## 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

### Citations:

Zbl 0702.35084; Zbl 0072.10101
Full Text:

### References:

 [1] C. Baiocchi: Soluzioni ordinarie e generalizzate del problema di Cauchy per equazioni differenziali astrate del secondo ordine in spazi di Hilbert. Ricerche Mat. (1967), 27-95. · Zbl 0157.21602 [2] S. Campanato: A Cordes type condition for non linear and variational systems. Rend. Accad. Naz. Sci. XL Mem. Mat. $$107^0$$ (1989), no. XIII, fasc. 20, 307-321. · Zbl 0702.35084 [3] S. Campanato: Non variational basic parabolic systems of second order. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Mem. (9) Mat. Appl. 2 (1991), 129-136. · Zbl 0749.35016 [4] S. Campanato: Sistemi differenziali del secondo ordine di tipo ellittico. Quaderno n. 1 del Dottorato di ricerca in Mat., Univ. di Catania (1991). [5] S. Campanato: A history of Cordes condition for second order elliptic operators. Res. Notes in Appl. Math. 29, Dedicated to E. Magenes, J. L. Lions and C. Baiocchi (eds.), 1993, pp. 319-325. · Zbl 0819.35029 [6] M. J. Greenberg: Lectures on algebraic topology. W. A. Benjamin Inc, New York, 1967. · Zbl 0169.54403 [7] A. Haraux: Non linear evolution equations-Global behavior of solutions. Lectures Notes in Math. 841, Springer, New York, 1981. · Zbl 0461.35002 [8] L. Herrmann: Periodic solutions of abstract differential equations: the Fourier method. Czechoslovak Math. J. 30 (105) (1980), 177-206. · Zbl 0445.35013 [9] J.L. Lions, E. Megenes: Problemes aux limites non homogenes et applications, vol. I. vol. 1, Dunod, Paris, 1968. [10] G.Prodi: Soluzioni periodiche di equazioni a derivate parziali di tipo iperbolico non lineari. Ann. Mat. Pura Appl. (4) 42 (1956), 25-49. · Zbl 0072.10101 [11] G. Prodi: Soluzioni periodiche dell’equazione delle onde con termine dissipativo non lineare. Rend. Sem. Mat. Univ. Padova 36 (1966), 37-49. · Zbl 0145.35601 [12] P. H. Rabinowitz: Periodic solutions of non linear hyperbolic partial differential equations. Comm. Pure Appl. Math. XX (1967), 145-205. · Zbl 0152.10003 [13] A. Tarsia: Sistemi parabolici non variazionali con soluzioni verificanti una condizione di periodicita’ generalizzata. Rend. Circ. Mat. Palermo (2) XLII (1993), 135-154. · Zbl 0826.35049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.