A fixed point theorem for Hardy-Rogers type on generalized fractional differential equations

Abstract

In this research paper, we introduce a generalization of Hardy-Rogers type contraction in a metric like space. Moreover, we apply this technique to investigate the existence and uniqueness of solutions for the classical boundary value problems and generalized fractional boundary value problems through deducing the main properties of the related Green functions. The main result of this paper is to establish the modified conditions of Hardy-Roger's fixed point theorem and introduce some advanced applications.

Keywords

• Hardy-Rogers-type contractions
• fixed points
• metric-like spaces
• generalized fractional differential equations

References

1. [1] M. Abbas et. al., fixed point of T-Hardy-Rogers contractive mappings in partially ordered partial metric spaces, Inter. j. math. sci., vol. 2012,Articale ID 313675,11.
2. [2] M.S. Abdo, S.K. Panchal, Caputo fractional integro-differential equation with nonlocal conditions in Banach space}, Int. J. Appl. Math. (IJAM), (2019), 32(2), 279-288.
3. [3] M.S. Abdo, H.A. Wahash and S.K. Panchal, \textit{ Positive solution of a fractional differential equation with integral boundary conditions}, Journal of Applied Mathematics and Computational Mechanics,{17} (2018), 5-15.
4. [4] M.S. Abdo, S.K. Panchal, A.M. Saeed,\textit{Fractional boundary value problem with $\psi $-Caputo fractional derivative}, Proceedings- Math. Sci.,{129}, No 5 (2019), 65.
5. [5] M.S. Abdo, A.G. Ibrahim and S.K. Panchal, \textit{Nonlinear implicit fractional differential equation involving $\psi $-Caputo fractional derivative}. Nonlinear implicit Proceedings of the Jangjeon Mathematical Society, 22 (3), (2019), 387-400.
6. [6] M. Alfuraidan, M. Bachar, M. A. Khamsi.,\textit{A graphical version of Reich's fixed point theorem}, Journal on nonlinear science and applications, 9(2016), 3931-3938.
7. [7] R. Almeida, \textit{A Caputo fractional derivative of a function with respect to another function}, Communications in Nonlinear Science and Numerical Simulation, 44(2017), 460-481.
8. [8] R. Almeida, A. B. Malinowska and M. T. Monteiro, \textit{Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications}, Mathematical Methods in the Applied Sciences, 41(2018), 336-352.

9. [9] B. Alqahtani, A. Fulga, F. Jarad, and E. Karapınar, \textit{Nonlinear F-contractions on b-metric spaces and differential equations in the frame of fractional derivatives with Mittag–Leffler kernel} Chaos, Solitons \& Fractals, 128, (2019) 349-354.‏
10. [10] H. Aydi, E. Karapinar and A. Francisco, \textit{ W-Interpolative Ciric-Reich-Rus Type contractions}, E-mathematics, .mdpi., 7(1).57(2019).
11. [11] S. Balasubramainam., \textit{An extended Reich fixed point theorem.}, arXiv:1301.4578v9[math.FA] 2014.
12. [12] S. Banach, \textit{Sur Les operations' dand Les ensembles abstrait et Leur application aux equations}, integrals Fundam. Math., (3)133-181, 1922.
13. [13] S. Belmor, C. Ravichandran, and F. Jarad, \textit{Nonlinear generalized fractional differential equations with generalized fractional integral conditions}, Journal of Taibah University for Science, 14(1), (2020) 114-123.‏
14. [14] C. Chifu, G.Patrusel., \textit{fixed point results for multi valued Hardy-Rogers contractions in b-metric spaces}, Faculty of sciences and mathematics, University of Nic, Serbia, 31:8(2017),2499-2507.
15. [15] G. E. Hardy and T. D. Rogers, \textit{A generalization of fixed point theorem of Reich}, Canada. Math. Bull. Vol. 16(2),1973.
16. [16] F. Jarad, T. Abdeljawad, S. Rashid, and Z. Hammouch, \textit{ More properties of the proportional fractional integrals and derivatives of a function with respect to another function} Advances in Difference Equations, 2020(1), (2020), 1-16.‏
17. [17] R. Kannan., \textit{some remarks on fixed points}., Bull Calcutta Math.Soc. 60(1960),71-76.
18. [18] A. A. Kilbas , Shrivastava H. M. and Trujillo J. J., {Theory and Applications of Fractional Differential Equations}, Elsevier, Amsterdam (2006).
19. [19] K. D. Kucche, J. J. Nieto, V. Venktesh, \textit {Theory of Nonlinear Implicit Fractional Differential Equations}, Differential Equations and Dynamical Systems, 28(2020), 1-17.
20. [20] S. Kumar, Tiwari and K. Das, \textit{Cone metric spaces and fixed point theorems for generalized T-Reich contraction with c-distance} , vol. 2017, (1-9)2017.
21. [21] T. Lazar, G. Mot, S. Szentesi, \textit{The theory of Reichs' fixed point theorem for multivalued}, Fixed point theory and applicatios: 178421(2010).
22. [22] A. Nastasi and P. V. Filomate, \textit{Ageneralization of Riech's fixed point theorem for multi-valued mappings}.,vol.31, No.11(2017), pp. 3295-3305.
23. [23] V. Olisama, J. Olalern and H. Akewe., \textit{ Best proximity point results for Hardy-Rogers p-proximal cyclic contraction in uniform spaces}, fixed point theory and applications., 18(2018).
24. [24] M. Rangamma and P. Rama Bhadra,\textit{Hardy and Rogers type contractive condition and common fixed point theorem in cone-2-metric space for a family of self-maps}., Global journal of pure and applied mathematics.Vol. 12, 3(2016), pp.2375-2385.
25. [25] S. Reich., \textit{Kannan's fixed point theorem}, Bull, Univ. Mat. Italiana., (4)4(1971),1-11.
26. [26] M. Shoaib, T. Abdeljawad, M. Sarwar, and F. Jarad, \textit{Fixed Point Theorems for Multi-Valued Contractions in $ b $-Metric Spaces With Applications to Fractional Differential and Integral Equations}, IEEE Access, 7, (2019) 127373-127383.‏
27. [27] V. Rhymend, R. Hemavathyy., \textit{Common fixed point theorem for T-Hardy-Rogers contraction mapping in a cone metric space}, International mathematical forum, 5, 2010, no.30, 1495-1506.
28. [28] P. Saipara, K. Khammahawong., \textit{fixed point theorem for a generalized almost Hardy-Rogers- type F-contractive on metric-like spaces}.Mathematical methods in the applied sciences. 2019.
29. [29] B. Sharbu, A. Geremew, A. Baerhaue, \textit{A common fixed point theorem for Reich type co-cyclic contraction in dislocated quasi metric space}., Ethiopian journal of sciences and Technology, vol 10.No. 2(2017).
30. [30] C. Yu-Qing, \textit{On a fixed point problem of Reich}, JSTOR, American mathematical society, vol. 124, No. 10(1996) pp.3085-3088.