A study of nonlinear Langevin equation involving two fractional orders in different intervals

This paper studies a nonlinear Langevin equation involving two fractional orders α∈(0,1] and β∈(1,2] with three-point boundary conditions. The contraction mapping principle and Krasnoselskii’s fixed point theorem are applied to prove the existence of solutions for the problem. The existence results for a three-point third-order nonlocal boundary value problem of nonlinear ordinary differential equations follow as a special case of our results. Some illustrative examples are also discussed.     

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.     

Existence of solutions for fractional differential equations with nonlocal and average type integral boundary conditions

In this paper, we establish sufficient conditions for the existence and uniqueness of solutions for a boundary value problem of fractional differential equations with nonlocal and average type integral boundary conditions. The Leray–Schauder nonlinear alternative, Krasnoselskii’s fixed point theorem and Banach’s fixed point theorem together with Hölder inequality are applied to construct proofs for the main results. Examples illustrating the obtained results are also presented.     

Approximate Controllability of Fractional Neutral Stochastic Evolution Equations with Nonlocal Conditions

In this paper, the approximate controllability of neutral stochastic fractional differential equations involving nonlocal initial conditions is studied. By using Sadovskii’s fixed point theorem with stochastic analysis theory, we derive a new set of sufficient conditions for the approximate controllability of semilinear fractional stochastic differential equations with nonlocal conditions under the assumption that the corresponding linear system is approximately controllable. Finally, an application to a fractional partial stochastic differential equation with nonlocal initial condition is provided to illustrate the obtained theory.     

Existence of global solutions for second order impulsive abstract partial differential equations

In this paper, we study the existence of global solutions for a class of second order impulsive abstract functional differential equations. The results are obtained by using Leray-Schauder’s Alternative fixed point theorem. An application is provided to illustrate the theory.     

Existence and controllability results for fractional semilinear differential inclusions

In this paper, we prove the existence and controllability results for fractional semilinear differential inclusions involving the Caputo derivative in Banach spaces. The results are obtained by using fractional calculation, operator semigroups and Bohnenblust–Karlin’s fixed point theorem. At last, an example is given to illustrate the theory.     

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.     

Study in Fractional Differential Equations by Means of Topological Degree Methods

In this article, we study some class of fractional differential equations involving the Caputo fractional derivative. By using a fixed point theorem on topological degree for condensing maps via a priori estimate method, some sufficient conditions for the existence of solutions are presented. Uniqueness and data dependence results of solutions are also deduced.     

Fractional order differential equations on an unbounded domain

We are concerned with the existence of bounded solutions of a boundary value problem on an unbounded domain for differential equations involving the Caputo fractional derivative. Our results are based on a fixed point theorem of Schauder combined with the diagonalization method.     

Ulam–Hyers Stability for the Darboux Problem for Partial Fractional Differential and Integro-differential Equations via Picard Operators

In the present paper we investigate some uniqueness and Ulam’s type stability concepts of fixed point equations due to Rus, for the Darboux problem of partial differential and integro-differential equations involving the Caputo fractional derivative. Our results are obtained by using weakly Picard operators theory.     

Existence of solutions for anti-periodic boundary value problems

By using Schauder’s fixed point theorem and the lower and upper solutions method, we discuss the anti-periodic boundary value problem for a class of second-order differential equations. Some sufficient conditions for the existence of solutions are obtained.     

Existence and stability results for semilinear systems of impulsive stochastic differential equations with fractional Brownian motion

Some results on the existence and uniqueness of mild solution for a system of semilinear impulsive differential equations with infinite fractional Brownian motions are proved. The approach is based on Perov's fixed point theorem and a new version of Schaefer's fixed point theorem in generalized Banach spaces. The relationship between mild and weak solutions and the exponential stability of mild solutions are investigated as well. The abstract theory is illustrated with an example.     

Controllability of nonlinear fractional dynamical systems

In this paper we establish a set of sufficient conditions for the controllability of nonlinear fractional dynamical systems. The results are obtained by using the recently derived formula for solution representation of systems of fractional differential equations and the application of the Schauder fixed point theorem. Examples are provided to illustrate the results.     

Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order

In this paper, the existence of solutions of an anti-periodic fractional boundary value problem for nonlinear fractional differential equations is investigated. The contraction mapping principle and Leray-Schauder’s fixed point theorem are applied to establish the results.     

An extension of Krasnoselskii’s fixed point theorem and its application to nonlocal problems for implicit fractional differential systems with uncertainty

The traditional Krasnoselskii’s fixed point theorem in Banach spaces does not reproduce the rich and varied forms of operator equations in abstract spaces which are not linear structure. Consequently, its applications to integral equations and differential equations have met many obstacles. The present alternative Krasnoselskii’s fixed point theorem in generalized semilinear Banach spaces overcomes this deficiency and opens up for profitable investigation such as differential systems with uncertainty. An application to the existence of solutions of nonlocal problems for fuzzy implicit fractional differential systems under Caputo generalized Hukuhara differentiability with demonstrated example is given to validate the effectiveness of our theoretical results.     

Exponential Stability of Positive Conformable BAM Neural Networks with Communication Delays

In this paper, we consider a class of nonlinear differential equations with delays described by conformable fractional derivative. This type of differential equations can be used to describe dynamics of various practical models including biological and artificial neural networks with heterogeneous time-varying delays. By novel comparison techniques via fractional differential and integral inequalities, we prove under assumptions involving the order-preserving property of nonlinear vector fields that, with nonnegative initial states and inputs, the system state trajectories are always nonnegative for all time. This feature, called positivity, induces a special character, namely the monotonicity of the system. We then derive tractable conditions in terms of linear programming and prove, by utilizing the Brouwer''s fixed point theorem and comparisons induced by the monotonicity, that the system possesses a unique positive equilibrium point which attracts exponentially all state trajectories. An application to the exponential stability of fractional linear time-delay systems is also discussed. Numerical examples with simulations are given to illustrate the theoretical results.     

Existence,uniqueness, and multiplicity results on positive solutions for a class of Hadamard-type fractional boundary value problem on an infinite interval

This paper focuses on a class of Hadamard-type fractional differential equation with nonlocal boundary conditions on an infinite interval. New existence, uniqueness, and multiplicity results of positive solutions are obtained by using Schauder's fixed point theorem, Banach's contraction mapping principle, the monotone iterative method, and the Avery-Peterson fixed point theorem. Examples are included to illustrate our main results.     

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.     

Stability in functional differential equations established using fixed point theory

In this paper we consider a nonlinear scalar delay differential equation with variable delays and give some new conditions for the boundedness and stability by means of Krasnoselskii’s fixed point theory. A stability theorem with a necessary and sufficient condition is proved. The results in [T.A. Burton, Stability by fixed point theory or Liapunov’s theory: A comparison, Fixed Point Theory 4 (2003) 15–32; T.A. Burton, T. Furumochi, Asymptotic behavior of solutions of functional differential equations by fixed point theorems, Dynamic Systems and Applications 11 (2002) 499–519; B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Analysis 63 (2005) e233–e242] are improved and generalized. Some examples are given to illustrate our theory.     

On solvability of differential equations with the Riesz fractional derivative

We consider the solvability of fractional differential equations involving the Riesz fractional derivative. Our approach basically relies on the reduction of the problem considered to the equivalent nonlinear mixed Volterra and Cauchy-type singular integral equation and on the theory of fractional calculus. By establishing a compactness property of the Riemann–Liouville fractional integral operator on Lebesgue spaces and using the well-known Krasnoselskii's fixed point theorem, an existence of at least one solution is gleaned. An example is finally included to show the applicability of the theory.