Yige Zhao

Positive solutions for boundary value problems of nonlinear fractional differential equations

In this paper, we consider the existence and multiplicity of posi- tive solutions for the nonlinear fractional dierential equation boundary-value problem D 0+u(t) = f(t,u(t)), 0 < t < 1 u(0) + u 0 (0) = 0, u(1) + u 0 (1) = 0 where 1 < 2 is a real number, and D 0+ is the Caputos fractional deriva- tive, and f : (0,1)◊(0,+1) ! (0,+1) is continuous. By means of a fixed-point theorem on cones, some existence and multiplicity results of positive solutions are obtained.

Positive Solutions to Boundary Value Problems of Nonlinear Fractional Differential Equations

We study the existence of positive solutions for the boundary value problem of nonlinear fractional differential equations 𝐷𝛼0

Theory of fractional hybrid differential equations

In this paper, we develop the theory of fractional hybrid differential equations involving Riemann-Liouville differential operators of order 0

Existence of solutions for a singular system of nonlinear fractional differential equations

In this paper, we establish the existence of positive solutions for a singular system of nonlinear fractional differential equations. The differential operator is taken in the standard Riemann-Liouville sense. By using Greens function and its corresponding properties, we transform the derivative systems into equivalent integral systems. The existence is based on a nonlinear alternative of Leray-Schauder type and Krasnoselskiis fixed point theorem in a cone.

Uniqueness of positive solutions for boundary value problems of singular fractional differential equations

In this article, we study the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem where 3 < α ≤ 4 is a real number, is the Riemann–Liouville fractional derivative and f : (0, 1] × [0, +∞) → [0, +∞) is continuous, (i.e. f is singular at t = 0). Our analysis relies on a fixed point theorem in partially ordered sets. As an application, an example is presented to illustrate the main results.

Positive solutions for a coupled system of nonlinear differential equations of mixed fractional orders

AbstractIn this article, we study the existence of positive solutions for a coupled system of nonlinear differential equations of mixed fractional orders where 2 < α ≤ 3, 3 < β ≤ 4, , are the standard Riemann-Liouville fractional derivative, and f, g : [0, 1] × [0, +∞) → [0, +∞) are given continuous functions, f(t, 0) ≡ 0, g(t, 0) ≡ 0. Our analysis relies on fixed point theorems on cones. Some sufficient conditions for the existence of at least one or two positive solutions for the boundary value problem are established. As an application, examples are presented to illustrate the main results.

Existence on positive solutions for boundary value problems of singular nonlinear fractional differential equations

In this paper, we study the existence of positive solutions for the singular nonlinear fractional differential equation boundary value problem D^α<inf>0+</inf>u(t) + f(t, u(t)) = 0, 0 < t < 1, u(0) = u(1) = u′(0) = 0, where 2 < a ≤ 3 is a real number, D^α<inf>0+</inf> is the Riemann-Liouville fractional derivative, and f : (0, 1] × [0,+∞) → [0,+∞) is continuous, lim<inf>t→0+</inf> f(t, ·) = +∞ (i.e., f is singular at t = 0). Our analysis rely on nonlinear alternative of Leray-Schauder type and Krasnoselskii fixed point theorem on a cone. As an application, an example is presented to illustrate the main results.

On the existence and uniqueness of solutions for initial value problem of nonlinear fractional differential equations

In this paper, we study the existence and uniqueness of solutions for the nonlinear fractional differential equation initial value problem Dαu(t) = f(t, u(t)), t∈ (0, T], t2−αu(t)| t=0 = b 1 , t2−αu′(t)| t=0 = b 2 , where 1 < a < 2 is a real number, Dα is the Riemann-Liouville fractional derivative. Using the monotone iterative method and the lower and upper solution method, some existence and uniqueness criteria are established, which extend the given results in recent literary works.