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MTH641 Functional Analysis GDB Fall 2020 Solution / Discussion

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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

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