Abstract
By constructing a suitable projection scheme and using the coincidence degree theory of Mawhin, we study the existence of solutions for conjugate boundary value problems with functional boundary conditions at resonance with operatorname{dim}operatorname{Ker}L = 1. Examples are given to illustrate our main results.
Highlights
1 Introduction In the past years, the conjugate boundary value problems at nonresonance have been studied by many authors [1,2,3,4,5,6,7,8,9,10,11,12,13]
Du and Ge [15] investigated the existence of solutions for the (n – 1, 1) conjugate boundary value problems at resonance x(n)(t) = f x, x(t), x (t), . . . , x(n–1)(t) + e(t), a.e. t ∈ [0, 1], m–2 x(0) = αix(ξi), i=1 x (0) = x (0) = · · · = x(n–2)(0) = 0, x(1) = x(η)
Using Mawhin’s continuation theorem [17], Zhao and Liang [18] studied the existence of solutions for the second-order nonlinear boundary value problem
Summary
The conjugate boundary value problems at nonresonance have been studied by many authors [1,2,3,4,5,6,7,8,9,10,11,12,13]. Du and Ge [15] investigated the existence of solutions for the (n – 1, 1) conjugate boundary value problems at resonance x(n)(t) = f x, x(t), x (t), . ⎧ ⎨x (t) = f (t, x(t), x (t)), 0 < t < 1, ⎩Γ1(x) = 0, Γ2(x) = 0, where Γ1(x) and Γ2(x) are linear bounded operators Their results generalize a number of recent works such as multipoint and integral boundary value problems. Motivated by the literature cited, we study the following conjugate boundary value problems with functional boundary conditions at resonance and dim Ker L = 1:. The operator L : dom L ⊂ X → Y is a Fredholm operator of index zero, and the linear continuous projectors P : X → X and Q : Y → Y can be defined by (Pφ)(x) = φ(1)Φ(x), Qu
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