Nonlinear functional random differential equations in Banach algebras

Abstract

In this paper the existence as well as the existence of the extremal solutions for first order nonlinear perturbed functional random differential equations is proved under
 mixed Lipschitz, compactness and monotonic conditions.

Nonlinear Functional Differential Equation in Banach Space: a Study of Its Solution

“…We use the following hybrid theorem as tool for proving the existence of solution of FONFDE (1.1). Theorem (2.4): (Dhage [7]) Let 𝑋 be a Banach algebra and let 𝐴, 𝐵 ∶ 𝑋 → 𝑋 be operators satisfying (a) 𝐴 is a 𝒟 − Lipschitzian with a 𝒟 − function 𝜇, (b) 𝐵 is compact and continuous, and (c) 𝑀𝜇(𝑟) < 𝑟 whenever 𝑟 > 0, where 𝑀 = ‖𝐵(𝑋)‖ = 𝑠𝑢𝑝{‖𝐵𝑥‖: 𝑥 ∈ 𝑋}.…”

“…It was observed that, very few reports are available on the study of nonlinear functional differential equations in Banach space. We referred the hybrid fixed point theorems in this study [4][5][6]9]. The FONFDE (1.1) contributes to the area of functional differential equations, newly.…”

An Outlook on Hybrid Fractional Modeling of a Heat Controller with Multi-Valued Feedback Control

“…Putting L * = ω , then we get L * M * 0 < 1. Consequently, the Dhage hybrid fixed point theorem's [19] presumptions hold, and if either condition (a) or (b) is valid, then Dhage's hybrid fixed point theorem [19] is justified. In order for µ = AµBϑ + Cµ, let µ ∈ X and ν ∈ S be random elements.…”

On a fractional integral equation of periodic functions involving Weyl–Riesz operator in Banach algebras

“…Recently, the topic of nonlinear integral and differential equations in Banach algebras is received the attention of several authors [1][2][3]. Integral equations are often applicable in the theory of radiative transfer, kinetic theory of gases, in the theory of neutron transport and in the traffic theory.…”