tailieunhanh - Existence of solutions for a first-order nonlocal boundary value problem with changing-sign nonlinearity
This work is concerned with the existence of positive solutions to a nonlinear nonlocal first-order multipoint problem. Here the nonlinearity is allowed to take on negative values, not only positive values. | Turk J Math (2015) 39: 556 – 563 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Existence of solutions for a first-order nonlocal boundary value problem with changing-sign nonlinearity ˙ Erbil C ¸ ETIN, Fatma Serap TOPAL∗ ˙ Department of Mathematics, Ege University, Bornova, Izmir, Turkey Received: • Accepted/Published Online: • Printed: Abstract: This work is concerned with the existence of positive solutions to a nonlinear nonlocal first-order multipoint problem. Here the nonlinearity is allowed to take on negative values, not only positive values. Key words: Positive solution, nonlinear boundary condition, sign-changing problem 1. Introduction In this paper, we are interested in the existence of positive solutions for the following first-order m-point nonlocal boundary value problem: ∑n y ′ (t) + p(t)y(t) = i=1 fi (t, y(t)), t ∈ [0, 1], ∑m y(0) = y(1) + j=1 gj (tj , y(tj )), () () where p : [0, 1] → [0, ∞) is continuous, the nonlocal points satisfy 0 ≤ t1 0 such that fi (t, y) > −M for all ∫1 (t, y) ∈ [0, 1] × [0, ∞) and 0 (fi + M )ds > 0 . If there exist positive constants r and R such that r > 2M γ 2 and the following conditions are satisfied: ∫1 1 − e− 0 (A1 )fi (t, u) ≤ fi (t, v) ≤ 2n (A2 ) where γ = 1 − e− e 1 − e− 1+e 558 − ∫1 0 ∫1 0 ∫1 ∫1 0 0 p(ξ)dξ p(ξ)dξ p(ξ)dξ p(ξ)dξ e− ∫1 0 p(ξ)dξ R − M, t ∈ [0, 1], ∫1 r ≤ u ≤ v ≤ R, 2 r 1 − e− 0 p(ξ)dξ ≤ gj (t, u) ≤ gj (t, v) ≤ R, t ∈ [0, 1], m 2m p(ξ)dξ r ≤ u ≤ v ≤ R, 2 , then the boundary value problem () − () has positive solutions. ˙ and TOPAL/Turk J Math C ¸ ETIN Proof First we consider the following boundary value problem: ′ u (t) + p(t)u(t) = n ∑ t ∈ [0, 1], Fi (t, ux (t)), () i=1 u(0) = u(1) + m ∑ gj (tj , ux (tj )), () j=1 where Fi (t, ux (t)) = fi (t, ux (t)) + M and ux (t) = max{(u − x)(t), 0} such that x(t) = M ω(t) and ω(t) .
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