Some existence results and stability concepts for partial fractional random integral equations with multiple delay

References

[1] S. Abbas and M. Benchohra, Fractional order Riemann–Liouville integral equations with multiple time delays, Appl. Math. E-Notes 12 (2012), 79–87. Search in Google Scholar

[2] S. Abbas and M. Benchohra, Fractional order integral equations of two independent variables, Appl. Math. Comput. 227 (2014), 755–761. 10.1016/j.amc.2013.10.086Search in Google Scholar

[3] S. Abbas and M. Benchohra, Ulam–Hyers stability for the Darboux problem for partial fractional differential and integro-differential equations via Picard operators, Results Math. 65 (2014), no. 1–2, 67–79. 10.1007/s00025-013-0330-xSearch in Google Scholar

[4] S. Abbas, M. Benchohra and T. Diagana, Existence and attractivity results for some fractional order partial integro-differential equations with delay, Afr. Diaspora J. Math. 15 (2013), no. 2, 87–100. Search in Google Scholar

[5] S. Abbas, M. Benchohra and G. M. N’Guérékata, Topics in Fractional Differential Equations, Dev. Math. 27, Springer, New York, 2012. 10.1007/978-1-4614-4036-9Search in Google Scholar

[6] S. Abbas, M. Benchohra and G. M. N’Guerekata, Advanced Fractional Differential and Integral Equations, Math. Res. Dev. Ser., Nova Science Publishers, New York, 2015. Search in Google Scholar

[7] S. Abbas, M. Benchohra and A. Petruşel, Ulam stability for partial fractional differential inclusions via Picard operators theory, Electron. J. Qual. Theory Differ. Equ. 2014 (2014), Paper No. 51. 10.14232/ejqtde.2014.1.51Search in Google Scholar

[8] S. Abbas, M. Benchohra, M. Rivero and J. J. Trujillo, Existence and stability results for nonlinear fractional order Riemann–Liouville Volterra–Stieltjes quadratic integral equations, Appl. Math. Comput. 247 (2014), 319–328. 10.1016/j.amc.2014.09.023Search in Google Scholar

[9] S. Abbas, M. Benchohra and S. Sivasundaram, Ulam stability for partial fractional differential inclusions with multiple delays and impulses via Picard operators, Nonlinear Stud. 20 (2013), no. 4, 623–641. Search in Google Scholar

[10] J. Appell, Implicit functions, nonlinear integral equations, and the measure of noncompactness of the superposition operator, J. Math. Anal. Appl. 83 (1981), no. 1, 251–263. 10.1016/0022-247X(81)90261-4Search in Google Scholar

[11] J. M. Ayerbe Toledano, T. Domínguez Benavides and G. López Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Oper. Theory Adv. Appl. 99, Birkhäuser, Basel, 1997. 10.1007/978-3-0348-8920-9Search in Google Scholar

[12] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus. Models and Numerical Methods, Ser. Complex. Nonlinearity Chaos 3, World Scientific Publishing, Hackensack, 2012. 10.1142/8180Search in Google Scholar

[13] J. Banaś and B. Rzepka, On existence and asymptotic stability of solutions of a nonlinear integral equation, J. Math. Anal. Appl. 284 (2003), no. 1, 165–173. 10.1016/S0022-247X(03)00300-7Search in Google Scholar

[14] J. Banaś and T. Zaja̧c, Solvability of a functional integral equation of fractional order in the class of functions having limits at infinity, Nonlinear Anal. 71 (2009), no. 11, 5491–5500. 10.1016/j.na.2009.04.037Search in Google Scholar

[15] J. Banaś and T. Zaja̧c, A new approach to the theory of functional integral equations of fractional order, J. Math. Anal. Appl. 375 (2011), no. 2, 375–387. 10.1016/j.jmaa.2010.09.004Search in Google Scholar

[16] A. T. Bharucha-Reid, Random Integral Equations, Math. Sci. Eng. 96, Academic Press, New York, 1972. Search in Google Scholar

[17] M. F. Bota-Boriceanu and A. Petruşel, Ulam–Hyers stability for operatorial equations, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 57 (2011), no. 1, 65–74. 10.2478/v10157-011-0003-6Search in Google Scholar

[18] D. Bothe, Multivalued perturbations of m-accretive differential inclusions, Israel J. Math. 108 (1998), 109–138. 10.1007/BF02783044Search in Google Scholar

[19] T. A. Burton and T. Furumochi, A note on stability by Schauder’s theorem, Funkcial. Ekvac. 44 (2001), no. 1, 73–82. Search in Google Scholar

[20] L. P. Castro and A. Ramos, Hyers–Ulam–Rassias stability for a class of nonlinear Volterra integral equations, Banach J. Math. Anal. 3 (2009), no. 1, 36–43. 10.15352/bjma/1240336421Search in Google Scholar

[21] A. Chlebowicz, M. A. Darwish and K. Sadarangani, Existence and asymptotic stability of solutions of a functional integral equation via a consequence of Sadovskii’s theorem, J. Funct. Spaces 2014 (2014), Article ID 324082. 10.1155/2014/324082Search in Google Scholar

[22] M. A. Darwish, J. Banaś and E. O. Alzahrani, The existence and attractivity of solutions of an Urysohn integral equation on an unbounded interval, Abstr. Appl. Anal. 2013 (2013), Article ID 147409. 10.1155/2013/147409Search in Google Scholar

[23] M. A. Darwish, J. Henderson and 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 (2011), no. 3, 539–553. 10.4134/BKMS.2011.48.3.539Search in Google Scholar

[24] M. A. Darwish and K. Sadarangani, On existence and asymptotic stability of solutions of a functional-integral equation of fractional order, J. Convex Anal. 17 (2010), no. 2, 413–426. Search in Google Scholar

[25] H. W. Engl, A general stochastic fixed-point theorem for continuous random operators on stochastic domains, J. Math. Anal. Appl. 66 (1978), no. 1, 220–231. 10.1016/0022-247X(78)90279-2Search in Google Scholar

[26] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer, Berlin, 1981. 10.1007/BFb0089647Search in Google Scholar

[27] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224. 10.1073/pnas.27.4.222Search in Google Scholar PubMed PubMed Central

[28] D. H. Hyers, G. Isac and T. M. Rassias, Stability of Functional Equations in Several Variables, Progr. Nonlinear Differential Equations Appl. 34, Birkhäuser, Boston, 1998. 10.1007/978-1-4612-1790-9Search in Google Scholar

[29] S. Itoh, Random fixed-point theorems with an application to random differential equations in Banach spaces, J. Math. Anal. Appl. 67 (1979), no. 2, 261–273. 10.1016/0022-247X(79)90023-4Search in Google Scholar

[30] S.-M. Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001. Search in Google Scholar

[31] S.-M. Jung, A fixed point approach to the stability of a Volterra integral equation, Fixed Point Theory Appl. 2007 (2007), Article ID 57064. 10.1155/2007/57064Search in Google Scholar

[32] S.-M. Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis, Springer Optim. Appl. 48, Springer, New York, 2011. 10.1007/978-1-4419-9637-4Search in Google Scholar

[33] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 204, Elsevier Science, Amsterdam, 2006. Search in Google Scholar

[34] G. S. Ladde and V. Lakshmikantham, Random Differential Inequalities, Math. Sci. Eng. 150, Academic Press, New York, 1980. Search in Google Scholar

[35] L. Liu, F. Guo, C. Wu and Y. Wu, Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl. 309 (2005), no. 2, 638–649. 10.1016/j.jmaa.2004.10.069Search in Google Scholar

[36] V. Lupulescu, D. O’Regan and G. ur Rahman, Existence results for random fractional differential equations, Opuscula Math. 34 (2014), no. 4, 813–825. 10.7494/OpMath.2014.34.4.813Search in Google Scholar

[37] B. G. Pachpatte, On Fredholm type integral equation in two variables, Differ. Equ. Appl. 1 (2009), no. 1, 27–39. 10.7153/dea-01-02Search in Google Scholar

[38] T. P. Petru and M.-F. Bota, Ulam–Hyers stability of operatorial inclusions in complete gauge spaces, Fixed Point Theory 13 (2012), no. 2, 641–649. Search in Google Scholar

[39] T. P. Petru, A. Petruşel and J.-C. Yao, Ulam–Hyers stability for operatorial equations and inclusions via nonself operators, Taiwanese J. Math. 15 (2011), no. 5, 2195–2212. 10.11650/twjm/1500406430Search in Google Scholar

[40] I. Podlubny, Fractional Differential Equations, Math. Sci. Eng. 198, Academic Press, San Diego, 1999. Search in Google Scholar

[41] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297–300. 10.1090/S0002-9939-1978-0507327-1Search in Google Scholar

[42] I. A. Rus, Picard operators and applications, Sci. Math. Jpn. 58 (2003), no. 1, 191–219. Search in Google Scholar

[43] I. A. Rus, Remarks on Ulam stability of the operatorial equations, Fixed Point Theory 10 (2009), no. 2, 305–320. Search in Google Scholar

[44] I. A. Rus, Ulam stability of ordinary differential equations, Stud. Univ. Babeş-Bolyai Math. 54 (2009), no. 4, 125–133. Search in Google Scholar

[45] J. L. Strand, Random ordinary differential equations, J. Differential Equations 7 (1970), 538–553. 10.1016/0022-0396(70)90100-2Search in Google Scholar

[46] C. P. Tsokos and W. J. Padgett, Random Integral Equations with Applications to Life Sciences and Engineering, Math. Sci. Eng. 108, Academic Press, New York, 1974. Search in Google Scholar

[47] S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts Pure Appl. Math. 8, Interscience Publishers, New York, 1960. Search in Google Scholar

[48] A. N. Vityuk and A. V. Golushkov, Existence of solutions of systems of partial differential equations of fractional order, Nelīnīĭnī Koliv. 7 (2004), no. 3, 328–335. 10.1007/s11072-005-0015-9Search in Google Scholar

[49] J. Wang, L. Lv and Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ. 2011 (2011), Paper No. 63. 10.14232/ejqtde.2011.1.63Search in Google Scholar

[50] J. Wang, L. Lv and Y. Zhou, New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 6, 2530–2538. 10.1016/j.cnsns.2011.09.030Search in Google Scholar

[51] W. Wei, X. Li and X. Li, New stability results for fractional integral equation, Comput. Math. Appl. 64 (2012), no. 10, 3468–3476. 10.1016/j.camwa.2012.02.057Search in Google Scholar

[52] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific Publishing, Hackensack, 2014. 10.1142/9069Search in Google Scholar