MR2589663 Jagmohan TYAGI and Venkataramanarao RAGHAVENDRA Jagmohan TYAGI and Venkataramanarao RAGHAVENDRA Funkcialaj Ekvacioj. Serio Internacia 52 2009 395--410 http://fe.math.kobe-u.ac.jp/FE/FullPapers/52-3/52_395.pdf http://www.ams.org/mathscinet-getitem?mr=MR2589663 In this study, we establish the existence of the solution of a class of first order implicit integro-differential equation of type $F(t,(Kx)(t),x(t),x'(t))=0$, where $F:J\times\mathbb{R}^3\longrightarrow\mathbb{R}$, $K:C(J)\longrightarrow C(J)$ defined as $(Kx)(t)=\int_0^t k(t,\,s)x(s)ds$, $\forall x\in C(J)$, using the concept of viscosity solution. Implicit differential equations, Integro-differential equations. 34K05, 34A09, 45J05. 52-139 2009 Jagmohan TYAGI and Venkataramanarao RAGHAVENDRA Jagmohan TYAGI and Venkataramanarao RAGHAVENDRA 1 Bardi, M.; Capuzzo Dolecetta, Italo Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations Birkhäuser 1997 MR1484411

2 Carl, S.; Heikkilä, S. J. Differential Equations 148 1998 100--121 MR1637525

3 Crandall, M. G.; Evans, L. C.; Lions, P. L. Trans. Amer. Math. Soc. 282 1984 487--502 MR0732102

4 Crandall, M. G.; Ishii, H.; Lions, P. L. Bull. Amer. Math. Soc. 27 1992 1--67 MR1118699

5 Fečkan, M. Math. Slovaca 48 1998 35--42 MR1635231

6 Frigon, M.; Kaczynski, T. J. Math. Anal. Appl. 179 1993 317--326 MR1249822

7 Heikkilä, S.; Kumpulainen, M.; Seikkala, S. Dynam. Systems Appl. 7 1998 237--244 MR1626310

8 Ishii, H. Duke Math. J. 55 1987 369--384 MR0894587

9 Li, D. J. Math. Anal. Appl. 258 2001 591--616 MR1835561

10 Seikkala, S.; Heikkilä, S. Nonlinear Anal. 30 1997 1771--1780 MR1490092