Periodic solutions of abstract functional differential equations with state‐dependent delay

In this paper, we are concerned with the existence of solutions of systems determined by abstract functional differential equations with infinite and state‐dependent delay. We establish the existence of mild solutions and the existence of periodic solutions. Our results are based on local Lipschitz conditions of the involved functions. We apply our results to study the existence of periodic solutions of a partial differential equation with infinite and state‐dependent delay. Copyright © 2016 John Wiley & Sons, Ltd.     

Note on periodic solutions of fractional differential equations

The existence and nonexistence of periodic solutions are discussed for fractional differential equations by varying the lower limits of Caputo derivatives. The developed approach is illustrated on several examples.     

Almost automorphic solutions for abstract functional differential equations

For abstract linear functional differential equations with an almost automorphic forcing term, we establish a result on the existence of almost automorphic solutions, which extends the classical theorem due to Massera on the existence of periodic solutions for linear periodic ordinary differential equations.     

The existence of mild solutions for impulsive fractional partial differential equations

This paper is concerned with the existence of mild solutions for a class of impulsive fractional partial semilinear differential equations. Some errors in Mophou (2010) [2] are corrected, and some previous results are generalized.     

Variational solutions for partial differential equations driven by a fractional noise

In this article we develop an existence and uniqueness theory of variational solutions for a class of nonautonomous stochastic partial differential equations of parabolic type defined on a bounded open subset D⊂R^d and driven by an infinite-dimensional multiplicative fractional noise. We introduce two notions of such solutions for them and prove their existence and their indistinguishability by assuming that the noise is derived from an L²(D)-valued fractional Wiener process W^H with Hurst parameter , whose covariance operator satisfies appropriate integrability conditions, and where γ∈(0,1] denotes the Hölder exponent of the derivative of the nonlinearity in the stochastic term of the equations. We also prove the uniqueness of solutions when the stochastic term is an affine function of the unknown random field. Our existence and uniqueness proofs rest upon the construction and the convergence of a suitable sequence of Faedo-Galerkin approximations, while our proof of indistinguishability is based on certain density arguments as well as on new continuity properties of the stochastic integral we define with respect to W^H.     

Polynomial solutions of certain differential equations arising in physics

Conditions for the existence of polynomial solutions of certain second‐order differential equations have recently been investigated by several authors. In this paper, a new algorithmic procedure is given to determine necessary and sufficient conditions for a differential equation with polynomial coefficients containing parameters to admit polynomial solutions and to compute these solutions. The effectiveness of this approach is illustrated by applying it to determine new solutions of several differential equations of current interest. A comparative analysis is given to demonstrate the advantage of this algorithmic procedure over existing software. Copyright © 2012 John Wiley & Sons, Ltd.     

Weak solutions for stochastic differential equations with additive fractional noise

In this paper, we show the existence of a weak solution for a stochastic differential equation driven by an additive fractional Brownian motion with Hurst parameter , and a discontinuous drift. The proof of this result is based on the Girsanov theorem for the fractional Brownian motion.     

Correctness of Cauchy-type problems for abstract differential equations with fractional derivatives

We prove the uniform correctness of a Cauchy-type problem with two fractional derivatives and a bounded operator A. We propose a criterion for the uniform correctness of unbounded operator A.     

Boundary problems for fractional differential equations

In this paper, the existence of solutions of fractional differential equations with nonlinear boundary conditions is investigated. The monotone iterative method combined with lower and upper solutions is applied. Fractional differential inequalities are also discussed. Two examples are added to illustrate the results.     

Mild solutions of evolution quantum stochastic differential equations with nonlocal conditions

We study existence of a unique mild solution of evolution quantum stochastic differential equations with nonlocal conditions under the strong topology. Using the method of successive approximations, we do not need to transform the nonlocal problem to a fixed point form. The evolution operator A generates a family of semigroup that are continuous. Nonlocal conditions allow additional measurements of certain phenomena that cannot be captured by the traditional initial conditions. We show that under some given conditions, the mild solution is unique and also stable. The method applied here is much easier when compared with previous methods used in literature.     

Nonsmooth analysis and fractional differential equations

In this paper we study Euler solutions, strong and weak invariance of solutions for fractional differential equations.     

Mild solutions of non‐Lipschitz stochastic integrodifferential evolution equations

In this work, we study the existence and uniqueness of mild solutions for stochastic partial integrodifferential equations under local non‐Lipschitz conditions on the coefficients. Our analysis makes use of the theory of resolvent operators as developed by R. Grimmer as well as a stopping time technique. Our results complement and improve several earlier related works. An example is provided to illustrate the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Infinitely many solutions for fractional Schrodinger-Maxwell equations

In this paper using fountain theorems we study the existence of infinitely many solutions for fractional Schr\"{o}dinger-Maxwell equations \[\begin{cases} (-\Delta)^\alpha u+\lambda V(x)u+\phi u=f(x,u)-\mu g(x)|u|^{q-2}u, \text{ in } \mathbb R^3,\(-\Delta)^\alpha \phi=K_\alpha u^2, \text{ in } \mathbb R^3, \end{cases}\] where $\lambda,\mu >0$ are two parameters, $\alpha\in (0,1]$, $K_\alpha=\frac{\pi^{-\alpha}\Gamma(\alpha)}{\pi^{-(3-2\alpha)/2}\Gamma((3-2\alpha)/2)}$ and $(-\Delta)^\alpha$ is the fractional Laplacian. Under appropriate assumptions on $f$ and $g$ we obtain an existence theorem for this system.     

On stable iterative solutions for a class of boundary value problem of nonlinear fractional order differential equations

In this article, sufficient conditions for the existence of extremal solutions to nonlinear boundary value problem (BVP) of fractional order differential equations (FDEs) are provided. By using the method of monotone iterative technique together with upper and lower solutions, conditions for the existence and approximation of minimal and maximal solutions to the BVP under consideration are constructed. Some adequate results for different kinds of Ulam stability are investigated. Maximum error estimates for the corresponding solutions are given as well. Two examples are provided to illustrate the results.     

Extremal solutions of Cauchy problems for abstract fractional differential equations

In this paper, we study the extremal solutions of Cauchy problems for abstract fractional differential equations. Some definitions such as L ¹-Lipschitz-like, L ¹-Carathéodory-like and L ¹-Chandrabhan-like are introduced. By virtue of the singular integral inequalities with several nonlinearities due to Medved’, the properties of solutions are given. By using a hybrid fixed point theorem due to Dhage, existence results for extremal solutions are established. Finally, we present an example to illustrate our main results.