Existence results for some differential equations with deviating argument

Abstract

In this paper we shall establish sufficient conditions for the existence of solutions of some differential equation and its solvability in C L , subset of the Banach space (C[a, b], •). The main tool used in our study is the nonexpansive operator technique.

Sufficient conditions on existence of solution for nonlinear fractional iterative integral equation

“…In our study we employ non-expansive operator technique as essential tools, and include the principle of contraction and Schaefer's fixed point theorem. The new result generalizes the previous results, see [1,3,10,14,18]. Further we also establish some related existence results for some integral equations with a modified argument of fractional order, see the details in [6].…”

On simple iterative fractional order differential equations

“…We popularize and extends the classic model based on the concept of fractional calculus and the most befitting formula is the iterative fractional integral and differential equations. Many authors have studied this area including (see [1,2,3,4,5,6,7,8,9,10]). The first work in this field was that of E, Eder [11] in 1984, which gave the existence and unique solutions of equation w (τ) = w(w(τ))…”

“…Furthermore, Lauran [3] investigated the non-autonomous equation using Picard's successive approximation:…”

Existence and approximation of solutions of fractional order iterative differential equations

“…In fact, integer order iterative differential equations give a good approach to searching for approximate solutions and have been discussed by several authors [1][2][3][4][5][6][7][8][9][10][11] owing to their wide applications in engineer control and computational mathematics.…”

“…Thereafter, fractional massspring-damper system [35][36][37][38] which involving fractional Nutting's law and Hooke's law are studied by more and more researchers. Motivated by the wide application of fractional massspring-damper system [35][36][37][38] and the papers [10,11,33], we will apply the technique of nonexpansive mappings and explore suitable work spaces via powerful norms to study the following generalized fractional mass-spring-damper system. More precisely, we firstly study the existence and approximation of solutions of fractional order iterative differential equations of the form:…”