MTH641 Functional Analysis GDB Fall 2020 Solution / Discussion
Tags:
Share the GDB Question & Discuss Here....
Stay touched with this discussion, Solution idea will be uploaded as soon as possible in replies here before the due date.
Solution:
We are interested in the study of the existence of continuous solution of the following nonlinear Fredholm integral equation,
(1.1)
Where Usually the proof of the existence of a solution of (1.1) starts with some condition on the function as well as the limits of integration a,b and the function f(t).
Based on these condition ,a Banach space is chosen in such a way that the existence problem is converted into a fixed-point problem for an operator over this Banach space.In the first case , we use some conditions on the function g(t,s,x) and we required that )be bounded w.r.t. x. Then we used Schaefer’s fixed- point theorem and prove the existence of a solution belonging. In the second case, we replace the strong condition that is bounded w.r.t. x by a weaker condition. To prove the existence of a continuous solution of (1.1) in this case , we introduce a new norm over the space and use Schauder’s fixed- point thermo .
In the second part of this work, we study the following nonlinear Volterra eqution,,
where The main tool in the proof of the existence of a solution of (1.2) is the Leray-Schauder principle combined with Gronwall’s inequality.Also, we prove the uniqueness of the solution of (1.2) by showing that there exists an such that is a contraction on some closed ball of containing the possible solutions of (1.2) .We prove the existence of continuous solution of (1.1).In the second part , we investigate the existence and uniqueness of the solution of the nonlinear Volterra equation(1.2)
MTH641 Functional Analysis GDB Fall 2020 Solution
https://youtu.be/LBDvvZnEVDc
© 2021 Created by + M.Tariq Malik. Powered by
Promote Us | Report an Issue | Privacy Policy | Terms of Service
We are user-generated contents site. All product, videos, pictures & others contents on site don't seem to be beneath our Copyrights & belong to their respected owners & freely available on public domains. We believe in Our Policy & do according to them. If Any content is offensive in your Copyrights then please email at m.tariqmalik@gmail.com with copyright detail & We will happy to remove it immediately.
Management: Admins ::: Moderators
Awards Badges List | Moderators Group
All Members | Featured Members | Top Reputation Members | Angels Members | Intellectual Members | Criteria for Selection
Become a Team Member | Safety Guidelines for New | Site FAQ & Rules | Safety Matters | Online Safety | Rules For Blog Post