COERCIVITY OF LINEAR FUNCTIONALS ON FINITE ...

COERCIVITY OF LINEAR FUNCTIONALS ON FINITE DIMENSIONAL SPACES AND ITS APPLICATION TO DISCRETE BVPS CHRISTOPHER S. GOODRICH Abstract. We consider linear functionals having the form y 7→

n X

αi y (ξi ) ,

i=1

for αi ∈ R and ξi ∈ [1, b + 1]N0 for some integer b > 0, which can occur as nonlocal boundary elements in a discrete boundary value problem. We demonstrate that when y is restricted to a particular cone, it follows that this functional satisfies a coercivity condition, which is readily computable in terms of the Green’s function associated to the boundary value problem. Finally, we prove that the coercivity constant we construct by means of our new cone is superior to an approach utilizing a Harnack-like inequality, and we illustrate this claim by means of some examples.

1. Introduction In this paper we consider the summation equation, for t ∈ [0, b + 2]N0 := {0, 1, 2, . . . , b + 2} with b ≥ 1 an integer, y(t) = γ1 (t)H1

n X i=1

! αi y (ξi )

+ γ2 (t)H2

m X i=1

! βi y (ζi )

+λ

b X

G(t, s)f (s, y(s + 1)).

(1.1)

s=0

Later in Section 2 we impose some conditions on the various functions and constants appearing in (1.1), but let us just mention for now that the functions involved are assumed to be continuous, and we also wish to emphasize that while H1 and H2 could be nonlinear maps, they need not be and, in fact, our results can accommodate piecewise linear and linear maps, as explained later. One interest in an abstract summation equation such as (1.1) is that solutions of this equation can be associated to solutions of various boundary value problems (BVPs) by selecting the maps 2010 Mathematics Subject Classification. Primary: 26D15, 39A12, 39A70. Secondary: 47H07, 47H11. Key words and phrases. Summation equation; coercivity; discrete calculus; positive solution; nonlocal boundary value problem. 1

2

C. S. GOODRICH

t 7→ γ1 (t), γ2 (t) and (t, s) 7→ G(t, s) in particular ways. Thus, (1.1) provides a convenient way to study existence of solution of discrete BVPs. For example, if we put b+2−t t and γ2 (t) := b+2 b+2    t(b+1−s) , t≤s b+2 G(t, s) := ,   (s+1)(b+2−t) , t ≥ s + 1 b+2 γ1 (t) :=

(1.2)

where G : [0, b + 2]N0 × [0, b]N0 → [0, +∞), then it is a simple matter (see Erbe and Peterson [3, 4]) to argue that a solution of (1.1) is, in this case, a solution of the nonlocal BVP −∆2 y(t) = f (t, y(t + 1)), t ∈ Nb0 := {0, 1, . . . , b} ! ! n m X X y(0) = H1 αi y (ξi ) , y(b + 2) = H2 βi y (ζi ) , i=1

(1.3)

i=1

n m for {ξi }ni=1 , {ζi }m i=1 ⊆ [1, b + 1]N and {αi }i=1 , {βi }i=1 ⊆ R. But, due to the generality with which

we study (1.1), many different discrete BVPs can be recovered from summation equation (1.1). We wish to emphasize that, as will be seen later, the nonlocal conditions in (1.3) can, in some cases, be either linear or piecewise linear ; they do not necessarily need to be nonlinear. A key contribution of this article is to use a novel cone, K, to demonstrate that when restricted n m X X to K the linear functionals y 7→ αi y (ξi ) and y 7→ βi y (ζi ) are coercive. Of particular note i=1

i=1

is that the coercivity constants thus obtained are particularly useful in the context of a BVP such as (1.3) and, as we shall demonstrate, provide a superior methodology relative to some other techniques that have been utilized in the context of BVPs with nonlocal boundary conditions of the form in (1.3), for example. Roughly speaking, the cone that we introduce and utilize in this work is ( K :=

y ∈ B : y(t) ≥ 0,

n X

αi y (ξi ) ≥ min

s∈[0,b]N0

i=1 m X i=1

! n 1 X αi G (ξi , s) kyk, G(s) i=1

βi y (ζi ) ≥ min

s∈[0,b]N0

! ) m 1 X βi G (ζi , s) kyk G(s) i=1

(1.4)

COERCIVITY AND DISCRETE BOUNDARY VALUE PROBLEMS

3

where B ∼ = Rb+3 is the collection of maps {y : [0, b + 2]N0 → R} and, for each fixed s ∈ [0, b]N0 , G(s) :=

max

G(t, s).

(1.5)

t∈[0,b+2]N0

Technically, in (1.4) it may be the case that the minimum must be taken over a set slightly different than [0, b]N0 ; the exact conditions are specified in Section 2. One of the reasons for desiring some sort of coercivity condition is so that we have some hope of imposing asymptotic growth conditions on H1 and H2 , conditions which require that we can control the size of H1 and H2 in terms of the magnitude of their arguments. Along these lines, an alternative approach, which was originated by the author in the setting of differential equations [5] and then subsequently developed in the continuous setting (see, for example, [7, 8, 10]) is to replace the coercivity conditions in (1.4) with the Harnack-like inequality y(t) ≥ γkyk,

min

(1.6)

t∈[m1 ,m2 ]N

for some integers 0 < m1 ≤ m2 ≤ b + 1 and γ ∈ (0, 1], where γ is known from initial data. Then n [ a coercivity condition can be obtained by assuming, for example, that [m1 , m2 ]N ⊇ {ξi }, and i=1

writing n X i=1

" αi y (ξi ) =

n X

# (αi − εi ) y (ξi ) +

| i=1

{z

:=ϕ1 (y)

}

n X

εi y (ξi ),

|i=1 {z

:=ϕ2 (y)

(1.7)

}

for some sequence {εi }ni=1 ⊆ [0, +∞). If it can be shown that n X

(αi − εi ) G (ξi , s) ≥ 0,

(1.8)

i=1

for each s ∈ [0, b]N0 , then the condition ϕ1 (y) ≥ 0 can be incorporated into a cone of the form K0 := y ∈ B : y(t) ≥ 0, min y(t) ≥ γkyk, ϕ1 (y) ≥ 0 , and, furthermore, we get t∈[m1 ,m2 ]

from (1.6) and (1.8) the coercivity condition n X i=1

αi y (ξi ) = ϕ1 (y) + ϕ2 (y) ≥ ϕ2 (y) ≥

γ

n X i=1

! εi

kyk,

(1.9)

4

C. S. GOODRICH

for y ∈ K0 . Then (1.7)–(1.9) provide a coercivity condition and ensure that the functional is nonnegative and coercive when restricted to K0 . The key fact that we show in this paper is that n

n

i=1

i=1

X 1 X C0 := min αi G (ξi , s) ≥ γ εi =: C00 , s∈[0,b]N0 G(s)

(1.10)

C0 can be very large depending upon the value of C00 n b and the nonlocal evaluation sequences, {ξi }i=1 and {ζi }m i=1 . In fact, as we show by means of and that the ratio of coercivity constants

examples, the constant C0 in (1.10) can be several orders of magnitude greater than the constant C00 in (1.10). Thus, the method we introduce here can induce a substantial change in the strength of the coercivity constant, as we shall show in Section 3 – see Examples 3.6 and 3.8. Within this context, while it may not be apparent why a larger coercivity constant would be preferred, we shall demonstrate in Section 3 that this augmentation can be very useful in the applicability of certain existence theorems for problems like (1.1) and (1.3). In fact, the difference can be extreme – see, for example, Example 3.8. In addition, we wish to emphasize that by using the new cone K, we never need to identify a Harnack-like inequality of the form (1.6) and, thus, never need to identify or utilize an m1 , m2 , and decomposition procedure as in (1.6)–(1.9). Thus, this approach is more elegant and mathematically clean, we believe. To conclude this introduction, we briefly mention that there is a substantial literature on the study of nonlocal boundary problems. In particular, the study of nonlocal boundary value problems with linear, affine, or nonlinear boundary conditions has been considerable in the last several years – see, for example, Anderson [1], Cabada, et al. [2], Goodrich [6, 9], Infante, et al. [12, 13, 14], Jankowski [15], Karakostas [16], Webb and Infante [17], and Yang [18, 19]. Finally, within the context of boundary value problems, we wish to emphasize that an unusual aspect of our results, indeed facilitated by the coercivity of the linear functionals in (1.1), is that we provide existence theorems for (1.1) (and thus BVPs) in the case where λ is unrestricted (i.e., it does not need to be small), and even in the case where f has no growth condition imposed (see Corollary 3.2). Thus, we believe this to be an interesting consequence of the coercivity conditions introduced in this work since they somewhat naturally produce a growth condition even when H1 and H2 are not necessarily nonlinear maps.


5

2. Preliminaries We begin by stating some notations that are used throughout. Notation 2.1. Henceforth, we utilize the following notational conventions. • Given a continuous function f : X ×[0, +∞) → [0, +∞), with X ⊆ R some finite set, for m and feM , respectively, the quantities real numbers 0 ≤ a < b ≤ +∞ we denote by fe[a,b] [a,b] m := fe[a,b]

min (t,y)∈X×[a,b]

M := f (t, y) and fe[a,b]

max

f (t, y).

(t,y)∈X×[a,b]

• As already mentioned in Section 1, given integers r0 ≤ r1 , we denote, respectively, by Nr0 and Nrr10 the nonempty sets Nr0 := {r0 , r0 + 1, r0 + 2, . . . } and Nrr10 := {r0 , r0 + 1, . . . , r1 }. Given an interval [a, b] such that −∞ ≤ a < b ≤ +∞ by [a, b]Nr0 we denote the set [a, b]Nr0 := [a, b] ∩ Nr0 . • For ρ > 0, we put Ωρ := {y ∈ K : kyk < ρ}, where throughout this work, k · k denotes the usual max norm on the space B, which was defined in Section 1. We next list the assumptions that we impose on problem (1.1). A1: The map G : [0, b + 2]N0 × [0, b]N0 → [0, +∞) satisfies G(s) < +∞ for each s ∈ [0, b]N0 , where s 7→ G(s) was defined in (1.5). A2: Assume that (1) f : [0, b]N0 × [0, +∞) → [0, +∞) is continuous and satisfies f (t, y) = 0, y→+∞ y lim

uniformly for t ∈ [0, b]N0 ; (2) H1 , H2 : [0, +∞) → [0, +∞) are continuous; and (3) γ1 , γ2 : [0, b + 2]N0 → [0, +∞) are continuous. A3: There exist numbers A1 , A2 , B1 , B2 ∈ (0, +∞) such that lim

z→0+

Hi (z) > Ai and z

lim

z→+∞

Hi (z) < Bi , for each i = 1, 2. z

A4: Define the set S0 ⊆ [0, b]N0 by S0 := {s ∈ [0, b]N0 : G(s) 6= 0} .

6

C. S. GOODRICH m n Then with {ξi }ni=1 , {ζi }m i=1 ⊆ [1, b + 1]N and {αi }i=1 , {βi }i=1 ⊆ R, we assume that the

quantities n

1 X C0 := min αi G (ξi , s) s∈S0 G(s) i=1

and

m

D0 := min s∈S0

1 X βi G (ζi , s) G(s) i=1

satisfy C0 , D0 ∈ (0, +∞). A5: For each j = 1, 2 it holds that n X

αi γj (ξi ) ≥ C0 kγj k and

i=1

m X

βi γj (ζi ) ≥ D0 kγj k.

i=1

Remark 2.2. Due to (A4) the results here potentially can accommodate vanishing kernels. Note, also, that the numbers αi and βi can be negative so long as conditions (A4)–(A5) are satisfied – see, for instance, Example 3.6. Remark 2.3. As the examples demonstrate each of these conditions is easy to check and arises quite naturally in discrete BVPs. Note that, as in (A1), we require the kernel, G, to be nonnegative. Thus, we do not treat here the possibility of a sign-changing kernel; we leave this possibility for future work. The cone that we use in this work, as suggested in Section 1, is ! ( n n X 1 X αi G (ξi , s) kyk, K := y ∈ B : y(t) ≥ 0, αi y (ξi ) ≥ min s∈S0 G(s) i=1

i=1

m X

βi y (ζi ) ≥ min s∈S0

i=1

! ) m 1 X βi G (ζi , s) kyk , G(s)

(2.1)

i=1

which is nearly (1.4) save but for the slight change in the sets over which the minima are taken – see the proof of Lemma 2.4. Note K is a cone and that K 6= ∅ nor is it trivial; this is due to condition (A5), seeing as γ1 , γ2 ∈ K and that we assume in Section 3 the existence of an index i0 ∈ {1, 2} such that kγi0 k = 6 0. With this in hand, we define the operator T : B → B by ! ! n m b X X X (T y)(t) := γ1 (t)H1 αi y (ξi ) + γ2 (t)H2 βi y (ζi ) + λ G(t, s)f (s, y(s + 1)). (2.2) i=1

i=1

s=0


7

We show next that T (K) ⊆ K, the key point being the verification of the coercivity conditions imbedded in (2.1).

Lemma 2.4. Let T be defined as in (2.2). If conditions (A1) and (A4)–(A5) hold, then T (K) ⊆ K.

Proof. It is trivial that (T y)(t) ≥ 0, for each t ∈ [0, b + 2]N0 , whenever y ∈ K. On the other hand, for arbitrary but fixed y ∈ K we notice that

n X

αi (T y) (ξi ) ≥ C0 kγ1 kH1

i=1

n X

m X

! + C0 kγ2 kH2

αi y (ξi )

i=1

! βi y (ζi )

i=1

+λ

b X n X

αi G (ξi , s) f (s, y(s + 1))

s=0 i=1

= C0 kγ1 kH1

n X

m X

! + C0 kγ2 kH2

αi y (ξi )

i=1

!

(2.3)

βi y (ζi )

i=1

+λ

n XX

αi G (ξi , s) f (s, y(s + 1)),

s∈S0 i=1

where to obtain the inequality we have used the fact that by means of condition (A5) it holds that γ1 , γ2 ∈ K, whereas to obtain the equality we have used that if s0 ∈ / S0 , then G (t, s0 ) = 0, for each t ∈ [0, b + 2]N2 . Then since by condition (A4) we have # n 1 X αi G (ξi , s) G(s)f (s, y(s + 1)) αi G (ξi , s) f (s, y(s + 1)) = G(s) i=1 i=1 s∈S0 " # n X 1 X ≥ min αi G (ξi , s) G(s)f (s, y(s + 1)) s∈S0 G(s) i=1 s∈S0 | {z }

n XX s∈S0

"

X

=C0

= C0

b X s=0

G(s)f (s, y(s + 1)),

(2.4)

8

C. S. GOODRICH

it follows upon putting (2.4) into (2.3) that " ! n n X X αi y (ξi ) + kγ2 kH2 αi (T y) (ξi ) ≥ C0 kγ1 kH1 i=1

i=1

m X

! βi y (ζi )

i=1

+λ

b X

# G(s)f (s, y(s + 1))

s=0

≥ C0 kT yk, which proves that T y satisfies the coercivity condition with respect to the first functional in K. By essentially repeating the above steps but with respect to the second functional, we conclude that T y ∈ K, which completes the proof of the lemma.

We next state a simple technical lemma that will be used in the proof of Theorem 3.1. Since its proof is sufficiently similar to that of [10, Lemma 3.2], we omit the proof for the sake of brevity. M Lemma 2.5. Assume that f : Nb0 × [0, +∞) → [0, +∞) satisfies condition (A2.1), and let fe[0,ρ] M fe[0,ρ] be defined as in Notation 2.1. Then it holds that lim = 0. ρ→+∞ ρ

Finally, we recall the index theoretic result that we utilize in this paper. Further details may be found in either Guo and Lakshmikantham [11] or Zeidler [20]. Lemma 2.6. Let D be a bounded open set and, with K a cone in a Banach space X, suppose 6 K. Let D1 be open in X with D1 ⊆ D ∩ K. Assume that both that D ∩ K = 6 ∅ and that D ∩ K = T : D ∩ K → K is a compact map such that T x 6= x for x ∈ K ∩ ∂D. If iK (T, D ∩ K) = 1 and iK (T, D1 ∩ K) = 0, then T has a fixed point in (D ∩ K) \ D1 ∩ K . Moreover, the same result holds if iK (T, D ∩ K) = 0 and iK (T, D1 ∩ K) = 1. 3. Main Results and Discussion We begin this section by presenting a representative existence result for problem (1.1). We then demonstrate that the coercivity constants obtained by way of our new cone are superior to the Harnack inequality approach described in Section 1. Finally, we conclude by providing some brief examples to illustrate concretely the improvement that can be obtained by means of the new cone.


9

Theorem 3.1. Fix λ ∈ (0, +∞) and assume that conditions (A1)–(A5) hold. Suppose, in addition, both that γ1 (t0 ) A1 C0 + γ2 (t0 ) A2 D0 > 1 for some t0 ∈ [0, b + 2]N0

(3.1)

and that C0 > kγ1 kB1

n X

!2 |αi |

n X

+ kγ2 kB2

i=1

! |αi |

i=1

m X

! |βi | .

(3.2)

i=1

Then there exists at least one positive solution to problem (1.1). Proof. We first show that for ρ1 > 0 sufficiently small, it holds that iK (T, Ωρ1 ) = 0. To this end, we notice that by condition (A3) there exists ρ∗1 > 0 such that Hi (z) > Ai z for z ∈ (0, ρ∗1 ). Thus, putting M0 := max

( n X

|αi | ,

i=1

m X

) |βi |

> 0,

i=1

we see that if y ∈ Ω ρ∗1 , then M0

H1

n X

! αi y (ξi )

> A1

i=1

n X

αi y (ξi )

i=1

and H2

m X i=1

! βi y (ζi )

> A2

m X

βi y (ζi ) .

i=1

ρ∗1

. By condition (3.1) there is j0 ∈ {1, 2} such that kγj0 k > 0, and by condition M0 (A5) we have that γj0 ∈ K. We now show that y 6= T y + µe for all µ ≥ 0 and y ∈ ∂Ωρ1 , Put ρ1 :=

where e(t) := γj0 (t). For contradiction, assume that there exists y ∈ ∂Ωρ1 and µ ≥ 0 such that y = T y + µe. Then, in light of the coercivity conditions as well as inequality (3.1), we obtain y(t) ≥ (T y)(t) ≥ [γ1 (t)A1 C0 + γ2 (t)A2 D0 ] kyk, whereupon setting t = t0 , we obtain y (t0 ) > kyk, which is a contradiction. Thus, iK (T, Ωρ1 ) = 0. On the other hand, we show that for some ρ2 > ρ1 we have iK (T, Ωρ2 ) = 1. So, by condition ρ∗2 (A3) there exists ρ∗2 > 0 such that Hi (z) < Bi z whenever z ≥ ρ∗2 . Put ρ2 := min {C0 , D0 }

10

C. S. GOODRICH

with ρ2 selected sufficiently large such that ρ2 > ρ1 . By the coercivity conditions, we see that if kyk ≥ ρ2 , then min

( n X

αi y (ξi ) ,

m X

i=1

) ≥ ρ∗2

βi y (ζi )

i=1

so that n X

H1

n X

! αi y (ξi )

< B1

i=1

αi y (ξi )

i=1

and H1

m X

! βi y (ζi )

< B2

m X

i=1

βi y (ζi ) .

i=1

We now show that µy 6= T y for each µ ≥ 1 and y ∈ ∂Ωρ2 . Thus, for contradiction assume that there exists y ∈ ∂Ωρ2 and µ ≥ 1 such that µy = T y. Noticing then that µαi y (ξi ) = αi (T y) (ξi ), for i ∈ Nn1 , and then summing both sides over i, using the coercivity condition, and recalling that µ ≥ 1 yields 

n X

C0 kyk ≤ kγ1 kB1

!2 |αi |

n X

+ kγ2 kB2

i=1

m X

! |αi |

i=1

! |βi |  kyk

i=1 M + λfe[0,ρ 2]

n X b X

αi G (ξi , s) ,

i=1 s=0

where we have used condition (A4) to deduce that

n X

αi G (ξi , s) ≥ 0 for each s. Hence,

i=1

C0 ≤ kγ1 kB1

n X

!2 |αi |

+ kγ2 kB2

i=1

n X

! |αi |

i=1

m X

! |βi |

+

M b X n X λfe[0,ρ 2]

i=1

ρ2

αi G (ξi , s) .

Now, by inequality (3.2), there exists ε > 0 such that !2 ! m ! n n X X X C0 > kγ1 kB1 |αi | + kγ2 kB2 |αi | |βi | + ε. i=1

i=1

(3.3)

s=0 i=1

(3.4)

i=1

Since condition (A2.1) holds, we have by Lemma 2.5 that for ρ2 sufficiently large, M fe[0,ρ 2]

ρ2

0 such that

C0 > kγ1 kB1

n X i=1

!2 |αi |

+ kγ2 kB2

n X

m X

! |αi |

i=1

! |βi |

+

M b X n X λfe[0,ρ 2]

i=1

ρ2

αi G (ξi , s) .

(3.6)

s=0 i=1

Then problem (1.1) has at least one positive solution. Remark 3.3. Observe that in each of Theorem 3.1 and Corollary 3.2 we see that the larger the value of C0 , the more flexibility in each of conditions (3.1), (3.2), and (3.6). This is one of the reasons, as shown in the examples, that we seek a “stronger” coercivity constant. Recall from Section 1 that in light of (1.7)–(1.9) the coercivity constant we obtain in the old, n X Harnack inequality approach is the number γ εi . In our next result, we demonstrate that γ

n X

i=1

εi is always a lower bound for the new coercivity constant C0 (or D0 , as the case may be).

i=1

In fact, as the examples momentarily shall show, C0 can, in fact, be much larger. Theorem 3.4. Consider the functional y 7→

n X

αi y (ξi ), and assume that conditions (A1) and

i=1

(A4)–(A5) hold. Assume that there is γ ∈ (0, 1] and integers 0 < m1 ≤ m2 ≤ b + 1 such that n [ mint∈[m1 ,m2 ]N G(t, s) ≥ γG(s), for each s ∈ [0, b]N0 , where [m1 , m2 ]N ⊇ {ξi }. Finally, assume that there is

{εi }ni=1

⊆ [0, +∞) such that

n X

i=1

(αi − εi ) G (ξi , s) ≥ 0, for each s ∈ [0, b]N0 . Then

i=1 n

n

i=1

i=1

X 1 X min αi G (ξi , s) ≥ γ εi . s∈S0 G(s)

12

C. S. GOODRICH

Proof. Essentially we proceed as in (1.7)–(1.9) from the introduction. Indeed, under the hypotheses of this theorem we have, similar to (1.7)–(1.9) in Section 1, that ! " n # n n n X X X X αi G (ξi , s) = εi G(s). (αi − εi ) G (ξi , s) + εi G (ξi , s) ≥ γ | {z } i=1 i=1 i=1 i=1 ≥γG(s) {z } |

(3.7)

≥0

Letting s ∈ S0 and dividing both sides of (3.7) by G(s) we obtain that min

s∈S0

n

n

i=1

i=1

X 1 X αi G (ξi , s) ≥ γ εi , G(s)

and this completes the proof.

Remark 3.5. Note that in the statement of Theorem 3.4 it need only hold that [m1 , m2 ]N0 contains the point ξi if and only if εi 6= 0. To conclude this paper, we now provide a couple of examples. In each example we utilize the maps in (1.2). While Theorem 3.1 applies to a wide collection of BVPs due to the general nature of summation equation (1.1), the conjugate problem (1.3) with nonlocal boundary conditions is an important special case, and so to keep the paper succinct we focus on this one case. Example 3.6. Suppose that b = 100. Further suppose that (with n = m = 2) n X i=1 m X i=1

1 1 αi y (ξi ) = y(10) − y(3) 2 4 (3.8) 8 1 βi y (ζi ) = y(8) − y(5). 3 25

Then in light of the formula for the Green’s function G in (1.2) it  85   4(101−s) ,    1 1 1 17(11s−7) G(10, s) − G(3, s) = 4(s+1)(101−s) ,  G(s) 2 4     17 , 4(s+1)

can be shown that s ∈ N20 s ∈ N93

,

s ∈ N100 10

(s + 1)(101 − s) 17 , from which we easily calculate C0 := ; 102 404 22 note that here S0 = N100 . 0 . A similar calculation reveals that D0 = 7575 using the fact (see [3]) that G(s) :=


13

If we now select

H1 (z) :=

  25z,  25 +

H2 (z) := then we may put A1 := 24, A2 :=

0≤z≤1 1 60 (z

− 1),

z>1

1 z, 20

1 1 1 , B1 := , and B2 := . Inequality (3.1) holds since 21 59 19

γ1 (0) A1 C0 + γ2 (0) A2 D0 =

160672 > 1. 159075

(3.9)

Likewise, inequality (3.2) holds since kγ1 kB1

n X

!2 |αi |

+ kγ2 kB2

i=1

n X

! |αi |

i=1

m X

! |βi |

i=1

=

15839 < C0 . 448400

(3.10)

Therefore, due to (3.9)–(3.10) we conclude that given any λ ∈ (0, +∞) • if f verifies condition (A2.1), then Theorem 3.1 implies the existence of at least one positive solution to problem (1.1); whereas • if, instead, inequality (3.6) holds for some ρ2 > 0, then Corollary 3.2 implies the existence of at least one positive solution to problem (1.1). Moreover, in each case by the correspondence identified in Section 1, we obtain a positive solution to the difference equation −∆2 y(t) = f (t, y(t + 1)), t ∈ N100 0 , equipped with the boundary conditions

y(0) =

   25 y(10) − 2

  y(102) =

1 120 y(10)

25 4 y(3),

−

1 240 y(3)

+

1499 60 ,

1 2 y(10)

− 41 y(3) ∈ [0, 1]

1 2 y(10)

− 41 y(3) ∈ (1, +∞)

1 2 y(8) − y(5). 60 125

Finally, we compare the preceding calculations with what happens in the Harnack inequality approach. In this case, for example, we could write, for some ε > 0, 1 1 1 1 y(10) − y(3) = − ε y(10) − y(3) + εy(10). 2 4 2 4

14

C. S. GOODRICH

Since, per the discussion in Section 1, particularly inequality (1.8), we must have 1 1 − ε G(10, s) − G(3, s) ≥ 0, 2 4 for each s ∈ [0, 100]N0 , it can be shown that ε
0. We will show briefly that for certain regions of the 2 C0 parameter space determined by the 4-tuple (a1 , a2 , ξ1 , ξ2 ), the ratio 0 may be extremely large; C0 here we denote by C00 the coercivity constant obtained from using a Harnack-like inequality. where 0 < ξ2 < ξ1 ≤

To see that this claim holds, straightforward calculations reveal that for (3.11) we have ( 1 1 a1 (b + 2 − ξ1 ) − a2 (b + 2 − ξ2 ) , [a1 ξ1 − a2 ξ2 ] , C0 := min b+1 b+1 1 a1 (ξ2 + 1) (b + 2 − ξ1 ) − a2 ξ2 (b + 1 − ξ2 ) (ξ2 + 1) (b + 1 − ξ2 )

)

and, again using calculations similar to the second-to-last paragraph of Example 3.6, that ( 1 ξ1 1 0 C0 < min a1 (b + 2 − ξ1 ) − a2 (b + 2 − ξ2 ) , [a1 ξ1 − a2 ξ2 ] , b+1 b + 2 − ξ1 ξ1 ) 1 a1 (ξ2 + 1) (b + 2 − ξ1 ) − a2 ξ2 (b + 1 − ξ2 ) . (ξ2 + 1) (b + 2 − ξ1 ) Thus, for example, in the subset of the parameter space for which the minimum in each case is the first quantity, we obtain that C0 := C00

C0 C00

(ξ1 , b) >

b + 2 − ξ1 . ξ1

For fixed ξ1 , we thus see that lim

b→+∞

C0 = +∞. C00

This means that for b large the improvement afforded by C0 can be considerable, as we have already seen in Example 3.6. Finally, as one may easily show, C0 > C00 may hold in other regions of the parameter space, too. Acknowledgements. The author would like to thank the two anonymous referees for their useful comments, which improved the presentation of the manuscript. References [1] D. R. Anderson, Existence of three solutions for a first-order problem with nonlinear nonlocal boundary conditions, J. Math. Anal. Appl. 408 (2013), 318–323

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[2] A. Cabada, J. A. Cid, G. Infante, A positive fixed point theorem with applications to systems of Hammerstein integral equations, Bound. Value Probl. (2014), 10 pp. [3] L. Erbe, A. Peterson, Positive solutions for a nonlinear differential equation on a measure chain, Math. Comput. Modelling 32 (2000), 571–585 [4] L. Erbe, A. Peterson, Eigenvalue conditions and positive solutions, J. Difference Equ. Appl. 6 (2000), 165–191 [5] C. S. Goodrich, On nonlocal BVPs with boundary conditions with asymptotically sublinear or superlinear growth, Math. Nachr. 285 (2012), 1404–1421 [6] C. S. Goodrich, On semipositone discrete fractional boundary value problems with nonlocal boundary conditions, J. Difference Equ. Appl. 19 (2013), 1758–1780 [7] C. S. Goodrich, Semipositone boundary value problems with nonlocal, nonlinear boundary conditions, Adv. Differential Equations 20 (2015), 117–142 [8] C. S. Goodrich, Coupled systems of boundary value problems with nonlocal boundary conditions, Appl. Math. Lett. 41 (2015), 17–22 [9] C. S. Goodrich, Systems of discrete fractional boundary value problems with nonlinearities satisfying no growth conditions, J. Difference Equ. Appl. 21 (2015), 437–453 [10] C. S. Goodrich, On nonlinear boundary conditions involving decomposable linear functionals, Proc. Edinb. Math. Soc. (2) 58 (2015), 421–439 [11] D. Guo, V. Lakshmikantham, Nonlinear problems in abstract cones, Academic Press, Boston, 1988 [12] G. Infante, Nonlocal boundary value problems with two nonlinear boundary conditions, Commun. Appl. Anal. 12 (2008), 279–288 [13] G. Infante, P. Pietramala, M. Tenuta, Existence and localization of positive solutions for a nonlocal BVP arising in chemical reactor theory, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 2245–2251 [14] G. Infante, P. Pietramala, Multiple nonnegative solutions of systems with coupled nonlinear boundary conditions, Math. Methods Appl. Sci. 37 (2014), 2080–2090 [15] T. Jankowski, Positive solutions to fractional differential equations involving Stieltjes integral conditions, Appl. Math. Comput. 241 (2014), 200–213 [16] G. L. Karakostas, Existence of solutions for an n-dimensional operator equation and applications to BVPs, Electron. J. Differential Equations (2014), No. 71, 17 pp. [17] J. R. L. Webb, G. Infante, Positive solutions of nonlocal boundary value problems: a unified approach, J. Lond. Math. Soc. (2) 74 (2006), 673–693 [18] Z. Yang, Positive solutions to a system of second-order nonlocal boundary value problems, Nonlinear Anal. 62 (2005), 1251–1265 [19] Z. Yang, Positive solutions of a second-order integral boundary value problem, J. Math. Anal. Appl. 321 (2006), 751–765


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[20] E. Zeidler, Nonlinear Functional Analysis and Its Applications, I: Fixed-Point Theorems, Springer, New York, 1986

Department of Mathematics, Creighton Preparatory School, Omaha, NE 68114 USA E-mail address, Christopher S. Goodrich: [email protected]