Well-Posedness Results for Nonlocal Hilfer Fractional Coupled Boundary Value Problem

Authors

  • Priti Bansal Department of Mathematics, NIILM University, Kaithal (Haryana)136027, India
  • Manjeet Singh Department of Mathematics, NIILM University, Kaithal (Haryana)136027, India

DOI:

https://doi.org/10.31305/rrijm.2025.v10.n2.001

Keywords:

Hilfer fractional derivative, Coupled boundary value problem, Nonlocal boundary conditions, Existence and uniqueness, UH stability

Abstract

In this paper, we analyze qualitative results for existence and uniqueness for solution of the nonlocal coupled boundary value problem (BVP) involving Hilfer fractional derivative. To find our results, Banach’s contraction mapping principle have been applied. Also, we will establish stability results of Ulam-Hyers (UH) type for our solution. Illustrative examples are also have been presented to justify our results.

References

Agarwal R.P.,Belmekki M.and Benchohra M.:Existence results for semilinear functional differential inclusions involving Riemann-Liouville fractional derivative ,Dyn.Continuos Discrete Impuls. Syst.,(2008 to appear).

Ait Dads E., Benchohra M. and Hamani S. :Impulsive fractional differential inclusions involving the Caputo fractional derivative,Fract.Calc.Appl.Anal.,(2008 ,to appear ).

Asawasamrit S., Kijjathanakorn A., Ntouyas S. K. and Tariboon J.: Nonlocal boundary value problems for Hilfer fractional differential equations, Bulletin of the Korean Mathematical Society, 55:6 (2018), 1639-1657.

Asawasamrit S., Nithiarayaphaks W., Ntouyas S. K. and Tariboon J.: Existence and Stability Analysis for Fractional Differential Equations with Mixed Nonlocal Conditions, Mathematics, 7:117 (2019).

Benchohra M.,Hamani S.,Nieto J.J. and Slimani B.A. :Existence results for differential inclusions with fractional order and impulses,(2008 submitted).

Diethelm K.: The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2010.

Denardo,Eric V.: Contraction Mappings in the Theory Underlying Dynamic Programming,SIAM Review.9(2),165-177

Deimling K.: Nonlinear Functional Analysis. Springer-Verlag, New York, 1985.

Dhawan K., Vats R.K. and Agarwal, R.P.: Qualitative analysis of couple fractional differential equations involving Hilfer Derivative, An. St. Univ. Ovidius Constanta, 30:1 (2022), 191–217.

Dhawan K., Vats R.K. and Vijaykumar V.: Analysis of Neutral Fractional Differential Equation via the Method of Upper and Lower Solution, Qualitative Theory of Dynamical Systems, 22 (2023), 1-15.

Erturk Suat V.,Ali A.,Shah K.,Kumar K. and Abdeljawad T.: Existence and stability results for nonlocal boundary value problems of fractional order,Boundary Value Problems ,2022:25 ,(2022) .

Furati K.M., Kassim M.D. and Tatar N.E.: Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64 (2012), 1616-1626.

Gu H. and Trujillo J. J.: Existence of mild solution for evolution equation with Hilfer fractional derivative , Appl.Math. Comput., 257(2015), 344-354.

Heymans N and Podlubny I.: Physical interpretation of initial conditions for fractional differential equations with Reimann-Liouville fractional derivatives,Rheol.Acta,textbf45(5),(2006),765-772.

Hilfer R.: Applications of fractional calculus in Physics. World Scientific, Singapore, 2000.

Hilfer R.:Threefold introduction to fractional derivatives,In Anomalous Transport:Foundations and Applications,2008,17-73.

Hilfer R.: Experimental evidence for fractional time evolution in glass forming materials, Chemical Physics, 284 (2002), 399-408.

Hilfer R., Luchko Y. and Tomovski Z.: Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives, Fractional Calculus and Applied Analysis, 12 (2009), 299-318.

Krasnoselskii M. A.: Two remarks on the method of successive approximations, Uspekhi Matematicheskikh Nauk, 10 (1955), 123-127.

Kilbas A. A., Srivastava H. M. and Trujillo J. J.: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands,204,(2006).

Lakshmikantham V., Leela S. and Devi J. V.: Theory of Fractional Dynamic Systems. Cambridge, UK, 2009.

Metzler F.,Schick,W,Killian H.G. and Nonnenmacher,T.F. :Relaxation in filled polymers:A fractional calculus approach ,J.Chem. Phys.,103,(1995),7180-7186.

Miller K. S. and Ross B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, New York, NY, USA, 1993.

Nain A., Vats R. and Kumar A.: Coupled fractional differential equations involving Caputo–Hadamard derivative with nonlocal boundary conditions. Math Meth Appl Sci., 44 (2020), 4192-4204.

Nain A.K., Vats R.K. and Verma S.K.: Existence and uniqueness results for positive solutions of Hadamard type fractional BVP, Journal of Interdisciplinary Mathematics, 22(5), (2019), 697-710.

Nain A.K., Vats R.K. and Kumar A.: Caputo-Hadamard fractional differential equation with impulsive boundary conditions, Journal of Mathematical Modeling, 9(1), (2021), 93-106.

Oldham K.B.,Spanier J.:The Fractional Calculus, Academic Press, New York (1974).

Podlubny I.: Geometric and Physical interpretation of fractional integration and fractional differentiation,Fract.Calculus Appl.Anal.,textbf5,(2002),367-386.

Rus I. A.: Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J.Math., 26 (2010), 103-107.

Subashini R. and Ravichandran C.: On the results of nonlocal Hilfer fractional semilinear differential inclusions,Proceedings of the jangjeon mathematical society, 22(2), 2019,249-267.

Subashini R.,Jothimani K.,Saranya S. and Ravichandran C.: On the results of Hilfer fractional derivative with nonlocal conditions,International Journal of Pure and Applied Mathematics, 118(11), 2018,277-289.

Samko S.G., Kilbas A.A. and Marichev O.I.: Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Yver-don, 1993.

Verma S.K., Vats R.K. and Nain A.K.: Existence and uniqueness results for a fractional differential equation with nonlocal boundary conditions, Boletim da Sociedade Paranaense de Matematica, 40 (2022), 1-7.

Vivek D., Kanagarajan K. and Sivasundaram, S.: Dynamics and stability results for Hilfer fractional type thermistor problem, Fractal Fract. 1(1), (2017), 1-14.

Vivek D., Kanagarajan K. and Elsayed E.M.: Some existence and stability results for Hilfer-fractional implicit differential equations with nonlocal conditions, Mediterr. J. Math., 15, (2018), 1-21.

Wongcharoen A., Ahmed B., Ntouyas S. K. and Tariboon J.: Three-point boundary value problems for the Langevin equation with the Hilfer fractional derivative, Advances in Mathematical Physics, 2020 (2020), 9606428:1-11.

Wongcharoen A, Ntouyas S. K. and Tariboon J.: On coupled systems for Hilfer fractional equations with nonlocal integral boundary conditions, J.Math, 2020 (2020),2875152.

Wang J. and Zhang Y.: Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl.Math.Comput., 266 (2015), 850-859.

Zhou Y.: Baic Theory of Fractional Differential Equations. World Scientific:Singapore, 2014.

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Published

18-02-2025

How to Cite

Bansal, P., & Singh, M. (2025). Well-Posedness Results for Nonlocal Hilfer Fractional Coupled Boundary Value Problem. RESEARCH REVIEW International Journal of Multidisciplinary, 10(2), 01–16. https://doi.org/10.31305/rrijm.2025.v10.n2.001