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Some topological properties preserved by nearness between operators and applications to P.D.E. (English) Zbl 0879.47039

The author deals with the theory of the so-called near operators introduced by S. Campanato [Rend. Acad. Naz. Sci. Detta XL, V. Ser. 13, No.1, 307-321 (1989; Zbl 0702.35084)] which have a connection with monotone operators. Basic results proved in Sections 1 and 2 say that if an operator \(B\) has (1) an open range or (2) a dense range then these properties are preserved for an operator \(A\) which is near \(B\).
An operator \(\phi \:\Omega_1 \subset \mathcal B\to \mathcal B\) (\(\mathcal B\) is a Banach space) is said to be near the identity \(I_{\mathcal B}\) if there exist \(\alpha >0\) and \(k\in (0,1)\) such that \[ |y_1-y_2-\alpha [\phi (y_1)-\phi (y_2)]|\leq k|y_1-y_2|,\;y_1,y_2\in \Omega_1. \] The operator \(\phi\) near \(I_{\mathcal B}\) maps open sets to open sets. It is injective and Lipschitzian, but such operators do not possess the open range property (unless \(\mathcal B\) is finite-dimensional).
Let \(A\) and \(B\) be two operators defined on a set \(\mathcal X\) with values in a Banach space \(\mathcal B\). The operator \(A\) is called near \(B\) if the operator \(\phi \: B(\mathcal X)\to A(\mathcal X)\), (correctly) defined by the formula \(\phi (y)=A(x),\;x\in B^{-1}(y)\), is near \(I_{\mathcal B}\). \(A\) is near \(B\) if and only if there exist \(\alpha >0\) and \(k\in (0,1)\) such that \[ |B(x_1)-B(x_2)-\alpha [A(x_1)-A(x_2)]|\leq k|B(x_1)-B(x_2)|,\;x_1,x_2\in \mathcal X. \] In Section 3 the author applies the first property of near operators to the proof of the local existence of solutions of the nonlinear elliptic system \[ \begin{cases} u\in H^2\cap H_0^1(\Omega , \mathbb R^N),\\ \sum_{i,j=1}^n A_{ij}(x,u)D_{ij}u=f(x),&x\in \Omega , \end{cases} \] where \(A_{ij}\) are \(N\times N\) matrices of functions satisfying certain conditions, \(f\in L_2(\Omega ,\mathbb R^N)\) with a small norm, \(n\leq 3\). The results of Campanato are used and generalized.
In Section 4 the theory of near operators is for the first time successfully applied to hyperbolic problems. By means of the above mentioned second property of near operators the author proves the existence of periodic solutions to abstract differential equation \(u''+F(t,u')+\mathcal A u+cu=g\), where \(\mathcal A\) is a linear symmetric operator, \(c\) a constant, \(g\) and \(F\) are periodic in time \(t\) and satisfy some additional assumptions. At the end he shows how the result can be applied to the wave equation with a nonlinear damping (similar assumptions as in G. Prodi [Ann. Mat. Pura Appl., IV. Ser. 42, 25-49 (1956; Zbl 0072.10101)].
Reviewer: L.Herrmann (Praha)

MSC:

47H99 Nonlinear operators and their properties
35J60 Nonlinear elliptic equations
35L70 Second-order nonlinear hyperbolic equations
35B10 Periodic solutions to PDEs
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References:

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