Some qualitative analyses of neutral functional delay differential equation with generalized Caputo operator. (English) Zbl 1476.34159

Boutiara, Abdellatif; Matar, Mohammed M.; Kaabar, Mohammed K. A.; Martínez, Francisco; Etemad, Sina; Rezapour, Shahram

J. Funct. Spaces 2021, Article ID 9993177, 13 p. (2021).

Summary: In this paper, a new class of a neutral functional delay differential equation involving the generalized \(\psi\)-Caputo derivative is investigated on a partially ordered Banach space. The existence and uniqueness results to the given boundary value problem are established with the help of the Dhage’s technique and Banach contraction principle. Also, we prove other existence criteria by means of the topological degree method. Finally, Ulam-Hyers type stability and its generalized version are studied. Two illustrative examples are presented to demonstrate the validity of our obtained results.

MSC:

34K37	Functional-differential equations with fractional derivatives
34K40	Neutral functional-differential equations
34K30	Functional-differential equations in abstract spaces
34K10	Boundary value problems for functional-differential equations
47N20	Applications of operator theory to differential and integral equations
34K27	Perturbations of functional-differential equations

References:

[1]	Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0789.26002
[2]	Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and applications of fractional differential equations, 204 of North-Holland Mathematics Sudies (2006), Amsterdam: Elsevier Science B.V., Amsterdam · Zbl 1092.45003
[3]	Podlubny, I., Fractional Differential Equations (1999), New York: Academic Press, New York · Zbl 0918.34010
[4]	Boutiara, A.; Guerbati, K.; Benbachir, M., Caputo-Hadamard fractional differential equation with three-point boundary conditions in Banach spaces, AIMS Mathematics, 5, 1, 259-272 (2020) · Zbl 1484.34018
[5]	Boutiara, A.; Guerbati, K.; Benbachir, M., Measure of noncompactness for nonlinear Hilfer fractional differential equation in Banach spaces, Ikonion Journal of Mathematics, 1, 2, 456-465 (2019)
[6]	Boutiara, A.; Guerbati, K.; Benbachir, M., Caputo type fractional differential equation with nonlocal Erdélyi-Kober type integral boundary conditions in Banach spaces, Surveys in Mathematics and its Applications, 15, 399-418 (2020) · Zbl 1463.34253
[7]	Matar, M. M.; Abbas, M. I.; Alzabut, J.; Kaabar, M. K. A.; Etemad, S.; Rezapour, S., Investigation of the p-Laplacian nonperiodic nonlinear boundary value problem via generalized Caputo fractional derivatives, Advances in Difference Equations, 2021, 1 (2021) · Zbl 1487.34028
[8]	Mohammadi, H.; Baleanu, D.; Etemad, S.; Rezapour, S., Criteria for existence of solutions for a Liouville-Caputo boundary value problem via generalized Gronwall’s inequality, Journal of Inequalities and Applications, 2021, 1 (2021) · Zbl 1504.34012
[9]	Martnez, F.; Martnez, I.; Kaabar, M. K. A.; Ortz-Munuera, R.; Paredes, S., Note on the conformable fractional derivatives and integrals of complex-valued functions of a real variable, IAENG International Journal of Applied Mathematics, 50, 3, 609-615 (2020)
[10]	Martnez, F.; Martnez, I.; Kaabar, M. K. A.; Paredes, S., New results on complex conformable integral, AIMS Mathematics, 5, 6, 7695-7710 (2020) · Zbl 1484.30002
[11]	Alqahtani, B.; Aydi, H.; Karapinar, E.; Rakocevic, V., A solution for Volterra fractional integral equations by hybrid contractions, Mathematics, 7, 8, 694 (2019)
[12]	Afshari, H.; Kalantari, S.; Karapinar, E., Solution of fractional differential equations via coupled fixed point, Electronic Journal of Differential Equations, 2015, 286 (2015) · Zbl 1328.47081
[13]	Benbernou, S.; Gala, S.; Ragusa, M. A., On the regularity criteria for the 3D magnetohydrodynamic equations via two components in terms of BMO space, Mathematicsl Methods in the Applied Sciences, 37, 15, 2320-2325 (2014) · Zbl 1303.35068
[14]	Mahmudov, N.; Matar, M. M., Existence of mild solution for hybrid differential equations with arbitrary fractional order, TWMS Journal of Pure and Applied Mathematics, 8, 2, 160-169 (2017) · Zbl 1388.34007
[15]	Matar, M. M., Existence of solution for fractional neutral hybrid differential equations with finite delay, Rocky Mountain Journal of Mathematics, 50, 6 (2020) · Zbl 1458.34134
[16]	Baleanu, D.; Etemad, S.; Rezapour, S., On a fractional hybrid integro-differential equation with mixed hybrid integral boundary value conditions by using three operators, Alexandria Engineering Journal, 59, 5, 3019-3027 (2020)
[17]	Matar, M. M., Qualitative properties of solution for hybrid nonlinear fractional differential equations, Afrika Matematika, 30, 7-8, 1169-1179 (2019) · Zbl 1449.34029
[18]	Matar, M. M., Approximate controllability of fractional nonlinear hybrid differential systems via resolvent operators, Journal of Mathematics, 2019 (2019) · Zbl 1437.34010
[19]	Zhao, Y.; Sun, S.; Han, Z.; Li, Q., Theory of fractional hybrid differential equations, Computers & Mathematcs with Applications, 62, 3, 1312-1324 (2011) · Zbl 1228.45017
[20]	Almeida, R., A Caputo fractional derivative of a function with respect to another function, Communications in Nonlinear Science and Numerical Simulation, 44, 460-481 (2017) · Zbl 1465.26005
[21]	Abdo, M. S.; Panchal, S. K.; Saeed, A. M., Fractional boundary value problem with \(\psi \)-Caputo fractional derivative, Proceedings Mathematical Sciences, 129, 5 (2019) · Zbl 1426.34003
[22]	Almeida, R.; Malinowska, A. B.; Monteiro, M. T. T., Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications, Mathematicsl Methods in the Applied Sciences, 41, 1, 336-352 (2018) · Zbl 1384.34010
[23]	Almeida, R., Functional differential equations involving the \(\psi \)-Caputo fractional derivative, Fractal and Fractional, 4, 2, 1-8 (2020)
[24]	Baitiche, Z.; Derbazi, C.; Matar, M. M., Ulam stability for nonlinear-Langevin fractional differential equations involving two fractional orders in the \(\psi \)-Caputo sense, Applicable Analysis, 1-16 (2021) · Zbl 07584247
[25]	Jarad, F.; Abdeljawad, T.; Baleanu, D., On the generalized fractional derivatives and their Caputo modification, The Journal of Nonlinear Sciences and Applications, 10, 5, 2607-2619 (2017) · Zbl 1412.26006
[26]	Abbas, M. I.; Alessandra Ragusa, M., Solvability of Langevin equations with two Hadamard fractional derivatives via Mittag-Leffler functions, Applicable Analysis, 1-15 (2021) · Zbl 1500.34004
[27]	Akdemir, A. O.; Butt, S. I.; Nadeem, M.; Ragusa, M. A., New general variants of Chebyshev type inequalities via generalized fractional integral operators, Mathematics, 9, 2, 122 (2021)
[28]	Dhage, B. C., A new monotone iteration principle in the theory of nonlinear first order integro-differential equations, Nonlinear Studies, 22, 2015, 397-417 (2010) · Zbl 1329.45011
[29]	Dhage, B. C., Partially condensing mappings in partially ordered normed linar spaces and applications to functional integral equations, Tamkang Journal of Mathematics, 45, 4, 397-426 (2014) · Zbl 1343.45004
[30]	Dhage, B. C., Hybrid fixed point theory in partially ordered normed linear spaces and applications to fractional integral equations, Differential Equations & Applications, 5, 2, 155-184 (2013) · Zbl 1279.45005
[31]	Dhage, B. C.; Khurpe, G. T.; Shete, A. Y.; Salunke, J. N., Existence and approximate solutions for nonlinear hybrid fractional integro-differential equations, International Journal of Analysis and Applications, 11, 2, 157-167 (2016) · Zbl 1379.65099
[32]	Ardjouni, A.; Djoudi, A.; Department of Mathematics, University of Annaba, Approximating solutions of nonlinear hybrid Caputo fractional integro-differential equations via Dhage iteration principle, Ural Mathematical Journal, 5, 1, 3-12 (2019) · Zbl 1463.45028
[33]	Shah, K.; Ali, A.; Khan, R. A., Degree theory and existence of positive solutions to coupled systems of multi-point boundary value problems, Boundary Value Problem, 2016, 1, article 43 (2016) · Zbl 1339.34013
[34]	Sher, M.; Shah, K.; Fečkan, M.; Khan, R. A., Qualitative analysis of multi-terms fractional order delay differential equations via the topological degree theory, Mathematics, 8, 2, 218 (2020)
[35]	Ullah, A.; Shah, K.; Abdeljawad, T.; Khan, R. A.; Mahariq, I., Study of impulsive fractional differential equation under Robin boundary conditions by topological degree method, Boundary Value Problem, 2020, 1, article 98 (2020) · Zbl 1495.34012
[36]	Shah, K.; Hussain, W., Investigating a class of nonlinear fractional differential equations and its Hyers-Ulam stability by means of topological degree theory, Numerical Functional Analysis and Optimization, 40, 12, 1355-1372 (2019) · Zbl 1425.34025
[37]	Shah, K.; Khan, R. A., Existence and uniqueness results to a coupled system of fractional order boundary value problems by topological degree theory, Numerical Functional Analysis and Optimization, 37, 7, 887-899 (2016) · Zbl 06648989
[38]	Zada, M. B.; Shah, K.; Khan, R. A., Existence theory to a coupled system of higher order fractional hybrid differential equations by topological degree theory, International Journal of Applied and Computational Mathematics, 4, 4, 102 (2018) · Zbl 1400.34015
[39]	Ulam, S. M., A Collection of the Mathematical Problems (1960), New York: Interscience, New York · Zbl 0086.24101
[40]	Hyers, D., On the stability of the linear functional equation, Proceedings of the National Academy of Sciences, 27, 4, 222-224 (1941) · JFM 67.0424.01
[41]	Ahmad, B.; Matar, M. M.; El-Salmy, O. M., Existence of solutions and Ulam stability for Caputo type sequential fractional differential equations of order \(\alpha\in\left( 2 , 3\right)\), International Journal of Analysis and Applications, 15, 1, 86-101 (2017) · Zbl 1382.34004
[42]	Ameen, R.; Jarad, F.; Abdeljawad, T., Ulam stability for delay fractional differential equations with a generalized Caputo derivative, Filomat, 32, 15, 5265-5274 (2018) · Zbl 07552840
[43]	Boutiara, A.; Etemad, S.; Hussain, A.; Rezapour, S., The generalized U-H and U-H stability and existence analysis of a coupled hybrid system of integro-differential IVPs involving \(\varphi \)-Caputo fractional operators, Advances in Difference Equations, 2021, 1 (2021) · Zbl 1494.34023
[44]	Matar, M. M.; Abu Skhail, E. S., On stability of nonautonomous perturbed semilinear fractional differential systems of order, Journal of Mathematics, 2018 (2018) · Zbl 1487.34029
[45]	Agarwal, R. P.; Meehan, M.; O’Regan, D., Fixed Point Theory and Applications, Cambridge Tracts in Mathematics (2001), Cambridge: Cambridge University Press, Cambridge · Zbl 0960.54027
[46]	Cho, Y. J.; Chen, Y.-Q., Toplogical Degree Theory and its Applications (2006), New York: Tylor and Francis, New York
[47]	Deimling, K., Nonlinear Functional Analysis (1985), Berlin: Springer, Berlin · Zbl 0559.47040
[48]	Isaia, F., On a nonlinear integral equation without compactness, Acta Mathematica Universitatis Comenianae, 75, 233-240 (2006) · Zbl 1164.45305
[49]	Hallaci, A.; Boulares, H.; Ardjouni, A., Existence and uniqueness for delay fractional differential equations with mixed fractional derivatives, Open Journal of Mathematical Analysis, 4, 2, 26-31 (2020)