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Nonlocal problem for fractional evolution equations of mixed type with the measure of noncompactness. (English) Zbl 1274.35400
Summary: A general class of semilinear fractional evolution equations of mixed type with nonlocal conditions on infinite dimensional Banach spaces is concerned. Under more general conditions, the existence of mild solutions and positive mild solutions is obtained by utilizing a new estimation technique of the measure of noncompactness and a new fixed point theorem with respect to convex-power condensing operator.

MSC:
35R11Fractional partial differential equations
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References:
[1] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, Wiley, New York, NY, USA, 1993. · Zbl 0943.82582 · doi:10.1007/BF01048101
[2] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. · Zbl 1056.93542 · doi:10.1109/9.739144
[3] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. · Zbl 1206.26007 · doi:10.1016/S0304-0208(06)80001-0
[4] S. D. Eidelman and A. N. Kochubei, “Cauchy problem for fractional diffusion equations,” Journal of Differential Equations, vol. 199, no. 2, pp. 211-255, 2004. · Zbl 1068.35037 · doi:10.1016/j.jde.2003.12.002
[5] V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2677-2682, 2008. · Zbl 1161.34001 · doi:10.1016/j.na.2007.08.042
[6] R. P. Agarwal, V. Lakshmikantham, and J. J. Nieto, “On the concept of solution for fractional differential equations with uncertainty,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 6, pp. 2859-2862, 2010. · Zbl 1188.34005 · doi:10.1016/j.na.2009.11.029
[7] M. A. Darwish and S. K. Ntouyas, “Boundary value problems for fractional functional differential equations of mixed type,” Communications in Applied Analysis, vol. 13, no. 1, pp. 31-38, 2009. · Zbl 1179.26020
[8] M. A. Darwish and S. K. Ntouyas, “Semilinear functional differential equations of fractional order with state-dependent delay,” Electronic Journal of Differential Equations, vol. 2009, no. 38, pp. 1-10, 2009. · Zbl 1167.26302 · emis:journals/EJDE/Volumes/2009/38/abstr.html · eudml:130407
[9] M. A. Darwish and S. K. Ntouyas, “On initial and boundary value problems for fractional order mixed type functional differential inclusions,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1253-1265, 2010. · Zbl 1189.34029 · doi:10.1016/j.camwa.2009.05.006
[10] M. A. Darwish and S. K. Ntouyas, “On a quadratic fractional Hammerstein-Volterra integral equation with linear modification of the argument,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 11, pp. 3510-3517, 2011. · Zbl 1228.45006 · doi:10.1016/j.na.2011.02.035
[11] M. A. Darwish, J. Henderson, and D. O’Regan, “Existence and asymptotic stability of solutions of a perturbed fractional functional-integral equation with linear modification of the argument,” Bulletin of the Korean Mathematical Society, vol. 48, no. 3, pp. 539-553, 2011. · Zbl 1220.45011 · doi:10.4134/BKMS.2011.48.3.539 · http://www.mathnet.or.kr/mathnet/kms_content.php?no=406070
[12] M. M. El-Borai, “Some probability densities and fundamental solutions of fractional evolution equations,” Chaos, Solitons and Fractals, vol. 14, no. 3, pp. 433-440, 2002. · Zbl 1005.34051 · doi:10.1016/S0960-0779(01)00208-9
[13] M. M. El-Borai, “Semigroups and some nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 149, no. 3, pp. 823-831, 2004. · Zbl 1046.34079 · doi:10.1016/S0096-3003(03)00188-7
[14] J. Wang, Y. Zhou, and M. Fe\vckan, “Abstract Cauchy problem for fractional differential equations,” Nonlinear Dynamics, vol. 71, no. 4, pp. 685-700, 2013. · Zbl 1268.34034 · doi:10.1007/s11071-012-0452-9
[15] J. Wang, Y. Zhou, and M. Medved’, “On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay,” Journal of Optimization Theory and Applications, vol. 152, no. 1, pp. 31-50, 2012. · Zbl 06028533 · doi:10.1007/s10957-011-9892-5
[16] J. Wang, Y. Zhou, and W. Wei, “Optimal feedback control for semilinear fractional evolution equations in Banach spaces,” Systems & Control Letters, vol. 61, no. 4, pp. 472-476, 2012. · Zbl 1250.49035 · doi:10.1016/j.sysconle.2011.12.009
[17] T. Diagana, G. M. Mophou, and G. M. N’Guérékata, “On the existence of mild solutions to some semilinear fractional integro-differential equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2010, no. 58, pp. 1-17, 2010. · Zbl 1211.34094 · emis:journals/EJQTDE/2010/201058.html
[18] R. Wang, T. Xiao, and J. Liang, “A note on the fractional Cauchy problems with nonlocal initial conditions,” Applied Mathematics Letters, vol. 24, no. 8, pp. 1435-1442, 2011. · Zbl 1251.45008 · doi:10.1016/j.aml.2011.03.026
[19] F. Li, J. Liang, and H. K. Xu, “Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions,” Journal of Mathematical Analysis and Applications, vol. 391, no. 2, pp. 510-525, 2012. · Zbl 1242.45009 · doi:10.1016/j.jmaa.2012.02.057
[20] Y. Zhou and F. Jiao, “Nonlocal Cauchy problem for fractional evolution equations,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 4465-4475, 2010. · Zbl 1260.34017 · doi:10.1016/j.nonrwa.2010.05.029
[21] J. Wang, Y. Zhou, W. Wei, and H. Xu, “Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1427-1441, 2011. · Zbl 1228.45015 · doi:10.1016/j.camwa.2011.02.040
[22] J. Wang, M. Fe\vckan, and Y. Zhou, “On the new concept of solutions and existence results for impulsive fractional evolution equations,” Dynamics of Partial Differential Equations, vol. 8, no. 4, pp. 345-361, 2011. · Zbl 1264.34014
[23] J. Wang, Z. Fan, and Y. Zhou, “Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces,” Journal of Optimization Theory and Applications, vol. 154, no. 1, pp. 292-302, 2012. · Zbl 1252.93028 · doi:10.1007/s10957-012-9999-3
[24] Y. Chang, V. Kavitha, and M. Mallika Arjunan, “Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 11, pp. 5551-5559, 2009. · Zbl 1179.45010 · doi:10.1016/j.na.2009.04.058
[25] K. Balachandran and J. Y. Park, “Nonlocal Cauchy problem for abstract fractional semilinear evolution equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 10, pp. 4471-4475, 2009. · Zbl 1213.34008 · doi:10.1016/j.na.2009.03.005
[26] K. Balachandran and J. J. Trujillo, “The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 12, pp. 4587-4593, 2010. · Zbl 1196.34007 · doi:10.1016/j.na.2010.02.035
[27] L. Byszewski and V. Lakshmikantham, “Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space,” Applicable Analysis, vol. 40, no. 1, pp. 11-19, 1991. · Zbl 0694.34001 · doi:10.1080/00036819008839989
[28] L. Byszewski, “Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem,” Journal of Mathematical Analysis and Applications, vol. 162, no. 2, pp. 494-505, 1991. · Zbl 0748.34040 · doi:10.1016/0022-247X(91)90164-U
[29] L. Byszewski, “Application of properties of the right-hand sides of evolution equations to an investigation of nonlocal evolution problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 33, no. 5, pp. 413-426, 1998. · Zbl 0933.34064 · doi:10.1016/S0362-546X(97)00594-4
[30] H. Liu and J.-C. Chang, “Existence for a class of partial differential equations with nonlocal conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3076-3083, 2009. · Zbl 1185.34112 · doi:10.1016/j.na.2009.02.035
[31] J. Wang and W. Wei, “A class of nonlocal impulsive problems for integrodifferential equations in Banach spaces,” Results in Mathematics, vol. 58, no. 3-4, pp. 379-397, 2010. · Zbl 1209.34095 · doi:10.1007/s00025-010-0057-x
[32] P. Chen and Y. Li, “Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions,” Results in Mathematics, 2012. · doi:10.1007/s00025-012-0230-5
[33] V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, Pergamon Press, New York, NY, USA, 1981. · Zbl 0456.34002
[34] D. Guo, “Solutions of nonlinear integro-differential equations of mixed type in Banach spaces,” Journal of Applied Mathematics and Simulation, vol. 2, no. 1, pp. 1-11, 1989. · Zbl 0681.45011
[35] L. Liu, C. Wu, and F. Guo, “Existence theorems of global solutions of initial value problems for nonlinear integro-differential equations of mixed type in Banach spaces and applications,” Computers & Mathematics with Applications, vol. 47, no. 1, pp. 13-22, 2004. · Zbl 1050.45003 · doi:10.1016/S0898-1221(04)90002-8
[36] L. Liu, F. Guo, C. Wu, and Y. Wu, “Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 309, no. 2, pp. 638-649, 2005. · Zbl 1080.45005 · doi:10.1016/j.jmaa.2004.10.069
[37] J. X. Sun and X. Y. Zhang, “The fixed point theorem of convex-power condensing operator and applications to abstract semilinear evolution equations,” Acta Mathematica Sinica, vol. 48, no. 3, pp. 439-446, 2005 (Chinese). · Zbl 1124.34342
[38] H. B. Shi, W. T. Li, and H. R. Sun, “Existence of mild solutions for abstract mixed type semilinear evolution equations,” Turkish Journal of Mathematics, vol. 35, no. 3, pp. 457-472, 2011. · Zbl 1241.34085 · http://mistug.tubitak.gov.tr/bdyim/abs.php?dergi=mat&rak=0905-29
[39] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, vol. 60 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980. · Zbl 0441.47056
[40] K. Deimling, Nonlinear Functional Analysis, Springer, New York, NY, USA, 1985. · Zbl 0559.47040
[41] H. P. Heinz, “On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 7, no. 12, pp. 1351-1371, 1983. · Zbl 0528.47046 · doi:10.1016/0362-546X(83)90006-8
[42] Y. X. Li, “Existence of solutions to initial value problems for abstract semilinear evolution equations,” Acta Mathematica Sinica, vol. 48, no. 6, pp. 1089-1094, 2005 (Chinese). · Zbl 1124.34341
[43] D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Orlando, Fla, USA, 1988. · Zbl 0661.47045
[44] J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer, London, UK, 2006. · Zbl 1097.47038
[45] Y. X. Li, “The positive solutions of abstract semilinear evolution equations and their applications,” Acta Mathematica Sinica, vol. 39, no. 5, pp. 666-672, 1996 (Chinese). · Zbl 0870.47040
[46] I. Vasile, Fixed Point Theory, D. Reidel Publishing, Dordrecht, The Netherlands, 1981. · Zbl 0465.47035
[47] R. Nagel, W. Arendt, A. Grabosch et al., One Parameter Semigroups of Positive Operators, vol. 1184 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1986.
[48] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin, Germany, 1983. · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1