### AIMS Mathematics

2021, Issue 4: 4119-4141. doi: 10.3934/math.2021244
Research article Special Issues

# Existence and stability results for $\psi$-Hilfer fractional integro-differential equation with mixed nonlocal boundary conditions

• Received: 19 December 2020 Accepted: 01 February 2021 Published: 05 February 2021
• MSC : 26A33, 34A08, 34A12, 34B15

• In this paper, we discuss the existence, uniqueness and stability of boundary value problems for $\psi$-Hilfer fractional integro-differential equations with mixed nonlocal (multi-point, fractional derivative multi-order and fractional integral multi-order) boundary conditions. The uniqueness result is proved via Banach's contraction mapping principle and the existence results are established by using the Krasnosel'ski${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$'s fixed point theorem and the Larey-Schauder nonlinear alternative. Further, by using the techniques of nonlinear functional analysis, we study four different types of Ulam's stability, i.e., Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability. Some examples are also constructed to demonstrate the application of main results.

Citation: Weerawat Sudsutad, Chatthai Thaiprayoon, Sotiris K. Ntouyas. Existence and stability results for $\psi$-Hilfer fractional integro-differential equation with mixed nonlocal boundary conditions[J]. AIMS Mathematics, 2021, 6(4): 4119-4141. doi: 10.3934/math.2021244

### Related Papers:

• In this paper, we discuss the existence, uniqueness and stability of boundary value problems for $\psi$-Hilfer fractional integro-differential equations with mixed nonlocal (multi-point, fractional derivative multi-order and fractional integral multi-order) boundary conditions. The uniqueness result is proved via Banach's contraction mapping principle and the existence results are established by using the Krasnosel'ski${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$'s fixed point theorem and the Larey-Schauder nonlinear alternative. Further, by using the techniques of nonlinear functional analysis, we study four different types of Ulam's stability, i.e., Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability. Some examples are also constructed to demonstrate the application of main results.

 [1] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, New York: Springer, 2010. [2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of the Fractional Differential Equations, Amsterdam: Elseiver, 2006. [3] V. Lakshmikantham, S. Leela, J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge: Cambridge Scientific Publishers, 2009. [4] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, New York: John Wiley, 1993. [5] I. Podlubny, Fractional differential equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, New York: Academic Press, 1998. [6] B. Ahmad, A. Alsaedi, S. K. Ntouyas, J. Tariboon, Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities, Switzerland: Springer International Publishing, 2017. [7] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives Theory and Applications, New York: Gordon and Breach, 1993. [8] Y. Zhou, J. R. Wang, L. Zhang, Basic Theory of Fractional Differential Equations, Singapore: World Scientific, 2014. [9] R. Hilfer, Applications of Fractional Calculus in Physics, Singapore: World Scientific, 2000. [10] R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, Chem. Phys., 284 (2002), 399–408. doi: 10.1016/S0301-0104(02)00670-5 [11] R. Hilfer, Y. Luchko, Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Liouvill fractional derivatives, Fractional Calculus Appl. Anal., 12 (2009), 299–318. [12] K. M. Furati, M. D. Kassim, N. E. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64 (2012), 1616–1626. doi: 10.1016/j.camwa.2012.01.009 [13] H. Gu, J. J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput., 257 (2015), 344–354. [14] J. Wang, Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput., 266 (2015), 850–859. [15] C. Thaiprayoon, W. Sudsutad, S. K. Ntouyas, Mixed nonlocal boundary value problem for implicit fractional integro-differential equations via $\psi$-Hilfer fractional derivative, Adv. Difference Equations, 2021 (2021), 1–24. [16] S. Asawasamrit, A. Kijjathanakorn, S. K. Ntouyas, J. Tariboon, Nonlocal boundary value problems for Hilfer fractional differential equations, Bull. Korean Math. Soc., 55 (2018), 1639–1657. [17] W. Saengthong, E. Thailert, S. K. Ntouyas, Existence and uniqueness of solutions for system of Hilfer-Hadamard sequential fractional differential equations with two point boundary conditions, Adv. Difference Equations, 2019 (2019), 1–16. [18] J. Vanterler da C. Sousa, E. C. Oliveira, On the $\psi$-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72–91. doi: 10.1016/j.cnsns.2018.01.005 [19] J. P. Kharade, K. D. Kucche, On the impulsive implicit $\psi$‐Hilfer fractional differential equations with delay, Math. Meth. Appl. Sci., 2019 (2019), 1–15. [20] J. Vanterler da C. Sousa, E. Capelas de Oliveira, Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation, Appl. Math. Letters, 81 (2018), 50–56. doi: 10.1016/j.aml.2018.01.016 [21] J. Vanterler da C. Sousa, K. D. Kucche, E. Capelas de Oliveira, On the Ulam-Hyers stabilities of the solutions of $\psi$-Hilfer fractional differential equation with abstract Volterra operator, Math. Methods Appl. Sci., 42 (2019), 3021–3032. doi: 10.1002/mma.5562 [22] J. Vanterler da C. Sousa, K. D. Kucche, E. Capelas de Oliveira, Stability of $\psi$-Hilfer impulsive fractional differential equations, Appl. Math. Letters, 88 (2019), 73–80. doi: 10.1016/j.aml.2018.08.013 [23] J. Vanterler da C. Sousa, E. Capelas de Oliveira, K. D. Kucche, On the Fractional Functional Differential Equation with Abstract Volterra Operator, Bull. Braz. Math. Soc. New Series, 50 (2019), 803–822. doi: 10.1007/s00574-019-00139-y [24] J. Vanterler da C. Sousa, E. Capelas de Oliveira, On the Ulam-Hyers-Rassias stability for nonlinear fractional differential equations using the $\psi$–Hilfer operator, J. Fixed Point Theory Appl., 20 (2018), 1–21. doi: 10.1007/s11784-018-0489-6 [25] K. D. Kucche, Ashwini D. Mali, Initial time difference quasilinearization method for fractional differential equations involving generalized Hilfer fractional derivative, Comput. Appl. Math., 39 (2020), 1–33. doi: 10.1007/s40314-019-0964-8 [26] J. Vanterler da C. Sousa, K. D. Kucche, E. Capelas de Oliveira, On the Ulam‐Hyers stabilities of the solutions of $\psi$‐Hilfer fractional differential equation with abstract Volterra operator, Math. Meth. Appl. Sci., 42 (2019), 1–12. doi: 10.1002/mma.5148 [27] J. Vanterler da C. Sousa, E. Capelas de Oliveira, Leibniz type rule: $\psi$-Hilfer fractional operator, Commun. Nonlinear Sci. Numer. Simul., 77 (2019), 305–311. doi: 10.1016/j.cnsns.2019.05.003 [28] J. Vanterler da C. Sousa, M. Benchohra, G. M. N'Guérékata, Attractivity for differential equations of fractional order and $\psi$-Hilfer type, Fractional Calculus Appl. Anal., 23 (2020), 1188–1207. doi: 10.1515/fca-2020-0060 [29] J. Vanterler da C. Sousa, F. Jarad, Th. Abdeljawad, Existence of mild solutions to Hilfer fractional evolution equations in Banach space, Ann. Funct. Anal., 12 (2021), 1–16. doi: 10.1007/s43034-020-00089-3 [30] A. Mali, K. Kucche, Nonlocal boundary value problem for generalized Hilfer implicit fractional differential equations, Math Meth Appl Sci., 2020 (2020), 1–24. [31] S. K. Ntouyas, D. Vivek, Existence and uniqueness results for sequential $\psi$-Hilfer fractional differential equations with multi-point boundary conditions, Acta Math. Univ. Comenianae, 2021 (2021), 1–15. [32] A. Granas, J. Dugundji, Fixed Point Theory, New York: Springer, 2003. [33] M. A. Krasnosel'ski${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$, Two remarks on the method of successive approximation, Uspekhi Mat. Nauk., 10 (1955), 123–127. [34] I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math., 26 (2010), 103–107.
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