Existence of mild solutions for a class of non-autonomous evolution equations with nonlocal initial conditions
-
2221
Downloads
-
3515
Views
Authors
Tahira Jabeen
- Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan.
Vasile Lupulescu
- Constantin Brancusi University, Republicii 1, 210152 Targu-Jiu, Romania.
Abstract
Using the techniques of measures of noncompactness and Schauder fixed point theorem, we present some existence results
for mild solutions of a class of nonlocal evolution equations involving causal operators. Moreover, we obtain the compactness
of the set of global mild solutions. An example is given to show the efficiency and usefulness of the results.
Share and Cite
ISRP Style
Tahira Jabeen, Vasile Lupulescu, Existence of mild solutions for a class of non-autonomous evolution equations with nonlocal initial conditions, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 1, 141--153
AMA Style
Jabeen Tahira, Lupulescu Vasile, Existence of mild solutions for a class of non-autonomous evolution equations with nonlocal initial conditions. J. Nonlinear Sci. Appl. (2017); 10(1):141--153
Chicago/Turabian Style
Jabeen, Tahira, Lupulescu, Vasile. "Existence of mild solutions for a class of non-autonomous evolution equations with nonlocal initial conditions." Journal of Nonlinear Sciences and Applications, 10, no. 1 (2017): 141--153
Keywords
- Non-autonomous evolution equation
- nonlocal condition
- evolution semigroups
- causal operator
- mild solution.
MSC
- 34A12
- 34A30
- 34G10
- 47D06
- 47H30
References
-
[1]
R. P. Agarwal, S. Arshad, V. Lupulescu, D. O’Regan, Evolution equations with causal operators, Differ. Equ. Appl., 7 (2015), 15–26.
-
[2]
R. P. Agarwal, J. Banaś, B. C. Dhage, S. D. Sarkate, Attractivity results for a nonlinear functional integral equation, Georgian Math. J., 18 (2011), 1–19.
-
[3]
S. Aizicovici, H.-W. Lee, Nonlinear nonlocal Cauchy problems in Banach spaces, Appl. Math. Lett., 18 (2005), 401–407.
-
[4]
S. Aizicovici, M. McKibben, Existence results for a class of abstract nonlocal Cauchy problems, Nonlinear Anal., 39 (2000), 649–668.
-
[5]
S. Aizicovici, V. Staicu, Multivalued evolution equations with nonlocal initial conditions in Banach spaces, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 361–376.
-
[6]
K. Balachandran, J. J. Trujillo, The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces, Nonlinear Anal., 72 (2010), 4587–4593.
-
[7]
D. Baleanu, H. Jafari, H. Khan, S. J. Johnston, Results for mild solution of fractional coupled hybrid boundary value problems, Open Math., 13 (2015), 601–608.
-
[8]
M. Benchohra, E. P. Gatsori, S. K. Ntouyas, Existence results for semi-linear integrodifferential inclusions with nonlocal conditions, Rocky Mountain J. Math., 34 (2004), 833–848.
-
[9]
M. Benchohra, J. Henderson, S. K. Ntouyas, Existence results for impulsive multivalued semilinear neutral functional differential inclusions in Banach spaces, J. Math. Anal. Appl., 263 (2001), 763–780.
-
[10]
M. Benchohra, G. M. N’Guérékata, D. Seba, Measure of noncompactness and nondensely defined semilinear functional differential equations with fractional order, Cubo, 12 (2010), 35–48.
-
[11]
I. Benedetti, N. V. Loi, L. Malaguti, Nonlocal problems for differential inclusions in Hilbert spaces, Set-Valued Var. Anal., 22 (2014), 639–656.
-
[12]
I. Benedetti, L. Malaguti, V. Taddei, I. I. Vrabie, Semilinear delay evolution equations with measures subjected to nonlocal initial conditions, Ann. Mat. Pura Appl., 195 (2016), 1639–1658.
-
[13]
L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162 (1991), 494–505.
-
[14]
L. Byszewski, V. Lakshmikantham, Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. Anal., 40 (1991), 11–19.
-
[15]
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.
-
[16]
T. Cardinali, P. Rubbioni, On the existence of mild solutions of semilinear evolution differential inclusions, J. Math. Anal. Appl., 308 (2005), 620–635.
-
[17]
T. Cardinali, P. Rubbioni, Corrigendum and addendum to ''On the existence of mild solutions of semilinear evolution differential inclusions'', [J. Math. Anal. Appl., 308 (2005), 620–635], J. Math. Anal. Appl., 438 (2016), 514–517.
-
[18]
P.-Y. Chen, Y.-X. Li, Q. Li, Existence of mild solutions for fractional evolution equations with nonlocal initial conditions, Ann. Polon. Math., 110 (2014), 13–24.
-
[19]
P.-Y. Chen, Y.-X. Li , Nonlocal problem for fractional evolution equations of mixed type with the measure of noncompactness, Abstr. Appl. Anal., 2013 (2013), 12 pages.
-
[20]
C. Corduneanu, Functional equations with causal operators, Stability and Control: Theory, Methods and Applications, Taylor & Francis, London (2002)
-
[21]
R. F. Curtain, H. Zwart, An introduction to infinite-dimensional linear systems theory, Texts in Applied Mathematics, Springer-Verlag,, New York (1995)
-
[22]
A. Debbouche, D. Baleanu, R. P. Agarwal, Nonlocal nonlinear integrodifferential equations of fractional orders, Bound. Value Probl., 2012 (2012 ), 10 pages.
-
[23]
R. E. Edwards, Functional analysis, Theory and applications, Holt, Rinehart and Winston, New York-Toronto- London (1965)
-
[24]
Z.-B. Fan, G. Li, Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Funct. Anal., 258 (2010), 1709–1727.
-
[25]
X.-L. Fu, K. Ezzinbi, Existence of solutions for neutral functional differential evolution equations with nonlocal conditions, Nonlinear Anal., 54 (2003), 215–227.
-
[26]
M. A. E. Herzallah, D. Baleanu, Existence of a periodic mild solution for a nonlinear fractional differential equation, Comput. Math. Appl., 64 (2012), 3059–3064.
-
[27]
L.-Y. Hu, Y. Ren, R. Sakthivel, Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays, Semigroup Forum, 79 (2009), 507–514.
-
[28]
S.-C. Ji, G. Li, Existence results for impulsive differential inclusions with nonlocal conditions, Comput. Math. Appl., 62 (2011), 1908–1915.
-
[29]
J.-F. Jiang, D.-Q. Cao, H.-T. Chen, The fixed point approach to the stability of fractional differential equations with causal operators, Qual. Theory Dyn. Syst., 15 (2016), 3–18.
-
[30]
J.-F. Jiang, C. F. Li, D.-Q. Cao, H.-T. Chen, Existence and uniqueness of solution for fractional differential equation with causal operators in Banach spaces, Mediterr. J. Math., 12 (2015), 751–769.
-
[31]
M. Kamenskii, V. Obukhovskii, P. Zecca, Condensing multivalued maps and semilinear differential inclusions in Banach spaces, De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin (2001)
-
[32]
T. D. Ke, V. Obukhovskii, N.-C. Wong, J.-C. Yao, On semilinear integro-differential equations with nonlocal conditions in Banach spaces, Abstr. Appl. Anal., 2012 (2012 ), 26 pages.
-
[33]
M. Kisielewicz, Multivalued differential equations in separable Banach spaces, J. Optim. Theory Appl., 37 (1982), 231– 249.
-
[34]
K. Kuratowski, Sur les espaces complets, Fund. Math., 15 (1930), 301–309.
-
[35]
V. G. Kurbatov, Functional-differential operators and equations, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht (1999)
-
[36]
V. Lakshmikantham, S. Leela, Z. Drici, F. A. McRae, Theory of causal differential equations, Atlantis Studies in Mathematics for Engineering and Science, Atlantis Press, Paris; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2009)
-
[37]
J. Liang, J. H. Liu, T.-J. Xiao, Nonlocal Cauchy problems for nonautonomous evolution equations, Commun. Pure Appl. Anal., 5 (2006), 529–535.
-
[38]
Y. P. Lin, J. H. Liu, Semilinear integrodifferential equations with nonlocal Cauchy problem, Nonlinear Anal., 26 (1996), 1023–1033.
-
[39]
Q. Liu, R. Yuan, Existence of mild solutions for semilinear evolution equations with non-local initial conditions, Nonlinear Anal., 71 (2009), 4177–4184.
-
[40]
A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Verlag, Basel (1995)
-
[41]
V. Lupulescu, Causal functional differential equations in Banach spaces, Nonlinear Anal., 69 (2008), 4787–4795.
-
[42]
M. Mahdavi, Y.-Z. Li , Linear and quasilinear equations with abstract Volterra operators, Volterra equations and applications, Arlington, TX, (1996), 325–330, Stability Control Theory Methods Appl., Gordon and Breach, Amsterdam (2000)
-
[43]
G. M. Mophou, Existence and uniqueness of mild solutions to impulsive fractional differential equations, Nonlinear Anal., 72 (2010), 1604–1615.
-
[44]
G. M. Mophou, G. M. N’Guérékata, Existence of the mild solution for some fractional differential equations with nonlocal conditions, Semigroup Forum, 79 (2009), 315–322.
-
[45]
G. M. Mophou, G. M. N’Guérékata, Mild solutions for semilinear fractional differential equations, Electron. J. Differential Equations, 2009 (2009 ), 9 pages.
-
[46]
G. M. N’Guérékata, A Cauchy problem for some fractional abstract differential equation with non local conditions, Nonlinear Anal., 70 (2009), 1873–1876.
-
[47]
S. K. Ntouyas, P. Ch. Tsamatos, Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl., 210 (1997), 679–687.
-
[48]
A. Paicu, I. I. Vrabie, A class of nonlinear evolution equations subjected to nonlocal initial conditions, Nonlinear Anal., 72 (2010), 4091–4100.
-
[49]
N. S. Papageorgiou, Mild solutions of semilinear evolution inclusions and optimal control, Indian J. Pure Appl. Math., 26 (1995), 189–216.
-
[50]
A. Pazy , Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, Springer-Verlag, New York (1983)
-
[51]
H. Tanabe, Equations of evolution, Translated from the Japanese by N. Mugibayashi and H. Haneda, Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, Mass.-London (1979)
-
[52]
I. I. Vrabie , \(C_0\)-semigroups and applications, North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam (2003)
-
[53]
R.-N. Wang, K. Ezzinbi, P.-X. Zhu, Non-autonomous impulsive Cauchy problems of parabolic type involving nonlocal initial conditions, J. Integral Equations Appl., 26 (2014), 275–299.
-
[54]
J.-R. Wang, Y. Zhou, M. Fečkan, Abstract Cauchy problem for fractional differential equations, Nonlinear Dynam., 71 (2013), 685–700.
-
[55]
S. Xie, Existence of solutions for nonlinear mixed type integrodifferential functional evolution equations with nonlocal conditions, Abstr. Appl. Anal., 2012 (2012 ), 11 pages.
-
[56]
X.-M. Xue, Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces, Nonlinear Anal., 70 (2009), 2593–2601.
-
[57]
A. Yagi, Abstract parabolic evolution equations and their applications, Springer Monographs in Mathematics, Springer- Verlag, Berlin (2010)
-
[58]
Y.-L. Yang, J.-R. Wang, On some existence results of mild solutions for nonlocal integrodifferential Cauchy problems in Banach spaces, Opuscula Math., 31 (2011), 443–455.
-
[59]
T. Zhu, C. Song, G. Li, Existence of mild solutions for abstract semilinear evolution equations in Banach spaces, Nonlinear Anal., 75 (2012), 177–181.