Fractional \(p\)-Laplacian differential equations with multi-point boundary conditions in Banach spaces. (English) Zbl 1538.34032
Summary: This paper is devoted to the existence and the Hyers-Ulam stability of solutions for certain types of nonlinear differential equations involving the Liouville-Caputo fractional derivative with multi-point boundary conditions and the \(p\)-Laplacian operator in Banach spaces. The existence of solutions is derived with the help of the well-known Darbo’s fixed point theorem. Finally, an illustrative example is presented to show the validity of the obtained results.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 26A33 | Fractional derivatives and integrals |
| 46E15 | Banach spaces of continuous, differentiable or analytic functions |
| 47H10 | Fixed-point theorems |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 34G20 | Nonlinear differential equations in abstract spaces |
| 47H08 | Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc. |
Keywords:
Liouville-Caputo derivative; \(p\)-Laplacian operator; multi-point boundary conditions; fractional differential equations (FDE); Banach spaces; Darbo’s fixed point theorem; Hausdorff measure of non-compactness (HMNC)References:
| [1] | Abbas, S.; Benchohra, M.; Hamidi, N.; Henderson, J., Caputo-Hadamard fractional differential equations in Banach spaces, Fract. Calc. Appl. Anal., 21, 1027-1045 (2018) · Zbl 1434.34006 · doi:10.1515/fca-2018-0056 |
| [2] | Agarwal, RP; Benchohra, M.; Seba, D., On the application of measure of noncompactness to the existence of solutions for fractional differential equations, Results Math., 55, 221-230 (2009) · Zbl 1196.26009 · doi:10.1007/s00025-009-0434-5 |
| [3] | Aghajani, A.; Pourhadi, E.; Trujillo, JJ, Application of measure of noncompactness to a Cauchy problem for fractional differential equations in Banach spaces, Fract. Calc. Appl. Anal., 16, 962-977 (2013) · Zbl 1312.47095 · doi:10.2478/s13540-013-0059-y |
| [4] | Araci, A., Şen, E., Açikgöz, M., Srivastava, H.M.: Existence and uniqueness of positive and nondecreasing solutions for a class of fractional boundary value problems involving the \(p\)-Laplacian operator. Adv. Differ. Equ. 2015 (Article ID 40), 1-12 (2015) · Zbl 1346.35207 |
| [5] | Ayerbe Toledano, JM; Domínguez Benavides, T.; López Acedo, G., Measures of Noncompactness in Metric Fixed Point Theory. Springer Book Series on Operator Theory: Advances and Applications (1997), Basel: Birkhäuser Verlag, Basel · Zbl 0885.47021 |
| [6] | Bates, PW, On Some Nonlocal Evolution Equations Arising in Materials Science. Nonlinear Dynamics and Evolution Equations. Fields Institute Communications, 13-52 (2006), Providence: American Mathematical Society, Providence · Zbl 1101.35073 |
| [7] | Baitiche, Z.; Guerbati, K.; Benchohra, M., Weak solutions for nonlinear fractional differential equations with integral and multi-point boundary conditions, Pan Amer. Math. J., 29, 86-100 (2019) |
| [8] | Banaś, J., Goebel, K,: Measures of Noncompactness in Banach Spaces. Marcel Dekker, New York (1980) · Zbl 0441.47056 |
| [9] | Bednarz, A.; Byszewski, L., On abstract nonlocal Cauchy problem, Czasopismo Tech., 20, 11-17 (2015) |
| [10] | Benchohra, M.; Henderson, J.; Seba, D., Measure of noncompactness and fractional differential equations in Banach spaces, Commun. Appl. Anal., 12, 419-428 (2008) · Zbl 1182.26007 |
| [11] | Bothe, D., Multivalued perturbations of \(m\)-accretive differential inclusions, Israel J. Math., 108, 109-138 (1998) · Zbl 0922.47048 · doi:10.1007/BF02783044 |
| [12] | Boutiara, A., Mixed fractional differential equation with nonlocal conditions in Banach spaces, J. Math. Model., 9, 451-463 (2021) · Zbl 1499.34034 |
| [13] | Boutiara, A.; Benbachir, M.; Guerbati, K., Caputo type fractional differential equation with nonlocal Erdélyi-Kober type integral boundary conditions in Banach spaces, Surv. Math. Appl., 15, 399-418 (2020) · Zbl 1463.34253 |
| [14] | Boutiara, A.; Guerbati, K.; Benbachir, M., Caputo-Hadamard fractional differential equation with three-point boundary conditions in Banach spaces, AIMS Math., 5, 259-272 (2020) · Zbl 1484.34018 |
| [15] | 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. Adv. Differ. Equ. 2021 (Article ID 95), 1-21 (2021) · Zbl 1494.34023 |
| [16] | Byszewski, L., Theorem about existence and uniqueness of continuous solutions of nonlocal problem for nonlinear hyperbolic equation, Appl. Anal., 12, 173-180 (1991) · Zbl 0725.35060 · doi:10.1080/00036819108840001 |
| [17] | Byszewski, L., Existence and uniqueness of a classical solution to a functional-differential abstract nonlocal Cauchy problem, J. Appl. Math. Stochast. Anal., 12, 91-97 (1999) · Zbl 0934.34067 · doi:10.1155/S1048953399000088 |
| [18] | Chaharlang, M.M., Ragusa, M.A., Razani, A.: A sequence of radially symmetric weak solutions for some nonlocal elliptic problem in \(R^N\). Mediterr. J. Math. 17 (Article ID 53), 1-12 (2020) · Zbl 1437.35285 |
| [19] | Darbo, G., Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova, 24, 84-92 (1955) · Zbl 0064.35704 |
| [20] | Derbazi, C., Hammouche, H., Benchohra, M.: Weak solutions for some nonlinear fractional differential equations with fractional integral boundary conditions in Banach spaces. J. Nonlinear Funct. Anal. 2019 (Article ID 7), 1-11 (2019) · Zbl 1459.34014 |
| [21] | Goodrich, CS; Ragusa, MA, Hölder continuity of weak solutions of \(p\)-Laplacian PDEs with VMO coefficients, Nonlinear Anal. Theory Meth. Appl., 185, 336-355 (2019) · Zbl 1418.35058 · doi:10.1016/j.na.2019.03.015 |
| [22] | Heinz, HR, On the behavior of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., 7, 1351-1371 (1983) · Zbl 0528.47046 · doi:10.1016/0362-546X(83)90006-8 |
| [23] | Hilfer, R., Applications of Fractional Calculus in Physics (2000), Singapore: World Scientific Publishing Company, Singapore · Zbl 0998.26002 |
| [24] | Khan, H., Li, Y., Chen, W., Baleanu, D., Khan, A.: Existence theorems and Hyers-Ulam stability for a coupled system of fractional differential equations with \(p\)-Laplacian operator. Bound. Value Probl. 2017 (Article ID 157), 1-16 (2017) · Zbl 1483.35320 |
| [25] | Kilbas, AA; Srivastava, HM; Trujillo, JJ, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies (2006), Amsterdam: Elsevier Science BV, Amsterdam · Zbl 1092.45003 |
| [26] | Leibenson, LS, General problem of the movement of a compressible fluid in a porous medium, Izv. Akad. Nauk Kirg. SSR Ser. Biol. Nauk., 9, 7-10 (1983) |
| [27] | Liu, X.; Jia, M.; Xiang, X., On the solvability of a fractional differential equation model involving the \(p\)-Laplacian operator, Comput. Math. Appl., 64, 3267-3275 (2012) · Zbl 1268.34020 |
| [28] | Lu, H., Han, Z., Sun, S.: Multiplicity of positive solutions for Sturm-Liouville boundary value problems of fractional differential equations with \(p\)-Laplacian. Bound. Value Probl. 2014 (Article ID 26), 1-17 (2014) · Zbl 1304.34010 |
| [29] | Merzoug, I.; Guezane-Lakoud, A.; Khaldi, R., Existence of solutions for a nonlinear fractional \(p\)-Laplacian boundary value problem, Rend. Circ. Mat. Palermo (Ser. II), 69, 1099-1106 (2020) · Zbl 1465.34033 · doi:10.1007/s12215-019-00459-4 |
| [30] | Miller, KS; Ross, B., An Introdsction to Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication (1993), New York: Wiley, New York · Zbl 0789.26002 |
| [31] | Oldham, KB, Fractional differential equations in electrochemistry, Adv. Eng. Softw., 41, 9-12 (2010) · Zbl 1177.78041 · doi:10.1016/j.advengsoft.2008.12.012 |
| [32] | Ourraoui, A.; Ragusa, MA, An existence result for a class of \(p(x)\)-anisotropic type equations, Symmetry, 13, Article ID 633, 1-12 (2021) |
| [33] | Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering (1999), New York: Academic Press, New York · Zbl 0924.34008 |
| [34] | Sabatier, J.; Agrawal, OP; Machado, JAT, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering (2007), Dordrecht: Springer, Dordrecht · Zbl 1116.00014 · doi:10.1007/978-1-4020-6042-7 |
| [35] | Schwabik, S.; Guoju, Y., Topics in Banach Spaces Integration. Series in Real Analysis (2005), Singapore: World Scientific Publishing Company, Singapore · Zbl 1088.28008 · doi:10.1142/5905 |
| [36] | Shatanawi, W., Boutiara, A., Abdo, M.S., Jeelani, M.B., Abodayeh, K.: Nonlocal and multiple-point fractional boundary value problem in the frame of a generalized Hilfer derivative. Adv. Differ. Equ. 2021 (Article ID 294), 1-19 (2021) · Zbl 1494.34058 |
| [37] | Srivastava, HM, Fractional-order derivatives and integrals: Introductory overview and recent developments, Kyungpook Math. J., 60, 73-116 (2020) · Zbl 1453.26008 |
| [38] | Srivastava, HM, An introductory overview of fractional-calculus operators based upon the Fox-Wright and related higher transcendental functions, J. Adv. Engrg. Comput., 5, 135-166 (2021) · doi:10.55579/jaec.202153.340 |
| [39] | Srivastava, HM, Some parametric and argument variations of the operators of fractional calculus and related special functions and integral transformations, J. Nonlinear Convex Anal., 22, 1501-1520 (2021) · Zbl 1515.26013 |
| [40] | Srivastava, HM; da Costa Sousa, JV, Multiplicity of solutions for fractional-order differential equations via the \(\kappa (x)\)-Laplacian operator and the Genus theory, Fractal Fract., 6, Article ID 294, 1-27 (2022) |
| [41] | Srivastava, HM; Das, A.; Hazarika, B.; Mohiuddine, SA, Existence of solutions of infinite systems of differential equations of general order with boundary conditions in the spaces \(c_0\) and \(\ell_1\) via the measure of noncompactness, Math. Methods Appl. Sci., 41, 3558-3569 (2018) · Zbl 06923678 · doi:10.1002/mma.4845 |
| [42] | Tan, J.; Cheng, C., Existence of solutions of boundary value problems for fractional differential equations with \(p\)-Laplacian operator in Banach spaces, Numer. Funct. Anal. Optim., 38, 738-753 (2017) · Zbl 1372.34021 · doi:10.1080/01630563.2017.1293091 |
| [43] | Tan, J., Li, M.: Solutions of fractional differential equations with \(p\)-Laplacian operator in Banach spaces. Bound. Value Probl. 2018 (Article ID 15), 1-13 (2018) · Zbl 1483.34018 |
| [44] | Tarasov, VE, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media (2010), Beijing: Heidelberg & Higher Education Press, Beijing · Zbl 1214.81004 · doi:10.1007/978-3-642-14003-7 |
| [45] | Wang, JR; Lv, L.; Zhou, Y., Boundary value problems for fractional differential equations involving Caputo derivative in Banach spaces, J. Appl. Math. Comput., 38, 209-224 (2012) · Zbl 1296.34032 · doi:10.1007/s12190-011-0474-3 |
| [46] | Yang, C., Yan, J.: Positive solutions for third-order Sturm-Liouville boundary value problems with \(p\)-Laplacian. Comput. Math. Appl. 59, 2059-2066 (2010) · Zbl 1189.34056 |
| [47] | Zeidler, E., Nonlinear Functional Analysis and Its Applications. Nonlinear Monotone Operators (1989), Berlin: Springer, Berlin |
| [48] | Zhai, C., Guo, C.: Positive solutions for third-order Sturm-Liouville boundary-value problems with \(p\)-Laplacian. Electron. J. Differ. Equ. 2009 (Article ID 154), 1-9 (2009) · Zbl 1186.34035 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
