# Positive Fixed Points for Semipositone Operators in Ordered Banach Spaces and Applications

Authors: Zengqin Zhao; Xinsheng Du
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The theory of semipositone integral equations and semipositone ordinary differential equations has been emerging as an important area of investigation in recent years, but the research on semipositone operators in abstract spaces is yet rare. By employing a well-known fixed point index theorem and combining it with a translation substitution, we study the existence of positive fixed points for a semipositone operator in ordered Banach space. Lastly, we apply the results to Hammerstein integral equations of polynomial type.


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## Body

## 1. Introduction

Existence of fixed points for positive operators have been studied by many authors; see [1–9] and their references. The theory of semipositone integral equations and semipositone ordinary differential equations has been emerging as an important area of investigation in recent years (see [10–17]). But the research on semipositone operators in abstract spaces are yet rare up to now.Inspired by a number of semipositone problems for integral equations and ordinary differential equations, we study the existence of positive fixed points for semipositone operators in ordered Banach spaces. Then the results are applied to Hammerstein integral equations of polynomial type.LetE be a real Banach space with the norm ∥·∥, P a cone of E, and “≤” the partial ordering defined by P, θ denoting the zero element of E, P+=P∖{θ}, [a,b]={x∈E∣a≤x≤b}.Recall that coneP is said to be normal if there exists a positive constant N such that θ≤x≤y implies ∥x∥≤N∥y∥, the smallest N is called the normal constant of P. An element z∈E is called the least upper bound (i.e., supremum) of set D⊂E, if it satisfies two conditions: (i) x≤z for any x∈D; (ii) x≤y, x∈D implies z≤y. We denote the least upper bound of D by supD, that is, z=supD.Definition 1.
ConeP⊂E is said to be minihedral if  sup{x,y} exists for each pair of elements x,y∈E. For any x in E we define x+=sup{x,θ}.Definition 2 (see [1, 3]).
LetEi be real Banach spaces, Pi cones of Ei, i=1,2,  T:P1→P2, and α∈R. Then we say T is α-convex if and only if T(tu)≤tαTu for all (u,t)∈P1×(0,1).Definition 3.
LetEi be real Banach spaces, Pi cones of Ei, and i=1,2. P1⊂D⊂E1, T:D→E2. T is said to be nondecreasing if x1≤x2(x1,x2∈D) implies Tx1≤Tx2; T is said to be positive if Tx∈P2 for any x∈P1; T is said to be semipositone if (i)  there exists an element x0∈P1 such that F(x0)∉P2 and (ii) there exists an element q∈E2 such that Tx+q∈P2 for any x∈P1.In order to prove the main results, we need the following lemma which is obtained in [18].Lemma 4.
LetE be a real Banach space and Ω a bounded open subset of E, with θ∈Ω, and A:Ω¯∩Q→Q is a completely continuous operator, where Q is a cone in E.(i)
Suppose thatAu≠μu, for all u∈∂Ω∩Q, μ≥1, then the fixed point index i(A,Ω∩Q,Q)=1.(ii)
Suppose thatAu≰u, for all u∈∂Ω∩Q, then i(A,Ω∩Q,Q)=0.The research on ordered Banach spaces, cones, fixed point index, and the above lemma can be seen in [18, 19].


## 2. Main Results and Their Proofs

Theorem 5.
LetEi be Banach space, Pi⊂Ei cones, and i=1,2. Suppose that operator A:E1→E2 can be expressed as A=BF, where the cone Pi and the operator F and B satisfy the following conditions:(H1)
whenP1 is normal and minihedral, P2 is normal;(H2)
whenF:E1→E2 is continuous, there exist g∈P2+, q∈E2, a nondecreasing α-convex operator G:P1→P2, (α>1), and a bounded functional H:P1→[0,+∞) such that
(1)Gu≤Fu+q≤H(u)g,∀u∈P1;(H3)
whenB:E2→E1 is linear completely continuous, there exists e∈P1+ such that
(2)Bx≥∥Bx∥e∀x∈P2;Ge>θ;(H4)
when there exists a positive numberr0 such that
(3)θ<Bq<r0e,h(r0N)∥Bg∥<r0N,
with h(t)=maxu∈P1,∥u∥≤tH(u), N is the normal constant of P1. Then A has a fixed point w∈P1+.Proof.
Forq in (H2) and e in (H3), we define that
(4)x0=Bq,Pe={u∈P1∣u≥∥u∥e},(5)Ku=B(F([u-x0]+)+q),∀u∈P1.
Clearly, Pe⊂P1 is a normal cone of E1. Since the cone P1 is minihedral, [u-x0]+ makes sense. By (H4) and (4), we know that
(6)x0<r0e≤y∥y∥r0,∀y∈Pe+.
From the condition (H3) and (4), we know that x0∈Pe⊂P1, and hence u-x0≤u and
(7)θ≤[u-x0]+≤u,∀u∈P1.
By (7), we have [u-x0]+∈P1, using (H2) we know that
(8)F([u-x0]+)+q≥G([u-x0]+),∀u∈P1+.
That is, F([u-x0]+)+q∈P2. This and (2) and (5) imply Ku∈Pe, for all u∈P1. Hence,
(9)K(Pe)⊂Pe.
Suppose thatD is a bounded set of Pe, L is a positive number satisfying ∥u∥≤L, for all u∈D. By (7) and normality of P1, we obtain that
(10)∥[u-x0]+∥≤N∥u∥≤NL,∀u∈D.
Therefore, (H2) implies that F([u-x0]+)∈[-q,h(NL)g], u∈D. Since P2 is normal, the order interval [-q,h(NL)g] is a bounded set of E2; therefore, {F([u-x0]+)∣u∈D} is a bounded set of E2. This together with (9), continuity of F, and the completely continuity of B, we obtain that K map Pe into Pe and is completely continuous.
For ther0 in (H4), we let Ωr0={u∈E1∣∥u∥<r0}. By (7) we know that
(11)∥[u-x0]+∥≤N∥u∥≤r0N,∀u∈Ωr0∩Pe.
Therefore, from (H2) we obtain that
(12)F([u-x0]+)+q≤H([u-x0]+)g≤h(r0N)g,q≤H([u-x0]+)g≤h(r0N)gg∀u∈Ωr0∩Pe,
where h(t) is as in (H4).
We prove that(13)Ku≠μu,∀u∈∂Ωr0∩Pe,μ≥1.
Assume there exist μ0∈(0,1] and z0∈∂Ωr0∩Pe, such that z0=μ0Kz0. Using (12) we have
(14)Kz0=B(F([z0-x0]+)+q)≤h(r0N)Bg,
hence
(15)r0=∥z0∥=∥μ0Kz0∥≤∥Kz0∥≤Nh(r0N)∥Bg∥
which contradicts the condition (3), thus (13) holds. By Lemma 4 we know
(16)i(K,Ωr0∩Pe,Pe)=1.
Takem0>0 such that m0<1/r0, and set
(17)R>max{2r0,(m0∥Bq∥)-1,r01-m0r0,R>max2N1/(α-1)((m0∥Bq∥)α∥BGe∥)-1/(α-1)r01-m0r0},
where r0 as in (3), N is the normal constant of P1. In the following, we prove
(18)u≱Ku,∀u∈∂ΩR∩Pe.
Assume there exists y1∈∂ΩR∩Pe such that y1≥Ky1. Using (6), we have x0<(y1/∥y1∥)r0=(y1/R)r0, thus it is obtained that
(19)y1>Rr0x0,y1-x0∈P1+,
by (17). From (17) we know R>r0/(1-m0r0), thus (R-r0)/r0≥m0R. This and (H3), (4), and (19) imply
(20)[y1-x0]+=y1-x0>(Rr0-1)Bq≥m0RBq≥m0R∥Bq∥e.
Byα-convexity of G we know
(21)G(su)≥sαG(u),∀u∈P1,s>1.
By (17) we know m0R∥Bq∥>1, hence (20) and (21) imply
(22)G([y1-x0]+)≥G(m0R∥Bq∥e)≥(m0R∥Bq∥)αGe.
This together with (5) and the condition (H2) imply
(23)y1≥Ky1=B(F([y1-x0]+)+q)≥B(G([y1-x0]+))≥(m0R∥Bq∥)αBGe.
This and (23) imply
(24)NR=N∥y1∥≥(m0R∥Bq∥)α∥BGe∥=Rα(m0∥Bq∥)α∥BGe∥,
therefore,
(25)N1/(α-1)((m0∥Bq∥)α∥BGe∥)-1/(α-1)≥R,
which contradicts (17), thus (18) holds. Using Lemma 4 we have
(26)i(K,ΩR∩Pe,Pe)=0.
By (16) and (26) and additivity of fixed point indexes we know that
(27)i(K,(ΩR∖Ωr0¯)⋂‍Pe,Pe)=-1.
Thus, K has a fixed point z on (ΩR∖Ωr0¯)⋂‍Pe. Hence,
(28)z=B(F([z-x0]+)+q),z∈Pe,r0≤∥z∥≤R.
Letw=z-x0. From (6) and ∥z∥≥r0 we know x0<(z/∥z∥)r0≤z, then [z-x0]+=w∈P1+. This together with (4) and (28) imply w=z-x0=BF(w)=A(w), so that w is a positive fixed point of A.


## 3. Corollary and Applications

From Theorem5 we obtain the following corollary.Corollary 6.
Suppose that conditions (H1), (H2), and (H3) hold, and in addition assume the following.(H5)
For anyx∈P2+, there exists a positive number Lx such that Bx≤Lxe.Then there exists a small enough λ*>0 such that u=λAu has a positive solution for any λ∈(0,λ*).Proof.
For any fixedr0>0, by (H5), we can all take λ-=λ-(r0), such that
(29)λBq<r0e,λh(r0N)∥Bg∥<r0N,∀λ∈(0,λ-),
hence (H4) holds. We take that
(30)F*(t,u)=λF(t,u),G*(u)=λG(u),q*(t)=λq(t),g*(t)=λg(t).
Then for λA=B(λF), the conditions in Theorem 5 are satisfied. Thus, λA has a positive fixed point, that is, u=λA has a positive solution, and the proof is complete.We consider the integral equation(31)u(x)=∫G‍k(x,y)(∑i=1m‍ai(y)u(y)αi+q(y)u(x)=×(u(y)γ-u(y)δ-w0)∑i=1m‍)dy,
where G is a bounded closed domain in Rn and αi≥0, ai(x), q(x)∈L(G,[0,∞)), i=1,2,…,m, k(x,y) is nonnegative continuous on G×G.Theorem 7.
Suppose that amongαi(i=1,2,…,m) there exists αi0>1 such that infx∈Gai0(x)>0, and there exist nontrivial nonnegative functions a(x), b(x)∈C(G), and a positive number c,γ,δ,w0 such that
(32)ca(x)b(y)≤k(x,y)≤a(x),k(x,y)≤b(y),∀x,y∈G,(33)γ>δ>0,0<w0≤1+mint∈[0,1]{tγ-tδ},(34)∫G‍q(y)dy<c,∫G‍b(y)·max(∑i=1m‍ai(y),q(y))dy<12-w0.
Then (31) has a nontrivial nonnegative solution in C(G).Proof.
Let the Banach spaceE1=C(G) with the sup norm ∥·∥,
(35)P1={u∈E1∣u(x)≥0,∀x∈G},(36)E2=L(G),P2={u∈E2∣u(x)≥0,∀x∈G},(37)e=ca(x),q=q(x),g(x)=max{q(x),∑i=1m‍ai(x)},(38)Gu=ai0(x)u(x)αi0,∀u(x)∈P1,(39)Fu=∑i=1m‍ai(x)u(x)αi+q(x)(u(x)γ-u(x)δ-w0),Fu=∑i=1m‍ai(x)u(x)αi+q(x)u(x)δ-w0∀u(x)∈P1,(40)Ju(x)={u(x)α,ifu(x)≤1,u(x)β,ifu(x)>1,,∀u(x)∈P1,
with α=min1≤i≤n{αi}, β=max1≤i≤n{αi},
(41)H(u)=∥Ju(x)+u(x)γ-u(x)δ-w0+1∥C,∀u(x)∈P1,(42)Bu=∫G‍k(x,y)u(y)dy,r0=1.
ThenP1⊂E1 is normal minihedral, the normal constant N=1,e∈P1+. P2 is a cone of E2, q, g∈P2+. G:P1→P2 is nondecreasing αi0-convex operator, and Ge>θ. F:P1→E2 is continuous; h:P1→[0,+∞).
It is known easily that(43)-1<mint∈[0,1]{tγ-tδ}≤tγ-tδ<0,t∈(0,1),
thus w0 exits in (33) and
(44)tγ-tδ-w0≤-w0,t∈[0,1].
By (33), (43), and γ>δ we have
(45)u(x)γ-u(x)δ-w0≥u(x)γ-u(x)δ-1-mint∈[0,1]{tγ-tδ}≥-1,∀u(x)∈P1+,
therefore
(46)u(x)γ-u(x)δ-w0+1≥0,∀u(x)∈P1+.
From (33), (39), and (44) we know easily that there exists u0∈P1 such that Fu∉P2. From (37)–(46), we obtain that
(47)Gu≤Fu+q=∑i=1m‍ai(x)u(x)αi+q(x)(u(x)γ-u(x)δ-w0+1)≤((Ju)(x)+u(x)γ-u(x)δ-w0+1)g(x)≤H(u)g(x),∀x∈G,u∈P1+.
Equations (32) and (42) imply that ∥Bu∥≤∫G‍b(y)u(y)dy, and hence
(48)Bu≥ca(x)∫G‍b(y)u(y)dy≥∥Bu∥e,∀u∈P1.
By (42), (32), (34), and (37), we obtain that
(49)Bq≤a(x)∫G‍q(y)dy<ca(x)=r0e.
By (41) we have h(r0N)=h(1)=max∥u∥≤1{H(u)}=2-w0. This and  (34) and (42) get that
(50)Bg=∫G‍k(x,y)g(y)dy≤∫G‍b(y)g(y)dy<12-w0=r0h(r0N).
From (35) and (36) we know that (H1) is satisfied. By (47) and (48) we obtain that (H2) and (H3) are satisfied. Equations (49) and (50) imply that (H4) is satisfied. Therefore, using Theorem 5, the integral equation (31) has a positive solution in C(G).



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