# On Solvability Theorems of Second-Order Ordinary Differential Equations with Delay

Authors: Nai-Sher Yeh
Journal: Abstract and Applied Analysis
 (2018)

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## Abstract

For eachx0∈[0,2π) and k∈N, we obtain some existence theorems of periodic solutions to the two-point boundary value problem u′′(x)+k2u(x-x0)+g(x,u(x-x0))=h(x) in (0,2π) with u(0)-u(2π)=u′(0)-u′(2π)=0 when g:(0,2π)×R→R is a Caratheodory function which grows linearly in u as u→∞, and h∈L1(0,2π) may satisfy a generalized Landesman-Lazer condition (1+sign(β))∫02πh(x)v(x)dx<∫v(x)>0gβ+(x)vx1-βdx+∫v(x)<0gβ-(x)vx1-βdx for all v∈N(L)\{0}. Here N(L) denotes the subspace of L1(0,2π) spanned by sin⁡kx and cos⁡kx, -1<β≤0, gβ+(x)=liminfu→∞(gx,uu/u1-β), and gβ-(x)=liminfu→-∞(gx,uu/u1-β).


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## Body

## 1. Introduction

Letx0∈[0,2π) and k∈N be fixed. We consider the following two-point boundary value problems:(1)u′′x+k2ux-x0+gx,ux-x0=hxin0,2π,u0-u2π=u′0-u′2π=0,(2)u′′x+k2ux-gx,ux-x0=-hxin0,2π,u0-u2π=u′0-u′2π=0,where h∈L1(0,2π) is given and g:(0,2π)×R→R is a Caratheodory function; that is, g(x,u) is continuous in u∈R, for a.e. x∈(0,2π), is measurable in x∈(0,2π) for all u∈R, and satisfies, for each r>0, the fact that there exists an ar∈L1(0,2π) such that (3)gx,u≤arxfor a.e. x∈(0,2π) and all u≤r. Concerning the growth condition of the nonlinear term g to (1) and (2), we assume that (H)
there exist constants-1<β≤0, r0>0, and a,b,c,d∈L1(0,2π),a,b≥0 and a(x)≤2k+1 for a.e. x∈(0,2π) with strict inequality on a positive measurable subset of (0,2π), such that for a.e. x∈(0,2π) and all u≥r0(4)cxu-β≤gx,u≤axu+bx;and for a.e.x∈(0,2π) and all u≤-r0(5)-axu-bx≤gx,u≤dxu-β;(G)
there exist constants-1<β≤0, r0>0, and a,b,c,d∈L1(0,2π),a,b≥0 and a(x)≤2k-1 for a.e. x∈(0,2π) with strict inequality on a positive measurable subset of (0,2π), such that for a.e. x∈(0,2π) and all u≥r0(6)cxu-β≤gx,u≤axu+bx;and for a.e.x∈(0,2π) and all u≤-r0(7)-axu-bx≤gx,u≤dxu-β;respectively, and a generalized Landesman-Lazer condition(8)0<∫vx>0gβ+xvx1-βdx+∫vx<0gβ-xvx1-βdx,for all v∈N(L)\{0}, may be satisfied. Here N(L) denotes the subspace of L1(0,2π) spanned by sin⁡kx and cos⁡kx, β∈R, gβ+(x)=liminfu→∞(gx,uu/u1-β), and gβ-(x)=liminfu→-∞(gx,uu/u1-β). Under assumptions and either with or without the Landesman-Lazer condition(9)∫02πhxvxdx<∫vx>0g0+vxdx+∫vx<0g0-vxdxfor all v∈N(L)\{0}, the solvability of the problem (1) has been extensively studied if the nonlinearity g(x,u) has at most linear growth in u as u→∞ (see [1–13] for the case x0=0 and [14–16] for the general case) or grows superlinearly in u in one of directions u→∞ and u→-∞ and may be bounded in the other (see [8, 17] for the case x0=0 and [14] for the general case when k=0). Based on the well-known Leray-Schauder continuation method (see [18, 19]), we obtain solvability theorems to (1) (resp., (2)) when g(x,u) satisfies (H) (resp., (G)) and either (8) with -1<β<0 or (9) with β=0 is satisfied, which extends the results of [15] for the nonresonance case, and has been established in [9] for the case x0=0 and g(x,u) grows sublinearly in u as u→∞ with -1<β≤1. Unfortunately, it is still unknown when k∈N,  g(x,u) grows linearly in u as u→∞ and the assumption of (8) is replaced by (10)∫02πhxvxdx=0<∫vx>0gβ+xvx1-βdx+∫vx<0gβ-xvx1-βdxfor all v∈N(L)\{0} with β>0. In the following we will make use of real Banach spaces Lp0,2π,C[0,2π] and Sobolev spaces W2,1(0,2π) and H1(0,2π). The norms of Lp(0,2π),C[0,2π] and H1(0,2π) are denoted by uLp,uC and uH1, respectively. By a solution of (1), we mean a periodic function u:R→R of period 2π which belongs to W2,1(0,2π) and satisfies the differential equation in (1) a.e. x∈(0,2π).


## 2. Existence Theorems

For eachv∈W2,1(0,2π) with v(0)-v(2π)=v′(0)-v′(2π)=0 and k∈N, we write v¯=∑0≤j≤kPjv,  v~=∑j>kPjv, and v⊥=∑0≤j≠kPjv. Here Pjv denotes the projection of v on the eigenspace of d2/dx2 spanned by sin⁡jx and cos⁡jx for j∈N∪{0}. Just as an application of [11, Lemma 2] or [1, Lemma 2.2], we can modify slightly the proof of [15, Lemma 1] to obtain the next lemma.Lemma 1.
Letk∈N∪{0} and Γ be a nonnegative L1(0,2π)-function such that for a.e. x∈(0,2π),Γ(x)≤2k+1 with strict inequality on a positive measurable subset of (0,2π). Then there exists a constant K1>0 such that (11)∫02πu¯x-x0-u~xu′′x+k2ux-x0+pxux-x0dx≥K1u⊥H12whenever p∈L1(0,2π) with 0≤p(x)≤Γ(x) for a.e. x∈(0,2π) and u∈W2,1(0,2π) is a periodic function of period 2π with u(0)-u(2π)=u′(0)-u′(2π)=0.Proof.
Just as in [20, Lemma 1], we can modify slightly the proof of [11, Lemma 2] or [1, Lemma 2.2] to obtain the fact that there exists a constant K1>0 such that (12)∫02πu~′x2-k2+pxu~x2dx+∫02πk2+pxu¯x2-u¯′x2dx≥2K1u⊥H12whenever p∈L1(0,2π) with 0≤p(x)≤Γ(x) for a.e. x∈(0,2π) and u∈W2,1(0,2π) with u(0)-u(2π)=u′(0)-u′(2π)=0. Let us extend u(x) and p(x)2π periodically in x to all ofR and then use the same notations for the periodic extensions as for the original functions. In this case, we have ∫02π(u~′(x))2dx=∫02π(u~′(x-x0))2dx and(13)∫02πu′′x+k2+pxux-x0u¯x-x0-u~xdx=∫02πu~′x2dx-∫02πu¯′xu¯′x-x0dx+12∫02πk2+pxu¯x-x02-u~x2-u~x-x02dx+12∫02πk2+pxu¯x-x0+u~x-x0-u~x2dx≥∫02πu~′x2dx-12∫02πu¯′x2+u¯′x-x02dx+12∫02πk2+pxu¯x-x02-u~x2-u~x-x02dx+12∫02πk2+pxu¯x-x0+u~x-x0-u~x2dx≥12∫02πu~′x2-k2+pxu~x2dx+12∫02πu~′x-x02-k2+pxu~x-x02dx+12∫02πk2+pxu¯x-x02-u¯′x-x02dx-12∫02πu¯′x2dx+12∫02πk2+pxu¯x-x0+u~x-x0-u~x2dx.Since ∫02π(u¯(x))2dx=∫02π(u¯(x-x0))2dx,  p(x)≥0 for a.e. x∈(0,2π), and ∫02πv¯(x)w~(x)=0 for all v,w∈W2,1(0,2π) with v(0)-v(2π)=v′(0)-v′(2π)=0 and w(0)-w(2π)=w′(0)-w′(2π)=0, we have ∫02π(u~′(x))2-(k2+p(x))(u~(x))2dx≥0 and (14)-12∫02πu¯′x2dx+12∫02πk2+pxu¯x-x0+u~x-x0-u~x2dx≥-12∫02πu¯′x2dx+12∫02πk2u¯x-x0+u~x-x0-u~x2dx=12-∫02πu¯′x2dx+k2∫02πu¯x-x02dx+12k2∫02πu~x-x0-u~x2dx≥0.Combining (12) with (13), we have (15)∫02πu¯x-x0-u~xu′′x+k2ux-x0+pxux-x0dx≥12∫02πu~′x-x02-k2+pxu~x-x02dx+12∫02πk2+pxu¯x-x02-u¯′x-x02dx≥K1u⊥H1x0,x0+2π2=K1u⊥H12.Lemma 2.
Letk∈N and Γ be a nonnegative L1(0,2π)-function such that for a.e. x∈(0,2π),Γ(x)≤2k-1 with strict inequality on a positive measurable subset of (0,2π). Then there exists a constant K2>0 such that (16)∫02πu¯¯x-x0-u~~xu′′x+k2ux-pxux-x0dx≥K2u⊥H12whenever p∈L1(0,2π) with 0≤p(x)≤Γ(x) for a.e. x∈(0,2π) and u∈W2,1(0,2π) is a periodic function of period 2π with u(0)-u(2π)=u′(0)-u′(2π)=0. Here v¯¯=∑0≤j<kPjv and v~~=∑j≥kPjv for each v∈W2,1(0,2π) with v(0)-v(2π)=v′(0)-v′(2π)=0.Theorem 3.
Letk∈N∪{0} and g:(0,2π)×R→R be a Caratheodory function satisfying (H). Then for each h∈L1(0,2π) problem (1) has a solution u, provided that either (8) with -1<β<0 or (9) with β=0 holds.Proof.
Letα∈R be fixed and 0<α<2k+1. We consider the boundary value problems (17)u′′x+k2ux-x0+1-tαux-x0+tgx,ux-x0=thxin0,2π,u0-u2π=u′0-u′2π=0for 0≤t≤1, which becomes the original problem when t=1. Since 0<α<2k+1, we observe from Lemma 1 that (17) has only a trivial solution when t=0. To apply the Leray-Schauder continuation method, it suffices to show that solutions to (17) for 0<t<1 have an a priori bound in H1(0,2π). To this end, let θ:R→R be a continuous function such that 0≤θ≤1,  θ(u)=0 for u≤r0, and θ(u)=1 for u≥2r0. We define e(x)=max⁡{ar0(x),b(x),cx,dx}, (18)g1x,u=min⁡gx,u+exu-β,axuθuifu≥0max⁡gx,u-exu-β,axuθuifu≤0,and g2(x,u)=g(x,u)-g1(x,u). Then g1,g2:(0,2π)×R→R are Caratheodory functions, such that for a.e. x∈(0,2π) and u∈R,  u≠0(19)0≤g1x,uu≤ax,(20)g2x,u≤exu-β+ex. If u is a possible solution to (17) for some 0<t<1, then using (19), (20), and Lemma 1, we have (21)0=∫02πu¯x-u~x-x0u′′x+k2ux-x0+1-tαux-x0+tg1x,ux-x0+tg2x,ux-x0-thxdx≥K1u⊥H12-eL1uC-β+eL1+hL1u¯C+u~C≥K1u⊥H12-C1uC-β+1u¯H1+u~H1,which implies that (22)u⊥H12≤C1K1uC-β+1u¯H1+u~H1≤C2uH1+uH11-βfor some constants C1,C2>0 independent of u. It remains to show that solutions to (17) for 0<t<1 have an a priori bound in H1(0,2π). We argue by contradiction and suppose that there exists a sequence {un} of periodic functions with period 2π and a corresponding sequence {tn} in (0,1) such that un is a solution to (17) with t=tn and unH1≥n for all n. Let vn=un/unH1; then vnH1=1 for all n∈N, and by (22) we have vn⊥H1→0 as n→∞. Since vnH1=1 and PkvnH1≤vnH1+vn⊥H1 for all n∈N, we have a bounded sequence {Pkvn} in H1(0,2π). For simplicity, we may assume that vn converges to v in H1(0,2π) for some v∈N(L) with vH1=1. In particular, vn→v in C[0,2π]. Clearly, v(·-x0)∈N(L) and v(·-x0)H1=vH1. It follows that un(x)→∞ for each x∈R with v(x)>0, and un(x)→-∞ for each x∈R with v(x)<0. Since ∫02πun′′(x)Pkun(x-x0)dx+∫02πk2un(x)Pkun(x-x0)dx=0 and Pkun(·)L22=Pkun(·-x0)L22, we have (23)∫02πun′′xPkunx-x0dx+∫02πk2unx-x0Pkunx-x0dx=∫02πun′′xPkunx-x0dx+∫02πk2unxPkunx-x0dx+∫02πk2unx-x0-unxPkunx-x0dx=∫02πk2unx-x0-unxPkunx-x0dx=∫02πk2Pkunx-x0-PkunxPkunx-x0dx≥0.Multiplying each side of (17) by Pkvn(x-x0), and then integrating them over [0,2π] when u=un and t=tn, we get (24)tn∫02πgx,unx-x0Pkvnx-x0dx≤1-tnα∫02πunx-x0Pkvnx-x0dx+tn∫02πgx,unx-x0Pkvnx-x0dx≤tn∫02πhxPkvnx-x0dx. By (19) and the assumption of -1<β≤0, we have(25)g1x,unx-x0Pkvnx-x0unH1β=g1x,unx-x0unx-x0unx-x0Pkvnx-x0unH1β≥g1x,unx-x0unx-x0-12unx-x0-Pkunx-x02unH1β-1≥-12axun⊥x-x02unH1β-1for a.e. x∈(0,2π). Combining (22) with (25), we get that g1(x,un(x-x0))Pkvn(x-x0)unH1β is bounded from below by an L1(0,2π)-function independent of n. By (20) and the assumption of -1<β≤0, we have (26)g2x,unx-x0Pkvnx-x0unH1β≤exunx-x0-β+exPkvnx-x0unH1β≤exvnx-x0-β+exPkvnx-x0for a.e. x∈(0,2π), In particular, g2(x,un(x-x0))Pkvn(x-x0)unH1β is bounded from below by an L1(0,2π)-function independent of n, which implies that g(x,un(x-x0))Pkvn(x-x0)unH1β is also so, ∫v(x-x0)=0liminfn→∞g(x,un(x-x0))Pkvn(x-x0)unH1βdx=0, and (27)gx,unx-x0Pkvnx-x0unH1β=gx,unx-x0unx-x0unx-x01-βPkvnx-x0sgnunx-x0vnx-x0βfor all n∈N with un(x-x0)≠0. Here sign(w)=1 if w>0, sign(w)=0 if w=0, and sign(w)=-1 if w<0. Applying Fatou’s lemma to the integral ∫02πg(x,un(x-x0))Pkvn(x-x0)unH1βdx, we have (28)∫vx-x0>0gβ+xvx-x01-βdx+∫vx-x0<0gβ-xvx-x01-βdx=∫vx-x0>0liminfn→∞⁡gx,unx-x0unx-x0unx-x01-βlimn→∞⁡Pkvnx-x0sgnunx-x0vnx-x0βdx+∫vx-x0<0liminfn→∞⁡gx,unx-x0unx-x0unx-x01-βlimn→∞⁡Pkvnx-x0sgnunx-x0vnx-x0βdx=∫vx-x0>0liminfn→∞⁡gx,unx-x0Pkvnx-x0unH1βdx+∫vx-x0<0liminfn→∞⁡gx,unx-x0Pkvnx-x0unH1βdx=∫vx-x0>0liminfn→∞⁡gx,unx-x0Pkvnx-x0unH1βdx+∫vx-x0<0liminfn→∞⁡gx,unx-x0Pkvnx-x0unH1βdx+∫vx-x0=0limn→∞⁡gx,unx-x0Pkvnx-x0unH1βdx=∫02πliminfn→∞⁡gx,unx-x0Pkvnx-x0unH1βdx≤liminfn→∞⁡∫02πgx,unx-x0Pkvnx-x0unH1βdx≤1+signβ∫02πhxvx-x0dx,which is a contradiction when either (8) with -1<β<0 or (9) with β=0 is satisfied. Hence, the proof of this theorem is complete.By slightly modifying the proof of Theorem3, we can apply Lemma 2 to obtain an existence theorem to (2) when condition (H) is replaced by (G) and either (8) with -1<β<0 or (9) with β=0 is satisfied, which has been established in [20] for the case x0=0 when (9) with β=0 is satisfied and in [9] for the case x0=0 when (8) with β=-1 is satisfied.Theorem 4.
Letk∈N and g:(0,2π)×R→R be a Caratheodory function satisfying (G). Then for each h∈L1(0,2π) problem (2) has a solution u, provided that either (8) with -1<β<0 or (9) with β=0 holds.



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