Second order nonlocal boundary value problems at resonance

Math. Nachr. 284, No. 7, 875 – 884 (2011) / DOI 10.1002/mana.200810841

Second order nonlocal boundary value problems at resonance

Daniel Franco∗1, Gennaro Infante∗∗2, and Mirosława Zima∗∗∗3

1 Departamento de Matematica Aplicada, UNED, ETSI Industriales, c/ Juan del Rosal 12, Madrid 28040, Spain2 Dipartimento di Matematica, Universita della Calabria, 87036 Arcavacata di Rende, Cosenza, Italy3 Institute of Mathematics, University of Rzeszow, Rejtana 16A, 35-310 Rzeszow, Poland

Received 1 May 2008, revised 9 July 2010, accepted 18 July 2010Published online 14 April 2011

Key words Positive solution, resonance, cone, semilinear equationMSC (2010) Primary: 34B18; Secondary: 34B10, 47H11

We present sufficient conditions for the existence of positive solutions for some second order boundary valueproblems at resonance. The boundary conditions that we study are quite general, involve a Stieltjes integral andinclude, as particular cases, multi-point and integral boundary conditions. Our results are based on a Leggett-Williams norm-type theorem due to O’Regan and Zima. We employ a general abstract approach which allowsus to improve and complement recent results in the literature.

c© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

In this paper we investigate the existence of positive solutions of some boundary value problems (BVPs) atresonance. In particular, we focus on the existence of positive solutions for the second order differential equation

−u′′(t) = f(t, u(t)), t ∈ (0, 1), (1.1)

where f : [0, 1] × R → R is continuous, subject to a type of nonlocal boundary conditions (BCs). Our boundaryvalue problems are at resonance, in the sense that, under the BCs, we study the linear equation −u′′(t) = 0,t ∈ (0, 1), which has non-trivial solutions.

Existence of solutions for nonlocal second order BVPs at resonance, under a variety of BCs, has been investi-gated by several authors, see for example [2], [8], [9], [12], [13], [18]–[20], [22], [23], [25], [32] and referencestherein. When the solvability of a BVP is concerned, a typical approach is to re-write the problem as a semilinearequation of the form Lu = Nu, where L is a linear Fredholm operator of index zero and N is a nonlinear map.L is usually chosen as Lu = −u′′ and N is the Nemytskiı operator associated with the nonlinearity f . For someresults on solvability of semilinear equations and their applications to BVPs, we refer the reader to the pioneeringwork of Mawhin [27] and to [4]–[6], [28], [33], [34].

Given the current interest in the existence of positive solutions for nonlocal non-resonant BVPs (see for exam-ple [1], [11], [10], [16], [17], [21], [26], [30], [35], [37]–[40] and references therein), it is rather natural to inves-tigate under which conditions the existence of positive solutions of a nonlocal BVP at resonance can be achieved.As far as we are aware of, this problem has been investigated only in few papers. In particular, Palamides [29]studied the existence of positive solutions for a nth-order m-point BVP via the Sperner Lemma. Bai and Fang [3]discussed the existence of positive solutions for the second order differential equation

(p(t)u′(t))′ = f(t, u(t), u′(t)), t ∈ (0, 1),

under the BCs

u′(0) = 0, u(1) = u(η), (1.2)

∗ e-mail: [email protected], Phone: +34 91 398 8134, Fax: +34 91 398 8104∗∗ Coordinating author: e-mail: [email protected], Phone: +39 0984 496474, Fax: +39 0984 496410∗∗∗ e-mail: [email protected], Phone: +48 17 872 11 71, Fax: +48 17 872 12 81

c© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

876 D. Franco, G. Infante, and M. Zima: Nonlocal BVPs at resonance

by means of a fixed point index theorem for semilinear A-proper maps due to Cremins [4]. Han [13] studied theexistence of positive solutions of the BVP (1.1)–(1.2) via the Krasnosel’skiı-Guo fixed point theorem in cones. ALeggett-Williams norm-type theorem for coincidences due to O’Regan and Zima [28] has been used by Infanteand Zima [15], who dealt with the existence of positive solutions of the second order differential Equation (1.1)under the BCs

u′(0) = 0, u(1) =m−2∑i=1

αiu(ηi). (1.3)

Webb and Zima [39] discussed the existence and multiplicity of positive solutions of the Equation (1.1) subjectto various nonlocal BCs via classical fixed point index theory. Their approach provides growth conditions on thenonlinearity f that are not directly comparable with the ones provided by our method.

Here we focus on the following BCs

u′(0) = 0, a1u(1) + a2u′(1) = α[u], (1.4)

where a1 > 0, a2 ≥ 0 and α[u] is a linear functional on C[0, 1] given by

α[u] =∫ 1

0u(s) dA(s),

involving a Riemann-Stieltjes integral, in particular A has bounded variation and dA is a positive measure.A special case of these BCs (namely when a2 = 0) has been studied, in the non-resonant case, by Webb and

Infante [36], and for resonant problems by Webb and Zima [39]. These conditions are quite general, since theyinclude as special cases

α[u] =m∑

i=1

αiu(ηi) and α[u] =∫ 1

0α(s)u(s) ds,

that is, multi-point and integral BCs. These are widely studied objects; see for example [7], [8], [15], [18], [19],[22], [32].

Note that the BVP (1.1)–(1.4) does have a unique solution when f ≡ 0. If instead of α[u] we write 1, then wedenote by γ(t) such a solution. It can be verified that

γ(t) ≡ 1a1

.

Throughout the paper we assume that α[γ] = 1. This leads to resonance situations in the sense that the equation−u′′(t) = 0, t ∈ (0, 1), under the BCs (1.4) has the non-trivial solutions u(t) = θ, where θ ∈ R is an arbitraryconstant.

Note that, in the case when a1 = 1, a2 = 0 and α[u] =∑m−2

i=1 αiu(ηi), m > 2, 0 < η1 < η2 < · · · <ηm−2 < 1, i.e., in the case of a m-point BC, our resonance assumption reads, as it is standard in the literature,∑m−2

i=1 αi = 1.The BVP (1.1)–(1.4) can be reduced to a coincidence equation of the form Lu = Nu, where L is a Fredholm

operator of index zero and N is a nonlinear operator.We point out that, in this case, the natural choice of Lu = −u′′ leads to a coincidence equation which involves

a Hammerstein integral operator, the kernel of which can be very complicated. This makes it difficult to verifythe required conditions.

Our idea here is to choose Lu(t) = u(t) − γ(t)α[u], so that N becomes a nonlinear integral operator witha much simpler kernel. This simplifies considerably the way nonlocal resonance problems can be handled. As faras we know, this approach has never been used before in the context of second order resonance BVPs.

For the solvability of our semilinear equation we use, as in [15], the Leggett-Williams norm-type theoremgiven in [28].

We improve the results in the literature in the sense that we can deal with more general BCs and we provide adifferent approach to earlier studied BVPs. This is illustrated in some examples.

c© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mn-journal.com

Math. Nachr. 284, No. 7 (2011) / www.mn-journal.com 877

2 Existence of positive solutions

Firstly, we recall some standard facts on Fredholm operators and cones in Banach spaces. Let X , Y be realBanach spaces. Consider a linear mapping L : dom L ⊂ X → Y and a nonlinear operator N : X → Y . Assumethat L is a Fredholm operator of index zero, that is, Im L is closed and dim KerL = codim Im L < ∞. Thisassumption implies that there exist continuous projections P : X → X and Q : Y → Y such that Im P = Ker Land Ker Q = ImL (see for example [27], [33]). Moreover, since dim ImQ = codim Im L, there exists anisomorphism J : Im Q → Ker L. Denote by LP the restriction of L to Ker P ∩ dom L. Clearly, LP is anisomorphism from Ker P ∩ dom L to Im L. Let KP denote its inverse. Then KP : Im L → Ker P ∩ dom L.

It is known (see [27], [33]) that the coincidence equation Lu = Nu is equivalent to

u = (P + JQN)u + KP (I − Q)Nu.

Let C be a cone in the Banach space X , that is, a closed and convex subset of X such that

(i) λx ∈ C for all x ∈ C and λ ≥ 0,

(ii) x, −x ∈ C implies x = 0.

It is well-known that C induces a partial order in X by

x y if and only if y − x ∈ C.

The following property is valid for every cone in a Banach space.

Lemma 2.1 ([31]) Let C be a cone in X . Then for every u ∈ C \{0} there exists a positive number σ(u) suchthat

‖x + u‖ ≥ σ(u)‖x‖

for all x ∈ C.

Let ρ : X → C be a retraction, that is, a continuous mapping such that ρ(x) = x for all x ∈ C. Set

Ψ := P + JQN + KP (I − Q)N,

and

Ψρ := Ψ ◦ ρ.

We make use of the following theorem.

Theorem 2.2 ([28]) Let C be a cone in X and let Ω1 , Ω2 be open bounded subsets of X with Ω1 ⊂ Ω2 andC ∩ (Ω2 \ Ω1) �= ∅. Assume that:

1◦ L is a Fredholm operator of index zero,

2◦ QN : X → Y is continuous and bounded and KP (I − Q)N : X → X is compact on every boundedsubset of X ,

3◦ Lu �= λNu for all u ∈ C ∩ ∂Ω2 ∩ dom L and λ ∈ (0, 1),

4◦ ρ maps subsets of Ω2 into bounded subsets of C,

5◦ dB

([I − (P + JQN)ρ]

∣∣Ker L

,Ker L ∩ Ω2 , 0)�= 0, where dB stands for the Brouwer degree,

6◦ there exists u0 ∈ C \ {0} such that ‖u‖ ≤ σ(u0)‖Ψu‖ for u ∈ C(u0) ∩ ∂Ω1 , where C(u0) = {u ∈ C :μu0 u for some μ > 0} and σ(u0) is such that ‖u + u0‖ ≥ σ(u0)‖u‖ for every u ∈ C,

7◦ (P + JQN)ρ(∂Ω2) ⊂ C,

8◦ Ψρ(Ω2 \ Ω1) ⊂ C.

Then the equation Lu = Nu has a solution in the set C ∩ (Ω2 \ Ω1).

www.mn-journal.com c© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


Let us denote by k(t, s) the Green’s function of the BVP (1.1)–(1.4) with α[u] ≡ 0 and f ≡ 0. We have

k(t, s) =

⎧⎪⎨⎪⎩

1 − t +a2

a1, 0 ≤ s ≤ t ≤ 1,

1 − s +a2

a1, 0 ≤ t ≤ s ≤ 1.

(2.1)

We can now associate to the BVP (1.1)–(1.4) the perturbed Hammerstein integral equation

u(t) = γ(t)α[u] +∫ 1

0k(t, s)f(s, u(s)) ds, (2.2)

where γ(t) ≡ 1/a1 and k(t, s) is defined above. These types of perturbed integral equations have been studiedbefore in [14, 36]. Note that it can be checked via a routine calculation that a solution of (2.2) would provide asolution of the BVP (1.1)–(1.4).

We make use of the following notation:

KA (s) =∫ 1

0k(t, s) dA(t), s ∈ [0, 1],

K(s) =∫ 1

0k(t, s) dt, s ∈ [0, 1],

and

G(t, s) =M

a1KA (s) + k(t, s) −K(s), t, s ∈ [0, 1],

where M is a positive constant. Observe that KA (s) =∫ 1

0 k(t, s) dA(t) ≥ 0 for s ∈ [0, 1] and KA is notidentically zero on [0, 1].

We can now prove the following result.

Theorem 2.3 Assume that there exist constants M > 0, κ > 0 and R > 0 such that:

(H1) κM max{KA (s) : s ∈ [0, 1]} ≤ a1 ,

(H2) G(t, s) ≥ 0 for t, s ∈ [0, 1],(H3) 1 − κG(t, s) ≥ 0 for t, s ∈ [0, 1],(H4) f(t, u) > −κu, for all (t, u) ∈ [0, 1] × [0, R],(H5) f(t, R) < 0 for all t ∈ [0, 1],(H6) there exist r ∈ (0, R), t0 ∈ [0, 1], a > 0, β ∈ (0, 1) and continuous functions g : [0, 1] → [0,∞),

h : (0, r] → [0,∞) such that f(t, u) ≥ g(t)h(u) for all t ∈ [0, 1] and u ∈ (0, r], and h(u)/ua isnon-increasing on (0, r] with

h(r)r

∫ 1

0G(t0 , s)g(s) ds ≥ 1 − β

βa.

Then the equation (2.2) has a positive solution on [0, 1].

P r o o f. Consider the Banach spaces

X = Y = C[0, 1]

with

‖u‖ = maxt∈[0,1]

|u(t)|.

We can write (2.2) as a coincidence equation

Lu = Nu,



where

Lu(t) = u(t) − γ(t)α[u], t ∈ [0, 1],

and

Nu(t) =∫ 1

0k(t, s)f(s, u(s)) ds.

We now show that the assumptions of Theorem 2.2 are satisfied. Let

dom L = X.

Then

Ker L = {u ∈ X : u(t) − γ(t)α[u] = 0, t ∈ [0, 1]}= {u ∈ X : u(t) = c, t ∈ [0, 1], c ∈ R}

and dim KerL = 1. Observe that if α[γ] �= 1 then Ker L = {0}.Since α[γ] = 1, we have

Im L = {y ∈ Y : α[y] = 0}.

Clearly, Im L is closed.Observe that Y = Y1 + ImL, where

Y1 = {y1 ∈ Y : y1 = γ(t)α[y], y ∈ Y }.

Indeed, for each y ∈ Y we have

α[y − y1 ] = α[y] − α[y1 ] = α[y] − α[γ]α[y] = 0,

which gives y − y1 ∈ Im L. Moreover, Y1 ∩ Im L = {0}. Hence Y = Y1 ⊕ Im L. Since dim Y1 = 1, we getcodim Im L = 1. Therefore L is Fredholm of index zero, so the assumption 1◦ is satisfied.

Define the projections P : X → X by

Pu(t) =∫ 1

0u(s) ds, t ∈ [0, 1],

and Q : Y → Y by

Qy(t) = γ(t)α[y], t ∈ [0, 1].

Then Im P = Ker L, Ker Q = Im L and Ker P ={u ∈ X :

∫ 10 u(s) ds = 0

}. Denote by LP the restriction of

L to Ker P ∩ dom L. Let

y(t) = u(t) − γ(t)α[u], t ∈ [0, 1],

for u ∈ Ker P . Then∫ 1

0y(s) ds =

∫ 1

0u(s) ds −

∫ 1

0γ(s)α[u] ds = −α[u]

∫ 1

0γ(s) ds.

Thus, for y ∈ Im L the inverse KP of LP is given by

KP y(t) = y(t) −∫ 1

0y(s) ds, t ∈ [0, 1].



For z ∈ Im Q define

J(z) = Mz.

Clearly, J is an isomorphism from Im Q to Ker L.Now, Equation (2.2), that is,

u(t) − γ(t)α[u] =∫ 1

0k(t, s)f(s, u(s)) ds,

is equivalent to

u = (P + JQN)u + KP (I − Q)Nu,

that is,

u(t) = (P + JQN)u(t) + KP Nu(t) − KP QNu(t)

= (P + JQN)u(t) + KP Nu(t) − γ(t)α[Nu] +∫ 1

0γ(s)α[Nu] ds

= (P + JQN)u(t) + KP Nu(t)

=∫ 1

0u(s) ds + Mγ(t)

∫ 1

0

[∫ 1

0k(s, τ)f(τ, u(τ)) dτ

]dA(s)

+∫ 1

0k(t, s)f(s, u(s)) ds −

∫ 1

0

[∫ 1

0k(s, τ)f(τ, u(τ)) dτ

]ds.

Clearly, the assumption 2◦ is satisfied. Consider the cone of nonnegative functions

C = {u ∈ X : u(t) ≥ 0 on [0, 1]}.

Let

Ω1 = {u ∈ X : r > |u(t)| > β‖u‖, t ∈ [0, 1]},Ω2 = {u ∈ X : ‖u‖ < R}

and ρu(t) = |u(t)| for u ∈ X . Then ρ is a retraction and maps subsets of Ω2 into bounded subsets of C.To verify 3◦ suppose that there exist u0 ∈ C ∩ ∂Ω2 ∩ dom L and λ0 ∈ (0, 1) such that

u0(t) − γ(t)α[u0 ] = λ0

∫ 1

0k(t, s)f(s, u0(s)) ds. (2.3)

Then

−u′′0 (t) = λ0f(t, u0(t)), t ∈ [0, 1]. (2.4)

Let t1 ∈ [0, 1] be such that u0(t1) = R. It follows from the boundary conditions, (H5) and the resonant assump-tion that t1 ∈ (0, 1). This gives

0 ≥ u′′0 (t1) = −λ0f(t1 , u0(t1))

which contradicts (H5). Therefore 3◦ is satisfied.To verify 5◦ it is enough to consider for u ∈ Ker L ∩ Ω2 (that is u(t) = c on [0, 1] with ‖u‖ < R), and

λ ∈ [0, 1] the mapping

H(u, λ)(t) = u(t) − λ

(∫ 1

0|u(s)| ds +

M

a1

∫ 1

0KA (s)f(s, |u(s)|) ds

).



Suppose H(u, λ) = 0 for u ∈ ∂Ω2 . Then, in view of (H1) and (H4), we get

c > λ

(|c| − κM

a1

∫ 1

0KA (s)|c| ds

)

= λ|c|(

1 − κM

a1

∫ 1

0KA (s) ds

)≥ 0.

This implies c > 0. Hence, if H(u, λ) = 0 for u ∈ ∂Ω2 , we would have by (H5)

0 ≤ R(1 − λ) = λM

a1

∫ 1

0KA (s)f(s,R) ds < 0,

which is a contradiction. As a result, H(u, λ) �= 0 for u ∈ ∂Ω2 and λ ∈ [0, 1], and therefore

dB (H(u, 0),Ker L ∩ Ω2 , 0) = dB (H(u, 1),Ker L ∩ Ω2 , 0).

This gives

dB

([I − (P + JQN)ρ]|Ker L ,Ker L ∩ Ω2 , 0

)= dB (H(c, 1),Ker L ∩ Ω2 , 0) �= 0.

By (H1) and (H4) for u ∈ ∂Ω2 we have

(P + JQN)ρu(t) =∫ 1

0|u(s)| ds +

M

a1

∫ 1

0KA (s)f(s, |u(s)|) ds

>

∫ 1

0|u(s)| ds − κM

a1

∫ 1

0KA (s)|u(s)| ds

=∫ 1

0

(1 − κM

a1KA (s)

)|u(s)| ds ≥ 0.

This means that 7◦ holds. Let u ∈ Ω2 \ Ω1 . From (H2)–(H4) we obtain

Ψρu(t) =∫ 1

0|u(s)| ds +

∫ 1

0G(t, s)f(s, |u(s)|) ds

≥∫ 1

0|u(s)| ds − κ

∫ 1

0G(t, s)|u(s)| ds

=∫ 1

0(1 − κG(t, s)) |u(s)| ds ≥ 0

which implies 8◦.Finally, we will show that 6◦ is satisfied. Let u0(t) ≡ 1 on [0, 1]. Then u0 ∈ C \ {0}, C(u0) = {u ∈ C :

u(t) > 0 on [0, 1]} and we can take σ(u0) = 1. Let u ∈ C(u0) ∩ ∂Ω1 . Then, in particular, u(t) > 0 on [0, 1],0 < ‖u‖ ≤ r and u(t) ≥ β‖u‖ on [0, 1].

Therefore, in view of (H6), we obtain for all u ∈ C(u0) ∩ ∂Ω1 ,

(Ψu)(t0) =∫ 1

0u(s) ds +

∫ 1

0G(t0 , s)f(s, u(s)) ds

≥ β‖u‖ +∫ 1

0G(t0 , s)g(s)h(u(s)) ds

= β‖u‖ +∫ 1

0G(t0 , s)g(s)

h(u(s))ua(s)

ua(s) ds

≥ β‖u‖ +h(r)ra

∫ 1

0G(t0 , s)g(s)ua(s) ds

≥ β‖u‖ +h(r)ra

∫ 1

0G(t0 , s)g(s)βa‖u‖a ds ≥ ‖u‖.

Thus, ‖u‖ ≤ σ(u0)‖Ψu‖ for all u ∈ C(u0) ∩ ∂Ω1 .



Remark 2.4 We point out, as in [15], that the conditions (H4) and (H5) cannot be used to prove the existenceof multiple positive solutions, due to an incompatibility of such requirements when nesting several Ωi’s.

Remark 2.5 Note that this result complements the ones in [36], where it is assumed that 0 ≤ α[γ] < 1.The case α[γ] = 1 was also studied in [39] by discussing an equivalent non-resonant equation −u′′ + ω2u =f(t, u) + ω2u. One of the sets of BCs dealt with in [39] was u′(0) = 0, u(1) = α[u], that is covered by (1.4).

3 Some examples

We now illustrate in two Examples the conditions that appear in Theorem 2.3.

Example 3.1 In the case of the three point BC

u′(0) = 0, u(1) = u(η), η ∈ (0, 1),

we have

KA (s) = k(η, s).

Take

G(t, s) = k(t, s) − 12(1 − s2) + MKA (s).

In order to satisfy (H2) we need to find M such that

G(t, s) = k(t, s) − 12(1 − s2) + Mk(η, s) ≥ 0.

If t ≤ s ≤ η, we have

G(t, s) = (1 − s) − 12(1 − s2) + M(1 − η)

= (1 − s)(

1 − 12(1 + s)

)+ M(1 − η) ≥ 0,

for every M ≥ 0.If t ≤ s and s ≥ η, we have

G(t, s) = (1 − s) − 12(1 − s2) + M(1 − s)

= (1 − s)(

1 − 12(1 + s)

)+ M(1 − s) ≥ 0,

for every M ≥ 0.If t ≥ s and s ≤ η, we have

G(t, s) = (1 − t) − 12(1 − s2) + M(1 − η) ≥ −1

2(1 − s2) + M(1 − η) ≥ 0,

for every M ≥ 12(1−η ) .

If t ≥ s ≥ η, we have

G(t, s) = (1 − t) − 12(1 − s2) + M(1 − s)

≥ −12(1 − s2) + M(1 − s)

= (1 − s)(

M − 12(1 + s)

)≥ 0,

for every M ≥ 1.



Thus we can choose

M ≥ max{

12(1 − η)

, 1}

.

In order to satisfy (H1) and (H3) we need to find κ > 0 such that 1 − κG(t, s) ≥ 0 and 1 − κMKA (s) ≥ 0.Clearly, that is possible since G and KA are bounded.

We finally provide an example where an integral BC is involved.

Example 3.2 Let us consider the BVP (1.1)–(1.4) with f(t, u) = 13

(t + 1

2

)(u2 − 4u + 1

), a1 = a2 = 1 and

α[u] =∫ 1

0 2t u(t) dt.In this case γ(t) ≡ 1,

KA (s) = α[k(·, s)] =∫ 1

02t k(t, s) dt =

13(4 − s3),

K(s) =12(3 − s2)

and

G(t, s) =M

3(4 − s3) + k(t, s) − 1

2(3 − s2).

We show that the assumptions of Theorem 2.3 are satisfied for M = 38 , κ = 1

2 , R = 1, r = 14 , t0 = 1, a = 1,

β = 0.987, g(t) = 13

(t + 1

2

)and h(u) = u2 − 4u + 1.

For M = 38 , G(t, s) ≥ 0 and we have that (H2) holds.

Since max{KA (s) : s ∈ [0, 1]} = 43 we get

κM max{KA (s) : s ∈ [0, 1]} =14,

and (H1) holds.It is easy to check that condition (H3) is satisfied.Since f(t, u) + 1

2 u is positive on [0, 1] × [0, 1], (H4) is satisfied.On the other hand, (H5) holds because f(t, 1) = − 2

3

(t + 1

2

).

Finally, all the assumptions for h and g in (H6) are satisfied and, by substitution, one obtains

h(r)r

∫ 1

0G(t0 , s)g(s) ds =

112

∫ 1

0

(18(4 − s3) + 1 − 1

2(3 − s2))(

s +12

)ds

=161

11520≥ 13

987=

1 − β

βa.

Acknowledgements D. Franco was partially supported by MEC (Spain) and FEDER, grant MTM2007-60679. A supportfrom TODEQ, project number MTK-CT-2005-030042, is gratefully acknowledged.

References

[1] R. P. Agarwal, D. O’Regan, and S. Stanek, Positive solutions of nonlocal singular boundary value problems, Glasg.Math. J. 46, 537–550 (2004).

[2] Y. An, Existence of solutions for a three-point boundary value problem at resonance, Nonlinear Anal. 65, 1633–1643(2006).

[3] C. Bai and J. Fang, Existence of positive solutions for three-point boundary value problems at resonance, J. Math. Anal.Appl. 291, 538–549 (2004).

[4] C. T. Cremins, A fixed-point index and existence theorems for semilinear equations in cones, Nonlinear Anal. 46, 789–806 (2001).

[5] C. T. Cremins, Existence theorems for semilinear equations in cones, J. Math. Anal. Appl. 265, 447–457 (2002).[6] C. T. Cremins, Existence theorems for weakly inward semilinear operators, Discrete Contin. Dyn. Syst. Suppl. 200–205

(2003).



[7] Z. Du, X. Lin, and W. Ge, Nonlocal boundary value problem of higher order ordinary differential equations at resonance,Rocky Mt J. Math. 36, 1471–1486 (2006).

[8] W. Feng and J. R. L. Webb, Solvability of three-point boundary value problems at resonance, Nonlinear Anal. 30,3227–3238 (1997).

[9] W. Feng and J. R. L. Webb, Solvability of m-point boundary value problems with nonlinear growth, J. Math. Anal.Appl. 212, 467–480 (1997).

[10] J. R. Graef and B. Yang, Positive solutions of a third order nonlocal boundary value problem, Discrete Contin. Dyn.Syst. Ser. S 1, 89–97 (2008).

[11] J. R. Graef and B. Yang, Positive solutions to a multi-point higher order boundary value problem, J. Math. Anal. Appl.316, 409–421 (2006).

[12] C. P. Gupta, Existence theorems for a second order m-point boundary value problem at resonance, Int. J. Math. Math.Sci. 18, 705–710 (1995).

[13] X. Han, Positive solutions for a three-point boundary value problem at resonance, J. Math. Anal. Appl. 336, 556–568(2007).

[14] G. Infante and J. R. L. Webb, Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equa-tions, Proc. Edinb. Math. Soc. 49, 637–656 (2006).

[15] G. Infante and M. Zima, Positive solutions of multi-point boundary value problems at resonance, Nonlinear Anal. 69,2458–2465 (2008).

[16] G. L. Karakostas and P. C. Tsamatos, Existence of multiple positive solutions for a nonlocal boundary value problem,Topol. Methods Nonlinear Anal. 19, 109–121 (2002).

[17] E. R. Kaufmann and N. Kosmatov, A singular three-point boundary value problem, Commun. Appl. Anal. 9, 429–441(2005).

[18] N. Kosmatov, A multi-point boundary value problem with two critical conditions, Nonlinear Anal. 65, 622–633 (2006).[19] N. Kosmatov, A symmetric solution of a multipoint boundary value problem at resonance, Abstr. Appl. Anal. 2006, Art.

ID 54121, 11 pp.[20] N. Kosmatov, Multi-point boundary value problems on an unbounded domain at resonance, Nonlinear Anal. 68, 2158–

2171 (2008).[21] K. Q. Lan, Properties of kernels and multiple positive solutions for three-point boundary value problems, Appl. Math.

Lett. 20, 352–357 (2007).[22] B. Liu, Solvability of multi-point boundary value problem at resonance. IV, Appl. Math. Comput. 143, 275–299 (2003).[23] B. Liu and Z. Zhao, A note on multi-point boundary value problems, Nonlinear Anal. 67, 2680–2689 (2007).[24] R. Ma, Positive Solutions of a Nonlinear Three-point Boundary-value Problem, Electronic Journal of Differential Equa-

tions Vol. 34 (Texas State University-San Marcos, Dept. Math., San Marcos, TX, 1999).[25] R. Ma, Multiplicity results for an m-point boundary value problem at resonance, Indian J. Math. 47, 15–31 (2005).[26] R. Ma, A survey on nonlocal boundary value problems, Appl. Math. E-Notes 7, 257–279 (2007).[27] J. Mawhin, Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings

in locally convex topological vector spaces, J. Differ. Equations, 12, 610–636 (1972).[28] D. O’Regan and M. Zima, Leggett-Williams norm-type theorems for coincidences, Arch. Math. 87, 233–244 (2006).[29] P. K. Palamides, Multi Point Boundary-value Problems at Resonance for n-order Differential Equations: Positive and

Monotone Solutions, Electronic Journal of Differential Equations Vol. 25 (Texas State University-San Marcos, Dept.Math., San Marcos, TX, 2004).

[30] P. K. Palamides and A. P. Palamides, A fundamentally simple approach to a singular boundary value problem, NonlinearAnal. 68, 3397–3409 (2008).

[31] W. V. Petryshyn, On the solvability of x ∈ Tx + λFx in quasinormal cones with T and F k-set contractive, NonlinearAnal. 5, 585–591 (1981).

[32] B. Przeradzki and R. Stanczy, Solvability of a multi-point boundary value problem at resonance, J. Math. Anal. Appl.264, 253–261 (2001).

[33] J. Santanilla, Some coincidence theorems in wedges, cones, and convex sets, J. Math. Anal. Appl. 105, 357–371 (1985).[34] J. R. L. Webb, Solutions of Semilinear Equations in Cones and Wedges, in: Proceedings of the World Congress of

Nonlinear Analysts ’92, Vol. I, Tampa, FL, 1992 (Walter de Gruyter & Co., Hawthorne, NI, USA, 1995), pp. 137–147.[35] J. R. L. Webb, Fixed point index and its application to positive solutions of nonlocal boundary value problems, in:

Seminar of Mathematical Analysis (Univ. Sevilla Secr. Publ., Seville, 2006), pp. 181–205.[36] J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems involving integral conditions,

NoDEA Nonlinear Differ. Equ. Appl. 15, 45–67 (2008).[37] J. R. L. Webb and K. Q. Lan, Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value

problems of local and nonlocal type, Topol. Methods Nonlinear Anal. 27, 91–115 (2006).[38] J. R. L. Webb, G. Infante, and D. Franco, Positive solutions of nonlinear fourth-order boundary-value problems with

local and non-local boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 138, 427–446 (2008).[39] J. R. L. Webb and M. Zima, Multiple positive solutions of resonant and non-resonant nonlocal boundary value problems,

Nonlinear Anal. 71, 1369–1378 (2009).[40] M. Zima, Fixed point theorem of Leggett-Williams type and its application, J. Math. Anal. Appl. 299, 254–260 (2004).


Documents

Second order nonlocal boundary value problems at resonance