Abstract

This work studies some two point impulsive boundary value problems composed by a fully differential equation, which higher order contains an increasing homeomorphism, by two point boundary conditions and impulsive effects. We point out that the impulsive conditions are given via multivariate generalized functions, including impulses on the referred homeomorphism. The method used apply lower and upper solutions technique together with fixed point theory. Therefore we have not only the existence of solutions but also the localization and qualitative data on their behavior. Moreover a Nagumo condition will play a key role in the arguments.

Highlights

  • In this article we study the following two point boundary value problem composed by the one-dimensional φ-Laplacian equation (φ(u (t))) + q(t)f (t, u(t), u (t), u (t)) = 0, t ∈ J, (1)where (A1) φ is an increasing homeomorphism such that φ(0) = 0 and φ(R) = R,(A2) q ∈ C([0, 1]) with q > 0 and 1 0 q(s)ds < ∞, f ∈ C([0, 1] ×R3, R), together the boundary conditions u(0) = A, u (0) = B, u (1) = C, A, B, C ∈ R

  • The paper is organized as it follows: Section 2 contains an uniqueness result for an associated problem to (1)-(3) and the definition of lower and upper solutions, with strict inequalities in some boundary and impulsive conditions

  • A function α(t) ∈ E with φ(α (t)) ∈ P C1[0, 1] is a lower solution of problem (1), (2), (3) if

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Summary

Introduction

Φ-Laplacian differential equations, Generalized impulsive conditions, Upper and lower solutions, Nagumo condition, Fixed point theory. The paper is organized as it follows: Section 2 contains an uniqueness result for an associated problem to (1)-(3) and the definition of lower and upper solutions, with strict inequalities in some boundary and impulsive conditions.

Results
Conclusion

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