Existence and Uniqueness of Positive Solutions for System of (p,q,r)-Laplacian Fractional Order Boundary Value Problems

Year 2021, Volume: 5 Issue: 1, 138 - 157, 31.03.2021

Kapula Rajendra Prasad Leela D Mahammad Khuddush

https://doi.org/10.31197/atnaa.703304

Cited By: 6

Abstract

In this paper the existence of unique positive solutions for system of (p,q,r)-Lapalacian Sturm-Liouville type two-point fractional order boundary vaue problems is established by an application of n-fixed point theorem of ternary operators on partially ordered metric spaces.

Keywords

Boundary value problem, Caputo fractional derivative, n-fixed point, Monotone mappings, Partially ordered complete metric spaces, Contractive

References

• [1] V. Berinde, M. Borcut, Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal., 74 (2011), 4889-4897.
• [2] M. Borcut, Tripled fixed point theorems for monotone mappings in partially ordered metric spaces, Carpathian J. of Math. 28(2012), no. 2, 215-222.
• [3] A. Cabada, G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value condi- tions, J. Math. Anal. Appl. 389, (2012), 403-411.
• [4] L. Cadariu, L. Gavruµa and P. Gavruµa, Weighted space method for the stability of some nonlinear equations, Appl. Anal. Discrete Math. 6(2012), no. 1, 126-139.
• [5] M.J. De Lemos, Turbulence in Porous Media: Modeling and Applications, Elsevier, (2012).
• [6] K. Diethelm, Lectures Notes in Mathematics, The Analysis of Fractional Di?erential Equations, Springer, Berlin, 2010.
• [7] S. M. Ege and F. S. Topal, Existence of positive solutions for fractional order boundary value problems, J. Applied Anal. Comp., 7(2) (2017) 702-712.
• [8] F. Haddouchi, Positive solutions of p-Laplacian fractional differential equations with fractional derivative boundary condi- tions, (2019). arXiv.1908.03966
• [9] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier B. V, Amsterdam, (2006).
• [10] D. Kumar, D. Baleanu, Editorial: fractional calculus and its applications in physics, Front. Phys., 7(6) (2019) 1-–2. DOI: 10.3389/fphy.2019.00081 [11] L. S. Leibenson, General problem of the movement of a compressible ?uid in a porous medium, Izvestiia Akademii Nauk Kirgizskoi SSR, 9 (1983) 7?10. [12] A.L. Ljung, V. Frishfelds, T.S. Lundström, B.D. Marjavaara, Discrete and continuous modeling of heat and mass transport in drying of a bed of iron ore pellets, Drying Technol, 30(7) (2012) 760-773.
• [13] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, USA, (1993).
• [14] I. Podlubny, Fractional Differential Equations, Academic Press, New York, (1999).
• [15] K.R. Prasad, M. Khuddush and D. Leela, Existence and uniqueness of solutions of system of neutral fractional order boundary value problems by tripled ?xed point theorem, J. Inter. Math. Virtual Inst., 10(1) (2020) 123-137.
• [16] K.R. Prasad, M. Khuddush and D. Leela, Existence of positive solutions for half-linear fractional order BVPs by application of mixed monotone operators, Creat. Math. Inform., 29(1) (2020) 65-80.
• [17] K.R. Prasad, M. Khuddush and M. Rashmita, Denumerably many positive solutions for fractional order boundary value problems, Creat. Math. Inform., 29(2) (2020) 191-203.
• [18] G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, (1993).
• [19] C.C. Tisdell, A. Zaidi, Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling, Nonlinear Anal., 68 (2008) 3504-3524.
• [20] L. Wang, C. Zhai, Unique solutions for new fractional differential equations with p-Laplacian and in?nite-point boundary conditions, Int. J. Dyn. Differ. Equ., 9(1) (2019) 1-13.
• [21] Y. Tian, Z. Bai and S. Sun, Positive solutions for a boundary value problem of fractional differential equation with p-Laplacian operator, Adv. di?er. Equ., 349 (2019).
• [22] T.K. Yuldashev, B.J. Kadirkulov, Nonlocal problem for a mixed type fourth order differential equation with Hilfer fractional operator, Ural Math. J., 6(2020) 153-167.
• [23] J. Zhang, R.P. Agarwal, and N. Jiang, N-fixed point theorems and N best proximity point theorems for generalized contraction in partially ordered metric spaces, J. Fixed Point Theory and Appl., 18 (2018).
• [24] K. Zhao and J. Liu, Multiple monotone positive solutions of integral BVPs for a higher-order fractional differential equation with monotone homomorphism, Adv. Di?erence Equ., 2016(2016), pp. 20.
• [25] Y. Zhao, H. Chen, L. Huang, Existence of positive solutions for nonlinear fractional functional differential equation, Comput. Math. Appl., 64 (10) (2012) 3456-3467.
• [26] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.