schaefer fixed point theorem and Krasnoselskii fixed point theorem, and theory of operators semigroup, some new results are obtained. At last, some examples are given to illustrate the theory.     

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

In this paper, by using the fixed point theory, we study the existence and uniqueness of initial value problems for nonlinear fractional differential equations and obtain a new result.     

非线性分数阶脉冲微分方程边值问题的解

通过Schauder不动点定理和Banach压缩映射原理得到了一类非线性分数阶脉冲微分方程边值问题解的存在性和唯一性结果.     

Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations

In this article, we study a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations. We obtain sufficient conditions for existence and uniqueness of positive solutions. We use the classical fixed point theorems such as Banach fixed point theorem and Krasnoselskii’s fixed point theorem for uniqueness and existence results. As in application, we provide an example to illustrate our main results.     

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.     

二阶混合单调型脉冲微分方程的初值问题

利用混合单调凝聚算子的耦合不动点定理,给出了二阶混合单调型脉冲微分方程的初值问题的解的存在唯一性及迭代逼近定理。     

Initial Boundary Value Problem for a Damped Nonlinear Hyperbolic Equation

In the paper, the existence and uniqueness of the generalized global solution and the classical global solution of the initial boundary value problems for the nonlinear hyperbolic equation u_{tt} + k_1u_{xxxx} + k_2u_{xxxxt} + g(u_{xx})_{xx} = f(x, t) are proved by Galerkin method and the sufficient conditions of blow-up of solution in finite time are given.     

关于Banach空间一阶非线性脉冲积分-微分方程初值问题解存在性的注记

设m(t)∈C[Jk,R ](k=1,2,…,m),且满足不等式m(t)<(L1 L2t)∫tn(s)ds L3t∫a m(s)ds ∑o0满足KaLs(eδ(L1 aL2)-1)相似文献   

Existence of solutions for fractional impulsive differential equations with p-Laplacian operator

We investigate the boundary value problems for nonlinear fractional impulsive differential equations with p-Laplacian operator. By applying some standard fixed point theorems, we obtain new results on the existence and uniqueness of solutions. Examples are given to show the applicability of our results.     

Upper and lower solutions method for impulsive partial hyperbolic differential equations with fractional order

In this paper we investigate the existence of solutions for a class of initial value problems for impulsive partial hyperbolic differential equations involving the Caputo fractional derivative by using the lower and upper solutions method combined with Schauder’s fixed point theorem.     

Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order

In this paper, we prove the existence and uniqueness of solutions for an anti-periodic boundary value problem of nonlinear impulsive differential equations of fractional order α∈(2,3] by applying some well-known fixed point theorems. Some examples are presented to illustrate the main results.     

Banach空间中脉冲积分-微分方程的初值问题

给出了 Banach空间中一阶线性脉冲积分 -微分方程初值问题解的存在唯一性的一个新证法 ,改进了已有结果 .利用它讨论了一阶非线性脉冲积分 -微分方程初值问题的解 ,所得结果大大推广了已有的相关结果 .     

Impulsive Boundary Value Problems for Two Classes of Fractional Differential Equation with Two Different Caputo Fractional Derivatives

We consider the impulsive boundary value problems for two classes of fractional differential equations with two different Caputo fractional derivatives and generalized boundary value conditions. Natural formulae of a solution for these problems are introduced, which can be regarded as a novelty item. Some sufficient conditions for existence and uniqueness of the solutions to this nonlinear equations are established by applying well-known Banach’s contraction mapping principle, Laplace transforms and some skills of inequalities. Finally, an example is given to illustrate the effectiveness of our results.     

An existence and uniqueness result for fractional order semilinear functional differential equations with nondense domain

In this paper, by using the Banach fixed point theorem, we study the existence and uniqueness of solutions of initial value problems for fractional order semilinear functional differential equations with nondense domain and obtain a new result.     

脉冲发展方程初值问题的正解

This paper deals with the existence of e-positive mild solutions（see Definition 1）for the initial value problem of impulsive evolution equation with noncompact semigroup u（t）＋ Au（t）= f（t,u（t））,t ∈ [0,＋∞）,t = tk,u-t=tk = Ik（u（tk））,k = 1,2,...,u（0）= x0 in an ordered Banach space E.By using operator semigroup theory and monotonic iterative technique,without any hypothesis on the impulsive functions,an existence result of e-positive mild solutions is obtained under weaker measure of noncompactness condition on nonlinearity of f.Particularly,an existence result without using measure of noncompaceness condition is presented in ordered and weakly sequentially complete Banach spaces,which is very convenient for application.An example is given to illustrate that our results are more valuable.