In this article, we discuss a new coupled system of fractional differential equations with integral boundary conditions $$\textstyle\begin{cases} D^{\alpha}u(t)+f(t,v(t))=a,\quad 0< t< 1,\\ D^{\beta}v(t)+g(t,u(t))=b,\quad 0< t< 1,\\ u(0)=0,\qquad u(1)=\int_{0}^{1} \phi(t)u(t)\,dt,\\ v(0)=0,\qquad v(1)=\int_{0}^{1} \psi(t)v(t)\,dt, \end{cases} $$
 where $1< \alpha,\beta\le2, f,g \in C([0,1]\times(-\infty,+\infty ),(-\infty,+\infty)), \phi,\psi\in L^{1}[0,1]$
 , $a,b$
 are constants and D denotes the usual Riemann-Liouville fractional derivative. Based upon a fixed point theorem of increasing φ-
 $(h,e)$
 -concave operators, we establish the existence and uniqueness of solutions for the new coupled system dependent on two constants. And then the obtained result is well demonstrated with the aid of an interesting example.

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Unique solutions for a new coupled system of fractional differential equations

This paper is concerned with the existence and uniqueness of solutions for a coupled system of Hadamard type fractional differential equations and integral boundary conditions. We emphasize that much work on fractional boundary value problems involves either Riemann-Liouville or Caputo type fractional differential equations. In the present work, we have considered a new problem which deals with a system of Hadamard differential equations and Hadamard type integral boundary conditions. The existence of solutions is derived from Leray-Schauder’s alternative, whereas the uniqueness of solution is established by Banach’s contraction principle. An illustrative example is also included.

A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations

Positive solutions of an abstract fractional semipositone differential system model for bioprocesses of HIV infection

In the this paper, we establish sufficient conditions for the existence and nonexistence of positive solutions to a general class of integral boundary value problems for a coupled system of fractional differential equations. The differential operator is taken in the Riemann-Liouville sense. Our analysis rely on Banach fixed point theorem, nonlinear differentiation of Leray-Schauder type and the fixed point theorems of cone expansion and compression of norm type. As applications, some examples are also provided to illustrate our main results.

Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions

Existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions

We consider the existence and uniqueness of positive solution to nonzero boundary values problem for a coupled system of fractional differential equations. The differential operator is taken in the standard Riemann-Liouville sense. By using Banach fixed point theorem and nonlinear differentiation of Leray-Schauder type, the existence and uniqueness of positive solution are obtained. Two examples are given to demonstrate the feasibility of the obtained results.

/pdf/positive-solution-to-nonzero-boundary-values-problem-for-a-3jc0c7a83j.pdf

Positive Solution to Nonzero Boundary Values Problem for a Coupled System of Nonlinear Fractional Differential Equations

We consider the existence and multiplicity of concave positive solutions for boundary value problem of nonlinear fractional differential equation with -Laplacian operator , , , , , where , , , denotes the Caputo derivative, and is continuous function, , , . By using fixed point theorem, the results for existence and multiplicity of concave positive solutions to the above boundary value problem are obtained. Finally, an example is given to show the effectiveness of our works.

/pdf/existence-of-concave-positive-solutions-for-boundary-value-4dzadz1h30.pdf

Zhigang Liu

Papers

Positive Solution to Nonzero Boundary Values Problem for a Coupled System of Nonlinear Fractional Differential Equations

Existence of Concave Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation with -Laplacian Operator