# American Institute of Mathematical Sciences

2015, 2015(special): 103-111. doi: 10.3934/proc.2015.0103

## Nonlocal problems in Hilbert spaces

 1 Department of Mathematics and Informatics, University of Perugia, Italy 2 Dept. of Engineering Sciences and Methods, University of Modena and Reggio Emilia, I-42100 3 Department of Physics, Informatics and Mathematics, University of Modena and Reggio Emilia, Italy

Received  September 2014 Revised  January 2015 Published  November 2015

An existence result for differential inclusions in a separable Hilbert space is furnished. A wide family of nonlocal boundary value problems is treated, including periodic, anti-periodic, mean value and multipoint conditions. The study is based on an approximation solvability method. Advanced topological methods are used as well as a Scorza Dragoni-type result for multivalued maps. The conclusions are original also in the single-valued setting. An application to a nonlocal dispersal model is given.
Citation: Irene Benedetti, Luisa Malaguti, Valentina Taddei. Nonlocal problems in Hilbert spaces. Conference Publications, 2015, 2015 (special) : 103-111. doi: 10.3934/proc.2015.0103
##### References:
 [1] J. Andres and L. Gorniewicz, Topological Fixed Point Principles for Boundary Value Problems, Kluwer, Dordrecht, 2003. [2] J. Andres, L. Malaguti and V. Taddei, A bounding function approach to multivalued boundary values problems, Dynam. Systems Appl. 16 (2007), 37-47. [3] I. Benedetti, N. V. Loi and L. Malaguti, Nonlocal problems for differential inclusions in Hilbert spaces, Set-Valued Var. Anal., 22 (2014), no. 3, 639-656. [4] I. Benedetti, L. Malaguti and V. Taddei, Nonlocal semilinear evolution equations without strong compactness: theory and applications, Bound. Value Probl., 2013:60, 18 pp. [5] I. Benedetti, L. Malaguti and V. Taddei, Semilinear evolution equations in abstract spaces and applications, Rend. Istit. Mat. Univ. Trieste, 44 (2012), 371-388. [6] I. Benedetti, L. Malaguti and V. Taddei, Two-points b.v.p. for multivalued equations with weakly regular r.h.s., Nonlinear Anal., 74 (2011), no. 11, 3657-3670. [7] I. Benedetti, V. Taddei and M. Väth, Evolution Problems with Nonlinear Nonlocal Boundary Conditions J. Dynam. Differential Equations 25 (2013), no. 2, 477-503. [8] A. Cwiszewski and W. Kryszewski, Constrained topological degree and positive solutions of fully nonlinear boundary value problems, J. Differential Equations 247 (2009), no. 8, 2235-2269. [9] K. Deimling, Multivalued Differential Equations, Walter de Gruyter & Co., Berlin, 1992. [10] M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, de Gruyter Series in Nonlinear Analysis and Applications 7, Walter de Gruyter & Co., Berlin, 2001. [11] K. Kuratowski and C. A. Ryll-Nardzewski, A general theorem on selectors. Bull. Acad. Polon. Sci. Sr. Sci. Math. Astronom. Phys. 13 (1965) 397-403. [12] A. Paicu and I. I. Vrabie, A class of nonlinear evolution equations subjected to nonlocal initial conditions, Nonlinear Anal., 72 (2010), no. 11, 4091-4100. [13] I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, 2nd Edition, Pitman Monographs and Surveys in Pure and Applied Mathematics, 75 Longman Scientific & Technical, Harlow, 1995.

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##### References:
 [1] J. Andres and L. Gorniewicz, Topological Fixed Point Principles for Boundary Value Problems, Kluwer, Dordrecht, 2003. [2] J. Andres, L. Malaguti and V. Taddei, A bounding function approach to multivalued boundary values problems, Dynam. Systems Appl. 16 (2007), 37-47. [3] I. Benedetti, N. V. Loi and L. Malaguti, Nonlocal problems for differential inclusions in Hilbert spaces, Set-Valued Var. Anal., 22 (2014), no. 3, 639-656. [4] I. Benedetti, L. Malaguti and V. Taddei, Nonlocal semilinear evolution equations without strong compactness: theory and applications, Bound. Value Probl., 2013:60, 18 pp. [5] I. Benedetti, L. Malaguti and V. Taddei, Semilinear evolution equations in abstract spaces and applications, Rend. Istit. Mat. Univ. Trieste, 44 (2012), 371-388. [6] I. Benedetti, L. Malaguti and V. Taddei, Two-points b.v.p. for multivalued equations with weakly regular r.h.s., Nonlinear Anal., 74 (2011), no. 11, 3657-3670. [7] I. Benedetti, V. Taddei and M. Väth, Evolution Problems with Nonlinear Nonlocal Boundary Conditions J. Dynam. Differential Equations 25 (2013), no. 2, 477-503. [8] A. Cwiszewski and W. Kryszewski, Constrained topological degree and positive solutions of fully nonlinear boundary value problems, J. Differential Equations 247 (2009), no. 8, 2235-2269. [9] K. Deimling, Multivalued Differential Equations, Walter de Gruyter & Co., Berlin, 1992. [10] M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, de Gruyter Series in Nonlinear Analysis and Applications 7, Walter de Gruyter & Co., Berlin, 2001. [11] K. Kuratowski and C. A. Ryll-Nardzewski, A general theorem on selectors. Bull. Acad. Polon. Sci. Sr. Sci. Math. Astronom. Phys. 13 (1965) 397-403. [12] A. Paicu and I. I. Vrabie, A class of nonlinear evolution equations subjected to nonlocal initial conditions, Nonlinear Anal., 72 (2010), no. 11, 4091-4100. [13] I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, 2nd Edition, Pitman Monographs and Surveys in Pure and Applied Mathematics, 75 Longman Scientific & Technical, Harlow, 1995.
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