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

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.

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.

Let 𝐸 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 cone 𝑃 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. Cone 𝑃⊂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]). Let 𝐸_𝑖 be real Banach spaces, 𝑃_𝑖 cones of 𝐸_𝑖, 𝑖=1,2,  𝑇∶𝑃₁→𝑃₂, and 𝛼∈𝑅. Then we say 𝑇 is 𝛼-convex if and only if 𝑇(𝑡𝑢)≤𝑡^𝛼𝑇𝑢 for all (𝑢,𝑡)∈𝑃₁×(0,1).

Definition 3. Let 𝐸_𝑖 be real Banach spaces, 𝑃_𝑖 cones of 𝐸_𝑖, 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. Let 𝐸 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 that 𝐴𝑢≠𝜇𝑢, for all 𝑢∈𝜕Ω∩𝑄, 𝜇≥1, then the fixed point index 𝑖(𝐴,Ω∩𝑄,𝑄)=1.(ii)Suppose that 𝐴𝑢≰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. Let 𝐸_𝑖 be Banach space, 𝑃_𝑖⊂𝐸_𝑖 cones, and 𝑖=1,2. Suppose that operator 𝐴∶𝐸₁→𝐸₂ can be expressed as 𝐴=𝐵𝐹, where the cone 𝑃_𝑖 and the operator 𝐹 and 𝐵 satisfy the following conditions:(H1) when 𝑃₁ is normal and minihedral, 𝑃₂ is normal;(H2) when 𝐹∶𝐸₁→𝐸₂ is continuous, there exist 𝑔∈𝑃⁺₂, 𝑞∈𝐸₂, a nondecreasing 𝛼-convex operator 𝐺∶𝑃₁→𝑃₂, (𝛼>1), and a bounded functional 𝐻∶𝑃₁→[0,+∞) such that 𝐺𝑢≤𝐹𝑢+𝑞≤𝐻(𝑢)𝑔,∀𝑢∈𝑃₁; (1)(H3) when 𝐵∶𝐸₂→𝐸₁ is linear completely continuous, there exists 𝑒∈𝑃⁺₁ such that 𝐵𝑥≥‖𝐵𝑥‖𝑒∀𝑥∈𝑃₂;𝐺𝑒>𝜃; (2)(H4) when there exists a positive number 𝑟₀ such that 𝜃<𝐵𝑞<𝑟₀𝑒,ℎ𝑟₀𝑁<𝑟𝐵𝑔⁰𝑁, (3) with ℎ(𝑡)=max_{𝑢∈𝑃1,‖𝑢‖≤𝑡}𝐻(𝑢), 𝑁 is the normal constant of 𝑃₁. Then 𝐴 has a fixed point 𝑤∈𝑃⁺₁.

Proof. For 𝑞 in (H2) and 𝑒 in (H3), we define that 𝑥₀=𝐵𝑞,𝑃_𝑒=𝑢∈𝑃₁‖𝑢‖𝑒,𝐹∣𝑢≥𝐾𝑢=𝐵𝑢−𝑥₀⁺+𝑞,∀𝑢∈𝑃₁. (4)(5) Clearly, 𝑃_𝑒⊂𝑃₁ is a normal cone of 𝐸₁. Since the cone 𝑃₁ is minihedral, [𝑢−𝑥₀]⁺ makes sense. By (H4) and (4), we know that 𝑥₀<𝑟₀𝑦𝑒≤𝑦𝑟₀,∀𝑦∈𝑃⁺_𝑒. (6)
From the condition (H3) and (4), we know that 𝑥₀∈𝑃_𝑒⊂𝑃₁, and hence 𝑢−𝑥₀≤𝑢 and 𝜃≤𝑢−𝑥₀⁺≤𝑢,∀𝑢∈𝑃₁. (7) By (7), we have [𝑢−𝑥₀]⁺∈𝑃₁, using (H2) we know that 𝐹𝑢−𝑥₀⁺+𝑞≥𝐺𝑢−𝑥₀⁺,∀𝑢∈𝑃⁺₁. (8) That is, 𝐹([𝑢−𝑥₀]⁺)+𝑞∈𝑃₂. This and (2) and (5) imply 𝐾𝑢∈𝑃_𝑒, for all 𝑢∈𝑃₁. Hence, 𝐾𝑃_𝑒⊂𝑃_𝑒. (9)
Suppose that 𝐷 is a bounded set of 𝑃_𝑒, 𝐿 is a positive number satisfying ‖𝑢‖≤𝐿, for all 𝑢∈𝐷. By (7) and normality of 𝑃₁, we obtain that 𝑢−𝑥₀⁺≤𝑁‖𝑢‖≤𝑁𝐿,∀𝑢∈𝐷. (10) 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 𝑃_𝑒 into 𝑃_𝑒 and is completely continuous.
For the 𝑟₀ in (H4), we let Ω_𝑟0={𝑢∈𝐸₁∣‖𝑢‖<𝑟₀}. By (7) we know that 𝑢−𝑥₀⁺≤𝑁‖𝑢‖≤𝑟₀𝑁,∀𝑢∈Ω_𝑟0∩𝑃_𝑒. (11) Therefore, from (H2) we obtain that 𝐹𝑢−𝑥₀⁺+𝑞≤𝐻𝑢−𝑥₀⁺𝑔≤ℎ𝑟₀𝑁𝑔,∀𝑢∈Ω_𝑟0∩𝑃_𝑒, (12) where ℎ(𝑡) is as in (H4).
We prove that 𝐾𝑢≠𝜇𝑢,∀𝑢∈𝜕Ω_𝑟0∩𝑃_𝑒,𝜇≥1. (13) Assume there exist 𝜇₀∈(0,1] and 𝑧₀∈𝜕Ω_𝑟0∩𝑃_𝑒, such that 𝑧₀=𝜇₀𝐾𝑧₀. Using (12) we have 𝐾𝑧₀=𝐵𝐹𝑧₀−𝑥₀⁺+𝑞≤ℎ𝑟₀𝑁𝐵𝑔, (14) hence 𝑟₀=𝑧₀=𝜇₀𝐾𝑧₀≤𝐾𝑧₀≤𝑁ℎ𝑟₀𝑁𝐵𝑔 (15) which contradicts the condition (3), thus (13) holds. By Lemma 4 we know 𝑖𝐾,Ω_𝑟0∩𝑃_𝑒,𝑃_𝑒=1. (16)
Take 𝑚₀>0 such that 𝑚₀<1/𝑟₀, and set 𝑅>max2𝑟₀,𝑚₀𝐵𝑞⁻¹,𝑟₀1−𝑚₀𝑟₀,𝑁^1/(𝛼−1)𝑚₀𝐵𝑞^𝛼‖𝐵𝐺𝑒‖^{−1/(𝛼−1)}, (17) where 𝑟₀ as in (3), 𝑁 is the normal constant of 𝑃₁. In the following, we prove 𝑢≱𝐾𝑢,∀𝑢∈𝜕Ω_𝑅∩𝑃_𝑒. (18) Assume there exists 𝑦₁∈𝜕Ω_𝑅∩𝑃_𝑒 such that 𝑦₁≥𝐾𝑦₁. Using (6), we have 𝑥₀<(𝑦₁/‖𝑦₁‖)𝑟₀=(𝑦₁/𝑅)𝑟₀, thus it is obtained that 𝑦₁>𝑅𝑟₀𝑥₀,𝑦₁−𝑥₀∈𝑃⁺₁, (19) by (17). From (17) we know 𝑅>𝑟₀/(1−𝑚₀𝑟₀), thus (𝑅−𝑟₀)/𝑟₀≥𝑚₀𝑅. This and (H3), (4), and (19) imply 𝑦₁−𝑥₀⁺=𝑦₁−𝑥₀>𝑅𝑟₀−1𝐵𝑞≥𝑚₀𝑅𝐵𝑞≥𝑚₀𝑅𝐵𝑞𝑒. (20)
By 𝛼-convexity of 𝐺 we know 𝐺(𝑠𝑢)≥𝑠^𝛼𝐺(𝑢),∀𝑢∈𝑃₁,𝑠>1. (21) By (17) we know 𝑚₀𝑅‖𝐵𝑞‖>1, hence (20) and (21) imply 𝐺𝑦₁−𝑥₀⁺≥𝐺𝑚₀𝑅𝐵𝑞𝑒≥𝑚₀𝑅𝐵𝑞^𝛼𝐺𝑒. (22) This together with (5) and the condition (H2) imply 𝑦₁≥𝐾𝑦₁=𝐵𝐹𝑦₁−𝑥₀⁺𝐺𝑦+𝑞≥𝐵₁−𝑥₀⁺≥𝑚₀𝑅𝐵𝑞^𝛼𝐵𝐺𝑒. (23) This and (23) imply 𝑁𝑅=𝑁𝑦₁≥𝑚₀𝑅𝐵𝑞^𝛼‖𝐵𝐺𝑒‖=𝑅^𝛼𝑚₀𝐵𝑞^𝛼‖𝐵𝐺𝑒‖, (24) therefore, 𝑁^1/(𝛼−1)𝑚₀𝐵𝑞^𝛼‖𝐵𝐺𝑒‖^{−1/(𝛼−1)}≥𝑅, (25) which contradicts (17), thus (18) holds. Using Lemma 4 we have 𝑖𝐾,Ω_𝑅∩𝑃_𝑒,𝑃_𝑒=0. (26)
By (16) and (26) and additivity of fixed point indexes we know that 𝑖𝐾,Ω_𝑅⧵Ω_𝑟0𝑃_𝑒,𝑃_𝑒=−1. (27) Thus, 𝐾 has a fixed point 𝑧 on (Ω_𝑅⧵Ω_𝑟0)⋂𝑃_𝑒. Hence, 𝑧=𝐵𝐹𝑧−𝑥₀⁺+𝑞,𝑧∈𝑃_𝑒,𝑟₀≤‖𝑧‖≤𝑅. (28)
Let 𝑤=𝑧−𝑥₀. From (6) and ‖𝑧‖≥𝑟₀ we know 𝑥₀<(𝑧/‖𝑧‖)𝑟₀≤𝑧, then [𝑧−𝑥₀]⁺=𝑤∈𝑃⁺₁. 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 Theorem 5 we obtain the following corollary.

Corollary 6. Suppose that conditions (H1), (H2), and (H3) hold, and in addition assume the following.(H5) For any 𝑥∈𝑃⁺₂, there exists a positive number 𝐿_𝑥 such that 𝐵𝑥≤𝐿_𝑥𝑒.Then there exists a small enough 𝜆^∗>0 such that 𝑢=𝜆𝐴𝑢 has a positive solution for any 𝜆∈(0,𝜆^∗).

Proof. For any fixed 𝑟₀>0, by (H5), we can all take 𝜆=𝜆(𝑟₀), such that 𝜆𝐵𝑞<𝑟₀𝑒,𝜆ℎ𝑟₀𝑁<𝑟𝐵𝑔⁰𝑁,∀𝜆∈0,𝜆, (29) hence (H4) holds. We take that 𝐹^∗(𝑡,𝑢)=𝜆𝐹(𝑡,𝑢),𝐺^∗(𝑢)(𝑢),𝑞=𝜆𝐺_∗(𝑡)=𝜆𝑞(𝑡),𝑔^∗(𝑡)=𝜆𝑔(𝑡). (30) 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 𝑢(𝑥)=_𝐺𝑘𝑥,𝑦^𝑚_𝑖=1𝑎_𝑖𝑦𝑢𝑦^𝛼𝑖𝑦×𝑢𝑦+𝑞_𝛾−𝑢𝑦^𝛿−𝑤₀𝑑𝑦, (31) where 𝐺 is a bounded closed domain in 𝑅^𝑛 and 𝛼_𝑖≥0, 𝑎_𝑖(𝑥), 𝑞(𝑥)∈𝐿(𝐺,[0,∞)), 𝑖=1,2,…,𝑚, 𝑘(𝑥,𝑦) is nonnegative continuous on 𝐺×𝐺.

Theorem 7. Suppose that among 𝛼_𝑖(𝑖=1,2,…,𝑚) there exists 𝛼_𝑖0>1 such that inf_𝑥∈𝐺𝑎_𝑖0(𝑥)>0, and there exist nontrivial nonnegative functions 𝑎(𝑥), 𝑏(𝑥)∈𝐶(𝐺), and a positive number 𝑐,𝛾,𝛿,𝑤₀ such that (𝑥)𝑏𝑦(𝑥),𝑘𝑦𝑐𝑎≤𝑘𝑥,𝑦≤𝑎𝑥,𝑦≤𝑏,∀𝑥,𝑦∈𝐺,𝛾>𝛿>0,0<𝑤₀≤1+min_[]𝑡∈0,1𝑡^𝛾−𝑡^𝛿,_𝐺𝑞𝑦𝑑𝑦<𝑐,_𝐺𝑏𝑦⋅max^𝑚_𝑖=1𝑎_𝑖𝑦𝑦1,𝑞𝑑𝑦<2−𝑤₀. (32)(33)(34) Then (31) has a nontrivial nonnegative solution in 𝐶(𝐺).

Proof. Let the Banach space 𝐸₁=𝐶(𝐺) with the sup norm ‖⋅‖, 𝑃₁=𝑢∈𝐸₁(𝑥),𝐸∣𝑢≥0,∀𝑥∈𝐺₂=𝐿(𝐺),𝑃₂=𝑢∈𝐸₂(𝑥),(𝑥)(𝑥),𝑔(𝑥)𝑞(𝑥),∣𝑢≥0,∀𝑥∈𝐺𝑒=𝑐𝑎,𝑞=𝑞=max^𝑚_𝑖=1𝑎_𝑖(𝑥),𝐺𝑢=𝑎_𝑖0(𝑥)𝑢(𝑥)^𝛼𝑖0,∀𝑢(𝑥)∈𝑃₁,𝐹𝑢=^𝑚_𝑖=1𝑎_𝑖(𝑥)𝑢(𝑥)^𝛼𝑖+𝑞(𝑥)𝑢(𝑥)^𝛾−𝑢(𝑥)^𝛿−𝑤₀,∀𝑢(𝑥)∈𝑃₁,(𝑥)=𝑢(𝑥)𝐽𝑢^𝛼(𝑥)𝑢(𝑥),if𝑢≤1,_𝛽(𝑥)(𝑥),if𝑢>1,,∀𝑢∈𝑃₁, (35)(36)(37)(38)(39)(40) with 𝛼=min_{1≤𝑖≤𝑛}{𝛼_𝑖}, 𝛽=max_{1≤𝑖≤𝑛}{𝛼_𝑖}, 𝐻(𝑢)=𝐽𝑢(𝑥)+𝑢(𝑥)^𝛾−𝑢(𝑥)^𝛿−𝑤₀+1_𝐶,∀𝑢(𝑥)∈𝑃₁,𝐵𝑢=_𝐺𝑘𝑥,𝑦𝑢𝑦𝑑𝑦,𝑟₀=1. (41)(42)
Then 𝑃₁⊂𝐸₁ is normal minihedral, the normal constant 𝑁=1,𝑒∈𝑃⁺₁. 𝑃₂ is a cone of 𝐸₂, 𝑞, 𝑔∈𝑃⁺₂. 𝐺∶𝑃₁→𝑃₂ is nondecreasing 𝛼_𝑖0-convex operator, and 𝐺𝑒>𝜃. 𝐹∶𝑃₁→𝐸₂ is continuous; ℎ∶𝑃₁→[0,+∞).
It is known easily that −1<min_[]𝑡∈0,1𝑡^𝛾−𝑡^𝛿≤𝑡^𝛾−𝑡^𝛿<0,𝑡∈(0,1), (43) thus 𝑤₀ exits in (33) and 𝑡^𝛾−𝑡^𝛿−𝑤₀≤−𝑤₀,𝑡∈[0,1]. (44) By (33), (43), and 𝛾>𝛿 we have 𝑢(𝑥)^𝛾−𝑢(𝑥)^𝛿−𝑤₀≥𝑢(𝑥)^𝛾−𝑢(𝑥)^𝛿−1−min_[]𝑡∈0,1𝑡^𝛾−𝑡^𝛿≥−1,∀𝑢(𝑥)∈𝑃⁺₁, (45) therefore 𝑢(𝑥)^𝛾−𝑢(𝑥)^𝛿−𝑤₀+1≥0,∀𝑢(𝑥)∈𝑃⁺₁. (46) From (33), (39), and (44) we know easily that there exists 𝑢₀∈𝑃₁ such that 𝐹𝑢∉𝑃₂. From (37)–(46), we obtain that 𝐺𝑢≤𝐹𝑢+𝑞=^𝑚_𝑖=1𝑎_𝑖(𝑥)𝑢(𝑥)^𝛼𝑖+𝑞(𝑥)𝑢(𝑥)^𝛾−𝑢(𝑥)^𝛿−𝑤₀≤(𝑥)(𝑥)+1𝐽𝑢)(+𝑢^𝛾−𝑢(𝑥)^𝛿−𝑤₀𝑔(𝑥)(𝑢)𝑔(𝑥)+1≤𝐻,∀𝑥∈𝐺,𝑢∈𝑃⁺₁. (47) Equations (32) and (42) imply that ‖𝐵𝑢‖≤∫_𝐺𝑏(𝑦)𝑢(𝑦)𝑑𝑦, and hence 𝐵𝑢≥𝑐𝑎(𝑥)_𝐺𝑏𝑦𝑢𝑦𝑑𝑦≥‖𝐵𝑢‖𝑒,∀𝑢∈𝑃₁. (48) By (42), (32), (34), and (37), we obtain that 𝐵𝑞≤𝑎(𝑥)_𝐺𝑞𝑦𝑑𝑦<𝑐𝑎(𝑥)=𝑟₀𝑒. (49) By (41) we have ℎ(𝑟₀𝑁)=ℎ(1)=max_{‖𝑢‖≤1}{𝐻(𝑢)}=2−𝑤₀. This and (34) and (42) get that 𝐵𝑔=_𝐺𝑘𝑔𝑦≤𝑥,𝑦𝑑𝑦_𝐺𝑏𝑦𝑔𝑦1𝑑𝑦<2−𝑤₀=𝑟₀ℎ𝑟₀𝑁. (50) 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).

Acknowledgments

The authors are very grateful to the referee for his or her valuable suggestions. This research was supported by the National Natural Science Foundation of China (10871116), the Doctoral Program Foundation of Education Ministry of China (20103705120002), Shandong Provincial Natural Science Foundation, China (ZR2012AM006), and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province.