Journal of Difference Equations and Applications, 2002 Vol. 8 (11), pp. 1073–1083
Fixed Point Theorem of Cone Expansion and Compression of Functional Type DOUGLAS R. ANDERSONa and RICHARD I. AVERYb,* a
Department of Mathematics and Computer Science, Concordia College, Moorhead, MN 56562 USA; bCollege of Natural Sciences, Dakota State University, Madison, SD 57042 USA (Received 1 August 2001; Revised 9 October 2001; In final form 9 October 2001)
Dedicated to Allan Peterson on the occasion of his 60th birthday. The fixed point theorem of cone expansion and compression of norm type is generalized by replacing the norms with two functionals satisfying certain conditions to produce a fixed point theorem of cone expansion and compression of functional type. We conclude with an application verifying the existence of a positive solution to a discrete second-order conjugate boundary value problem. Keywords: Fixed point theorems; Boundary value problem; Difference equations
AMS Subject Classification: 47H10; 39A10
*Corresponding author. E-mail:
[email protected] ISSN 1023-6198 print/ISSN 1563-5120 online q 2002 Taylor & Francis Ltd DOI: 10.1080/10236190290015344
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1. PRELIMINARIES There are many fixed point theorems. See Ref. [7] for an introduction to the study and applications of fixed point theorems. In this paper we will generalize the fixed point theorem of cone expansion and compression of norm type. The generalization allows the user to choose two functionals that satisfy certain conditions which are used in place of the norm. In applications to boundary value problems the functionals will typically be the minimum or maximum of the function over a specific interval. Hence in boundary value problem applications the functionals usually do not satisfy the triangle inequality property of a norm. The flexibility of using functionals instead of norms allows the theorem to be used in a wider variety of situations. In particular, in applications to boundary value problems it allows for improved sufficiency conditions for the existence of a positive solution. In this section we will state the definitions that are used in the remainder of the paper. Definition 1 Let E be a real Banach space. A nonempty closed convex set P , E is called a cone if it satisfies the following two conditions: i) x [ P; l $ 0 implies lx [ P; ii) x [ P; 2x [ P implies x ¼ 0: Every cone P , E induces an ordering in E given by x # y if and only if
y 2 x [ P:
Definition 2 An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets. Definition 3 A map a is said to be a nonnegative continuous functional on a cone P of a real Banach space E if
a : P ! ½0; 1Þ is continuous.
FIXED POINT THEOREM
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Let a and g be nonnegative continuous functionals on P; then, for positive real numbers r and R, we define the following sets: Pðg; RÞ ¼ {x [ P : gðxÞ , R}
ð1Þ
Pðg; a; r; RÞ ¼ {x [ P : r , aðxÞ and gðxÞ , R}:
ð2Þ
and
Definition 4 Let D be a subset of a real Banach space E. If r : E ! D is continuous with rðxÞ ¼ x for all x [ D; then D is a retract of E, and the map r is a retraction. The convex hull of a subset D of a real Banach space X is given by ( ) n n X X convðDÞ ¼ li xi : xi [ D; li [ ½0; 1; li ¼ 1; and n [ N : i¼1
i¼1
The following theorem is due to Dugundji and a proof can be found in Ref. [4, p. 44]. Theorem 5
For Banach spaces X and Y, let D , X be closed and let F :D!Y
be continuous. Then F has a continuous extension F~ : X ! Y such that ~ FðXÞ , convðFðDÞÞ: Corollary 6 Every closed convex set of a Banach space is a retract of the Banach space. Note that it follows from Corollary 6 that a cone P of a real Banach space E is a retract of E.
2. FIXED POINT INDEX The following theorem, which establishes the existence and uniqueness of the fixed point index, is from Ref. [5, pp. 82 –86]; an elementary proof can be found in Ref. [4, pp. 58&238]. The proof of the generalization of the
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fixed point theorem of norm type in the next section will invoke the properties of the fixed point index.
Theorem 7 Let X be a retract of a real Banach space E. Then, for every bounded relatively open subset U of X and every completely continuous ! X which has no fixed points on ›U (relative to X ), there operator A : U exists an integer i(A,U,X ) satisfying the following conditions: (G1) Normality: iðA; U; XÞ ¼ 1 if Ax ; y0 [ U for any x [ U (G2) Additivity: iðA; U; XÞ ¼ iðA; U 1 ; XÞ þ iðA; U 2 ; XÞ whenever U1 and U2 are disjoint open subsets of U such that A has no fixed points on 2 ðU 1 < U 2 Þ; U (G3) Homotopy Invariance: iðHðt; ·Þ; U; XÞ is independent of t [ ½0; 1 whenever H : ½0; 1 £ U ! X is completely continuous and Hðt; xÞ – x for any ðt; xÞ [ ½0; 1 £ ›U; (G4) Permanence: iðA; U; XÞ ¼ iðA; U > Y; YÞ if Y is a retract of X and , Y; AðUÞ (G5) Excision: iðA; U; XÞ ¼ iðA; U 0 ; XÞ whenever U0 is an open subset of U such that A has no fixed points in U 2 U 0 ; (G6) Solution: If iðA; U; XÞ – 0; then A has at least one fixed point in U. Moreover, iðA; U; XÞ is uniquely defined.
3. FIXED POINT THEOREMS The proof of the following fixed point results can be found in Ref. [5, pp. 88– 89]. Lemma 8 Let P be a cone in a real Banach space E, V a bounded open ! P a completely continuous subset of E with 0 [ V; and A : P > V operator. If Ax – mx for all x [ P > ›V and m $ 1; then iðA; P > V; PÞ ¼ 1:
FIXED POINT THEOREM
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Lemma 9 Let P be a cone in a real Banach space E, V a bounded open ! P a completely continuous operator. If subset of E, and A : P > V i) infkAxk . 0 and P>›V ii) Ax – mx for all x [ P > ›V and m [ ð0; 1; then iðA; P > V; PÞ ¼ 0: The following theorem is a generalization of the fixed point theorem of cone expansion and compression of norm type.
Theorem 10 Let P be a cone in a real Banach space E, and let a and g be nonnegative continuous functionals on P. Assume Pðg; a; r; RÞ as in Eq. (2) is a nonempty bounded subset of P, A : Pðg; a; r; RÞ ! P is a completely continuous operator with infkAxk . 0;
x[›Pðg;a;r;RÞ
and Pða; rÞ # Pðg; RÞ for these sets as in Eq. (1). If one of the two conditions (H1) aðAxÞ # r for all x [ ›Pða; rÞ; gðAxÞ $ R for all x [ ›Pðg; RÞ; and for all y [ ›Pða; rÞ; z [ ›Pðg; RÞ; l $ 1; and m [ ð0; 1 the functionals satisfy the properties
aðlyÞ $ laðyÞ;
gðmzÞ # mgðzÞ; and að0Þ ¼ 0
or (H2) aðAxÞ $ r for all x [ ›Pða; rÞ; gðAxÞ # R for all x [ ›Pðg; RÞ; and for all y [ ›Pða; rÞ; z [ ›Pðg; RÞ; l [ ð0; 1 and m $ 1 the functionals satisfy the properties
aðlyÞ # laðyÞ;
gðmzÞ $ mgðzÞ; and gð0Þ ¼ 0
is satisfied, then A has at least one positive fixed point x* such that r # aðx * Þ and gðx * Þ # R:
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Proof If there exists an x [ ›Pðg; a; r; RÞ such that Ax ¼ x; then there is nothing to prove; thus suppose that Ax – x for all x [ ›Pðg; a; r; RÞ: By Dugundji’s Theorem (Theorem 5), A has a completely continuous extension A : Pðg; RÞ ! P: Suppose condition (H1) is satisfied; the proof when (H2) is satisfied is nearly identical and will be omitted. Claim 1: Ay – ly for all y [ ›Pða; rÞ and l $ 1: To the contrary, suppose there exists a y0 [ ›Pða; rÞ and l0 . 1 (since A has no fixed points on the boundary) such that Ay0 ¼ l0 y0 : Then
aðAy0 Þ ¼ aðl0 y0 Þ $ l0 aðy0 Þ . aðy0 Þ ¼ r; which is a contradiction of the first part of (H1). Note that 0 [ Pða; rÞ by assumption, hence by Lemma 8 iðA; Pða; rÞ; PÞ ¼ 1: Claim 2: Az – mz for all z [ ›Pðg; RÞ and m [ ð0; 1: Suppose to the contrary that there exists a z0 [ ›Pðg; RÞ and m0 [ ð0; 1 such that Az0 ¼ m0 z0 : Then
gðAz0 Þ ¼ gðm0 z0 Þ # m0 gðz0 Þ , gðz0 Þ ¼ R; which is a contradiction of the second part of (H1). Hence by Lemma 9 iðA; Pðg; RÞ; PÞ ¼ 0 since we have assumed that infkAxk . 0:
x[›Pðg;RÞ
FIXED POINT THEOREM
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Thus by the additivity (G2) of the fixed point index iðA; Pðg; a; r; RÞ; PÞ ¼ iðA; Pðg; RÞ; PÞ 2 iðA; Pða; rÞ; PÞ ¼ 21 – 0; and by the solution property (G6) of the fixed point index the operator A has a fixed point x * [ Pðg; a; r; RÞ: Thus in any event the operator A has a fixed point x* such that r # aðx * Þ
and gðx * Þ # R:
A
4. APPLICATION Throughout the remainder of this paper we apply the above result to a second-order difference equation; see Ref. [6] for an introduction to the general subject, and [1 –3] for representative examples of applying fixed point theorems to difference equations. Consider the discrete second-order nonlinear conjugate boundary value problem, D2 xðt 2 1Þ þ f ðxðtÞÞ ¼ 0
for all t [ ½a þ 1; b þ 1
xðaÞ ¼ 0 ¼ xðb þ 2Þ
ð3Þ ð4Þ
where f : R ! R is continuous and f is nonnegative for x $ 0; R denotes the real numbers. Here we assume a and b are integers with b . a þ 2; and we define the discrete interval ½a; b þ 2 by ½a; b þ 2 U {a; a þ 1; . . .; b þ 1; b þ 2}: (Other intervals may be discrete or continuous; the type of interval will be obvious by context.) Define the Banach space (with sup norm) E ¼ {xjx : ½a; b þ 2 ! R} and the corresponding cone 9 x is concave; symmetric; > > = xðtÞ $ 0 for all t [ ½a; b þ 2 and P ¼ x [ E > xðtÞ $ t2a kxk for all t [ ½a; t > > > ; : t2a 8 > >
