Lyapunov‐type inequalities for a class of fractional boundary value problems with integral boundary conditions

In this paper, Lyapunov‐type inequalities are derived for a class of fractional boundary value problems with integral boundary conditions. As an application, we obtain a lower bound for the eigenvalues of corresponding equations.     

Positive solution for four‐point nonlocal boundary value problems of fractional order

In this paper, we consider the existence of positive solution for four‐point nonlocal boundary value problems of fractional order. By means of some fixed point theorems, some results on the existence and multiplicity of positive solutions are obtained. Furthermore, we provide a representative example to illustrate a possible application of the established results. Copyright © 2013 John Wiley & Sons, Ltd.     

Lyapunov type inequalities for a fractional two‐point boundary value problem

In this paper, new Lyapunov‐type inequalities are obtained for the case when one is dealing with a class of fractional two‐point boundary value problems. As an application of this result, we obtain a lower bound for the eigenvalues of corresponding equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Lyapunov inequalities for nabla Caputo boundary value problems

ABSTRACT

We will establish uniqueness of solutions to boundary value problems involving the nabla Caputo fractional difference under two-point boundary conditions and give an explicit expression for the Green's functions for these problems. Using the Green's functions for specific cases of these boundary value problems, we will then develop Lyapunov inequalities for certain nabla Caputo BVPs.     

Periodic boundary value problems for higher‐order fractional differential systems

Approximation of solutions of fractional differential systems (FDS) of higher orders is studied for periodic boundary value problem (PBVP). We propose a numerical‐analytic technique to construct a sequence of functions convergent to the limit function, which is a solution of the given PBVP, if the corresponding determined equation has a root. We also study scalar fractional differential equations (FDE) with asymptotically constant nonlinearities leading to Landesman‐Lazer–type conditions.     

On a difference scheme of fourth order of accuracy for the Bitsadze–Samarskii type nonlocal boundary value problem

The Bitsadze–Samarskii type nonlocal boundary value problem for the differential equation in a Hilbert space H with the self‐adjoint positive definite operator A with a closed domain D(A) ? H is considered. Here, f(t) be a given abstract continuous function defined on [0,1] with values in H, φ and ψ be the elements of D(A), and λ_j are the numbers from the set [0,1]. The well‐posedness of the problem in Hölder spaces with a weight is established. The coercivity inequalities for the solution of the nonlocal boundary value problem for elliptic equations are obtained. The fourth order of accuracy difference scheme for approximate solution of the problem is presented. The well‐posedness of this difference scheme in difference analogue of Hölder spaces is established. For applications, the stability, the almost coercivity, and the coercivity estimates for the solutions of difference schemes for elliptic equations are obtained. Mathematical Methods in the Applied Sciences. Copyright © 2012 John Wiley & Sons, Ltd.     

Existence of solutions of periodic‐type boundary value problems for multi‐term fractional differential equations

Results on the existence of solutions of a periodic‐type boundary value problem of singular multi‐term fractional differential equations with the nonlinearity depending on are established and being singular at t = 0 and t = 1. The analysis relies on the well‐known fixed‐point theorems. An example is given to illustrate the efficiency of the main theorems. Copyright © 2013 John Wiley & Sons, Ltd.     

On positive solutions to fractional sum boundary value problems for nonlinear fractional difference equations

In this paper, we study a new class of 3‐point boundary value problems of nonlinear fractional difference equations. Our problems contain difference and fractional sum boundary conditions. Existence and uniqueness of solutions are proved by using the Banach fixed‐point theorem, and existence of the positive solutions is proved by using the Krasnoselskii's fixed‐point theorem. Copyright © 2015 John Wiley & Sons, Ltd.     

On a nonlocal problem for the Laplace equation in the unit ball with fractional boundary conditions

In this paper, we investigate the correct solvability for the Laplace equation with a nonlocal boundary condition in the unit ball. The considered boundary operator is of fractional order. This problem is a generalization of the well‐known Bitsadze–Samarskii problem. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence and Ulam's type stability results for a class of fractional boundary value problems on a star graph

This paper is concerned with a nonlinear fractional boundary value problem on a star graph. By using a transformation, the suggested problem is converted into an equivalent system of fractional boundary value problem. Schaefer's fixed point theorem and Banach's contraction principle is used to establish its existence and uniqueness results. Further, different kinds of Ulam's type stability results for the proposed problem have been discussed. Finally, two examples are presented to illustrate the application of the obtained results.     

Three symmetric positive solutions for second-order nonlocal boundary value problems

Using the Leggett-Williams fixed point theorem,we will obtain at least three symmetric positive solutions to the second-order nonlocal boundary value problem of the form u(t)+g(t)f(t,u(t))=0,0相似文献   

THE SINGULARLY PERTURBED NONLOCAL BOUNDARY VALUE PROBLEMS FOR HIGHER ORDER NONLINEAR ELLIPTIC EQUATIONS

In this paper,a class of singular perturbation of noidocal boundary value problems forelliptic partial differentia[ equations of higher order is considered by using the differential in-equalities. The uniformly valid asymptotic expansion of solution is obtained.     

Two-point boundary value problems for finite fractional difference equations

In this paper, we introduce a two-point boundary value problem for a finite fractional difference equation. We invert the problem and construct and analyse the corresponding Green's function. We then provide an application and obtain sufficient conditions for the existence of positive solutions for a two-point boundary value problem for a nonlinear finite fractional difference equation.     

一类分数阶微分方程积分边值问题正解的分歧性

利用分歧方法和拓扑度理论,研究了一类带参数的分数阶微分方程积分边值问题正解的存在性.根据格林函数的性质,得到了系统正解的存在的若干充分条件.最后,通过数值例子验证了所得结果的有效性.     

Impulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative

In this paper, we investigate the existence of solutions of the periodic boundary value problem for nonlinear impulsive fractional differential equation involving Riemann-Liouville sequential fractional derivative by using monotone iterative method. An example is presented to illustrate our main result.     

一类分数阶微分方程组积分边值问题的正解

利用锥拉伸和压缩不动点定理,研究了一类具有Riemann-Liouvile分数阶积分条件的分数阶微分方程组边值问题.结合该问题相应Green函数的性质,获得了其正解的存在性条件,并给出了一些应用实例.     

Weighted Hermite–Hadamard–Mercer type inequalities for convex functions

In this paper, first, we prove the weighted Hermite–Hadamard–Mercer inequalities for convex functions, after we establish some new weighted inequalities connected with the right‐sides of weighted Hermite–Hadamard–Mercer type inequalities for differentiable functions whose derivatives in absolute value at certain powers are convex. The results presented here would provide extensions of those given in earlier works.     

Well-posedness of general nonlocal nonhomogeneous boundary value problems for pseudodifferential equations with partial derivatives

In the article we study the questions of well-posedness of general nonlocal boundary value problems for pseudodifferential equations in the Besov-type limit spaces.     

On a singular nonlocal time fractional order mixed problem with a memory term

We use the priori estimate method to prove the existence and uniqueness of a solution as well as its dependence on the given data of a singular time fractional mixed problem having a memory term. The considered fractional equation is associated with a nonlocal condition of integral type and a Neuman condition. Our results develop and show the efficiency and effectiveness of the energy inequalities method for the time fractional order differential equations with a nonlocal condition.