ORIGINAL ARTICLE Toward the existence and uniqueness of solutions of second-order fuzzy volterra integro-differential equations with fuzzy kernel T. Allahviranloo • M. Khezerloo • O. Sedaghatfar • S. Salahshour Received: 7 June 2011 / Accepted: 17 January 2012 / Published online: 2 February 2012 Ó Springer-Verlag London Limited 2012 Abstract In this paper, we study existence and uniqueness of solutions of second-order fuzzy integro-differential equa- tions with fuzzy kernel under strongly generalized differen- tiability. To this end, four cases are considered to show the existence of the fuzzy solution mentioned equation. Some theorems are proved, and the uniqueness of the fuzzy solution is discussed step by step. Finally, the illustrated examples are solved to investigate the conditions of theorems. Keywords Second-order fuzzy  Volterra integro- differential equations  Fuzzy valued functions  Continuous solution 1 Introduction The fuzzy differential and integral equations are important part of the fuzzy analysis theory, and they have the important value of theory and application in control theory. Seikkala in [19] defined the fuzzy derivative and then, some generalizations of that have been investigated in [6, 16–18, 20, 22]. Consequently, the fuzzy integral which is the same as that of Dubois and Prade in [8] and, by means of the extension principle of Zadeh, showed that the fuzzy initial value problem x 0 ðtÞ¼ f ðt; xðtÞÞ; xð0Þ¼ x 0 has an unique fuzzy solution when f satisfies the generalized Lipschitz condition, which guarantees an unique solution of the deterministic initial value problem. Kaleva [11] studied the Cauchy problem of fuzzy differential equation, charac- terized those subsets of fuzzy sets in which the Peano theo- rem is valid. Bede et al. in [4, 5] have introduced a more general definition of the derivative for fuzzy mappings, enlarging the class of differentiable. Allahviranlo et al. [2] proposed Euler method to obtain the fuzzy solutions of hybrid fuzzy differential equations under generalized Hukuhara differentiability and in [1], a novel operator method is introduced for solving fuzzy linear differential equations under the assumption of strongly generalized differentiability. Park et al. in [13] have considered the existence of solution of fuzzy integral equation in Banach space, and Subrahmaniam and Sudarsanam in [21] have proved the existence of solution of fuzzy functional equations. Park and Jeong [12, 14] studied existence of solution of fuzzy integral equations of the form xðtÞ¼ f ðtÞþ Z t 0 kðt; s; xðsÞÞ ds; t  0; where f(t) and x(t) are fuzzy valued functions and k is a crisp function on real numbers. The existence and uniqueness of solutions of fuzzy Volterra integro- differential equations of the second kind using strongly generalized differentiability have been discussed by Hajighasemi et al. in [10]. We try to improve [10] and we use the proposed method in [10] to prove the existence and uniqueness of solutions of second-order fuzzy Volterra integro-differential equations. In both papers, we consider equations with fuzzy kernel but in fuzzy studies, discussion T. Allahviranloo (&)  M. Khezerloo  O. Sedaghatfar Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran e-mail: tofigh@allahviranloo.com M. Khezerloo e-mail: khezerloo_m@yahoo.com S. Salahshour Department of Mathematics, Mobarakeh Branch, Islamic Azad University, Mobarakeh, Iran 123 Neural Comput & Applic (2013) 22 (Suppl 1):S133–S141 DOI 10.1007/s00521-012-0849-x