Positive solutions for the initial value problems of fractional evolution equation. (English) Zbl 1294.35189
Summary: This paper discusses the existence of positive solutions for the initial value problem of fractional evolution equation with noncompact semigroup \(D^q u(t)+Au(t)=f(t,u(t)), t\geq 0; u(0)=u_0\) in a Banach space \(X\), where \(D^q\) denotes the Caputo fractional derivative of order \(q \in (0, 1)\), \(A : D(A) \subset X \to X\) is a closed linear operator, \(-A\) generates an equicontinuous \(C_0\) semigroup, and \(F : [0, \infty) \times X \to X\) is continuous. In the case where \(f\) satisfies a weaker measure of noncompactness condition and a weaker boundedness condition, the existence results of positive and saturated mild solutions are obtained. Particularly, an existence result without using measure of noncompactness condition is presented in ordered and weakly sequentially complete Banach spaces. These results are very convenient for application. As an example, we study the partial differential equation of parabolic type of fractional order.
Editorial remark: A very similar paper is reference [20] by H. Yang and the first author [Abstr. Appl. Anal. 2013, Article ID 428793, 7 p. (2013; Zbl 1291.35430)].
Editorial remark: A very similar paper is reference [20] by H. Yang and the first author [Abstr. Appl. Anal. 2013, Article ID 428793, 7 p. (2013; Zbl 1291.35430)].
MSC:
| 35R11 | Fractional partial differential equations |
| 34A08 | Fractional ordinary differential equations |
| 34G20 | Nonlinear differential equations in abstract spaces |
| 35B09 | Positive solutions to PDEs |
Citations:
Zbl 1291.35430References:
| [1] | K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0789.26002 |
| [2] | I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008 |
| [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, Amsterdam, The Netherlands, 2006. · Zbl 1092.45003 · doi:10.1016/S0304-0208(06)80001-0 |
| [4] | V. Lakshmikantham, S. Leela, and J. Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cambridge, UK, 2009. · Zbl 1188.37002 |
| [5] | K. Diethelm, The Analysis of Fractional Differential Equations, vol. 2004 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2010. · Zbl 1083.65064 · doi:10.1007/978-3-642-14574-2 |
| [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. M. El-Borai, “Some probability densities and fundamental solutions of fractional evolution equations,” Chaos, Solitons & Fractals, vol. 14, no. 3, pp. 433-440, 2002. · Zbl 1005.34051 · doi:10.1016/S0960-0779(01)00208-9 |
| [8] | M. M. El-Borai, “The fundamental solutions for fractional evolution equations of parabolic type,” Journal of Applied Mathematics and Stochastic Analysis, no. 3, pp. 197-211, 2004. · Zbl 1081.34053 · doi:10.1155/S1048953304311020 |
| [9] | 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 |
| [10] | M. El-Borai, K. El-Nadi, and E. El-Akabawy, “Fractional evolution equations with nonlocal conditions,” Journal of Applied Mathematics and Mechanics, vol. 4, no. 6, pp. 1-12, 2008. |
| [11] | 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 |
| [12] | J. Wang and Y. Zhou, “A class of fractional evolution equations and optimal controls,” Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp. 262-272, 2011. · Zbl 1214.34010 · doi:10.1016/j.nonrwa.2010.06.013 |
| [13] | Y. Zhou and F. Jiao, “Existence of mild solutions for fractional neutral evolution equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1063-1077, 2010. · Zbl 1189.34154 · doi:10.1016/j.camwa.2009.06.026 |
| [14] | J. Wang, Y. Zhou, and W. Wei, “A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 10, pp. 4049-4059, 2011. · Zbl 1223.45007 · doi:10.1016/j.cnsns.2011.02.003 |
| [15] | Z. Tai and X. Wang, “Controllability of fractional-order impulsive neutral functional infinite delay integrodifferential systems in Banach spaces,” Applied Mathematics Letters, vol. 22, no. 11, pp. 1760-1765, 2009. · Zbl 1181.34078 · doi:10.1016/j.aml.2009.06.017 |
| [16] | A. Debbouche and D. Baleanu, “Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1442-1450, 2011. · Zbl 1228.45013 · doi:10.1016/j.camwa.2011.03.075 |
| [17] | G. M. Mophou, “Existence and uniqueness of mild solutions to impulsive fractional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1604-1615, 2010. · Zbl 1187.34108 · doi:10.1016/j.na.2009.08.046 |
| [18] | N. Nyamoradi, D. Baleanu, and T. Bashiri, “Positive solutions to fractional boundary value problems with nonlinear boundary conditions,” Abstract and Applied Analysis, vol. 2013, Article ID 579740, 20 pages, 2013. · Zbl 1296.34027 · doi:10.1155/2013/579740 |
| [19] | D. Baleanu, H. Mohammadi, and Sh. Rezapour, “Positive solutions of an initial value problem for nonlinear fractional differential equations,” Abstract and Applied Analysis, vol. 2012, Article ID 837437, 7 pages, 2012. · Zbl 1242.35215 · doi:10.1155/2012/837437 |
| [20] | H. Yang and Y. Liang, “Positive solutions for the initial value problem of fractional evolution equations,” Abstract and Applied Analysis, vol. 2013, Article ID 428793, 7 pages, 2013. · Zbl 1291.35430 · doi:10.1155/2013/428793 |
| [21] | K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985. · Zbl 0559.47040 |
| [22] | D. Guo and J. Sun, Ordinary Differential Equations in Abstract Spaces, Shandong Science and Technology, Jinan, China, 1989. |
| [23] | 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 |
| [24] | 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. · Zbl 1124.34341 |
| [25] | B. N. Sadovskiĭ, “On a fixed point principle,” Funkcional’nyi Analiz i ego Prilo\vzenija, vol. 1, no. 2, pp. 74-76, 1967. · Zbl 0165.49102 |
| [26] | D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1981. · Zbl 0456.35001 |
| [27] | Y. H. Du, “Fixed points of increasing operators in ordered Banach spaces and applications,” Applicable Analysis, vol. 38, no. 1-2, pp. 1-20, 1990. · Zbl 0671.47054 · doi:10.1080/00036819008839957 |
| [28] | A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983. · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1 |
| [29] | J. Liang, J. H. Liu, and T.-J. Xiao, “Nonlocal problems for integrodifferential equations,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 15, no. 6, pp. 815-824, 2008. · Zbl 1163.45010 |
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