Some new existence results and stability concepts for fractional partial random differential equations

• Autorzy: Abbas S. , Benchohra M. , Darwish M. A.

Identyfikatory

DOI

10.7862/rf.2016.1

• Języki publikacji: EN

Abstrakty

EN

In the present paper we provide some existence results and Ulam’s type stability concepts for the Darboux problem of partial fractional random differential equations in Banach spaces, by applying the measure of noncompactness and a random fixed point theorem with stochastic domain.

Słowa kluczowe

EN

random differential equations; left-sided mixed Riemann-Liouville integral; Caputo fractional-order derivative; Banach space; Darboux problem; Ulam stability

PL

równanie różniczkowe; pochodna Caputo; przestrzeń Banacha; problem Darbouxa

• Wydawca: Oficyna Wydawnicza Politechniki Rzeszowskiej
• Czasopismo: Journal of Mathematics and Applications
• Rocznik: 2016
• Tom: Vol. 39
• Strony: 5--22
• Opis fizyczny: Bibliogr. 51 poz.

Twórcy

autor

Abbas S.

• Laboratory of Mathematics, University of Saïda, P.O. Box 138, Saïda 20000, Algeria

autor

Benchohra M.

• Laboratory of Mathematics, University of Sidi Bel-Abbes, P.O. Box 89, Sidi Bel-Abbes 22000, Algeria

autor

Darwish M. A.

• Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University, Jeddah, Saudi Arabia
• Department of Mathematics, Faculty of Science, Damanhour University, Damanhour, Egypt

Bibliografia

• [1] S. Abbas, M. Benchohra, Partial hyperbolic differential equations with finite delay involving the Caputo fractional derivative, Commun. Math. Anal. 7 (2009) 62-72.
• [2] S. Abbas, M. Benchohra, Fractional order partial hyperbolic differential equations involving Caputo’s derivative, Stud. Univ. Babes-Bolyai Math. 57 (4) (2012) 469479.
• [3] S. Abbas, M. Benchohra, A. Cabada, Partial neutral functional integro-differential equations of fractional order with delay, Bound. Value Prob. 2012 (2012) 128, 15 pp.
• [4] S. Abbas, M. Benchohra, G.M. N’Guerekata, Topics in Fractional Differential Equations, Springer, New York, 2012.
• [5] S. Abbas, M. Benchohra, G.M. N’Guerekata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015.
• [6] S. Abbas, M. Benchohra, J.J. Nieto, Ulam stabilities for impulsive partial fractional differential equations, Acta Univ. Palacki. Olomuc. 53 (1) (2014) 5-17.
• [7] S. Abbas, M. Benchohra, S. Sivasundaram, Ulam stability for partial fractional differential inclusions with multiple delay and impulses via Picard operators, J. Nonlinear Stud. 20 (4) (2013) 623-641.
• [8] S. Abbas, M. Benchohra, A.N. Vityuk, On fractional order derivatives and Darboux problem for implicit differential equations, Frac. Calc. Appl. Anal. 15 (2012) 168-182.
• [9] J. Appell, Implicit functions, nonlinear integral equations, and the measure of noncompactness of the superposition operator, J. Math. Anal. Appl. 83 (1981) 251-263.
• [10] J.M. Ayerbee Toledano, T. Dominguez Benavides, G. Lopez Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Operator Theory, Advances and Applications, vol. 99, Birkhauser, Basel, Boston, Berlin, 1997.
• [11] D. Baleanu, R.P. Agarwal, H. Mohammadi, Sh. Rezapour, Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces, Bound. Value Prob. 2013 (2013) 112, 10 pp.
• [12] D. Bothe, Multivalued perturbation of m-accretive differential inclusions, Isr. J. Math. 108 (1998) 109-138.
• [13] M.F. Bota-Boriceanu, A. Petrusel, Ulam-Hyers stability for operatorial equations and inclusions, Analele Univ. I. Cuza Iasi 57 (2011) 65-74.
• [14] T.A. Burton, T. Furumochi, A note on stability by Schauder’s theorem, Funkcial. Ekvac. 44 (2001) 73-82.
• [15] L.P. Castro, A. Ramos, Hyers-Ulam-Rassias stability for a class of Volterra integral equations, Banach J. Math. Anal. 3 (2009) 36-43.
• [16] M.A. Darwish, J. Henderson, D. O’Regan, Existence and asymptotic stability of solutions of a perturbed fractional functional-integral equation with linear modification of the argument, Bull. Korean Math. Soc. 48 (3) (2011) 539-553.
• [17] M.A. Darwish, J. Henderson, Nondecreasing solutions of a quadratic integral equation of Urysohn-Stieltjes type, Rocky Mountain J. Math. 42 (2) (2012) 545566.
• [18] M.A. Darwish, J. Banas, Existence and characterization of solutions of nonlinear Volterra-Stieltjes integral equations in two vriables, Abstr. Appl. Anal. 2014, Art. ID 618434, 11 pp.
• [19] B.C. Dhage, S.V. Badgire, S.K. Ntouyas, Periodic boundary value problems of second order random differential equations, Electron. J. Qual. Theory Diff. Equ. 21 (2009) 1-14.
• [20] H.W. Engl, A general stochastic fixed-point theorem for continuous random operators on stochastic domains, J. Math. Anal. Appl. 66 (1978) 220-231.
• [21] X. Han, X. Ma, G. Dai, Solutions to fourth-order random differential equations with periodic boundary conditions, Electron. J. Differential Equations 235 (2012) 1-9.
• [22] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
• [23] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27 (1941) 222-224.
• [24] D.H. Hyers, G. Isac, Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, 1998.
• [25] S. Itoh, Random fixed point theorems with applications to random differential equations in Banach spaces, J. Math. Anal. Appl. 67 (1979) 261-273.
• [26] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.
• [27] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, 2011.
• [28] S.-M. Jung, A fixed point approach to the stability of a Volterra integral equation, Fixed Point Theory Appl. 2007 (2007), Article ID 57064, 9 pp.
• [29] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
• [30] L. Liu, F. Guo, C. Wu, Y. Wu, Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl. 309 (2005) 638-649.
• [31] H. Monch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal., Theory Methods Appl. 4 (1980) 985-999.
• [32] B.G. Pachpatte, Analytic Inequalities. Recent Advances. Atlantis Studies in Mathematics, 3. Atlantis Press, Paris, 2012.
• [33] B.G. Pachpatte, Multidimensional Integral Equations and Inequalities. Atlantis Studies in Mathematics for Engineering and Science, 9. Atlantis Press, Paris, 2011.
• [34] T.P. Petru, M.-F. Bota, Ulam-Hyers stabillity of operational inclusions in complete gauge spaces, Fixed Point Theory 13 (2012) 641-650.
• [35] T.P. Petru, A. Petrusel, J.-C. Yao, Ulam-Hyers stability for operatorial equations and inclusions via nonself operators, Taiwanese J. Math. 15 (2011) 2169-2193.
• [36] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
• [37] I. Podlubny, I. Petras, B.M. Vinagre, P. O’Leary, L. Dorcak, Analogue realizations of fractional-order controllers. Fractional order calculus and its applications, Nonlinear Dynam. 29 (2002) 281-296.
• [38] Th.M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297-300.
• [39] I.A. Rus, Ulam stability of ordinary differential equations, Studia Univ. Babes-Bolyai, Math. LIV (4) (2009) 125-133.
• [40] I.A. Rus, Remarks on Ulam stability of the operatorial equations, Fixed Point Theory 10 (2009) 305-320.
• [41] T.T. Soong, Random Differential Equations in Science and Engineering. Academic Press, New York, 1973.
• [42] V.E. Tarasov, Fractional Dynamics. Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Heidelberg, 2010.
• [43] S.M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 1968.
• [44] A.N. Vityuk, A.V. Golushkov, Existence of solutions of systems of partial differential equations of fractional order, Nonlinear Oscil. 7 (3) (2004) 318-325.
• [45] A.N. Vityuk, A.V. Mikhailenko, The Darboux problem for an implicit fractional-order differential equation, J. Math. Sci. 175 (4) (2011) 391-401.
• [46] A.N. Vityuk, A.V. Mikhailenko, Darboux problem for differential equation with mixed regularized derivative of fractional order. Nonlinear Stud. 20 no. 4 (2013) 571-580.
• [47] J. Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ. 63 (2011) 1-10.
• [48] J. Wang, L. Lv, Y. Zhou, New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 25302538.
• [49] W. Wei, X. Li, X. Li, New stability results for fractional integral equation, Comput. Math. Appl. 64 (2012) 3468-3476.
• [50] S. Zhang, J. Sun, Existence of mild solutions for the impulsive semilinear nonlocal problem with random effects, Advances in Difference Equations 19 (2014) 1-11.
• [51] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).