The object of investigation of the paper is a special type of functional differential equations containing the maximum value of the unknown function over a past time interval. An improved algorithm of the monotone-iterative technique is suggested to nonlinear differential equations with “maxima.” The case when upper and lower solutions of the given problem are known at different initial time is studied. Additionally, all initial value problems for successive approximations have both initial time and initial functions different. It allows us to construct sequences of successive approximations as well as sequences of initial functions, which are convergent to the solution and to the initial function of the given initial value problem, respectively. The suggested algorithm is realized as a computer program, and it is applied to several examples, illustrating the advantages of the suggested scheme.

## References

*Computers and Mathematics with Applications*, vol. 59, no. 2, pp. 977–993, 2010. MR2575586 R. P. Agarwal, D. R. Anderson, and A. Zafer, “Interval oscillation criteria for second-order forced de-lay dynamic equations with mixed nonlinearities,”

*Computers and Mathematics with Applications*, vol. 59, no. 2, pp. 977–993, 2010. MR2575586

*Applied Mathematics Letters*, vol. 13, no. 5, pp. 99–105, 2000. MR1760270 10.1016/S0893-9659(00)00040-9 M. U. Akhmetov and A. Zafer, “Successive approximation method for quasilinear impulsive differential equations with control,”

*Applied Mathematics Letters*, vol. 13, no. 5, pp. 99–105, 2000. MR1760270 10.1016/S0893-9659(00)00040-9

*Abstract and Applied Analysis*, vol. 2010, Article ID 417869, 20 pages, 2010. MR2720031 1209.34080 J. Bastinec, L. Berezansky, J. Diblik, and Z. Smarda, “On the critical case in oscillation for differential equations with a single delay and with several delays,”

*Abstract and Applied Analysis*, vol. 2010, Article ID 417869, 20 pages, 2010. MR2720031 1209.34080

*Journal of Mathematical Analysis and Applications*, vol. 314, no. 2, pp. 391–411, 2006. MR2185238 1101.34057 10.1016/j.jmaa.2005.03.103 L. Berezansky and E. Braverman, “On stability of some linear and nonlinear delay differential equations,”

*Journal of Mathematical Analysis and Applications*, vol. 314, no. 2, pp. 391–411, 2006. MR2185238 1101.34057 10.1016/j.jmaa.2005.03.103

*Applied Mathematics Letters*, vol. 21, no. 6, pp. 636–640, 2008. MR2412391 10.1016/j.aml.2007.07.017 L. Berezansky and E. Braverman, “Positive solutions for a scalar differential equation with several de-lays,”

*Applied Mathematics Letters*, vol. 21, no. 6, pp. 636–640, 2008. MR2412391 10.1016/j.aml.2007.07.017

*Abstract and Applied Analysis*, vol. 2011, Article ID 178568, 19 pages, 2011. MR2800075 1222.34089 L. Berezansky and E. Braverman, “New stability conditions for linear differential equations with several delays,”

*Abstract and Applied Analysis*, vol. 2011, Article ID 178568, 19 pages, 2011. MR2800075 1222.34089

*Journal of Mathematical Analysis and Applications*, vol. 274, no. 1, pp. 349–373, 2002. MR1936703 1025.34062 10.1016/S0022-247X(02)00311-6 J. Diblik, “Asymptotic convergence criteria of solutions of delayed functional differential equations,”

*Journal of Mathematical Analysis and Applications*, vol. 274, no. 1, pp. 349–373, 2002. MR1936703 1025.34062 10.1016/S0022-247X(02)00311-6

*Nonlinear Analysis*, vol. 64, no. 6, pp. 1153–1170, 2006. MR2200483 J. Diblik and N. Koksch, “Existence of global solutions of delayed differential equations via retract approach,”

*Nonlinear Analysis*, vol. 64, no. 6, pp. 1153–1170, 2006. MR2200483

*Abstract and Applied Analysis*, vol. 2011, Article ID 754701, 16 pages, 2011. MR2739690 1210.34106 J. Diblik and M. Kudelcikova, “Existence and asymptotic behavior of positive solutions of functional differential equations of delayed type,”

*Abstract and Applied Analysis*, vol. 2011, Article ID 754701, 16 pages, 2011. MR2739690 1210.34106

*Journal of Mathematical Analysis and Applications*, vol. 331, no. 2, pp. 1361–1370, 2007. MR2313718 1125.34059 10.1016/j.jmaa.2006.10.001 J. Diblik and M. Ruzickova, “Convergence of the solutions of the equation $\stackrel{\cdot\,\!}{y}(t)=\beta (t)[y(t-\delta )-y(t-\tau )]$ in the critical case,”

*Journal of Mathematical Analysis and Applications*, vol. 331, no. 2, pp. 1361–1370, 2007. MR2313718 1125.34059 10.1016/j.jmaa.2006.10.001

*Advances in Difference Equations*, vol. 2010, Article ID 749852, 12 pages, 2010. MR2660941 1203.34122 J. Diblik, M. Ruzickova, and Z. Suta, “Asymptotical convergence of the solutions of a linear differential equation with delays,”

*Advances in Difference Equations*, vol. 2010, Article ID 749852, 12 pages, 2010. MR2660941 1203.34122

*Extracta Mathematicae*, vol. 13, no. 3, pp. 313–326, 1998. MR1695501 0987.34009 D. Franco, J. J. Nieto, and Y. V. Rogovchenko, “Periodic boundary value problem for nonlinear first order ordinary differential equations with impulses at fixed moments,”

*Extracta Mathematicae*, vol. 13, no. 3, pp. 313–326, 1998. MR1695501 0987.34009

*Fields Institute Communications*, vol. 29, pp. 181–194, 2001. MR1821781 I. Gyori and F. Hartung, “Stability in delay perturbed differential and difference equations,”

*Fields Institute Communications*, vol. 29, pp. 181–194, 2001. MR1821781

*Journal of Mathematical Analysis and Applications*, vol. 190, no. 2, pp. 391–401, 1995. MR1318401 0823.34064 10.1006/jmaa.1995.1083 D. D. Bainov and S. G. Hristova, “Monotone-iterative techniques of Lakshmikantham for a boundary value problem for systems of differential equations with `maxima',”

*Journal of Mathematical Analysis and Applications*, vol. 190, no. 2, pp. 391–401, 1995. MR1318401 0823.34064 10.1006/jmaa.1995.1083

*Journal of Mathematical Analysis and Applications*, vol. 87, no. 2, pp. 454–459, 1982. MR658024 0523.34057 10.1016/0022-247X(82)90134-2 S. W. Du and V. Lakshmikantham, “Monotone iterative technique for differential equations in a Ba-nach space,”

*Journal of Mathematical Analysis and Applications*, vol. 87, no. 2, pp. 454–459, 1982. MR658024 0523.34057 10.1016/0022-247X(82)90134-2

*Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations*, vol. 181 of

*Monographs and Textbooks in Pure and Applied Mathematics*, Marcel Dekker, New York, NY, USA, 1994. MR1280028 S. Heikkila and V. Lakshmikantham,

*Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations*, vol. 181 of

*Monographs and Textbooks in Pure and Applied Mathematics*, Marcel Dekker, New York, NY, USA, 1994. MR1280028

*Applied Mathematics Letters*, vol. 21, no. 8, pp. 828–834, 2008. MR2436173 10.1016/j.aml.2007.09.006 V. Lakshmikantham and A. S. Vatsala, “General uniqueness and monotone iterative technique for fractional differential equations,”

*Applied Mathematics Letters*, vol. 21, no. 8, pp. 828–834, 2008. MR2436173 10.1016/j.aml.2007.09.006

*Nonlinear Analysis*, vol. 32, no. 6, pp. 741–747, 1998. MR1612118 J. J. Nieto, Y. Jiang, and Y. Jurang, “Monotone iterative method for functional-differential equations,”

*Nonlinear Analysis*, vol. 32, no. 6, pp. 741–747, 1998. MR1612118

*Computers and Mathematics with Applications*, vol. 28, no. 1–3, pp. 243–254, 1994. MR1284239 P. Y. H. Pang and R. P. Agarwal, “Monotone iterative methods for a general class of discrete boundary value problems,”

*Computers and Mathematics with Applications*, vol. 28, no. 1–3, pp. 243–254, 1994. MR1284239

*Applicable Analysis*, vol. 81, no. 3, pp. 627–635, 2002. MR1929113 10.1080/0003681021000004348 T. Jankowski, “Differential inequalities with initial time difference,”

*Applicable Analysis*, vol. 81, no. 3, pp. 627–635, 2002. MR1929113 10.1080/0003681021000004348

*Journal of Mathematical Analysis and Applications*, vol. 291, no. 2, pp. 605–624, 2004. MR2039073 1054.45004 10.1016/j.jmaa.2003.11.036 T. Jankowski, “Delay integro-differential inequalities with initial time difference and applications,”

*Journal of Mathematical Analysis and Applications*, vol. 291, no. 2, pp. 605–624, 2004. MR2039073 1054.45004 10.1016/j.jmaa.2003.11.036

*Applied Mathematics and Computation*, vol. 217, no. 12, pp. 5923–5929, 2011. MR2770211 1216.37010 10.1016/j.amc.2010.12.097 A. Li, W. Lv, and J. Ye, “Exponential and global stability of nonlinear dynamical systems relative to initial time difference,”

*Applied Mathematics and Computation*, vol. 217, no. 12, pp. 5923–5929, 2011. MR2770211 1216.37010 10.1016/j.amc.2010.12.097

*Nonlinear Analysis. Real World Applications*, vol. 10, no. 2, pp. 1073–1080, 2009. MR2474282 10.1016/j.nonrwa.2007.11.023 A. Li, E. Feng, and S. Li, “Stability and boundedness criteria for nonlinear differential systems relative to initial time difference and applications,”

*Nonlinear Analysis. Real World Applications*, vol. 10, no. 2, pp. 1073–1080, 2009. MR2474282 10.1016/j.nonrwa.2007.11.023

*Journal of Inequalities and Applications*, vol. 4, no. 2, pp. 163–174, 1999. MR1733481 V. Lakshmikantham, S. Leela, and J. Vasundhara, “Another approach to the theory of differential inequalities relative to changes in the initial times,”

*Journal of Inequalities and Applications*, vol. 4, no. 2, pp. 163–174, 1999. MR1733481

*Journal of Inequalities and Applications*, vol. 3, no. 3, pp. 233–244, 1999. MR1732930 0933.34007 V. Lakshmikantham and A. S. Vatsala, “Differential inequalities with initial time difference and applications,”

*Journal of Inequalities and Applications*, vol. 3, no. 3, pp. 233–244, 1999. MR1732930 0933.34007

*Dynamic Systems and Applications*, vol. 10, no. 4, pp. 523–532, 2001. MR1870486 1022.34056 Y. Zhang and Z. Yi, “Theory of functional differential equation with initial data difference,”

*Dynamic Systems and Applications*, vol. 10, no. 4, pp. 523–532, 2001. MR1870486 1022.34056

*Dynamics of Continuous, Discrete and Impulsive Systems Series A*, vol. 12, no. 6, pp. 731–737, 2005. MR2178676 C. Yakar and M. D. Shaw, “A comparison result and Lyapunov stability criteria with initial time difference,”

*Dynamics of Continuous, Discrete and Impulsive Systems Series A*, vol. 12, no. 6, pp. 731–737, 2005. MR2178676

*Nonlinear Analysis*, vol. 71, no. 12, pp. 781–785, 2009. C. Yakar and M. Shaw, “Practical stability in terms of two measures with initial time difference,”

*Nonlinear Analysis*, vol. 71, no. 12, pp. 781–785, 2009.

*Dynamic Systems and Applications*, vol. 16, no. 3, pp. 497–506, 2007. MR2356334 1148.45007 B. Ahmad, “Integro-differential equations with initial time difference,”

*Dynamic Systems and Applications*, vol. 16, no. 3, pp. 497–506, 2007. MR2356334 1148.45007

*Hacettepe Journal of Mathematics and Statistics*, vol. 40, no. 2, pp. 331–340, 2011. MR2839198 1236.34009 C. Yakar and A. Yakar, “Monotone iterative technique with initial time difference for fractional differential equations,”

*Hacettepe Journal of Mathematics and Statistics*, vol. 40, no. 2, pp. 331–340, 2011. MR2839198 1236.34009

*Applied Mathematics and Computation*, vol. 87, no. 2-3, pp. 217–226, 1997. MR1468300 0904.65067 10.1016/S0096-3003(96)00285-8 A. Cabada and J. J. Nieto, “Rapid convergence of the iterative technique for first order initial value problems,”

*Applied Mathematics and Computation*, vol. 87, no. 2-3, pp. 217–226, 1997. MR1468300 0904.65067 10.1016/S0096-3003(96)00285-8

*Journal of Mathematical Analysis and Applications*, vol. 152, no. 1, pp. 131–155, 1990. MR1072931 0719.34129 10.1016/0022-247X(90)90096-X I. Gyori, “Global attractivity in delay differential equations using a mixed monotone technique,”

*Journal of Mathematical Analysis and Applications*, vol. 152, no. 1, pp. 131–155, 1990. MR1072931 0719.34129 10.1016/0022-247X(90)90096-X

*Nonlinear Analysis*, vol. 17, no. 1, pp. 21–29, 1991. MR1113447 I. Gyori and J. Turi, “Uniform approximation of a nonlinear delay equation on infinite intervals,”

*Nonlinear Analysis*, vol. 17, no. 1, pp. 21–29, 1991. MR1113447

*The Rocky Mountain Journal of Mathematics*, vol. 23, no. 2, pp. 609–618, 1993. MR1226191 10.1216/rmjm/1181072579 euclid.rmjm/1181072579 S. G. Hristova and D. D. Bainov, “Application of the monotone-iterative techniques of V. Lakshmikantham for solving the initial value problem for impulsive differential-difference equations,”

*The Rocky Mountain Journal of Mathematics*, vol. 23, no. 2, pp. 609–618, 1993. MR1226191 10.1216/rmjm/1181072579 euclid.rmjm/1181072579

*Journal of Mathematical and Physical Sciences*, vol. 25, no. 1, pp. 69–80, 1991. MR1109861 S. G. Hristova and D. D. Bainov, “Application of the monotone-iterative techniques of V. Lakshmikantham to the solution of the initial value problem for impulsive differential equations with `supremum',”

*Journal of Mathematical and Physical Sciences*, vol. 25, no. 1, pp. 69–80, 1991. MR1109861

*Monotone Iterative Techniques for Nonlinear Differential Equations*, Pitman, New York, NY, USA, 1985. MR855240 0658.35003 G. S. Ladde, V. Lakshmikantham, and A. S. Vatsala,

*Monotone Iterative Techniques for Nonlinear Differential Equations*, Pitman, New York, NY, USA, 1985. MR855240 0658.35003

*Applicable Analysis*, vol. 22, no. 3-4, pp. 227–233, 1986. MR860991 10.1080/00036818608839620 V. Lakshmikantham and B. G. Zhang, “Monotone iterative technique for delay differential equations,”

*Applicable Analysis*, vol. 22, no. 3-4, pp. 227–233, 1986. MR860991 10.1080/00036818608839620

*Nonlinear Analysis*, vol. 60, no. 3, pp. 429–442, 2005. MR2103944 Y. Xing, M. Han, and G. Zheng, “Initial value problem for first-order integro-differential equation of Volterra type on time scales,”

*Nonlinear Analysis*, vol. 60, no. 3, pp. 429–442, 2005. MR2103944

*Nonlinear Analysis*, vol. 71, no. 5-6, pp. 2087–2093, 2009. MR2524420 Sh. Zhang, “Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives,”

*Nonlinear Analysis*, vol. 71, no. 5-6, pp. 2087–2093, 2009. MR2524420

*Computers and Mathematics with Applications*, vol. 60, no. 10, pp. 2771–2778, 2010. MR2734318 A. Golev, S. Hristova, and A. Rahnev, “An algorithm for approximate solving of differential equations with `maxima',”

*Computers and Mathematics with Applications*, vol. 60, no. 10, pp. 2771–2778, 2010. MR2734318

*Applicable Analysis*, vol. 16, no. 3, pp. 187–194, 1983. MR712731 10.1080/00036818308839468 V. G. Angelov and D. D. Bainov, “On the functional-differential equations with `maxima',”

*Applicable Analysis*, vol. 16, no. 3, pp. 187–194, 1983. MR712731 10.1080/00036818308839468