Maximum and minimum solutions for a nonlocal \(p\)-Laplacian fractional differential system from eco-economical processes. (English) Zbl 1376.35083
Summary: This paper focuses on the maximum and minimum solutions for a fractional order differential system, involving a \( p\)-Laplacian operator and nonlocal boundary conditions, which arises from many complex processes such as ecological economy phenomena and diffusive interaction. By introducing new type growth conditions and using the monotone iterative technique, some new results about the existence of maximal and minimal solutions for a fractional order differential system is established, and the estimation of the lower and upper bounds of the maximum and minimum solutions is also derived. In addition, the iterative schemes starting from some explicit initial values and converging to the exact maximum and minimum solutions are also constructed.
MSC:
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35R11 | Fractional partial differential equations |
References:
| [1] | Caffarelli, L, Vazquez, J: Nonlinear porous medium flow with fractional potential pressure. Arch. Ration. Mech. Anal. 202, 537-565 (2011) · Zbl 1264.76105 · doi:10.1007/s00205-011-0420-4 |
| [2] | Ivancevic, VG, Ivancevic, TT: Geometrical Dynamics of Complex Systems: A Unified Modelling Approach to Physics, Control, Biomechanics, Neurodynamics and Psycho-Socio-Economical Dynamics. Springer, Berlin (2006) · Zbl 1092.53001 · doi:10.1007/1-4020-4545-X |
| [3] | Tarasov, V: Lattice model of fractional gradient and integral elasticity: long-range interaction of Grünwald-Letnikov-Riesz type. Mech. Mater. 70, 106-114 (2014) · doi:10.1016/j.mechmat.2013.12.004 |
| [4] | Yin, D, Duan, X, Zhou, X: Fractional time-dependent deformation component models for characterizing viscoelastic Poisson’s ratio. Eur. J. Mech. A, Solids 42, 422-429 (2013) · doi:10.1016/j.euromechsol.2013.07.010 |
| [5] | Paola, M, Pinnola, F, Zingales, M: Fractional differential equations and related exact mechanical models. Comput. Math. Appl. 66, 608-620 (2013) · Zbl 1381.74038 · doi:10.1016/j.camwa.2013.03.012 |
| [6] | Das, S, Maharatna, K: Fractional dynamical model for the generation of ECG like signals from filtered coupled Van-der Pol oscillators. Comput. Methods Programs Biomed. 122, 490-507 (2013) · doi:10.1016/j.cmpb.2013.08.012 |
| [7] | Sumelka, W: Fractional viscoplasticity. Mech. Res. Commun. 56, 31-36 (2014) · Zbl 1302.74035 · doi:10.1016/j.mechrescom.2013.11.005 |
| [8] | Hartley, T, Lorenzo, C, Qammer, H: Chaos in fractional order Chua’s system. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 42(8), 485-490 (1995) · doi:10.1109/81.404062 |
| [9] | Caputo, M: Free modes splitting and alterations of electro chemically polarizable media. Rend Fis. Accad. Lincei 4, 89-98 (1993) · doi:10.1007/BF03001421 |
| [10] | Zhang, X, Liu, L, Wu, Y: Multiple positive solutions of a singular fractional differential equation with negatively perturbed term. Math. Comput. Model. 55, 1263-1274 (2012) · Zbl 1255.34010 · doi:10.1016/j.mcm.2011.10.006 |
| [11] | Goodrich, C: Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions. Comput. Math. Appl. 61, 191-202 (2011) · Zbl 1211.39002 · doi:10.1016/j.camwa.2010.10.041 |
| [12] | Paola, M, Zingales, M: The multiscale stochastic model of fractional hereditary materials (FHM). Proc. IUTAM 6, 50-59 (2013) · doi:10.1016/j.piutam.2013.01.006 |
| [13] | El-Saka, H: The fractional-order SIS epidemic model with variable population size. J. Egypt. Math. Soc. 22, 50-54 (2014) · Zbl 1336.92078 · doi:10.1016/j.joems.2013.06.006 |
| [14] | Mongiovi, M, Zingales, M: A non-local model of thermal energy transport: the fractional temperature equation. Int. J. Heat Mass Transf. 67, 593-601 (2013) · doi:10.1016/j.ijheatmasstransfer.2013.07.037 |
| [15] | Miller, K, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) · Zbl 0789.26002 |
| [16] | Zhang, X, Han, Y: Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations. Appl. Math. Lett. 25, 555-560 (2012) · Zbl 1244.34009 · doi:10.1016/j.aml.2011.09.058 |
| [17] | Webb, J: Nonlocal conjugate type boundary value problems of higher order. Nonlinear Anal. 71, 1933-1940 (2009) · Zbl 1181.34025 · doi:10.1016/j.na.2009.01.033 |
| [18] | Zhang, X, Liu, L, Wu, Y: The uniqueness of positive solution for a singular fractional differential system involving derivatives. Commun. Nonlinear Sci. Numer. Simul. 18, 1400-1409 (2013) · Zbl 1283.34006 · doi:10.1016/j.cnsns.2012.08.033 |
| [19] | Zhang, X, Liu, L, Wu, Y: The eigenvalue problem for a singular higher fractional differential equation involving fractional derivatives. Appl. Math. Comput. 218, 8526-8536 (2012) · Zbl 1254.34016 |
| [20] | Zhang, X, Liu, L, Wu, Y: Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives. Appl. Math. Comput. 219, 1420-1433 (2012) · Zbl 1296.34046 |
| [21] | Liu, X, Jia, M, Xiang, X: On the solvability of a fractional differential equation model involving the p-Laplacian operator. Comput. Math. Appl. 64, 3267-3275 (2012) · Zbl 1268.34020 · doi:10.1016/j.camwa.2012.03.001 |
| [22] | Zhang, X, Liu, L, Wu, Y, Wiwatanapataphee, B: Nontrivial solutions for a fractional advection dispersion equation in anomalous diffusion. Appl. Math. Lett. 66, 1-8 (2017) · Zbl 1364.35429 · doi:10.1016/j.aml.2016.10.015 |
| [23] | Chen, T, Liu, W: An anti-periodic boundary value problem for the fractional differential equation with a p-Laplacian operator. Appl. Math. Lett. 25, 1671-1675 (2012) · Zbl 1248.35219 · doi:10.1016/j.aml.2012.01.035 |
| [24] | Jiang, W: Solvability of fractional differential equations with p-Laplacian at resonance. Appl. Math. Comput. 260, 48-56 (2015) · Zbl 1410.34020 |
| [25] | Chen, T, Liu, W, Liu, J: Solvability of periodic boundary value problem for fractional p-Laplacian equation. Appl. Math. Comput. 244, 422-431 (2014) · Zbl 1335.34012 |
| [26] | Wang, J, Xiang, H: Upper and lower solutions method for a class of singular fractional boundary value problems with p-Laplacian operator. Abstr. Appl. Anal. 2010, Article ID 971824 (2010) · Zbl 1209.34005 |
| [27] | Zhang, X, Mao, C, Liu, L, Wu, Y: Exact iterative solution for an abstract fractional dynamic system model for bioprocess. Qual. Theory Dyn. Syst. 16, 205-222 (2017) · Zbl 1454.34023 · doi:10.1007/s12346-015-0162-z |
| [28] | Zhang, X, Wu, Y, Caccetta, L: Nonlocal fractional order differential equations with changing-sign singular perturbation. Appl. Math. Model. 39, 6543-6552 (2015) · Zbl 1443.34014 · doi:10.1016/j.apm.2015.02.005 |
| [29] | Zhang, X, Liu, L, Wu, Y: Variational structure and multiple solutions for a fractional advection-dispersion equation. Comput. Math. Appl. 68, 1794-1805 (2014) · Zbl 1369.35108 · doi:10.1016/j.camwa.2014.10.011 |
| [30] | Zhang, X, Liu, L, Wiwatanapataphee, B, Wu, Y: The eigenvalue for a class of singular p-Laplacian fractional differential equations involving the Riemann-Stieltjes integral boundary condition. Appl. Math. Comput. 235, 412-422 (2014) · Zbl 1334.34060 |
| [31] | Guo, Z, Yuan, H: Pricing European option under the time-changed mixed Brownian-fractional Brownian model. Physica A 406, 73-79 (2014) · Zbl 1402.91786 · doi:10.1016/j.physa.2014.03.032 |
| [32] | Zhang, X, Liu, L, Wu, Y, Wiwatanapataphee, B: The spectral analysis for a singular fractional differential equation with a signed measure. Appl. Math. Comput. 257, 252-263 (2015) · Zbl 1338.34032 |
| [33] | Sun, F, Liu, L, Zhang, X, Wu, Y: Spectral analysis for a singular differential system with integral boundary conditions. Mediterr. J. Math. 13, 4763-4782 (2016) · Zbl 1349.34080 · doi:10.1007/s00009-016-0774-9 |
| [34] | Liu, L, Sun, F, Zhang, X, Wu, Y: Bifurcation analysis for a singular differential system with two parameters via to degree theory. Nonlinear Anal., Model. Control 22(1), 31-50 (2017) · Zbl 1420.34048 |
| [35] | Liu, L, Li, H, Liu, C, Wu, Y: Existence and uniqueness of positive solutions for singular fractional differential systems with coupled integral boundary value problems. J. Nonlinear Sci. Appl. 10, 243-262 (2017) · Zbl 1412.34099 · doi:10.22436/jnsa.010.01.24 |
| [36] | Guo, L, Liu, L, Wu, Y: Existence of positive solutions for singular fractional differential equations with infinite-point boundary conditions. Nonlinear Anal., Model. Control 21(5), 635-650 (2016) · Zbl 1420.34015 · doi:10.15388/NA.2016.5.5 |
| [37] | Liu, L, Zhang, X, Jiang, J, Wu, Y: The unique solution of a class of sum mixed monotone operator equations and its application to fractional boundary value problems. J. Nonlinear Sci. Appl. 9(5), 2943-2958 (2016) · Zbl 1492.47060 |
| [38] | Zhang, X, Liu, L, Wu, Y: The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium. Appl. Math. Lett. 37, 26-33 (2014) · Zbl 1320.35007 · doi:10.1016/j.aml.2014.05.002 |
| [39] | Podlubny, I: Fractional Differential Equations. Mathematics in Science and Engineering. Academic Press, New York (1999) · Zbl 0924.34008 |
| [40] | Kilbas, A, Srivastava, H, Trujillo, J: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) · Zbl 1092.45003 · doi:10.1016/S0304-0208(06)80001-0 |
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