Existence results for fractional order functional differential equations with infinite delay

The Banach fixed point theorem and the nonlinear alternative of Leray-Schauder type are used to investigate the existence of solutions for fractional order functional and neutral functional differential equations with infinite delay.     

Comparison results for functional differential equations with impulses

Comparison principles play an important role in the qualitative and quantitative study of differential equations. In this paper, we investigate a first order functional differential equations with impulses and establish new comparison results.     

Existence results for fractional order semilinear functional differential equations with nondense domain

In this paper, we establish sufficient conditions for existence and uniqueness of solutions for some nondensely defined semilinear functional differential equations involving the Riemann-Liouville derivative. Our approach is based on integrated semigroup theory, the Banach contraction principle, and the nonlinear alternative of Leray-Schauder type.     

Existence results of second order impulsive functional differential equations with infinite delay and fractional damping

In this paper we prove the existence, uniqueness, regularity and continuous dependence of mild solutions for second order impulsive functional differential equations with infinite delay and fractional damping in Banach spaces. We generalize the existence theorem of integer order differential equations to the fractional order case. The results obtained here improve and generalize some known results.     

Existence and exponential stability for neutral stochastic fractional differential equations with impulses driven by Poisson jumps

The paper is mainly concerned with a class of neutral stochastic fractional integro-differential equation with Poisson jumps. First, the existence and uniqueness for mild solution of an impulsive stochastic system driven by Poisson jumps is established by using the Banach fixed point theorem and resolvent operator. The exponential stability in the pth moment for mild solution to neutral stochastic fractional integro-differential equations with Poisson jump is obtained by establishing an integral inequality.     

Existence results for impulsive neutral functional differential equations with infinite delay

In this work, we prove the existence of mild solutions for impulsive partial neutral functional differential equations with infinite delay in a Banach space. The results are obtained by using the Krasnoselski–Schaefer type fixed point theorem.     

Existence results for partial neutral functional differential equations with state-dependent delay

In this paper we study the existence of mild solutions for a class of first order abstract partial neutral differential equations with state-dependent delay.     

Existence results for some partial functional differential equations with state-dependent delay

In this paper, we establish sufficient conditions for the existence of solutions for some partial functional differential equations with state-dependent delay; we assume that the linear part is not necessarily densely defined and satisfies the well-known Hille–Yosida conditions. Our approach is based on a nonlinear alternative of Leray–Schauder type and integrated semigroup theory. An application is provided to a reaction–diffusion equation with state-dependent delay.     

Stability of functional differential equations with impulses

In this paper, the stability of functional differential equations (FDE) with impulses is investigated. Some comparison theorems are given. Several Lyapunov-Razumikhin functions of partial components of the state variable x, which can be much easier constructed, are used so that the conditions ensuring that stability are simpler and less restrictive. The results improve and generalize the ones in the literature. An example is also given to illustrate the importance of our results.     

Existence of fractional differential equations

Consider the fractional differential equation

Dαx=f(t,x),     

Existence and uniqueness results for Cauchy problem of variable-order fractional differential equations

In this paper, the existence and uniqueness results of variable-order fractional differential equations (VOFDEs) are studied. The variable-order fractional derivative is defined in the Caputo sense, and the fractional order is a bounded function. The existence result of Cauchy problem of VOFDEs is obtained by constructing an iteration series which converges to the analytical solution. The uniqueness result is obtained by employing the contraction mapping principle. Since the variable-order fractional derivatives contain classical and fractional derivatives as special cases, many existence and uniqueness results of references are significantly generalized. Finally, we draw some conclusions of variable-order fractional calculus, and two examples are given for demonstrating the theoretical analysis.     

Existence results for a coupled system of nonlinear neutral fractional differential equations

Applying the monotone iterative method, we investigate the existence of solutions for a coupled system of nonlinear neutral fractional differential equations, which involves Riemann–Liouville derivatives of different fractional orders. As an application, an example is presented to illustrate the main results.     

Existence results for BVP of a class of Hilfer fractional differential equations

In this paper, we consider the existence and uniqueness of solutions to the nonlocal boundary value problem for semi-linear differential equations involving Hilfer fractional derivative. With the help of properties of Hilfer fractional calculus, Mittag-Leffler functions, and fixed point methods, we derive existence and uniqueness results. Finally, examples are given to illustrate our theoretical results.     

A class of nonlinear differential equations with fractional integrable impulses

In this paper, we introduce a new class of impulsive differential equations, which is more suitable to characterize memory processes of the drugs in the bloodstream and the consequent absorption for the body. This fact offers many difficulties in applying the usual methods to analysis and novel techniques in Bielecki’s normed Banach spaces and thus makes the study of existence and uniqueness theorems interesting. Meanwhile, new concepts of Bielecki–Ulam’s type stability are introduced and generalized Ulam–Hyers–Rassias stability results on a compact interval are established. This is another novelty of this paper. Finally, an interesting example is given to illustrate our theory results.     

Existence results of fractional differential equations with irregular boundary conditions and p-Laplacian operator

In this paper, we discuss the existence of solutions for irregular boundary value problems of nonlinear fractional differential equations with p-Laplacian operator $$\left \{ \begin{array}{l} {\phi}_p(^cD_{0+}^{\alpha}u(t))=f(t,u(t),u'(t)), \quad 0< t<1, \ 1< \alpha \leq2, \\ u(0)+(-1)^{\theta}u'(0)+bu(1)=\lambda, \qquad u(1)+(-1)^{\theta}u'(1)=\int_0^1g(s,u(s))ds,\\ \quad \theta=0,1, \ b \neq \pm1, \end{array} \right . $$ where \(^{c}D_{0+}^{\alpha}\) is the Caputo fractional derivative, ? _p (s)=|s| ^p?2 s, p>1, \({\phi}_{p}^{-1}={\phi}_{q}\) , \(\frac {1}{p}+\frac{1}{q}=1\) and \(f: [0,1] \times\mathbb{R} \times\mathbb {R} \longrightarrow\mathbb{R}\) . Our results are based on the Schauder and Banach fixed point theorems. Furthermore, two examples are also given to illustrate the results.     

Existence of solutions for fractional neutral functional differential equations driven by fBm with infinite delay

In this paper, we consider a class of fractional neutral stochastic functional differential equations with infinite delay driven by a cylindrical fractional Brownian motion (fBm) in a real separable Hilbert space. We prove the existence of mild solutions by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result.     

Existence results for nondensely defined impulsive neutral functional differential equations with infinite delay

In this paper, we study a class of impulsive neutral functional differential equations with infinite delay. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille–Yosida theorem. We give some sufficient conditions ensuring the existence of integral solutions and strict solutions. To illustrate our abstract results, we conclude this work with an example.