Monotone iterative technique and existence results for fractional differential equations

Using the method of upper and lower solutions, an existence result for IVP of Riemann–Liouville fractional differential equation is studied. Also, the monotone iterative technique is developed and the existence results for maximal and minimal solutions are obtained.     

Monotone iterative technique and existence results for fractional differential equations

Using the method of upper and lower solutions, an existence result for IVP of Riemann-Liouville fractional differential equation is studied. Also, the monotone iterative technique is developed and the existence results for maximal and minimal solutions are obtained.     

Nonstationary monotone iterative methods for nonlinear partial differential equations

A simple technique is given in this paper for the construction and analysis of monotone iterative methods for a class of nonlinear partial differential equations. With the help of the special nonlinear property we can construct nonstationary parameters which can speed up the iterative process in solving the nonlinear system. Picard, Gauss–Seidel, and Jacobi monotone iterative methods are presented and analyzed for the adaptive solutions. The adaptive meshes are generated by the 1-irregular mesh refinement scheme which together with the M-matrix of the finite element stiffness matrix lead to existence–uniqueness–comparison theorems with simple upper and lower solutions as initial iterates. Some numerical examples, including a test problem with known analytical solution, are presented to demonstrate the accuracy and efficiency of the adaptive and monotone properties. Numerical results of simulations on a MOSFET with the gate length down to 34 nm are also given.     

Mixed monotone iterative technique for semilinear impulsive fractional evolution equations

In this paper, we deals with the existence of mild $L$-quasi-solutions to the boundary value problem for a class of semilinear impulsive fractional evolution equations in an ordered Banach space $E$. Under a new concept of upper and lower solutions, a new monotone iterative technique on the initial value problem of impulsive fractional evolution equations has been established. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. As some application that illustrate our results, An example is also given.     

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 of solutions for nonlinear fractional stochastic differential equations

The fractional stochastic differential equations have wide applications in various fields of science and engineering. This paper addresses the issue of existence of mild solutions for a class of fractional stochastic differential equations with impulses in Hilbert spaces. Using fractional calculations, fixed point technique, stochastic analysis theory and methods adopted directly from deterministic fractional equations, new set of sufficient conditions are formulated and proved for the existence of mild solutions for the fractional impulsive stochastic differential equation with infinite delay. Further, we study the existence of solutions for fractional stochastic semilinear differential equations with nonlocal conditions. Examples are provided to illustrate the obtained theory.     

Mixed monotone iterative technique for Hilfer fractional evolution equations with nonlocal conditions

The purpose of this paper is concerned with the existence of mild $L$-quasi-solutions for Hilfer fractional evolution equations with nonlocal conditions in an ordered Banach spaces $E$. By employing mixed monotone iterative technique, measure of noncompactness and Sadovskii"s fixed point theorem, we obtain the existence of mild $L$-quasi-solutions for Hilfer fractional evolution equations with noncompact semigroups. Finally, an example is provide to illustrate the feasibility of our main results.     

Existence results for fractional functional differential equations with impulses

In this paper, we discuss some existence results for a class of multi-point boundary value problem for impulsive fractional functional differential equations. Some sufficient conditions are obtained by using suitable fixed point theorems. Examples are also given to illustrate our results.     

Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations

In this paper, we investigate existence and uniqueness of solutions for nonlinear impulsive fractional differential equations. By utilizing the well-known fixed point theorems, we obtain some sufficient conditions for the uniqueness and existence of the solutions. At the end of the paper, an example is given to illustrate our main results.     

Existence of positive solutions of nonlinear fractional differential equations

Existence of positive solutions for the nonlinear fractional differential equation D^su(x)=f(x,u(x)), 0<s<1, has been studied (S. Zhang, J. Math. Anal. Appl. 252 (2000) 804-812), where D^s denotes Riemann-Liouville fractional derivative. In the present work we study existence of positive solutions in case of the nonlinear fractional differential equation:

L(D)u=f(x,u),u(0)=0,0<x<1,     

A new approach to monotone iterative technique for solution of differential equations

We discuss a monotone interative technique for initial value problems. The novelty of the approach is the use of g-monotonicity, i.e. of monotonicity of the ratio of two functions. This is in contrast to the more usual notion of monotonicity     

Existence and uniqueness results for third-order nonlinear differential systems

Sufficient conditions are given for the existence of solutions to third order boundary value problems for nonlinear differential systems. Some appropriate conditions are given to guarantee that the Nagumo condition is satisfied. By constructing appropriate a lower solution-upper solution pair, a concept that is defined in this paper, the uniqueness result of the problem is also established. The emphasis here is that the differential systems has nonlinear dependence on all over order derivatives and the boundary conditions are nonlinear.     

Existence of positive solutions of nonlinear fractional delay differential equations

This paper investigates the existence and uniqueness of positive solutions for a class of nonlinear fractional delay differential equations. Using a nonlinear alternative of Leray-Schauder type, we show the existence of positive solutions for the equations in question.     

Existence of fractional differential equations

Consider the fractional differential equation

Dαx=f(t,x),     

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.     

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 of positive solution for some class of nonlinear fractional differential equations

In this paper, we investigate the multiple and infinitely solvability of positive solutions for nonlinear fractional differential equation Du(t)=t^νf(u), 0<t<1, where D=t^−βδD_β^γ−δ,δ, β>0, γ?0, 0<δ<1, ν>−β(γ+1). Our main work is to deal with limit case of f(s)/s as s→0 or s→∞ and Φ(s)/s, Ψ(s)/s as s→0 or s→∞, where Φ(s), Ψ(s) are functions connected with function f. In J. Math. Appl. 252 (2000) 804-812, we consider the existence of a positive solution for the particular case of Eq. (1.1), i.e., the Riemann-Liouville type (β=1, γ=0) nonlinear fractional differential equation, using the super-lower solutions method. Here, we devote to the existence of positive solution and multi-positive solutions for Eq. (1.1) by means of the fixed point theorems for the cone.     

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.     

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.