This paper is devoted to studying the maximal and minimal solutions for the interval-valued functional integro-differential equations (IFIDEs) under generalized Hukuhara differentiability by the method of upper and lower solutions and monotone iterative technique. Some examples are given to illustrate the results.
AgarwalR.P. and O’ReganD., Existence for set differential equations via mul-tivalued operator equations, Differential equations and Applications5 (2007), 1–5.
2.
AgarwalR.P., ArshadS., O’ReganD. and LupulescuV., A Schauder fixed point theorem in semilinear spaces and applications, Fixed Point Theory and Applications2013 (2013), 306.
3.
AgarwalR.P., ArshadS., O’ReganD. and LupulescuV., Fuzzy fractional integral equations under compactness type condition, Fract Calc Appl Anal15 (2012), 572–590.
4.
AnT.V., PhuN.D. and HoaN.V., A note on solutions of interval-valued Volterra integral equations, Journal of Integral Equations and Applications26 (2014), 1–16.
5.
ArshadS.and LupulescuV., On the fractional differential equations with uncertainty, Nonlinear Analysis: TMA74(11) (2011), 3685–3693.
6.
AllahviranlooT., AbbasbandyS., SedaghatfarO. and DarabiP., A New method for solving fuzzy integro-differential equation under generalized differentiability, J Neural Computing and Applications21(1) (2012), 191–196.
7.
AllahviranlooT., AmirteimooriA., KhezerlooM. and KhezerlooS., A new method for solving fuzzy Volterra integrodifferential equations, Australian Journal of Basic and Applied Sciences54 (2011), 154–164.
8.
AllahviranlooT., HajighasemiS., KhezerlooM., KhorasanyM. and SalahshourS., Existence and uniqueness of solutions of fuzzy Volterra integro-differential equations, IPMU II81 (2010), 491–500.
9.
AllahviranlooT. and GholamiS., Note on “Generalized Hukuhara differentiability of interval-valued functions and interval differential equations”, Journal of Fuzzy Set Valued Analysis2012 (2012). DOI: 10.5899/2012/jfsva-00135.
10.
BhaskarT.G., LakshmikanthamV. and DeviJ.V., Monotone iterative technique for functional differential equations with retardation and anticipation, Nonlinear Analysis: TMA66, 2237–2242.
11.
BedeB. and GalS.G., Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems151 (2005), 581–599.
12.
BedeB., RudasI.J. and BencsikA.L., First order linear fuzzy differential equations under generalized differentiability, Information Sciences177 (2007), 1648–1662.
13.
BedeB. and StefaniniL., Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems230 (2013), 119–141.
14.
BhaskarT.G. and LakshmikanthamV., Set differential equations and flow invariance, J Appl Anal82(2) (2003), 357–368.
15.
Chalco-CanoY., Rufián-LizanaA., Román-FloresH. and Jiménez-GameroM.D., Calculus for interval-valued functions using generalized Hukuhara derivative and applications, Fuzzy Sets and Systems219 (2013), 49–67.
16.
De BlasiF.S. and IervolinoF, Equazioni differenziali consoluzioni a valore compatto convesso. Journal of integral equations and applications, Boll Unione Mat Ital4(2) (1969), 194–501.
17.
De BlasiF.S., LakshmikanthamV and BhaskarT.G, An existence theorem for set differential inclusions in a semilinear metric space, Control and Cybernetics36(3) (2007), 571–582.
18.
DeviJ.V., Generalized monotone iterative technique for set differential equations involving causal operators with memory, Int J Adv Eng Sci Appl Math8(3) (2011), 74–83.
19.
DeviJ.V., SreedharCh.V. and NagamaniS., Monotone iterative technique for integro differential equations with retardation and anticipation, Commun Appl Anal14 (2010), 325–336.
20.
HaleJ.K., Theory of Functional Differential Equations, Springer, NewYork, 1977.
21.
HoaN.V. and PhuN.D., On maximal and minimal solutions for set-valued differential equations with feedback control, J Abstract and Applied Analysis2012 (2012), 1–11.
22.
HoaN.V., PhuN.D., TungT.T. and QuangL.T., Interval-valued functional integro-differential equations, Advcances in Difference Equations2014 (2014), 177.
23.
HoaN.V. and PhuN.D., Fuzzy functional integro-differential equations under generalized H-differentiability, Journal of Intelligent and Fuzzy Systems26 (2014), 2073–2085.
24.
HoaN.V., Fuzzy fractional functional differential equations under Caputo gH-differentiability, Communications in Nonlinear Science and Numerical Simulation22 (2015), 1134–1157.
25.
HoaN.V., Fuzzy fractional functional integral and differential equations, Fuzzy Sets and Systems280 (2015), 58–90. (SCI).
26.
HoaN.V., The initial value problem for interval-valued second-order differential equations under generalized H-differentiability, Information Sciences311 (2015), 119–148.
27.
HoaN.V., TriP.V. and DaoT.T., and I’van Zelinka, Some global existence results and stability theorem for fuzzy functional differential equations, Journal of Intelligent and Fuzzy Systems28 (2015), 393–409.
28.
HukuharaM., Inteǵration des applications mesurables dont la valeur est un compact convex, Funkcial Ekvac10 (1967), 205–229.
29.
KolmanovskiiV.B. and MyshkisA., Applied Theory of Functional Differential Equations. Kluwer Academic Publishers, Dordrecht, 1992.
30.
KuangY., Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.
31.
LakshmikanthamV., BhaskarT.G. and DeviJ.V., Theory of set differential equations in metric spaces, Cambridge Scientific Publisher, UK, 2006.
32.
LaddeG.S., LakshmikanthamV. and VatsalaA.S., Monotone Iterative Technique for Nonlinear Differential equations, Pittman Publishing Inc, London, 1985.
33.
LupulescuV., On a class of functional differential equations in Banach spaces, Electronic Journal of Qualitative Theory of Differential Equations64 (2010), 1–17.
34.
LupulescuV., On a class of fuzzy functional differential equations, Fuzzy Sets and Systems160 (2009), 1547–1562.
35.
LupulescuV., Initial value problem for fuzzy differential equations under dissipative conditions, Information Sciences178(23) (2008), 4523–4533.
36.
LupulescuV., Hukuhara differentiability of intervalvalued functions and interval differential equations on time scales, Information Sciences248 (2013), 50–67.
37.
LupulescuV., DongL.S., HoaN.V., Existence and uniqueness of solutions for random fuzzy fractional integral and differential equations, Journal of Intelligent and Fuzzy Systems29 (2015), 27–42.
38.
MalinowskiM.T., Interval Cauchy problem with a second type Hukuhara derivative, Information Sciences213 (2012), 94–105.
39.
MalinowskiM.T., Random fuzzy differential equations under generalized Lipschitz condition, Nonlinear Analysis RWA13(2) (2012), 860–881.
40.
MalinowskiM.T., On set differential equations in Banach spaces - a second type Hukuhara differentiability approach, Applied Mathematics and Computation219(1) (2012), 289–305.
41.
MalinowskiM.T., Second type Hukuhara differentiable solutions to the delay set-valued differential equations, Applied Mathematics and Computation218(18) (2012), 9427–9437.
42.
MalinowskiM.T., Interval differential equations with a second type Hukuhara derivative, Applied Mathematics Letters24 (2011), 2118–2123.
43.
MalinowskiM.T., Existence theorems for solutions to random fuzzy differential equations, Nonlinear Analysis TMA73(6) (2010), 1515–1532.
StefaniniL., A generalization of Hukuhara difference In: DuboisD, LubianoM.A, radeH, GilM.A, GrzegorzewskiP and HryniewiczO, (Eds), Soft Methods for Handling Variability and Imprecision, Series on Advances in Soft Computing, (vol 48), Springer, 2008.
48.
StefaniniL. and BedeB., Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Analysis TMA71 (2009), 1311–1328.
