Abstract
In this paper, we consider the non-periodic boundary value problem for a type of first order impulsive functional differential equation in Banach spaces. The existence of pulse in differential equations makes them an important area of investigation. We make use of fixed point index theory on the cone to prove existence of positive solutions. The conditions for existence of two and three positive solutions are given.
Highlights
Introduction and PreliminariesIn recent years, the theories of impulsive functional differential equations have been rapidly developed, and because such equations may exhibit several real world phenomena in physics, biology, engineering, and so forth (Bainov & Simeonov, 1993; Lakshmikantham, Bainov & Simeonov, 1989; Bainov & Hristova, 1993), they have received much attention (Ding, Mi & Han, 2005; Zhang & Liu, 2010),The periodic boundary value problem is an important research branch of the impulsive functional differential equations
Whereas the non periodic boundary value occurs more frequently in differential equations with pulse, researches are needed for the problem of existence of positive solutions and multiplicity of such equations
In this paper£we restrict our attention to the study of the following first order impulsive functional differential equations with non-periodic boundary value u′(t) + M2u(t) = f (t, ut), t ∈ J = [0, T ]
Summary
The theories of impulsive functional differential equations have been rapidly developed, and because such equations may exhibit several real world phenomena in physics, biology, engineering, and so forth (Bainov & Simeonov, 1993; Lakshmikantham, Bainov & Simeonov, 1989; Bainov & Hristova, 1993), they have received much attention (Ding, Mi & Han, 2005; Zhang & Liu, 2010),The periodic boundary value problem is an important research branch of the impulsive functional differential equations. & Weigao, 2002) about the existence of solutions and the multiplicity of positive solutions of the impulsive functional differential equations with periodic boundary value problems. Whereas the non periodic boundary value occurs more frequently in differential equations with pulse, researches are needed for the problem of existence of positive solutions and multiplicity of such equations. The approaches used for the investigation of existence of positive solutions for differential equations with impulse are monotone iterative technique and upper and lower solution method (Zhimin & She, 2002; Juan & Rosana, 2006; Luo & Jing, 2008; He & He, 2004). In (Zhao, 2010), Zhao studied the problem (1), the results are established using the fixed point index theorem on the cone, and they proved the existence of two solutions. Motivated by the results mentioned above, in this paper, we give the conditions of the existence of two positive solutions and three positive solutions of equations (1) using fixed point index theory on the cone
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