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Coupled fixed points of multivalued operators and first-order ODEs with state-dependent deviating arguments. (English) Zbl 1232.34091

This paper is concerned with the existence of absolutely continuous solutions of the problem

x ' (t)=f(t,x(t),x(τ(t,x(t),x))a.e.tI + =[t 0 ,t 0 +L],
x(t)=Λ(x)+k(t)a.e.tI - =[t 0 -r,t 0 ],

where t 0 , L>0, r0, k is a continuous function and f, τ, Λ are suitable functions not necessarily continuous. First, the authors prove an abstract result on the existence of coupled fixed points for multivalued operators. Then, they give two new theorems on the existence of coupled quasisolutions for the above initial value problem, which, in turn, yield two corresponding new theorems on the existence of unique solutions.

MSC:
34K05General theory of functional-differential equations
47N20Applications of operator theory to differential and integral equations
References:
[1]Dyki, A.; Jankowski, T.: Boundary value problems for ordinary differential equations with deviated arguments, J. optim. Theory appl. 135, No. 2, 257-269 (2007) · Zbl 1140.34027 · doi:10.1007/s10957-007-9248-3
[2]Gao, F.; Lu, S.; Zhang, W.: Existence and uniqueness of periodic solutions for a p-Laplacian Duffing equation with a deviating argument, Nonlinear anal. 70, No. 10, 3567-3574 (2009) · Zbl 1173.34341 · doi:10.1016/j.na.2008.07.014
[3]Jankowski, T.: On dynamic equations with deviating arguments, Appl. math. Comput. 208, No. 2, 423-426 (2009)
[4]Darwish, M. A.; Ntouyas, S. K.: Semilinear functional differential equations of fractional order with state-dependent delay, Electron. J. Differential equations 38 (2009)
[5]Morales, E. Hernández; Mckibben, M. A.; Henríquez, H. R.: Existence results for partial neutral functional differential equations with state-dependent delay, Math. comput. Modelling 49, No. 5–6, 1260-1267 (2009) · Zbl 1165.34420 · doi:10.1016/j.mcm.2008.07.011
[6]Walther, H. O.: A periodic solution of a differential equation with state-dependent delay, J. differential equations 244, No. 8, 1910-1945 (2008) · Zbl 1146.34048 · doi:10.1016/j.jde.2008.02.001
[7]Walther, H. O.: Algebraic-delay differential systems, state-dependent delay, and temporal order of reactions, J. dynam. Differential equations 21, No. 1, 195-232 (2009) · Zbl 1167.34026 · doi:10.1007/s10884-009-9129-6
[8]Zeng, Z.; Zhou, Z.: Multiple positive periodic solutions for a class of state-dependent delay functional differential equations with feedback control, Appl. math. Comput. 197, No. 1, 306-316 (2008) · Zbl 1145.34040 · doi:10.1016/j.amc.2007.07.085
[9]Hartung, F.; Krisztin, T.; Walther, H. O.; Wu, J.: Functional differential equations with state-dependent delays: theory and applications, Handbook of differential equations: ordinary differential equations, 435-545 (2006)
[10]Jankowski, T.: Existence of solutions of boundary value problems for differential equations in which deviated arguments depend on the unknown solution, Comput. math. Appl. 54, No. 3, 357-363 (2007)
[11]Heikkilä, S.; Lakshmikantham, V.: Monotone iterative techniques for discontinuous nonlinear differential equations, (1994)
[12]Sestelo, R. Figueroa; Pouso, R. López: Discontinuous first-order functional boundary value problems, Nonlinear anal. 69, No. 7, 2142-2149 (2008) · Zbl 1210.34088 · doi:10.1016/j.na.2007.07.051
[13]Moore, J.: Existence of multiple quasifixed points of mixed monotone operators by iterative techniques, Appl. math. Comput. 9, No. 2, 135-141 (1981) · Zbl 0502.47029 · doi:10.1016/0096-3003(81)90011-4
[14]Guo, D.; Lakshmikantham, V.: Coupled fixed points of nonlinear operators with applications, Nonlinear anal. 11, No. 5, 623-632 (1987) · Zbl 0635.47045 · doi:10.1016/0362-546X(87)90077-0
[15]Guo, D.; Lakshmikantham, V.: Nonlinear problems in abstract cones, Notes and reports in mathematics in science and engineering 5 (1988) · Zbl 0661.47045
[16]Aubin, J. P.; Cellina, A.: Differential inclusions, (1984)
[17]Cid, J. Á.; Pouso, R. L.: Ordinary differential equations and systems with time-dependent discontinuity sets, Proc. roy. Soc. Edinburgh sect. A 134, No. 4, 617-637 (2004) · Zbl 1079.34002 · doi:10.1017/S0308210500003383
[18]Figueroa, R.; Pouso, R. López: Coupled fixed points of multivalued operators and first-order odes with state-dependent deviating arguments · Zbl 1232.34091 · doi:10.1016/j.na.2011.07.010