# On a nonlinear Hadamard type fractional differential equation with p-Laplacian operator and strip condition

Volume 9, Issue 7, pp 5073--5081
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### Authors

Guotao Wang - School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, P. R. China. Taoli Wang - School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, P. R. China.

### Abstract

Under certain nonlinear growth conditions of the nonlinearity, we investigate the existence of solutions for a nonlinear Hadamard type fractional differential equation with strip condition and p-Laplacian operator. At the end, two examples are given to illustrate our main results.

### Share and Cite

##### ISRP Style

Guotao Wang, Taoli Wang, On a nonlinear Hadamard type fractional differential equation with p-Laplacian operator and strip condition, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 7, 5073--5081

##### AMA Style

Wang Guotao, Wang Taoli, On a nonlinear Hadamard type fractional differential equation with p-Laplacian operator and strip condition. J. Nonlinear Sci. Appl. (2016); 9(7):5073--5081

##### Chicago/Turabian Style

Wang, Guotao, Wang, Taoli. "On a nonlinear Hadamard type fractional differential equation with p-Laplacian operator and strip condition." Journal of Nonlinear Sciences and Applications, 9, no. 7 (2016): 5073--5081

### Keywords

• strip condition
• p-Laplacian operator
• fixed point.

•  34A08
•  34B10

### References

• [1] B. Ahmad, J. J. Nieto, The monotone iterative technique for three-point second-order integro differential boundary value problems with p-Laplacian, Bound. Value Probl., 2007 (2007), 9 pages

• [2] B. Ahmad, J. J. Nieto, A. Alsaedi, M. El-Shahed, A study of nonlinear Langevin equation involving two fractional orders in different intervals, Nonlinear Anal. Real World Appl., 13 (2012), 599--606

• [3] B. Ahmad, S. K. Ntouyas, A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations, Fract. Calc. Appl. Anal., 17 (2014), 348--360

• [4] B. Ahmad, S. K. Ntouyas, J. Tariboon, A study of mixed Hadamard and Riemann-Liouville fractional integro- differential inclusions via endpoint theory, Appl. Math. Lett., 52 (2016), 9--14

• [5] S. Aljoudi, B. Ahmad, J. J. Nieto, A. Alsaedi , A coupled system of Hadamard type sequential fractional differential equations with coupled strip conditions, Chaos Solitons Fractals, 91 (2016), 39--46

• [6] D. Averna, E. Tornatore , Ordinary $(p_1,..., p_m)$-Laplacian systems with mixed boundary value conditions, Nonlinear Anal. Real World Appl., 28 (2016), 20--31

• [7] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo , Fractional calculus: models and numerical methods, Series on Complexity, Nonlinearity and Chaos, World Scientific, Boston (2012)

• [8] D. Baleanu, O. G. Mustafa, R. P. Agarwal , An existence result for a superlinear fractional differential equation, Appl. Math. Lett., 23 (2010), 1129--1132

• [9] G. Bonanno, S. Heidarkhani, D. O'Regan, Multiple solutions for a class of Dirichlet quasilinear elliptic systems driven by a (P;Q)-Laplacian operator, Dynam. Systems Appl., 20 (2011), 89--100

• [10] A. Cabada, G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl., 389 (2012), 403-411

• [11] G. Cetin, F. S. Topal , Existence of solutions for fractional four point boundary value problems with p-Laplacian operator, J. Comput. Anal. Appl., 19 (2015), 892--903

• [12] A. Chadha, D. N. Pandey, Existence results for an impulsive neutral stochastic fractional integro-differential equation with infinite delay, Nonlinear Anal., 128 (2015), 149--175

• [13] T. Chen, W. Liu, Z. Hu , A boundary value problem for fractional differential equation with p-Laplacian operator at resonance, Nonlinear Anal., 75 (2012), 3210--3217

• [14] P. M. de Carvalho-Neto, G. Planas, Mild solutions to the time fractional Navier-Stokes equations in $\mathbb{R}^N$, J. Differential Equations, 259 (2015), 2948--2980

• [15] Y. Ding, Z. Wei, J. Xu, D. O'Regan, Extremal solutions for nonlinear fractional boundary value problems with p-Laplacian, J. Comput. Appl. Math., 288 (2015), 151--158

• [16] J. Hadamard , Essai sur l'etude des fonctions, donnees par leur developpment de Taylor, J. Math. Pures Appl., 8 (1892), 101--186

• [17] Z. Han, H. Lu, C. Zhang , Positive solutions for eigenvalue problems of fractional differential equation with generalized p-Laplacian, Appl. Math. Comput., 257 (2015), 526--536

• [18] W. Jiang, Solvability of fractional differential equations with p-Laplacian at resonance, Appl. Math. Comput., 260 (2015), 48--56

• [19] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo , Theory and applications of fractional differential equations, North- Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (2006)

• [20] V. Lakshmikantham, S. Leela, D. J. Vasundhara, Theory of fractional dynamic systems, Cambridge Academic Publishers, Cambridge (2009)

• [21] L. S. Leibenson, General problem of the movement of a compressible fluid in a porous medium, (Russian), Bull. Acad. Sci. URSS. Sér. Géograph. Géophys., 9 (1945), 7--10

• [22] C. Li, C. L. Tang, Three solutions for a class of quasilinear elliptic systems involving the (p; q)-Laplacian, Nonlinear Anal., 69 (2008), 3322--3329

• [23] S. Liang, J. Zhang, Existence and uniqueness of positive solutions for integral boundary problems of nonlinear fractional differential equations with p-Laplacian operator, Rocky Mountain J. Math., 44 (2014), 953--974

• [24] Q. Ma, R. Wang, J. Wang, Y. Ma, Qualitative analysis for solutions of a certain more generalized two-dimensional fractional differential system with Hadamard derivative, Appl. Math. Comput., 257 (2015), 436--445

• [25] I. Podlubny, Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego (1999)

• [26] M. Röeckner, R. Zhu, X. Zhu, Existence and uniqueness of solutions to stochastic functional differential equations in infinite dimensions, Nonlinear Anal., 125 (2015), 358--397

• [27] D. R. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, Cambridge University Press, London-New York (1974)

• [28] S. Suganya, M. Mallika Arjunan, J. J. Trujillo , Existence results for an impulsive fractional integro-differential equation with state-dependent delay , Appl. Math. Comput., 266 (2015), 54--69

• [29] G. Wang, Monotone iterative technique for boundary value problems of a nonlinear fractional differential equation with deviating arguments, J. Comput. Appl. Math., 236 (2012), 2425--2430

• [30] G. Wang, Explicit iteration and unbounded solutions for fractional integral boundary value problem on an infinite interval, Appl. Math. Lett., 47 (2015), 1--7

• [31] G. Wang, R. P. Agarwal, A. Cabada, Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations, Appl. Math. Lett., 25 (2012), 1019--1024

• [32] G. Wang, D. Baleanu, L. Zhang, Monotone iterative method for a class of nonlinear fractional differential equations, Fract. Calc. Appl. Anal., 15 (2012), 244--252

• [33] Y. Wang, C. Hou, Existence of multiple positive solutions for one-dimensional p-Laplacian, J. Math. Anal. Appl., 315 (2006), 144--153

• [34] J. Wang, Y. Zhang, On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives, Appl. Math. Lett., 39 (2015), 85--90

• [35] X. J. Yang, D. Baleanu, H. M. Srivastava, Local fractional integral transforms and their applications , Elsevier/ Academic Press, Amsterdam (2016)

• [36] X. J. Yang, H. M. Srivastava, J. A. Tenreiro Machado, A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow, Thermal Sci., 20 (2016), 753--756

• [37] X. J. Yang, J. A. Tenreiro Machado, J. J. Nieto, A new family of the local fractional PDEs, Fundamenta Informaticae, (accepted),

• [38] X. J. Yang, J. A. Tenreiro Machado, H. M. Srivastava, A new numerical technique for solving the local fractional diffusion equation: two-dimensional extended differential transform approach, Appl. Math. Comput., 274 (2016), 143--151

• [39] W. Yukunthorn, B. Ahmad, S. K. Ntouyas, J. Tariboon, On Caputo-Hadamard type fractional impulsive hybrid systems with nonlinear fractional integral conditions, Nonlinear Anal. Hybrid Syst., 19 (2016), 77--92

• [40] L. Zhang, B. Ahmad, G. Wang, The existence of an extremal solution to a nonlinear system with the right-handed Riemann-Liouville fractional derivative, Appl. Math. Lett., 31 (2014), 1--6

• [41] L. Zhang, B. Ahmad, G. Wang, Explicit iterations and extremal solutions for fractional differential equations with nonlinear integral boundary conditions, Appl. Math. Comput., 268 (2015), 388--392

• [42] L. Zhang, B. Ahmad, G. Wang, Successive iterations for positive extremal solutions of nonlinear fractional differential equations on a half-line, Bull. Aust. Math. Soc., 91 (2015), 116--128

• [43] L. Zhang, B. Ahmad, G. Wang, R. P. Agarwal, Nonlinear fractional integro-differential equations on unbounded domains in a Banach space, J. Comput. Appl. Math., 249 (2013), 51--56

• [44] X. Zhang, L. Liu, B. Wiwatanapataphee, Y. Wu, The eigenvalue for a class of singular p-Laplacian fractional differential equations involving the Riemann-Stieltjes integral boundary condition, Appl. Math. Comput., 235 (2015), 412--422

• [45] X. Zhang, L. Liu, Y. Wu, The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium, Appl. Math. Lett., 37 (2014), 26--33

• [46] Y. Zhou, Basic theory of fractional differential equations, World Scientific Publishing Co. Pte. Ltd., Hackensack (2014)