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A continuous regularization method of the first order for nonlinear monotone equations. (English. Russian original) Zbl 1148.47045

Russ. Math. 51, No. 1, 41-48 (2007); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2007, No. 1, 45-53 (2007).
Let \(X\) be a uniformly convex and uniformly smooth Banach space, \(A:X\rightarrow X^{\ast}\) be a monotone, boundedly Hölder continuous operator, and \(f\in X^{\ast}\). The main result of the paper is that, under suitable conditions, the normal solution of the equation \(Ax=f\) (i.e., the solution with minimum norm) is the norm limit of the trajectory of the first order problem \[ \begin{aligned} \frac{dJy(t)}{dt}+A(t)y(t)+\alpha(t)Jy(t) & =f(t),\\ y(t_{0}) & =y_{0}, \end{aligned} \] as \(t\rightarrow+\infty\), for any initial element \(y_{0}\). Here, \(J\) is the duality mapping, \(A(\cdot)\) and \(f(\cdot)\) are perturbations of \(A\) and \(f\), and \(\alpha(t)\) is a suitable differentiable positive decreasing function.

MSC:

47J06 Nonlinear ill-posed problems
47H05 Monotone operators and generalizations
47H14 Perturbations of nonlinear operators
34G10 Linear differential equations in abstract spaces
35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
47N20 Applications of operator theory to differential and integral equations
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References:

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