# Existence of solutions for fractional integral boundary value problems with $p(t)$-Laplacian operator

Volume 9, Issue 7, pp 5000--5010
• 1540 Views

### Authors

Tengfei Shen - College of Sciences, China University of Mining and Technology, Xuzhou 221116, P. R. China. Wenbin Liu - College of Sciences, China University of Mining and Technology, Xuzhou 221116, P. R. China.

### Abstract

This paper aims to investigate the existence of solutions for fractional integral boundary value problems (BVPs for short) with $p(t)$-Laplacian operator. By using the fixed point theorem and the coincidence degree theory, two existence results are obtained, which enrich existing literatures. Some examples are supplied to verify our main results.

### Share and Cite

##### ISRP Style

Tengfei Shen, Wenbin Liu, Existence of solutions for fractional integral boundary value problems with $p(t)$-Laplacian operator, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 7, 5000--5010

##### AMA Style

Shen Tengfei, Liu Wenbin, Existence of solutions for fractional integral boundary value problems with $p(t)$-Laplacian operator. J. Nonlinear Sci. Appl. (2016); 9(7):5000--5010

##### Chicago/Turabian Style

Shen, Tengfei, Liu, Wenbin. "Existence of solutions for fractional integral boundary value problems with $p(t)$-Laplacian operator." Journal of Nonlinear Sciences and Applications, 9, no. 7 (2016): 5000--5010

### Keywords

• Fractional differential equation
• boundary value problem
• $p(t)$-Laplacian operator
• fixed point theorem
• coincidence degree theory.

•  34A08
•  34B10
•  47N20
•  34B15

### References

• [1] B. Ahmad, R. P. Agarwal, Some new versions of fractional boundary value problems with slit-strips conditions, Bound. Value Probl., 2014 (2014), 12 pages

• [2] C. Bai , Existence of positive solutions for a functional fractional boundary value problem, Abstr. Appl. Anal., 2010 (2010), 13 pages

• [3] C. Bai , Existence of positive solutions for boundary value problems of fractional functional differential equations, Electron. J. Qual. Theory Differ. Equ., 2010 (2010), 14 pages

• [4] J. Bai, X. C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process., 16 (2007), 2492--2502

• [5] Z. Bai, Y. Zhang, Solvability of fractional three-point boundary value problems with nonlinear growth, Appl. Math. Comput., 218 (2011), 1719--1725

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

• [7] G. Chai , Positive solutions for boundary value problem of fractional differential equation with p-Laplacian operator, Bound. Value Probl., 2012 (2012), 20 pages

• [8] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383--1406

• [9] T. Chen, W. Liu , An anti-periodic boundary value problem for the fractional differential equation with a p- Laplacian operator, Appl. Math. Lett., 25 (2012), 1671--1675

• [10] M. De la Sen, About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory, Fixed Point Theory Appl., 2011 (2011), 19 pages

• [11] Z. Du, X. Lin, C. C. Tisdell , A multiplicity result for p-Lapacian boundary value problems via critical points theorem, Appl. Math. Comput., 205 (2008), 231--237

• [12] X. Fan, Q. Zhang, D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl., 302 (2005), 306--317

• [13] W. Ge, Boundary value problems for ordinary nonlinear differential equations, Science Press, Beijing, China (2007)

• [14] W. Ge, J. Ren, An extension of Mawhin's continuation theorem and its application to boundary value problems with a p-Laplacian, Nonlinear Anal., 58 (2004), 477--488

• [15] Z. Hu, W. Liu , Solvability for fractional order boundary value problems at resonance, Bound. Value Probl., 2011 (2011), 10 pages

• [16] Z. Hu, W. Liu, J. Liu, Existence of solutions of fractional differential equation with p-Laplacian operator at resonance, Abstr. Appl. Anal., 2014 (2014), 7 pages

• [17] J. B. Hu, G. P. Lu, S. B. Zhang, L. D. Zhao, Lyapunov stability theorem about fractional system without and with delay, Commun. Nonlinear Sci. Numer. Simul., 20 (2015), 905--913

• [18] W. Jiang, The existence of solutions to boundary value problems of fractional differential equations at resonance, Nonlinear Anal., 74 (2011), 1987--1994

• [19] B. Kawohl , On a family of torsional creep problems, J. Reine Angew. Math., 410 (1990), 1--22

• [20] 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)

• [21] J. C. Kuang, Applied Inequalities, Shandong Science and Technology Press, Jinan, China (2004)

• [22] 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 (1983), 7--10

• [23] J. S. Leszczynski, T. Blaszczyk , Modeling the transition between stable and unstable operation while emptying a silo , Granular Matter, 13 (2011), 429--438

• [24] R. L. Magin, Fractional calculus models of complex dynamics in biological tissues, Comput. Math. Appl., 59 (2010), 1586--1593

• [25] N. I. Mahmudov, S. Unul, Existence of solutions of fractional boundary value problems with p-Laplacian operator, Bound. Value Probl., 2015 (2015), 16 pages

• [26] J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations, Topological methods for ordinary differential equations, Lecture Notes in Math., Springer, Berlin (1993)

• [27] M. C. Pélissier, L. Reynaud, Étude d'un modèle mathématique d'écoulement de glacier, (French), C. R. Acad. Sci. Paris Sér. A, 279 (1974), 531--534

• [28] R. Rakkiyappan, J. Cao, G. Velmurugan, Existence and uniform stability analysis of fractional-order complex- valued neural networks with time delays, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 84--97

• [29] M. Růžička, Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics, Springer-Verlag, Berlin (2000)

• [30] E. Szymanek, The application of fractional order differential calculus for the description of temperature profiles in a granular layer, Adv. Theory Appl. Non-integer Order Syst., Springer Inter. Publ., Switzerland (2013)

• [31] Q. Zhang, Y. Wang, Z. Qiu , Existence of solutions and boundary asymptotic behavior of p(r)-Laplacian equation multi-point boundary value problems, Nonlinear Anal., 72 (2010), 2950--2973

• [32] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675--710