Abstract
By Leray-Schauder continuation theorem and the nonlinear alternative of Leray-Schauder type, the existence of a solution for an-point boundary value problem with impulses is proved.
Highlights
The main purpose of this paper is to get results on the solvability of the following boundary value problem (BVP):x (t) = f t,x(t),x (t), Δx tk = bkx tk, Δx tk = ckx tk, x (0) = 0, m−2 x(1) = aix ξi, i=1 (1.1)where ξi ∈ (0, 1), i = 1, 2, . . . , m − 2, 0 < ξ1 < ξ2 < · · · < ξm−2 < 1, ai ∈ R, i = 1, 2, . . . , m − 2, m−2 i=1 ai = t0 < t1 t2 tT tT+1Such problems without impulses effects have been solved before, for example, in [1,2,3]
The proofs are based on the Leray-Schauder continuation theorem [5] and the nonlinear alternative of Leray-Schauder type [6]
In order to define the concept of solution for BVP (1.1), we introduce the following spaces of functions: (i) PC[0, 1] = {u : [0, 1] → R, u is continuous at t = tk, u(tk+), u(tk−) exist, and u(tk−) = u(tk )}; (ii) PC1[0, 1] = {u ∈ PC[0, 1] : u is continuously differentiable at t = tk, u (0+), u, u exist and u = u}; (iii) PC2[0, 1] = {u ∈ PC1[0, 1] : u is twice continuously differentiable at t = tk}
Summary
The main purpose of this paper is to get results on the solvability of the following boundary value problem (BVP):. Such problems without impulses effects have been solved before, for example, in [1,2,3]. As far as we know the publication on the solvability of m-point problems with impulses is fewer [4]. Our main goal is to find condition for f , bk, ck, 1 ≤ k ≤ T, which guarantees the existence of at least one solution of problem (1.1). The proofs are based on the Leray-Schauder continuation theorem [5] and the nonlinear alternative of Leray-.
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