Toward the existence and uniqueness of solutions of second-order fuzzy volterra integro-differential equations with fuzzy kernel T. Allahviranloo Email author M. Khezerloo O. Sedaghatfar S. Salahshour Original Article

First Online: 02 February 2012 Received: 07 June 2011 Accepted: 17 January 2012 DOI :
10.1007/s00521-012-0849-x

Cite this article as: Allahviranloo, T., Khezerloo, M., Sedaghatfar, O. et al. Neural Comput & Applic (2013) 22: 133. doi:10.1007/s00521-012-0849-x
Abstract In this paper, we study existence and uniqueness of solutions of second-order fuzzy integro-differential equations with fuzzy kernel under strongly generalized differentiability. 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

References 1.

Allahviranloo T, Abbasbandy S, Salahshour S, Hakimzadeh A (2011) A new method for solving fuzzy linear differential equations. Computing 92:181–197

MathSciNet MATH CrossRef 2.

Allahviranloo T, Salahshour S (2011) Euler method for solving hybrid fuzzy differential equation. Soft Comput 15:1247–1253

MATH CrossRef 3.

Allahviranloo T, Kiani NA, Barkhordari M (2009) Toward the existence and uniqueness of solutions of second-order fuzzy differential equations. Inf Sci 179:1207–1215

MathSciNet MATH CrossRef 4.

Bede B, Rudas IJ, Attila L (2007) First order linear fuzzy differential equations under generalized differentiability. Inf Sci 177:3627–3635

CrossRef 5.

Bede B, Gal (2005) Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets Syst 151:581–599

6.

Chalco-cano Y, Roman-Flores H (2008) On new solution of fuzzy differential equations. Chaos, Solitons Fract 38:112–119

MathSciNet MATH CrossRef 7.

Diamond P, Kloeden P (1994) Metric spaces of fuzzy sets. World Scientific, Singapore

MATH CrossRef 8.

Dubois D, Prade H (1982) Towards fuzzy differential calculus, Part I: integration of fuzzy mappings, class of second-order. Fuzzy Sets Syst 8:1–17

MathSciNet MATH CrossRef 9.

Friedman M, Ma M, Kandel A (1999) Numerical solution of fuzzy differential and integral equations. Fuzzy Sets Syst 106:35–48

MathSciNet MATH CrossRef 10.

Hajighasemi S, Allahviranloo T, Khezerloo M, Khorasany M, Salahshour S (2010) Existence and uniqueness of solutions of fuzzy Volterra integro-differential equations. IPMU 2010, Part II, CCIS 81, pp 491–500

11.

Kaleva O (1990) The Cauchy problem for fuzzy differential equations. Fuzzy Sets Syst 35:389–396

MathSciNet MATH CrossRef 12.

Park JY, Jeong JU (1999) A note on fuzzy functional equations. Fuzzy Sets Syst 108:193–200

MathSciNet MATH CrossRef 13.

Park JY, Kwun YC, Jeong JU (1995) Existence of solutions of fuzzy integral equations in Banach spaces. Fuzzy Sets Syst 72:373–378

MathSciNet MATH CrossRef 14.

Park JY, Lee SY, Jeong JU (2000) On the existence and uniqueness of solutions of fuzzy Volterra-Fredholm integral equations. Fuzzy Sets Syst 115:425–431

MATH CrossRef 15.

Puri ML, Ralescu DA (1986) Fuzzy random variables. J Math Anal Appl 114:409–422

MathSciNet MATH CrossRef 16.

Puri ML, Ralescu DA (1983) Differentials of fuzzy functions. J Math Anal Appl 91:552–558

MathSciNet MATH CrossRef 17.

Rodriguez-Munize LJ, Lopez-Diaz M (2003) Hukuhara derivative of the fuzzy expected value. Fuzzy Sets Syst 138:593–600

CrossRef 18.

Rodriguez-Lipez R (2008) Comparison results for fuzzy differential equations. Inf Sci 178:1756–1779

CrossRef 19.

Seikkala S (1987) On the fuzzy initial value problem. Fuzzy Sets Syst 24:319–330

MathSciNet MATH CrossRef 20.

Stefanini L (2007) On the generalized LU-fuzzy derivative and fuzzy differential equations. IEEE international conference on fuzzy systems, art no. 4295453

21.

Subrahmaniam PV, Sudarsanam SK (1994) On some fuzzy functional equations. Fuzzy Sets Syst 64:333–338

CrossRef 22.

Zhang D, Feng W, Qiu J (2009) Global existence of solutions to fuzzy Volterra integral equations. ICIC Express Lett 3:707–711

© Springer-Verlag London Limited 2012

Authors and Affiliations T. Allahviranloo Email author M. Khezerloo O. Sedaghatfar S. Salahshour 1. Department of Mathematics, Science and Research Branch Islamic Azad University Tehran Iran 2. Department of Mathematics, Mobarakeh Branch Islamic Azad University Mobarakeh Iran