Existence of the Unique Nontrivial Solution for Mixed Fractional Differential Equations

In this paper, we consider the differential equations with right-sided Caputo and left-sided Riemann-Liouville fractional derivatives. Furthermore, the expression of Green’s function is derived, and its properties are investigated. By the fixed-point theorem for both φ − h , e -concave operators and mixed monotone operators, we get the existence and uniqueness of the solution, respectively. As applications, some examples are provided to illustrate our main results.

Existence and Uniqueness of Positive Solutions for a Class of Nonlinear Fractional Differential Equations with Singular Boundary Value Conditions

This paper focuses on a singular boundary value (SBV) problem of nonlinear fractional differential (NFD) equation defined as follows: D 0 + β υ τ + f τ , υ τ = 0 , τ ∈ 0,1 , υ 0 = υ ′ 0 = υ ″ 0 = υ ″ 1 = 0 , where 3 < β ≤ 4 , D 0 + β is the standard Riemann–Liouville fractional (RLF) derivative. The nonlinear function f τ , υ τ might be singular on the spatial and temporal variables. This paper proves that a positive solution to the SBV problem exists and is unique, taking advantage of Green’s function through a fixed-point (FP) theory on cones and mixed monotone operators.

Existence of positive solutions for a class of fractional differential equations with the derivative term via a new fixed point theorem

In this paper, we firstly establish the existence and uniqueness of solutions of the operator equation $A(x,x)+ B(x,x)+C(x)+e = x$ A ( x , x ) + B ( x , x ) + C ( x ) + e = x , where A and B are two mixed monotone operators, C is a decreasing operator, and $e\in P$ e ∈ P with $\theta \leq e \leq h$ θ ≤ e ≤ h . Then, using our abstract theorem, we prove a class of fractional boundary value problems with the derivative term to have a unique solution and construct the corresponding iterative sequences to approximate the unique solution.

Probabilistic $ (\omega, \gamma, \phi) $-contractions and coupled coincidence point results

<abstract><p>In this paper, we introduce the notion of probabilistic $ (\omega, \gamma, \phi) $-contraction and establish the existence coupled coincidence points for mixed monotone operators subjected to the introduced contraction in the framework of ordered Menger $ PM $-spaces with Had${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over z} }}$ić type $ t $-norm. As an application, a corresponding result in the setup of fuzzy metric space is also obtained.</p></abstract>

Existence of positive solutions for half-linear fractional order BVPs by application of mixed monotone operators

In this paper we developed tripled fixed point theorems of ternary operator on partially ordered metric spaces. As an application we established existence of positive solutions for half-linear fractional order boundary value problem.

Some Operations on Mixed Monotone Operator in Banach Spaces

This paper discusses some operations on mixed monotone operator in Banach space, especially addition an multiplication operations. We will prove the sum and product of two mixed monotone operators. The proof using some relevant definitions. The result is the sum o of them is a mixed monotone operator and the product is too if both satisfy some conditions

Positive Solutions for a Three-Point Boundary Value Problem of Fractional Q-Difference Equations

In this work, a three-point boundary value problem of fractional q-difference equations is discussed. By using fixed point theorems on mixed monotone operators, some sufficient conditions that guarantee the existence and uniqueness of positive solutions are given. In addition, an iterative scheme can be made to approximate the unique solution. Finally, some interesting examples are provided to illustrate the main results.

Coupled fixed point theorems in complete metric spaces endowed with a directed graph and application

The purpose of this paper is to present some existence results for coupled fixed point of a (φ,ψ) —contractive condition for mixed monotone operators in metric spaces endowed with a directed graph. Our results generalize the results obtained by Jain et al. in (International Journal of Analysis, Volume 2014, Article ID 586096, 9 pages). Moreover, we have an application to some integral system to support the results.