×

zbMATH — the first resource for mathematics

Continuity and differentiability of multivalued superposition operators with atoms and parameters. I. (English) Zbl 1237.47065
The article deals with a single- or multivalued nonlinear superposition operator \(S_f(\lambda,x):\;{\mathcal M}(S,U) \to {\mathcal M}(S,V)\), where \(U\) and \(V\) are quasi-pseudonormed spaces, \({\mathcal M}(S,U)\) and \({\mathcal M}(S,V)\) are spaces of measurable functions defined on \(S\) and taking values in \(U\) and \(V\), correspondingly, \(f:\Lambda \times S \times {\mathcal M}(S,U) \to {\mathcal M}(S,V)\) is a given single- or multivalued function, \[ S_f(\lambda,x) = \{\phi(s) \in f(\lambda,s,x(s)), \;\phi(\cdot) \in {\mathcal M}(S,V), \phi(\cdot) \text{ are constant on } S_i, \;i \in I\}, \] and, at last, \((S,\Sigma,\mu)\) is a complete \(\sigma\)-finite measure space and \(S_i \in \Sigma\) (\(i \in I\)) a fixed family of pairwise disjoint sets of positive measure (atoms). The first main result of this article gives natural conditions for the upper and lower semicontinuity of the operator \(S_f(\lambda,x)\) at a point \((\lambda_0,x_0)\). The other results are concerned with the differentiability properties of the operator \(S_f(\lambda,x)\). More precisely, the author describes conditions under which the following equations hold: \[ \lim_{x \to 0} \sup_{\lambda \in \Lambda} \frac{\sup \{y: y \in S_f(\lambda,x)\}}{\|x\|} = 0 \] or \[ \lim_{x \to 0,\, \lambda \to \lambda_0} \frac{\sup \{y: y \in S_f(\lambda,x)\}}{\|x\|} = 0. \] Apart from the main results which concern quasi-pseudonormed and ideal spaces, the author also considers the corresponding results in Orlicz and Lebesgue spaces.

MSC:
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
47H04 Set-valued operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E40 Spaces of vector- and operator-valued functions
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Appell, J., Upper estimates for superposition operators and some applications. Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), 149 - 159. · Zbl 0489.47017
[2] Appell, J., Über die Differenzierbarkeit des Superpositions-Operators in Orlicz- Räumen (in German). Math. Nachr. 123 (1985), 325 - 344. · Zbl 0537.47031
[3] Appell, J., de Pascale, E., Nguy\hat eñ, H. T. and Zabre\?ıko, P. P., Multi-valued superpositions. Dissertationes Math. (Rozprawy Mat.) 345 (1995), 1 - 97. · Zbl 0855.47037
[4] Appell, J., Nguy\hat eñ, H. T. and Zabre\?ıko, P. P., Multivalued superposition operators in ideal spaces of vector functions II. Indag. Math. 2 (1991)(4), 397 - 409. · Zbl 0748.47051
[5] Appell J. and Zabre\?ıko, P. P., Nonlinear Superposition Operators. Cambridge: Cambridge Univ. Press 1990.
[6] Aubin, J.-P. and Cellina, A., Differential Inclusions. Berlin: Springer 1984. · Zbl 0538.34007
[7] Babin, A. V. and Vishik, M. I., Attractors of Evolution Equations. Amsterdam: North-Holland 1992. · Zbl 0804.58003
[8] Borisovich, Yu. G., Gel’man, B. D., Myshkis, A. D. and Obukhovski\?ı, V. V., Introduction to the Theory of Multivalued Maps (in Russian). Voronezh: Izd. Voronezh. Gos. Univ. 1986. 123
[9] Eisner, J., Ku\check cera, M. and Recke, L., Smooth continuation of solutions and eigenvalues for variational inequalities based on the implicit function theorem. J. Math. Anal. Appl. 274 (2002), 159 - 180. · Zbl 1040.49006
[10] Eisner, J., Ku\check cera, M. and Väth, M., Degree and global bifurcation of ellip- tic equations with multivalued unilateral conditions. Nonlin. Anal. 64 (2006), 1710 - 1736. · Zbl 1111.47055
[11] Eisner, J., Ku\check cera, M. and Väth, M., Global bifurcation of a reaction-diffusion system with inclusions. J. Anal. Appl. 28 (2009), 373 - 409. · Zbl 1182.35025
[12] Eisner, J., Ku\check cera, M. and Väth, M., New bifurcation points for a reaction- diffusion system with two inequalities. J. Math. Anal. Appl. 365 (2010), 176 - 194. · Zbl 1185.35074
[13] Fletcher, P. and Lindgren, W. F., Quasi-Uniform Spaces. Lect. Notes Pure Appl. Math. 77. New York: Marcel Dekker 1982. · Zbl 0501.54018
[14] Krasnoselski\?ı, M. A., The continuity of a certain operator. Dokl. Akad. Nauk SSSR 77 (1951)(2), 185 - 188.
[15] Nguy\hat eñ, H. T., The theory of semimodules of infra-semiunits in ideal spaces of vector-valued functions, and its applications to integral operators (in Russian). Dokl. Akad. Nauk SSSR 317 (1991), 1303 - 1307; Engl. transl.: Soviet Math. Dokl. 43 (1991), 615 - 619. · Zbl 0752.46020
[16] Nguy\hat eñ, H. T. and Zabre\?ıko, P. P., Cones of vector functions in Orlicz spaces of vector functions (in Russian). Vesc\?ı Akad. Navuk BSSR, Ser. F\?ız.-Mat. Navuk 3 (1990), 30 - 34.
[17] Rudin, W., Functional Analysis. 14th ed., New Delhi: McGraw-Hill 1990. · Zbl 0867.46001
[18] \? Sen-Van (Wang Sheng-Wang), V., Differentiability of the Nemytski\?ı operator (in Russian). Dokl. Akad. Nauk SSSR 150 (1963)(5), 1198 - 1201; Engl. transl.: Soviet Math. Dokl. 150 (1963)(5), 834 - 837. · Zbl 0161.35002
[19] Vainberg, M. M., The operator of V. V. Nemytskij (in Russian). Ukrain. Mat. Zh. 7 (1955)(4), 363 - 378. · Zbl 0066.36306
[20] Valent, T., Boundary Value Problems of Finite Elasticity. New York: Springer 1988. · Zbl 0648.73019
[21] Väth, M., Ideal Spaces. Lect. Notes Math. 1664. Berlin: Springer 1997. · Zbl 0896.46018
[22] Väth, M., Integration Theory. A Second Course. Singapore: World Scientific 2002. · Zbl 1043.28001
[23] Väth, M., Continuity of single- and multivalued superposition operators in generalized ideal spaces of measurable vector functions. Nonlin. Funct. Anal. Appl. 11 (2006)(4), 607 - 646. · Zbl 1119.47060
[24] Väth, M., Generalized ideal spaces and applications to the superposition op- erator. In: Positivity IV - Theory and Applications (Proceedings Dresden (Germany) 2005; eds.: M. R. Weber et al.). Dresden: Technische Univ. Dresden 2006, pp. 147 - 154. M. Väth · Zbl 1128.47054
[25] Väth, M., Bifurcation for a reaction-diffusion system with unilateral obstacles with pointwise and integral conditions. Nonlin. Anal. Real World Appl. 12 (2011), 817 - 836. · Zbl 1225.35023
[26] Väth, M., Continuity and differentiability of multivalued superposition opera- tors with atoms and parameters II. Z. Anal. Anwend. 31 (2011) (to appear).
[27] Zabre\?ıko, P. P., Differentiability of nonlinear operators in Lp spaces (in Rus- sian). Dokl. Akad. Nauk SSSR 166 (1966)(5), 1039 - 1042; Engl. transl.: Soviet Math. Dokl. 166 (1966)(5), 224 - 228.
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.