Solvability criteria for second order generalized Sturm--Liouville problems at resonance.

1 Introduction

Consider the second order nonlinear equation

(p(t)u'(t))0--q(t)u(t) = f (t, [[integral].sup.t.sub.0] u(s)ds, u'(t)), t 2 (0, 1), (1.1)

subject to one of the following boundary conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

It is known [8] that the solutions of (1.1) with the m-point boundary conditions (1.2), (1.3) and (1.4) can be obtained via existence subject to the respective three-point boundary conditions

au(0)--bp(0)u'(0) = 0, cu(1) + dp(1)u'(1) = [[micro].sub.1]u([xi]), (1.5)

au(0)--bp(0)u'(0) = [[micro].sub.2]u([xi]), cu(1) + dp(1)u'(1) = 0, (1.6)

au(0)--bp(0)u'(0) = [[micro].sub.1]u([xi]), cu(1) + dp(1)u'(1) = [[micro].sub.2]u([xi]). (1.7)

In [6], Gupta studied some existence results for solutions of the boundary value problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [m-2.summation over (i=1)] [a.sub.i] = 1, where f(t, x, y) satisfies Caratheodory's conditions and e(t) is a function in [L.sup.1][0, 1]. Feng and Webb [4] considered the solvability of second order differential equations

u"(t) = f(t, u(t), u'(t)) + e(t), t [member of] (0, 1) (1.8)

with the three-point boundary conditions

u'(0) = 0, u(1) = [alpha]u([eta]), (1.9)

u(0) = 0, u(1) = [alpha]u([eta]) (1.10)

when [alpha] = 1 for (1.8)/(1.9) and [alpha] = 1/[eta] for (1.8)/(1.10) are at resonance.

There has been increasing interest in questions of the solvability of boundary value problems for ordinary differential equations at resonance. There were many excellent results on the existence of solutions for two-point or multipoint boundary value problems, for which the nonlinearity is only dependent of the first-order derivative. The main techniques used are the Leray-Schauder continuation theorem and the coincidence degree theory, see [3-5, 7, 9, 10, 13, 14] and references therein. The second-order nonlinear equation, for which the nonlinearity is involved with integration and first-order derivative, is a special case of an integro-differential equation. It is known that integro-differential equations arise from many fields of science, for example in applied areas which include engineering, mechanics, financial mathematics, etc. [1, 2, 11].

For the equation (1.1), with the generalized Sturm-Liouville boundary conditions, nothing is known regarding the solvability of this class of boundary value problems. So, in this paper, using coincidence degree theory of Mawhin type [12], we establish some solvability criteria for boundary value problem at resonance (1.1)/(1.5), (1.1)/(1.6) and (1.1)/(1.7), respectively. The problem (1.1)/(1.5) happens to be at resonance in the sense that the associated linear homogeneous boundary value problem

(p(t)u'(t))0 = 0, t [member of] (0, 1),

au(0)--bp(0)u'(0) = 0, cu(1) + dp(1)u'(1) = [[micro].sub.1]u([xi])

has u(t) = a [[integral].sup.t.sub.0] 1/p([tau])d[tau] + b as a nontrivial solution, while we assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This result implies that q(t)u(t) + f (t, [[integral].sup.t.sub.0] u(s)ds, u'(t)) [member of] [L.sup.1][0, 1] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If [[micro].sub.1] (a [[integral].sup.[xi].sub.0] 1/p([tau]) + b) [not equal to] ad + bc + ac [[integral].sup.1.sub.0] 1/p([tau]) then this problem has u(t) [equivalent to] 0 as its only solution. So we say that the boundary value problem (1.1)/(1.5) is at resonance when

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The cases such that the linear mapping Lu = (p(t)u0(t))0 is noninvertible are called resonance cases. Otherwise, they are called nonresonance cases.

Similarly, we can obtain that the problem (1.1)/(1.6) is at resonance when

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the problem (1.1)/(1.7) is at resonance when

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Throughout this paper, we use the following hypotheses:

([H.sub.1]) a, b, c, d [member of] R \ {0}, [[micro].sub.i] [not equal to] 0 (i = 1, 2), and [xi] 2 (0, 1) are given constants;

([H.sub.2]) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

([H.sub.3]) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The rest of the paper is organized as follows: Section 2 gathers together the definitions of Fredholm mapping of index zero and L-completely continuity, which will be useful in proving the main results. Using coincidence degree theory of Mawhin type, solvability criteria for the generalized Sturm-Liouville boundary value problems at resonance (1.1)/(1.5), (1.1)/(1.6) and (1.1)/(1.7) are established in Section 3 and Section 4, respectively.

2 Preliminaries

Definition 2.1. Let X and Z be normed spaces. A linear mapping L : DomL [subset] X [right arrow] Z is called a Fredholm mapping if the following two conditions hold:

(i) ker L has a finite dimension,

(ii) ImL is closed and has a finite codimension.

If L is a Fredholm mapping, its Fredholm index is the integral IndL = dim kerL--codimImL. In this paper, we are interested in a Fredholm mapping of index zero, that is, dim ker L = codimImL. From Definition 2.1, it follows that there exist continuous projections P : X [right arrow] X and Q : Z [right arrow] Z such that ImP = ker L, kerQ = ImL, X = ker L [direct sum] ker P, Z = ImL [direct sum] ImQ, and that the mapping

[L.sub.DomL[intersection]ker P] : DomL [intersection] ker P [right arrow] ImL

is invertible. We denote the inverse of [L.sub.|DomL[intersection]ker P] by KP : ImL [right arrow] DomL [intersection] ker P. The generalized inverse of L denoted by [K.sub.P,Q] : Z [right arrow] DomL [intersection] ker P is defined by [K.sub.P,Q] : [K.sub.P] (I--Q).

Definition 2.2. Let L : DomL [subset] X [right arrow] Z be a Fredholm mapping, let E be a metric space, and N : E [right arrow] Z be a mapping. N is called L-compact on E if QN : E [right arrow] Z and [K.sub.P,Q]N : E [right arrow] X are compact on E. In addition, we say that N is L-completely continuous if it is L-compact on every bounded E [subset] X.

Theorem 2.3 (see [12]). Let [OMEGA] X be open and bounded, L be a Fredholm mapping of index zero, and let N be L-compact on [[bar].[OMEGA]]. Assume that the following conditions are satisfied:

(i) Lu [not equal to] [lambda] N u for every (u, [lambda]) 2 ((DomL[intersection]kerL)[intersection] [partial derivative][OMEGA]) [x] (0, 1);

(ii) Nu [not member of] ImL for every u [member of] ker L [intersection] [partial derivative][OMEGA];

(iii) deg(Q[N.sub.|ker L|[intersection][partial derivative][OMEGA]ker L, 0) [not equal to] 0 with Q : Z [right arrow] Z a continuous projection such that ker Q = ImL.

Then the equation Lu = Nu has at least one solution in DomL \ .

3 Existence Results for (1.1)/(1.5)

Let X = [C.sup.1][0, 1] with the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and Z = [L.sup.1][0, 1] with the norm [[parallel]u[parallel].sub.1] = [integral].sup.1.sub.0] |u(t)| dt. We use the Sobolev space

[W.sup.2,1](0, 1)

= {u : [0, 1] [right arrow] R : u, u' are absolutely continuous on [0, 1], u" [member of] [L.sup.1][0, 1]}.

Define L to be the linear mapping from DomL [subset] X to Z with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by

Lu(t) = (p(t)u'(t))', u 2 DomL,

and define the mapping N : X [right arrow] Z by

Nu(t) = f (t, [[integral].sup.t.sub.0] u(s)ds, u0(t)) + q(t)u(t), t [member of] [0, 1].

For convenience, we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Lemma 3.1. The mapping L : DomL [subset] X [right arrow] Z is a Fredholm mapping of index zero when

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Furthermore, the linear continuous projection Q : Z [right arrow] Z can be defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the mapping [K.sub.P] : ImL [right arrow] DomL [intersection] ker P can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. It is clear that kerL = R. Let u 2 DomL, and z 2 Z and consider the linear equation

(p(t)u'(t))0 = z(t), t [member of](0, 1),

Taking the Cauchy integral from 0 to t, we obtain

u'(t) = p(0)u'(0)/p(t) + 1/p(t) [integral].sup.t.sub.0] z([tau])d[tau]

Again taking the Cauchy integral from 0 to t, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

which satisfies (1.5) if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

On the other hand, if (3.2) holds for some z [member of] Z, then we take u [member of] DomL as given by (3.1), (p(t)u'(t))0 = z(t) for t [member of] (0, 1), and (1.5) is satisfied. So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Further, we define the mapping Q : Z ! Z by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for z [member of] Z, and it is easy to check that Q : Z [right arrow] Z is a linear continuous projection. Furthermore, ImL = ker Q. Let z = (z--Qz) + Qz. Then z--Qz [member of] ker Q = ImL

and Qz [member of] ImQ, so Z = ImL + ImQ. If z [member of] ImL [intersection] ImQ, then z(t) = 0. Hence Z = ImL [direct sum] ImQ. From ker L = R, we obtain that

IndL = dim kerL--codimImL = dim kerL--dim ImQ = 0.

Thus L is a Fredholm mapping of index zero. Take P : X [right arrow] X as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and let u [member of] X be in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Obviously, ImP = ker L and X = ker L [??] ker P. Then the generalized inverse [K.sub.P] : ImL [right arrow] DomL [intersection] ker P is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.3)

In fact, for z [member of] ImL, we know

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

and for u [member of] DomL [intersection] ker P, we also know

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, in view of u [member of] DomL [intersection] ker P, Pu = u(0) + p(0)u'(0) [[intersection].sup.t.sub.0] 1/p([tau]) d[tau] = 0, and thus

([K.sub.P]L)u(t) = u(t). (3.5)

By (3.4) and (3.5), we obtain

KP = [([L.sub.|DomL[intersection]ker P).sup.-1].

The proof is complete.

Note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 3.2. Assume ([H.sub.1])-([H.sub.3). Suppose that

([A.sub.1]) There exists a constant M > 0 such that for u [member of] DomL, if |u'(t)| > M/p(t) for all t [member of] [0, 1], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

([A.sub.2]) There exist functions [alpha], [beta], [gamma], [theta] [member of] [L.sup.1][0, 1] and a constant [epsilon] [member of] [0, 1) such that for all (x, y) [member of] [R.sup.2] and t [member of] [0, 1] either

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.6)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

([A.sub.3]) There exists a constant [[bar].M] > 0 such that for any [omega] [member of] R, if |[omega]| > [[bar].M], then either

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.8)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.9)

Then for each q [member of] [L.sup.1][0, 1], the boundary value problem (1.1)/ (1.5) when

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

has at least one solution in C1[0, 1] provided that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For u [member of] [[OMEGA].sub.1], u [not member of] ker L, and Nu [member of] ImL = kerQ. Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

since Q Nu = 0. It follows from ([A.sub.1]) that there exists [t.sub.0] [member of] [0, 1] with |p([t.sub.0])u'(t0)| [less than or equal to] M.

From [[integral].sup.t.sub.0] (p(s)u'(s))'ds = p([t.sub.0])u'([t.sub.0])--p(0)u'(0), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.10)

Also, for u [member of] [[OMEGA].sub.1], observe that (I--P)u [member of] Im[K.sub.P] = DomL [intersection] ker P. Then using (3.3), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.11)

By (3.10) and (3.11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If (3.6) holds, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.12)

that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is easy to check that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. So, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.13)

Also, by (3.12) and (3.13), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [epsilon] [member of] [0, 1) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], there exists M1 > 0 such that [[parallel]u'[parallel].sub.[infinity]] [less than or equal to] [M.sub.1] for all u [member of] [[OMEGA].sub.1], and so there exists [M.sub.2] > 0 such that [[integral].sup.1.sub.0] |u(s)| ds [less than or equal to] [M.sub.2] for all u [member of] [OMEGA].sub.1]. Therefore, [[OMEGA].sub.1] is bounded. If (3.7) holds, then similar to the above arguments, we can derive the same conclusion. Now let

[[OMEGA].sub.2] = {u [member of] ker L : Nu [member of] ImL}.

If u [member of] [OMEGA].sub.2], then u(t) [equivalent to] e [member of] R, t [member of] [0, 1], Nu [member of] ImL = ker Q, that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

since Q Nu = 0. From (A1), we know [parallel]u[parallel] = |e| [less than or equal to] [[bar].M]. Thus [OMEGA].sub.2] is bounded. Define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where J : ker L [right arrow] ImQ is a linear isomorphism given by J(k) = k for all k [member of] R. If u(t) = k, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If [lambda] = 1, then [kappa] = 0 and in the case [lambda] [member of] [0, 1), if |k| > [[bar].M], in view of (3.8), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is a contradiction. If (3.9) holds, then we take

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where J is as above, similar to the above argument. Thus, in either case

[parallel]u[parallel] = [absolute value of k] [less than or equal to] [bar.M] for all u [member of] [[OMEGA].sub.3],

that is, [[OMEGA].sub.3] is bounded.

Let be a bounded open subset of X such that [[union].sup.3.sub.i=1] [[OMEGA].sub.i] [subset] [OMEGA]. By using the AscoliArzela theorem, we can prove that [K.sub.P] (I--Q)N : [right arrow] X is compact. Thus N is L-compact on [[bar].[OMEGA]]. Finally, it only remains to verify that the condition (iii) of Theorem 2.3 is fulfilled. We define a homotopy

H(u, [lambda]) = [+ or -] [lambda]Ju + (1 - [lambda])Q Nu.

According to the above argument, we have

H(u, [lambda]) [not equal to] 0 for u [member of] [partial derivative][OMEGA] [intersection] ker L.

Thus, by the degree property of homotopy invariance, we obtain

deg(Q[N.sub.ker L], [OMEGA] [intersection] ker L, 0)

= deg (H(x, 0), [OMEGA] [intersection] ker L, 0) = deg (H(x, 1), [OMEGA] [intersection] ker L, 0) = deg ([+ or -]J, [OMEGA] [intersection] ker L, 0) [not equal to] 0.

Then by Theorem 2.3, Lu = Nu has at least one solution in DomL [intersection] [[bar].[OMEGA]]. Therefore, the boundary value problem (1.1)/(1.5) has a solution in [C.sup.1][0, 1].

4 Existence Results for (1.1)/(1.6), (1.1)/(1.7)

In this section, we discuss existence of nontrivial solutions for (1.1)/(1.6) and (1.1)/(1.7), respectively. By using the same arguments as in Section 3, we easily show the following lemmas and theorems, and thus we omit their proofs. For the sake of convenience, we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Problem (1.1)/(1.6)

The mappings L and N are the same as in Section 3, and we set

DomL = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

ImL = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Lemma 4.1. The mapping L : DomL [subset] X [right arrow] Z is a Fredholm mapping of index zero when

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Furthermore, the continuous linear projection mapping Q : Z [right arrow] Z can be defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The mapping [K.sub.P] is the same as in Section 3.

Theorem 4.2. Assume ([H.sub.1])-([H.sub.3]). Suppose that the condition ([A.sub.2]) of Theorem 3.2 is satisfied and also assume

([A.sub.4]) There exists a constant M > 0 such that for u [member of] DomL, if |u'(t)| > M/p(t) for all t [member of] [0, 1], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

([A.sub.5]) There exists a constant [[bar].M] > 0 such that for any [omega] [member of] R, if |[omega]| > [[bar].M], then we have either

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or else

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then for each q [member of] [L.sup.1][0, 1], the boundary value problem (1.1)/ (1.6) when

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

has at least one solution in [C.sup.1][0, 1] provided that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Problem (1.1)/(1.7)

The mappings L and N are the same as in Section 3, and we also set

DomL = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

ImL = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Lemma 4.3. The mapping L : DomL [subset] X [right arrow] Z is a Fredholm mapping of index zero when

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Furthermore, the continuous linear projection mapping Q : Z [right arrow] Z can be defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The mapping [K.sub.P] is the same as in Section 3.

Theorem 4.4. Assume ([H.sub.1])-([H.sub.3]). Suppose that the condition ([A.sub.2]) of Theorem 3.2 is satisfied and also assume

([A.sub.6]) There exists a constant M > 0 such that for u [member of] DomL, if |u'(t)| > M/p(t) for all t [member of] [0, 1], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

([A.sub.7]) There exists a constant M > 0 such that for any [omega] [member of] R, if |[omega]| > [[bar].M], then we have either

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or else

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then for each q [member of] [L.sup.1][0, 1], the boundary value problem (1.1)/ (1.7) when

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

has at least a solution in [C.sup.1][0, 1] provided that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Acknowledgment

The author is grateful to the referees for their very valuable comments and suggestions.

Received October 22, 2007; Accepted February 25, 2008 Communicated by Gerhard Freiling

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Youwei Zhang

Hexi University

Department of Mathematics

Gansu 734000, PR China

ywzhang0288@163.com
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