MTH641 GDB Fall 2020 Solution idea:
We are interested in the study of the existence of continuous solution of the following nonlinear Fredholm integral equation,
x(t)=f(t)+∫_a^t 〖g(t,s,x(s))ds,-∞<a≤t≤b<+∞〗 (1.1)
Where f(t)∈∁([a,b]) Usually the proof of the existence of a solution of (1.1) starts with some condition on the function g(t,s,x) 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 (t,s,x)be bounded w.r.t. x. Then we used Schaefer’s fixed- point theorem and prove the existence of a solution belonging C([a,b]). In the second case, we replace the strong condition that g(t,s,x) 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 ‖f(t)‖μ over the space and use Schauder’s fixed- point thermo .
In the second part of this work, we study the following nonlinear Volterra eqution,
x(t)=f(t)+∫_a^t 〖g(t,s,x(s))ds,-∞<a≤t≤b<+∞ (2.1)
where f(t)∈∁([a,b]) 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 n∈N an such that is a contraction on some closed ball of C([a,b]) 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)
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