Conjugate points for fractional differential equations

Let b \u3e 0. Let 1 \u3c α ≤ 2. The theory of u 0-positive operators with respect to a cone in a Banach space is applied to study the conjugate boundary value problem for Riemann-Liouville fractional linear differential equations D 0+α u + λp(t)u = 0, 0 \u3c t \u3c b, satisfying the conjugate boundary conditions u(0) = u(b) = 0. The first extremal point, or conjugate point, of the conjugate boundary value problem is defined and criteria are established to characterize the conjugate point. As an application, a fixed point theorem is applied to give sufficient conditions for existence of a solution of a related boundary value problem for a nonlinear fractional differential equation

Fixed point theorems for the sum of three classes of mixed monotone operators and applications

In this paper we develop various new fixed point theorems for a class of operator equations with three general mixed monotone operators, namely A(x,x)+B(x,x)+C(x,x)=x on ordered Banach spaces, where A, B, C are the mixed monotone operators. A is such that for any t∈(0,1), there exists φ(t)∈(t,1] such that for all x,y∈P, A(tx,t−1y)≥φ(t)A(x,y); B is hypo-homogeneous, i.e. B satisfies that for any t∈(0,1), x,y∈P, B(tx,t−1y)≥tB(x,y); C is concave-convex, i.e. C satisfies that for fixed y, C(⋅,y):P→P is concave; for fixed x, C(x,⋅): P→P is convex. Also we study the solution of the nonlinear eigenvalue equation A(x,x)+B(x,x)+C(x,x)=λx and discuss its dependency to the parameter. Our work extends many existing results in the field of study. As an application, we utilize the results obtained in this paper for the operator equation to study the existence and uniqueness of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions

On the Nonlinear Impulsive Ψ\Psi--Hilfer Fractional Differential Equations

In this paper, we consider the nonlinear $\Psi$-Hilfer impulsive fractional differential equation. Our main objective is to derive the formula for the solution and examine the existence and uniqueness of results. The acquired results are extended to the nonlocal $\Psi$-Hilfer impulsive fractional differential equation. We gave an applications to the outcomes we procured. Further, examples are provided in support of the results we got.Comment: 2