Existence theory and stability analysis to the system of infinite point fractional order BVPs by multivariate best proximity point theorem

Document Type : Research Paper

Authors

1 Department of Mathematics, Dr. Lankapalli Bullayya College of Engineering, Resapuvanipalem, Visakhapatnam, 530013, India

2 Department of Applied Mathematics, College of Science and Technology, Andhra University, Visakhapatnam, 530003, India

Abstract

This paper deals with the existence of solutions to the system of nonlinear infinite-point fractional order boundary value problems by an application of n-best proximity point theorem in a complete metric space. Further, we study Hyers-Ulam stability of the addressed system. An appropriate example is given to demonstrate the established results.

Keywords

[1] M.A. Akcoglu, A pointwise ergodic theorem in Lp-spaces, Canad. J. Math. 27 (1975), no. 5, 1075–1082.
[2] S. Banach, Th´eorie des op´erations lin´eaires, Z. Subwencji funduszu kultury Narodowej, 1979.
[3] E. Berkson and T.A. Gillespie, Mean-boundedness and Littlewood-Paley for separation-preserving operators, Trans.
Amer. Math. Soc. 349 (1997), no. 3, 1169–1189.
[4] R.V. Chacon and U. Krengel, Linear modulus of a linear operator, Proc. Amer. Math. Soc. 15 (1964), no. 4,
553–559.
[5] Z. Faghih and M.B. Ghaemi, Characterizations of pseudo-differential operators on S
1
based on separationpreserving operators, J. Pseudo-Diff. Oper. Appl. 12 (2021), no. 1, 1–14.
[6] M.B. Ghaemi and M. Jamalpour Birgani, L
p
-boundedness, compactness of pseudo-differential operators on compact Lie groups, J. Pseudo-Differ. Oper. Appl. 8 (2017), no. 1, 1–11.
[7] M.B. Ghaemi, M. Jamalpour Birgani and M.W. Wong, Characterizations of nuclear pseudo-differential operators
on S
1 with applications to adjoints and products, J. Pseudo-Differ. Oper. Appl. 8 (2017), no. 2, 191–201.
[8] M.B. Ghaemi, E. Nabizadeh, M. Jamalpour Birgani and M.K. Kalleji, A study on the adjoint of pseudo-differential
operators on compact lie groups, Complex Var. Elliptic Equ. 63 (2018), no. 10, 1408–1420.
[9] C.H. Kan, Ergodic properties of Lamperti operators, Canad. J. Math. 30 (1978), no. 6, 1206–1214.
[10] J. Lamperti, On the isometries of certain function-spaces, Pacific J. Math. 8 (1958), no. 3, 459–466.
[11] W. Rudin, Real and complex analysis, McGraw-Hill Book Company, 1987.
[12] M. Ruzhansky and V. Turunen, Pseudo-differential operators and symmetries: Background analysis and advanced
topics, Vol. 2. Birkhauser and Boston, 2009.
[13] A. Tulcea, Ergodic properties of isometries in L
p
spaces 1 < p < ∞, Bull. Amer. Math. Soc. 70 (1964), no. 3,
366–371.
[14] A. Wolfgang, Spectral properties of Lamperti operators, Indiana Univ. Math. J. 32 (1983), no. 2, 199–215.[17] M.K.A. Kaabar, A. Refice, M.S. Souid, F. Mart´ınez, S. Etemad, Z. Siri and S. Rezapour, Existence and UHR
stability of solutions to the implicit nonlinear FBVP in the variable order settings, Math. 9 (2021), no. 14, 1693.
[18] M. K. A. Kaabar, M. Shabibi, J. Alzabut, S. Etemad, W. Sudsutad, F. Martinez, S. Rezapour, Investigation of
the fractional strongly singular thermostat model via fixed point techniques, Math. 9 (2021), no. 18, 2298.
[19] A. Khemphet, Best proximity coincidence point theorem for G-proximal generalized geraghty mapping in a metric
space with graph G, Thai J. Math. 18 (2020), no. 3, 1161–1171.
[20] M. Khuddush and K. R. Prasad, Infinitely many positive solutions for an iterative system of conformable fractional
order dynamic boundary value problems on time scales, Turk. J. Math. 46 (2022), no. 2, 338–359.
[21] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, NorthHolland Mathematics Studies, 204, Elsevier, North Holland, 2006.
[22] S. Komal, P. Kumam, K. Khammahawong and K. Sitthithakerngkiet, Best proximity coincidence point theorems
for generalized non-linear contraction mappings, Filomat 32 (2018), no. 19, 6753–6768.
[23] M. M. Matar, M. I. Abbas, J. Alzabut, M. K. Kaabar, S. Etemad, S. Rezapour, Investigation of the p-Laplacian
nonperiodic nonlinear boundary value problem via generalized Caputo fractional derivatives, Adv. Differ. Equ.
2021 (2021), no. 1, 1–8.
[24] K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equation, John Wiley,
New York, (1993).
[25] H. Mohammadi, M.K.A. Kaabar, J. Alzabut, A.G.M. Selvam and S. Rezapour, Complete model of CrimeanCongo hemorrhagic fever (CCHF) transmission cycle with nonlocal fractional derivative, J. Funct. Spaces 2021
(2021), 1–12.
[26] I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
[27] K. R. Prasad, M. Khuddush, D. Leela, Existence of solutions for n-dimensional fractional order hybrid BVPs
with integral boundary conditions by an application of n-fixed point theorem, J. Anal. 29 (2021), no. 3, 963–985.
[28] K. R. Prasad, M. Khuddush and D. Leela, Existence, uniqueness and Hyers–Ulam stability of a fractional order
iterative two-point boundary value problems, Afr. Mat. 32 (2021)no. 7, 1227–1237.
[29] K. R. Prasad, D. Leela and M. Khuddush, Existence and uniqueness of positive solutions for system of (p, q, r)-
Laplacian fractional order boundary value problems, Adv. Theory Nonlinear Anal. Appl. 5 (2021), 138–157.
[30] K. R. Prasad, M. Khuddush, D. Leela, Existence of solutions for fractional order BVPs by mixed monotone
ternary operator with perturbation on Banach spaces, J. Adv. Math. Stud. 14 (2021), no. 1, 109–125.
[31] J.B. Prolla, Fixed point theorems for set valued mappings and existence of best approximations, Numer. Funct.
Anal. Optim. 5 (1982-1983), 449–455.
[32] S. Reich, Approximate selections, best approximations, fixed points and invariant sets, J. Math. Anal. Appl. 62
(1978), 104–113.
[33] V.S. Raj, A best proximity point theorem for weakly contractive non-self mappings, Nonlinear Anal. 74 (2011),
no. 14, 4804–4808.
[34] S. Rezapour, A. Imran, A. Hussain, F. Mart´ınez, S. Etemad and M.K.A. Kaabar, Condensing functions and
approximate endpoint criterion for the existence analysis of quantum integro-difference FBVPs, Symmetry 13
(2021), no. 3, 469.
[35] Y. Rohen and N. Mlaiki, Tripled best proximity point in complete metric spaces, Open Math., 18 (2020), 204–210.
[36] H. Sahin, Best proximity point theory on vector metric spaces, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math.
Stat. 70 (2021), no. 1, 130–142.
[37] B. Samet, Some results on best proximity points, J. Optim. Theory Appl. 159 (2013), no. 1, 281–291.
[38] W. Shatanawi and A. Pitea, Best proximity point and best proximity coupled point in a complete metric space
with (P)-property, Filomat 29 (2015), no. 1, 63–74.
[39] H.M. Srivastava, Diabetes and its resulting complications: Mathematical modeling via fractional calculus, PublicHealth Open Access 4 (2020), no. 3, 1–5.
[40] H.M. Srivastava, K.M. Saad and M.M. Khader, An efficient spectral collocation method for the dynamic simulation
of the fractional epidemiological model of the Ebola virus, Chaos, Solitons and Fractals 140 (2020), 1–7.
[41] S. Muthaiah, J. Alzabut, D. Baleanu, M.E. Samei and A. Zada, Existence, uniqueness and stability analysis
of a coupled fractional-order differential systems involving Hadamard derivatives and associated with multi-point
boundary conditions, Adv. Diff. Equ. 2021 (2021), no. 1, 1–46.
Volume 13, Issue 2
July 2022
Pages 1713-1733
  • Receive Date: 16 January 2022
  • Revise Date: 21 May 2022
  • Accept Date: 27 May 2022