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.

show all references

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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