Abstract
This paper is concerned with the measure of noncompactness in the spaces of continuous functions and semilinear functional differential equations with nonlocal conditions in Banach spaces. The relationship between the Hausdorff measure of noncompactness of intersections and the modulus of equicontinuity is studied for some subsets related to the semigroup of linear operators in Banach spaces. The existence of mild solutions is obtained for a class of nonlocal semilinear functional differential equations without the assumption of compactness or equicontinuity on the associated semigroups of linear operators.
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References
Akmerov, R. R., Kamenski, M. I., Potapov, A. S., et al.: Meansure of Noncompactness and Condensing Operators, Birkhauser Verlag, Basel, 1992
Ayerbe Toledano, J. M., Dominguez Benavides, T., Lopez Acedo, G.: Meansure of Noncompactness in Metric Fixed Point Theory, Birkhauser Verlag, Basel, 1997
Bahuguna, D.: Existence, uniqueness and regularity of solutions to semilinear nonlocal functional differential problems. Nonlinear Anal., 57, 1021–1028 (2004)
Balachandran, K., Chandrasekaran, M.: Existence of solutions of a delay differential equation with nonlocal condition. Indian J. Pure. Appl. Math., 27, 443–449 (1996)
Banas, J., Goebel, K.: Measure of Noncompactness in Banach spaces, Lecture Notes in Pure and Applied Math. Vol. 60, Dekker, New York, 1980
Banas, J., Sadarangani, K.: On some mearsure of noncompactness in the space of continuous functions. Nonlinear Anal., 68, 377–383 (2008)
Bothe, D.: Multivalued perturbations of m-accretive differential inclusions. Isreal J. Math., 108, 109–138 (1998)
Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl., 162, 497–505 (1991)
Byszewski, L.: Existence and uniqueness of solutions of semilinear evolution nonlocal Cauchy problem. Zesz. Nauk. Pol. Rzes. Mat. Fiz., 18, 109–112 (1993)
Byszewski, L., Akca, H.: Existence of solutions of a semilinear functional-differential evolution nonlocal problem. Nonlinear Anal., 34, 65–72 (1998)
Byszewski, L., Lakshmikantham, V.: Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl. Anal., 40, 11–19 (1990)
Darbo, G.: Punti uniti in transformazioni a condominio non compatto. Rend. Sem. Mat. Univ. Padova, 24, 84–92 (1955)
Dong, Q., Fan, Z., Li, G.: Existence of solutions to nonlocal neutral functional differential and integrodifferential equations. Intern. J. Nonlinear Sci., 5(2), 140–151 (2008)
Dong, Q., Li, G., Zhang, J.: Quasilinear nonlocal intergrodifferential equations in Banach spaces. Electronic J. Diff. Equati., 2008(19), 1–8 (2008)
Dong, Q., Li, G.: Existence of solutions for semilinear differential equations with nonlocal conditions in Banach spaces. Electronic J. Quali. Theory Diff. Equati., 2009(47), 1–13 (2009)
Fan, Z., Dong, Q., Li, G.: Semilinear differential equations with nonlocal conditions in Banach spaces. Intern. J. Nonlinear Sci., 2(3), 131–139 (2006)
Guedda, L.: On the existence of mild solutions for neutral functional differential inclusions in Banach spaces. Electronic J. Quali. Theory of Diff. Equati., 2007(2), 1–15 (2007)
Heinz, H. P.: On the behavior of measure of noncompactness with respect to differentiation and integration of vector-valued functions. Nonlinear Anal. TMA, 7, 1351–1371 (1983)
Kamenskii, M., Obukhovskii, V., Zecca, P.: Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Ser. Nonlinear Anal. Appl., Vol.7, de Gruyter, Berlin, 2001
Kisielewicz, M.: Multivalued differential equations in separable Banach spaces. J. Optim. Theory Appl., 37, 231–249 (1982)
Kuratowski, K.: Sur les espaces complets. Fund. Math., 15, 301–309 (1930)
Li, F., N’Guerekata, G. M.: An existence result for neutral delay integrodifferential equations with fractional order and nonlocal conditions. Abstr. Appl. Anal., 2011, Artical ID 952782 (2011)
Ntouyas, S. K., Tsamatos, P. Ch.: Global existence for semilinear evolution equations with nonlocal conditions. J. Math. Anal. Appl., 210, 679–687 (1997)
Ntouyas, S. K., Tsamatos, P. Ch.: Global existence for semilinear evolution integrodifferential equations with delay and nonlocal conditions. Appl. Anal., 64, 99–105 (1997)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983
Xue, X.: Nonlinear differential equations with nonlocal conditions in Banach spaces. Nonlinear Anal., 63, 575–586 (2005)
Xue, X.: Semilinear nonlocal problems without the assumptions of compactness in Banach spaces. Anal. Appl., 8, 211–225 (2010)
Xue, X.: Lp theory for semilinear nonlocal problems with measure of noncompactness in separable Banach spaces. J. Fixed Point Theory Appl., 5, 129–144 (2009)
Zhu, L., Li, G.: Existence results of semilinear differential equations with nonlocal initial conditions in Banach spaces. Nonlinear Anal., 74, 5133–5140 (2011)
Zhu, L., Dong, Q., Li, G.: Impulsive differential equations with nonlocal conditions in general Banach spaces. Adv. Differential Equ., 2012(10), 1–11 (2012)
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Supported by National Natural Science Foundation of China (Grant Nos. 11271316 and 11201410) and Natural Science Foundation of Jiangsu Province (Grant No. BK2012260)
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Dong, Q.X., Li, G. Measure of noncompactness and semilinear nonlocal functional differential equations in Banach spaces. Acta. Math. Sin.-English Ser. 31, 140–150 (2015). https://doi.org/10.1007/s10114-015-3097-z
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DOI: https://doi.org/10.1007/s10114-015-3097-z
Keywords
- Measure of noncompactness
- equicontinuity
- differential equation
- nonlocal condition
- C 0-semigroup
- mild solution