Upper and lower solutions method for Caputo-Hadamard fractional differential inclusions

Abstract

In this paper, we use some background concerning multivalued functions and set-valued analysis, the fixed point theorem of Bohnenblust-Karlin and the method of upper and lower solutions to investigate the existence of solutions for a class of boundary value problem of functional differential inclusions involving the Caputo-Hadamard fractional derivative.

Initial Value Problem For Nonlinear Fractional Differential Equations With ψ-Caputo Derivative Via Monotone Iterative Technique

“…At the present day, different kinds of fixed point theorems are widely used as fundamental tools in order to prove the existence and uniqueness of solutions for various classes of nonlinear fractional differential equations for details, we refer the reader to a series of papers [24][25][26][27][28][29][30] and the references therein, but here we focus on those using the monotone iterative technique, coupled with the method of upper and lower solutions. This method is a very useful tool for proving the existence and approximation of solutions to many applied problems of nonlinear differential equations and integral equations (see [31][32][33][34][35][36][37][38][39][40][41][42]).…”

On coupled systems of fractional impulsive differential equations by using a new Caputo-Fabrizio fractional derivative

“…This implies that N is well defined. Clearly, fixed points of N are solutions of problem (1). We show that N is a contraction.…”

“…In Section 2, we introduce all the background material used in this paper such as definition of Caputo-Fabrizio derivatives of fractional order and some properties of generalized Banach spaces and fixed point theory. In Sections 3 and 4, using Perov's and Schaefer fixed point type theorems in generalized Banach spaces, we prove some existence and compactness results for problem (1). An example is given to demonstrate the application of our main results in section 4.…”

“…A fractional order derivative is important for developing mathematical models in many scientific and engineering disciplines (see [7]). Several qualitative results for different classes of differential equations with different types of fractional integral and derivatives were obtained in [1,3,17,23,25,27,28].…”

Fractional q-Difference Inclusions in Banach Spaces

“…Fractional differential equations and inclusions have attracted much more interest of mathematicians and physicists which provides an efficiency for the description of many practical dynamical arising in engineering, vulnerability of networks (fractional percolation on random graphs), and other applied sciences [1][2][3][4][5][6][7][8]. Recently, Riemann-Liouville and Caputo fractional differential equations with initial and boundary conditions are studied by many authors; [2,[9][10][11][12][13][14]. In [15][16][17][18] the authors present some interesting results for classes of fractional differential inclusions.…”