The monotonic property and stability of solutions of fractional differential equations

In this paper we improve on the monotone property of Lemma 1.7.3 in Lakshmikantham et al. (2009) [5] for the case g(t,u)=λu with a nonnegative real number λ. We also investigate the Mittag-Leffler stability of solutions of fractional differential equations by using the fractional comparison principle.     

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.     

Existence of global solutions to nonlinear fuzzy Volterra integro-differential equations

In this paper, we introduce new solutions for fuzzy differential equations as mixed solutions, and prove the existence and uniqueness of global solutions for fuzzy initial value problems involving integro-differential operators of Volterra type. One example is also given by applying mixed solution concept to fuzzy linear differential equations for obtaining their global solutions.     

Integral equations and initial value problems for nonlinear differential equations of fractional order

We discuss the solvability of integral equations associated with initial value problems for a nonlinear differential equation of fractional order. The differential operator is the Caputo fractional derivative and the inhomogeneous term depends on the fractional derivative of lower orders. We obtain the existence of at least one solution for integral equations using the Leray–Schauder Nonlinear Alternative for several types of initial value problems. In addition, using the Banach contraction principle, we establish sufficient conditions for unique solutions. Our approach in obtaining integral equations is the “reduction” of the fractional order of the integro-differential equations based on certain semigroup properties of the Caputo operator.     

Global solutions of nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces

In this paper, by using the generalization of Darbo’s fixed point theorem, we establish the existence of global solutions of an initial value problem for a class of second-order impulsive integro-differential equations of mixed type in a real Banach space. Our results generalize and improve on the results of Guo et al. [F. Guo, L.S. Liu, Y.H. Wu, P. Siew, Global solutions of initial value problems for nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces, Nonlinear Anal. 61 (2005) 1363–1382] in the sense that the conditions for existence of global solution in our theorem is simpler and less strict. To demonstrate the application of the theorem, we give the global solutions of two mixed boundary value problems for two classes of fourth order impulsive integro-differential equations.     

Almost periodic solutions for abstract impulsive differential equations

This paper studies the existence and uniqueness of exponentially stable almost periodic solutions for abstract impulsive differential equations in Banach space. The investigations are carried out by means of the fractional powers of operators. We construct an example to illustrate the feasibility of our results.     

Theory of fractional functional differential equations

In this paper, the basic theory for the initial value problems for fractional functional differential equations is considered, extending the corresponding theory of ordinary functional differential equations.     

Existence of positive solution for second-order nonlinear impulsive singular differential equations of mixed type in Banach spaces

In this paper, by constructing a closed convex set and using the fixed point theory of completely continuous operators, we investigate the existence of positive solutions for an initial value problem of second-order nonlinear impulsive singular integro-differential equations in a Banach space. The method used in this paper is different from that in the literature.     

A method for solving differential equations of fractional order

In this paper, we consider Caputo type fractional differential equations of order 0<α<1$0<\alpha <1$ with initial condition x(0)=_x0$x\left(0\right)=x_0$. We introduce a technique to find the exact solutions of fractional differential equations by using the solutions of integer order differential equations. Generalization of the technique to finite systems is also given. Finally, we give some examples to illustrate the applications of our results.     

Nonoscillation and positivity of solutions to first order state-dependent differential equations with impulses in variable moments

The state-dependent delay differential equation

     

Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative

In this paper, we shall discuss the properties of the well-known Mittag-Leffler function, and consider the existence and uniqueness of the solution of the periodic boundary value problem for a fractional differential equation involving a Riemann-Liouville fractional derivative by using the monotone iterative method.     

Positive solutions of two-point boundary value problems for systems of nonlinear second-order singular and impulsive differential equations

This paper is devoted to the study of multiple and single positive solutions of two-point boundary value problems for nonlinear second-order singular and impulsive differential systems. By constructing a cone _K1×_K2$K_1\times K_2$, which is the Cartesian product of two cones in the space C[0,1]$C[0,1]$, and computing the fixed-point index in the _K1×_K2$K_1\times K_2$, we establish the conditions for the existence of at least one or at least two positive solutions to two-point boundary value problems for systems of nonlinear second-order singular and impulsive differential equations.     

Numerical approach to differential equations of fractional order

In this paper, the variational iteration method and the Adomian decomposition method are implemented to give approximate solutions for linear and nonlinear systems of differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper presents a numerical comparison between the two methods for solving systems of fractional differential equations. Numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order.     

Monotone method for a system of nonlinear mixed type implicit impulsive integro-differential equations in Banach spaces

In this paper, by using a monotone iterative technique in the presence of lower and upper solutions, we discuss the existence of solutions for a new system of nonlinear mixed type implicit impulsive integro-differential equations in Banach spaces. Under wide monotonicity conditions and the noncompactness measure conditions, we also obtain the existence of extremal solutions and a unique solution between lower and upper solutions.     

Solving systems of fractional differential equations using differential transform method

This paper presents approximate analytical solutions for systems of fractional differential equations using the differential transform method. The fractional derivatives are described in the Caputo sense. The application of differential transform method, developed for differential equations of integer order, is extended to derive approximate analytical solutions of systems of fractional differential equations. The solutions of our model equations are calculated in the form of convergent series with easily computable components. Some examples are solved as illustrations, using symbolic computation. The numerical results show that the approach is easy to implement and accurate when applied to systems of fractional differential equations. The method introduces a promising tool for solving many linear and nonlinear fractional differential equations.     

The monotone iterative technique for impulsive hybrid set valued integro-differential equations

We develop the monotone method for impulsive hybrid set integro-differential equations in all its generality. Some interesting observations are presented.     

Existence and uniqueness results for impulsive functional differential equations with scalar multiple delay and infinite delay

In this paper, we discuss local and global existence and uniqueness results for first order impulsive functional differential equations with multiple delay. We shall rely on a nonlinear alternative of Leray–Schauder. For the global existence and uniqueness we apply a recent nonlinear alternative of Leray–Schauder type in Fréchet spaces, due to M. Frigon and A. Granas [Résultats de type Leray–Schauder pour des contractions sur des espaces de Fréchet, Ann. Sci. Math. Québec 22 (2) (1998) 161–168]. The goal of this paper is to extend the problems considered by A. Ouahab [Local and global existence and uniqueness results for impulsive differential equations with multiple delay, J. Math. Anal. Appl. 323 (2006) 456–472].