Complex Analysis in Locally Convex Spaces
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Complex analysis in locally convex spaces Home Complex analysis in locally convex spaces. Complex Analysis in Locally Convex Spaces. Read more. Nuclear Locally Convex Spaces.
Foundations of complex analysis in non locally convex spaces. Barrelled locally convex spaces. Topics in locally convex spaces. Moreover, since and are -osculating at , there are a function and such that. Choose such that. Therefore, for each such that , the operator from into is a contraction in the sense of [ 7 ].
Since is continuous at , we may further find such that. Set we have. This shows that.
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Then, by [ 7 , Theorem ], when , the operator has a unique fixed point , which is obviously a solution of 3. As an example of application of our main result, we study the stability of the solutions of an operator equation with respect to a parameter. Consider in the Hammerstein equation. In our case is a continuous linear operator on and is the so-called superposition operator. We have the following theorem.
Let be -bounded. Suppose that for each there exists such that the operator satisfies the Lipschitz condition. If is a solution of 4. Since the linear operator is -bounded, we can find a constant such that.
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If , then is clearly a solution of 4. Clearly the operator is continuous at. By the hypothesis made on the operator , there exists such that. Moreover, for each , we have and for any and. Then the result follows by Theorem 3. Now assume that is a solution of 4. Let be defined by.
The operator is continuous at and there exists such that. So the operators and are -osculating at. Further, assuming for some , we can find such that for any and. As before, the proof is completed by appealing to Theorem 3. Bogoljubov and N. Trombetta A: An implicit function theorem in complete -normed spaces. Trombetta A: -osculating operators in a space of continuous functions and applications.
Journal of Mathematical Analysis and Applications , 1 —