Existence of Solutions of MultiPoint BVPs for Impulsive Functional Differential Equations with Nonlinear Boundary Conditions
Authors
Yuji Liu
 Department of Mathematics, Guangdong University of Business Studies, Guangzhou, P. R. China.
Abstract
Two classes of multipoint BVPs for first order impulsive functional differential equations with nonlinear
boundary conditions are studied. Sufficient conditions for the existence of at least one solution to these
BVPs are established, respectively. Our results generalize and improve the known ones. Some examples are
presented to illustrate the main results.
Keywords
 Nonlinear multipoint boundary value problem
 first order impulsive functional differential equation
 fixedpoint theorem
 growth condition.
MSC
References

[1]
A. Cabada, The monotone method for first order problems with linear and nonlinear boundary conditions, Appl. Math. Comput. , 63 (1994), 163186.

[2]
D. Franco, J. J. Nieto, First order impulsive ordinary differential equations with antiperiodic and nonlinear boundary value conditions, Nonl. Anal. , 42 (2000), 163173.

[3]
D. Franco, J. J. Nieto, A new maximum principle for impulsive first order problems, Internat. J. Theoret. Phys., 37 (1998), 16071616.

[4]
R. Hakl, A. Lomtatidze, B. Puza, On a boundary value problem for first order scalar functional differential equations, Nonl. Anal. , 53 (2003), 391405.

[5]
Z. He, J. Yu, Periodic boundary value problems for first order impulsive ordinary differential equations, J. Math. Anal. Appl. , 272 (2002), 6778.

[6]
D. Jiang, J. J. Nieto, W. Zuo, On monotone method for first order and second order periodic boundary value problems and periodic solutions of functional differential equations, J. Math. Anal. Appl., 289 (2004), 691699.

[7]
G. S. Ladde, V. Lakshmikantham, A. S. Vatsala, Monotone iterative techniques for nonlinear differential equations, Pitman Advanced Publishing Program, (1985)

[8]
X. Li, X. Lin, D. Jiang, X. Zhang, Existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects, Nonl. Anal., 62 (2005), 683701.

[9]
X. Liu , Nonlinear boundary value problems for first order impulsive integradifferential equations, Appl. Anal. , 36 (1990), 119130.

[10]
J. J. Nieto, N. AlvarezNoriega, Periodic boundary value problems for nonlinear first order ordinary differential equations, Acta Math. Hungar, 71 (1996), 4958.

[11]
J. J. Nieto, R. RodriguezLopez, Periodic boundary value problem for nonLipschitzian impulsive functional differential equations, J. Math. Anal. Appl. , 318 (2006), 593610.

[12]
C. PiersonGorez, Impulsive differential equations of first order with periodic boundary conditions, Diff. Equs. Dyn. Systems, 11 (1993), 185196.

[13]
A. Cabada, The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems, J. Math. Anal. Appl. , 185 (1994), 302320.

[14]
A. Cabada, J. J. Nieto, D. Franco, S. I. Trofimchuk, A generalization of the monotone method for second order periodic boundary value problems with impulses at fixed points, Dynamics Contin. Discrete Impuls. System, 7 (2000), 145158.

[15]
A. S. Vatsala, Y. Sun, Periodic boundary value problems of impulsive differential equations, Appl. Anal. , 44 (1992), 145158.

[16]
D. Franco, R. L. Pouso, Nonresonance conditions and extremal solutions for first order impulsive problems under weak assumptions, ANZIAM J. , 44 (2003), 393407.

[17]
T. Jankowski, Existence of solutions of boundary value problems for differential equations with delayed arguments, J. Comput. Appl. Math. , 156 (2003), 239252.

[18]
D. Franco, J. J. Nieto, D. O'Regan, Existence of solutions for first order ordinary differential equations with nonlinear boundary conditions, Appl. Math. Letters, 153 (2004), 793802.

[19]
T. Jankowski, Existence of solutions of differential equations with nonlinear multipoint boundary conditions, Comput. Math. Appl. , 47 (2004), 10951103.

[20]
J. J. Nieto, Impulsive resonance periodic problems of first order, Appl. Math. Letters, 15 (2002), 489493.

[21]
J. J. Nieto , Periodic boundary value problems for first order impulsive ordinary differential equations, Nonl. Anal. , 51 (2002), 12231232.

[22]
D. Franco, J. J. Nieto , Maximum principles for periodic impulsive first order problems, J. Comput. Appl. Math. , 88 (1998), 144159.

[23]
Y. Liu, Further results on periodic boundary value problems for nonlinear first order impulsive functional differential equations, J. Math. Anal. Appl. , 327 (2007), 435452.

[24]
Y. Liu, W. Ge, Stability theorems and existence results for periodic solutions of nonlinear impulsive delay differential equations with variable coefficients , Nonl. Anal. , 57 (2004), 363399.

[25]
L. Kong, J. Sun, Nonlinear boundary value problem of first order impulsive functional differential equations, J. Math. Anal. Appl. , 318 (2006), 726741.

[26]
E. Liz, Existence and approximation of solutions for impulsive first order problems with nonlinear boundary conditions, Nonl. Anal., 25 (1995), 11911198.

[27]
X. Yang, J. Shen , Nonlinear boundary value problems for first order impulsive functional differential equations, Appl. Math. Comput. , 189 (2007), 19431952.

[28]
D. R. Smart, Fixed point theorems, Cambridge University Press, Cambridge (1980)

[29]
S. Tang, L. Chen, Global attractivity in a ''foodlimited'' population model with impulsive effects, J. Math. Anal. Appl. , 292 (2004), 211221.

[30]
J. J. Nieto, Basic theory for nonresonance impulsive periodic problems of first order, J. Math. Anal. Appl. , 205 (1997), 423433.

[31]
Z. Luo, Z. Jing , Periodic boundary value problem for firstorder impulsive functional differential equations, Comput. Math. Appl. , 55 (2008), 20942107.

[32]
R. E. Gaines, J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Math., 568, Springer, Berlin (1977)

[33]
T. Jankowski, Ordinary differential equations with nonlinear boundary conditions of antiperiodic type, Comput. Math. Appl., 47 (2004), 14191428.

[34]
J. J. Nieto, Differential inequalities for functional perturbations of first order ordinary differential equations, Appl. Math. Letters, 15 (2002), 173179.