Existence, uniqueness, and stability analysis of coupled random fractional boundary value problems with nonlocal conditions

Document Type : Research Paper


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

2 Department of Mathematics, Ecole Normale Sup\'{e}rieure, Taleb Abderrahmane de Laghouat, 03000 Laghouat, Algeria.

3 Department of Applied Mathematics, Andhra University, Visakhapatnam, 530003, India.


In this paper, the existence, uniqueness, compactness, and stability of a coupled random differential equations involving the Hilfer fractional derivatives with nonlocal boundary conditions are discussed. Arguments are discussed via some random fixed point theorems in a separable vector Banach spaces and Ulam type stability. Some examples are presented to ensure the abstract results.


  • [1] M. I. Abbas, Existence and uniqueness results for fractional differential equations with Riemann-Liouville frac- tional integral boundary conditions, Abstr. Appl. Anal. 290674 (2015), 1-6.
  • [2] S. Abbas, N. Al Arifi, M. Benchohra, and J. Graef, Random coupled systems of implicit Caputo-Hadamard fractional differential equations with multi-point boundary conditions in generalized Banach spaces, Dynam. Syst. Appl, 28(2) (2019), 229-350.
  • [3] S. Abbas, N. Al Arifi, M. Benchohra, and Y. Zhou, Random coupled Hilfer and Hadamard fractional differential systems in generalized Banach spaces, Mathematics, 7(3), (2019), 1-15.
  • [4] S.  Abbas,  M.  Benchohra,  and  G.  M.  N’Gu´er´e  kata,  Topics in fractional diferential equations,  Springer  Verlag, New York, 2012.
  • [5] S. Abbas, M. Benchohra, and J. Henderson, Coupled Caputo-Fabrizio fractional differential systems in generalized Banach spaces, Malaya Journal of Matematik, 9(1) (2021), 20-25.
  • [6] S. Abbas, M. Benchohra, J. E. Lazreg, and J. J. Nieto, On a coupled system of Hilfer and Hilfer-Hadamard fractional differential equations in Banach spaces, J. Nonlinear Funct. Anal, 12 (2018), 1-12.
  • [7] I. Altun, N. Hussain, M. Qasim, and H. H. Al-Sulami, A new fixed point result of Perov type and its application to a semilinear operator system, Mathematics, 7(11) (2019), 1-10.
  • [8] J. Alzabut, A. G. Selvam, D. Vignesh, and Y. Gholami, Solvability and stability of nonlinear hybrid ∆-difference equations of fractional order, Inter. J. Nonl. Sci. Num. Sim., 2021 Aug 20.
  • [9] E. H. Ait dads, M. Benyoub, and M. Ziane, Existernce results for Riemann- Liouville fractional evolution inclu- sions in Banach spaces, Afrika Matematika, 4 (2020), 709-715.
  • [10] G. Allaire and S. M. Kaber, Numerical linear algebra, Ser. Texts in Applied Mathematics, Springer, New York, 2008.
  • [11] S. Aljoudi, B. Ahmadand, and A. Alsaedi, Existence and uniqueness results for a coupled system of Caputo- Hadamard fractional differential equations with nonlocal Hadamard type integral boundary conditions, Fractal Fract., 4(13) (2020), 1-15.
  • [12] A. Baliki, J. J. Nieto, A. Ouahab, and M. L. Sinacer, Random semilinear system of differential equations with impulses, Fixed Point Theory and Applications, (2017), 1–29.
  • [13] A. T. Bharucha-Reid, Random Integral Equations, Academic Press, New York, (1972).
  • [14] M. Boumaaza, M. Benchohra, and J. Henderson, Random coupled Caputo-type modification of Erdelyi-Kober fractional differential systems in generalized Banach spaces with retarded and advanced arguments, 2021 (2021), 1–14.
  • [15] C. Burgos, J. C. Cortes, L. Villafuerte, and R. J. Villanueva, Solving random fractional second-order linear equations via the mean square Laplace transform: Theory and statistical computing, Applied Mathematics and Computation. 2022 Apr 1, 418:126846.
  • [16] K. Diethelm, Analysis of fractional differential equations, Springer, Berlin, 2010.
  • [17] A. M. El-Sayed AM, F. Gaafar, and M. El-Gendy, Continuous dependence of the solution of random fractional- order differential equation with nonlocal conditions, Fractional Differential Calculus, 1 (2017), 135-149.
  • [18] M. Q. Feng, X. M. Zhang, and W. G. Ge, New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions, Bound. Value Probl., 12(2011), Art. ID 720702, 1-20.
  • [19] C. Guendouz, J. E. Lazreg, J. J. Nieto, and A. Ouahab, Existence and compactness results for a system of fractional differential equations, Journal of Function Spaces, 5735140 (2020), 1-13.
  • [20] J. R. Graef, J. Henderson, and A. Ouahab, Topological methods for differential equations and inclusions, Mono- graphs and Research Notes in Mathematics Series Profile, CRC Press, Boca Raton, FL, 2019.
  • [21] Y. Hafssa and Z. Dahmani, Solvability for a sequential system of random fractional differential equations of Hermite type, Journal of Interdisciplinary Mathematics, 4(2022), 1-21.
  • [22] D. Henry, Geometric theory of semilinear parabolic differential equations, Springer, Berlin, New York, 1989.
  • [23] N. Heymans and I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheol. Acta., 45 (2006), 765-771.
  • [24] R. Hilfer, Applications of fractional calculus in Physics, World Scientific, Singapore, 2000.
  • [25] R. Hilfer, Y. Luchko, and Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal., 12(3) (2009), 299-318.
  • [26] S. Jabeen, Z. Zheng, M. U. Rehman, W. Wei, and J. Alzabut, Some fixed point results of weak-fuzzy graphical contraction mappings with application to integral equations, Mathematics, 9(5) (2021), 1-16.
  • [27] F. Jarad, S. Harikrishna, K. Shah, and K. Kanagarajan, Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative, Discrete and Continuous Dynamical Systems Series S, 13(3) (2020), 1-17.
  • [28] T. Jin, S. Gao, H. Xia, and H. Ding, Reliability analysis for the fractional-order circuit system subject to the uncertain random fractional-order model with Caputo type, Journal of Advanced Research, 32 (2021), 15-26.
  • [29] 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, Mathematics, 9(14) (2021), 1-23.
  • [30] M. K. A. Kaabar, M. Shabibi, J. Alzabut, S. Etemad, W. Sudsutad, F. Martinez, and S. Rezapour, Investigation of the fractional strongly singular thermostat model via fixed point techniques, Mathematics, 9(18) (2021), 2298.
  • [31] M. Kamenskii, V. Obukhovskii, and P. Zecca, Condensing multivalued maps and semilinear differential inclusions in Banach spaces, De Gruyter, Berlin, 2001.
  • [32] M. Khuddush, Existence of solutions to the iterative system of nonlinear two-point tempered fractional order boundary value problems, Advanced Studies: Euro-Tbilisi Mathematical Journal, 16(2) (2023), 97–114.
  • [33] 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., (2021). DOI: 10.3906/mat-2103-117
  • [34] M. Khuddush and K. R. Prasad, Existence, uniqueness and stability analysis of a tempered fractional order thermistor boundary value problems, The Journal of Analysis, (2022), 1-23.
  • [35] M. Khuddush and K. R. Prasad, Iterative system of nabla fractional fractional difference equations with two-point boundary conditions, Math. Appl., 11 (2022), 57-74.
  • [36] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
  • [37] G. S. Ladde and V. Lakshmikantham, Random Differential Inequalities, Academic Press, New York, (1980).
  • [38] 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. Equa., 2021(1) (2021), 1-46.
  • [39] S. Nageswara Rao, A. H. Msmali, M. S. Abdullah Ali, and H. Ahmadini, Existence and uniqueness for a system of Caputo-Hadamard fractional differential equations with multipoint boundary conditions, Journal of Function Spaces, 2020 (2020), 1-10.
  • [40] O. Nica, Initial-value problems for first-order differential systems with general nonlocal conditions, Elect. J. Differ. Equa., 2012 (2012), 1-15.
  • [41] D. O’Regan and R. Precup, Theorems of Leray-Schauder type and applications, Gordon and Breach, Amsterdam, 2021.
  • [42] K. R. Prasad, M. Khuddush, and 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., (2021).
  • [43] 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., (2021).
  • [44] 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(1) (2021), 138-157.
  • [45] A. I. Perov, On the Cauchy problem for a system of ordinary differential equations, Pviblizhen. Met. Reshen. Differ. Uvavn., 2 (1964), 115-134.
  • [46] A. I. Perov and A. V. Kibenko, On a certain general method for investigation of boundary value problems, Izvestiya Ros-siiskoi Akademii Nauk. Seriya Matematicheskaya, 30 (1996), 249-264.
  • [47] R. Precup and A. Viorel, Existence results for systems of nonlinear evolution equations, International Journal of Pure and Applied Mathematics, 47 (2008), 199-206.
  • [48] T. M. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carapathian Journal of Mathe- matics, 26 (2010), 103-107.
  • [49] I. Slimane and Z. Dahmani, A continuous and fractional derivative dependance of random differential equations with nonlocal conditions, Journal of Interdisciplinary Mathematics, 24(6) (2021), 1457-1470.
  • [50] R. S. Varga, Matrix Iterative Analysis, Springer Series in Computational Mathematics, Springer: Berlin, Germany, Volume 27, 2000.
  • [51] D. W. J. Victor and M. Khuddush, Existence of solutions for n-dimensional fractional order bvp with -point boundary conditions via the concept of measure of noncompactness, Advanced Studies: Ero-Tbilisi Mathematical Journal 15(1) (2022), 19-37.
  • [52] H. Yfrah, Z. Dahmani, L. Tabharit, and L. Abdelnebi, High order random fractional differential equations: exis- tence, uniqueness and data dependence, J. Interdisciplinary Mathematics, 24(8) (2021), 2121-2138.