Abstract
We investigate the effect of nonlocal conditions expressed by linear continuous mappings over the hypotheses which guarantee the existence of global mild solutions for functional-differential equations in a Banach space. A progressive transition from the Volterra integral operator associated to the Cauchy problem, to Fredholm type operators appears when the support of the nonlocal condition increases from zero to the entire interval of the problem. The results are extended to systems of equations in a such way that the system nonlinearities behave independently as much as possible and the support of the nonlocal condition may differ from one variable to another.
Authors
Tiziana Cardinali
Department of Mathematics and Informatics, University of Perugia, Perugia, Italy
Radu Precup
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
Paola Rubbioni
Department of Mathematics and Informatics, University of Perugia, Perugia, Italy
Keywords
Paper coordinates
T. Cardinali, R. Precup, P. Rubbioni, A unified existence theory for evolution equations and systems under nonlocal conditions, J. Math. Anal. Appl. 432 (2015), 1039-1057, https://doi.org/10.1016/j.jmaa.2015.07.019
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About this paper
Journal
Journal of Mathematical Analysis and Applications
Publisher Name
Elsevier
Print ISSN
Online ISSN
0022-247X
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[2] A. Ambrosetti, Un teorema di esistenza per le equazioni differenziali negli spazi di Banach, Rend. Semin. Mat. Univ. Padova, 39 (1967), pp. 349-360, Google Scholar
[3] J. Banas, K. Goebel, Measure of Noncompactness in Banach Spaces, Lect. Notes Pure Appl. Math., vol. 60, Marcel Dekker, New York (1980) Google Scholar
[4] M. Benchohra, S. Ntouyas, Nonlocal Cauchy problems for neutral functional differential and integrodifferential inclusions in Banach spaces, J. Math. Anal. Appl., 258 (2001), pp. 573-590 Google Scholar
[6] A. Boucherif, R. Precup, On nonlocal initial value problem for first order differential equations, Fixed Point Theory, 4 (2003), pp. 205-212 Google Scholar
[12] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin (1985), Google Scholar
[13] Z. Denkowski, S. Migorski, N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer, Boston (2003),Google Scholar
[14] Z. Fan, Impulsive problems for semilinear differential equations with nonlocal conditions Nonlinear Anal., 72 (2010), pp. 1104-1109 Google Scholar
[15] H.P. Heinz On the behaviour of measures of noncompactness with respect to differentiation and integration on vector-valued functions Nonlinear Anal., 7 (1983), pp. 1351-1371 sGoogle Scholar
[16] M. Kamenskii, V. Obukhovskii, P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Ser. Nonlinear Anal. Appl., vol. 7, De Gruyter, Berlin (2001) Google Scholar
[17] J. Liang, J.H. Liu, T.J. Xiao, Nonlocal Cauchy problems governed by compact operator families, Nonlinear Anal., 57 (1994), pp. 183-189, Google Scholar
[18] Y. Lin, J. Liu, Semilinear integrodifferential equations with nonlocal Cauchy problems, Nonlinear Anal., 26 (1996), pp. 1023-1033 Google Scholar
[19] H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal., 4 (1980), pp. 985-999 Google Scholar
[20] S.K. Ntouyas, P.Ch. Tsamatos, Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl., 210 (1997), pp. 679-687 Google Scholar
[21] D. O’Regan, R. Precup, Existence criteria for integral equations in Banach spaces, J. Inequal. Appl., 6 (2001), pp. 77-97, Google Scholar
[22] D. O’Regan, R. Precup, Theorems of Leray–Schauder Type and Applications, Gordon and Breach, Amsterdam (2001), Google Scholar
[23] R.S. Varga, Matrix Iterative Analysis, (second edition), Springer, Berlin (2000), Google Scholar
[24] J.R.L. Webb, G. Infante, Positive solutions of nonlocal initial boundary value problems involving integral conditions, NoDEA Nonlinear Differential Equations Appl., 15 (2008), pp. 45-67 Google Scholar