On Lipschitz Perturbations of a Self-Adjoint Strongly Positive Operator

Abstract

In this paper we study semilinear equations of the form 𝐴𝑢+𝜆𝐹(𝑢)=𝑓, where 𝐴 is a linear self-adjoint operator, satisfying a strong positivity condition, and 𝐹 is a nonlinear Lipschitz operator. As applications we develop Krasnoselskii and Ky Fan type approximation results for certain pair of maps and to illustrate the usability of the obtained results, the existence of solution of an integral equation is provided.

1. Introduction and Preliminaries

The study of abstract operator equations involving linear or nonlinear operators has generated over time useful instruments in the approach of some concrete equations. Therefore, we consider as interesting to present some aspects regarding the semilinear abstract operator equations in Hilbert spaces.

Let 𝐻 be a real Hilbert space endowed with the inner product ⟨⋅,⋅⟩ and the norm ‖⋅‖.

In [1, 2] semilinear equations of the form 𝐴𝑢−𝐹(𝑢)=0 were studied, where 𝐴∶𝐷(𝐴)⊂𝐻→𝐻 is a self-adjoint linear operator with the resolvent set 𝑅(𝐴) and 𝐹∶𝐻→𝐻 is a Gateaux differentiable gradient operator. If there exist real numbers 𝑎<𝑏 such that [𝑎,𝑏]⊂𝑅(𝐴) and ⟨𝐹(𝑢)(𝑣)⟩𝑎≤−𝐹,𝑢−𝑣‖𝑢−𝑣‖²≤𝑏, (1) for all 𝑢,𝑣∈𝐻, 𝑢≠𝑣 (i.e., 𝐹 interacts suitably with the spectrum of 𝐴), then it is proved in [2] that the equation 𝐴𝑢−𝐹(𝑢)=0 has exactly one solution.

The author in [3] presented an existence and uniqueness result for the semilinear equation 𝐴𝑢+𝐹(𝑢)=𝑓, where 𝐴∶𝐷(𝐴)⊂𝐻→𝐻 is a linear maximal monotone operator, satisfying a strong positivity condition, and the nonlinearity 𝐹∶𝐻→𝐻 is a Lipschitz operator.

Let 𝑋 be a real Banach space, ordered by a cone 𝐾. A cone 𝐾 is a closed convex subset of 𝑋 with 𝜆𝐾⊆𝐾  (𝜆≥0), and 𝐾∩(−𝐾)={0}. As usual 𝑥≤𝑦⇔𝑦−𝑥∈𝐾.

Definition 1. Let 𝑀 be a nonempty subset of an ordered Banach space 𝑋 with order ≤. Two mappings 𝑆,𝑇∶𝑀→𝑀 are said to be weakly isotone increasing if 𝑆𝑥≤𝑇𝑆𝑥 and 𝑇𝑥≤𝑆𝑇𝑥 hold for all 𝑥∈𝑀. Similarly, we say that 𝑆 and 𝑇 are weakly isotone decreasing if 𝑇𝑥≥𝑆𝑇𝑥 and 𝑆𝑥≥𝑇𝑆𝑥 hold for all 𝑥∈𝑀. The mappings 𝑆 and 𝑇 are said to be weakly isotone if they are either weakly isotone increasing or weakly isotone decreasing.

In our considerations, the following definition will play an important role. Let ℬ(𝑋) denote the collection of all nonempty bounded subsets of 𝑋 and 𝒲(𝑋) the subset of ℬ(𝑋) consisting of all weakly compact subsets of 𝑋. Also, let 𝐵_𝑟 denote the closed ball centered at 0 with radius 𝑟.

Definition 2 (see [4]). A function 𝜓∶ℬ(𝑋)→ℝ₊ is said to be a measure of weak noncompactness if it satisfies the following conditions.(1) The family ker(𝜓)={𝑀∈ℬ(𝑋)∶𝜓(𝑀)=0} is nonempty, and ker(𝜓) is contained in the set of relatively weakly compact sets of 𝑋.(2)𝑀₁⊆𝑀₂⇒𝜓(𝑀₁)≤𝜓(𝑀₂). (3)𝜓(co(𝑀))=𝜓(𝑀), where co(𝑀) is the closed convex hull of 𝑀.(4)𝜓(𝜆𝑀₁+(1−𝜆)𝑀₂)≤𝜆𝜓(𝑀₁)+(1−𝜆)𝜓(𝑀₂) for 𝜆∈[0,1].(5) If (𝑀_𝑛)_𝑛≥1 is a sequence of nonempty weakly closed subsets of 𝑋 with 𝑀₁ bounded and 𝑀₁⊇𝑀₂⊇⋯⊇𝑀_𝑛⊇⋯ such that lim_𝑛→∞𝜓(𝑀_𝑛)=0, then 𝑀_∞∶=⋂^∞_𝑛=1𝑀_𝑛 is nonempty.

The family ker𝜓 described in (1) is said to be the kernel of the measure of weak noncompactness 𝜓. Note that the intersection set 𝑀_∞ from (5) belongs to ker𝜓 since 𝜓(𝑀_∞)≤𝜓(𝑀_𝑛) for every 𝑛, and lim_𝑛→∞𝜓(𝑀_𝑛)=0. Also, it can be easily verified that the measure 𝜓 satisfies 𝜓𝑀^𝑤=𝜓(𝑀), (2) where 𝑀^𝑤 is the weak closure of 𝑀.

A measure of weak noncompactness 𝜓 is said to be regular if 𝜓(𝑀)=0iff𝑀isrelativelyweaklycompact, (3)subadditive if 𝜓𝑀₁+𝑀₂≤𝜓𝑀₁+𝜓𝑀₂, (4)homogeneous if 𝜓(𝜆𝑀)=|𝜆|𝜓(𝑀),𝜆∈ℝ, (5) and set additive (or has the maximum property) if 𝜓𝑀₁∪𝑀₂=max𝜓𝑀₁,𝜓𝑀₂. (6)

The first important example of a measure of weak noncompactness has been defined by de Blasi [5] as follows: 𝑤(𝑀)=inf{𝑟>0∶thereexists𝑊∈𝒲(𝑋)with𝑀⊆𝑊+𝐵_𝑟, (7) for each 𝑀∈ℬ(𝑋).

Notice that 𝑤(⋅) is regular, homogeneous, subadditive, and set additive (see [5]).

By a measure of noncompactness on a Banach space 𝑋, we mean a map 𝜓∶ℬ(𝑋)→ℝ₊ which satisfies conditions (1)–(5) in Definition 2 relative to the strong topology instead of the weak topology.

Definition 3. Let 𝑋 be a Banach space and 𝜓 a measure of (weak) noncompactness on 𝑋. Let 𝐴∶𝐷(𝐴)⊆𝑋→𝑋 be a mapping. If 𝐴(𝐷(𝐴)) is bounded and for every nonempty bounded subset 𝑀 of 𝐷(𝐴) with 𝜓(𝑀)>0, we have 𝜓(𝐴(𝑀))<𝜓(𝑀); then 𝐴 is called 𝜓-condensing. If there exists 𝑘, 0≤𝑘≤1, such that 𝐴(𝐷(𝐴)) is bounded and for each nonempty bounded subset 𝑀 of 𝐷(𝐴), we have 𝜓(𝐴(𝑀))≤𝑘𝜓(𝑀); then 𝐴 is called 𝑘-𝜓-contractive.

Definition 4 (see [6]). A map 𝐴∶𝐷(𝐴)→𝑋 is said to be ws-compact if it is continuous, and for any weakly convergent sequence (𝑥_𝑛)_𝑛∈ℕ in 𝐷(𝐴) the sequence (𝐴𝑥_𝑛)_𝑛∈ℕ has a strongly convergent subsequence in 𝑋.

Definition 5. A map 𝐴∶𝐷(𝐴)→𝑋 is said to be ww-compact if it is continuous, and for any weakly convergent sequence (𝑥_𝑛)_𝑛∈ℕ in 𝐷(𝐴) the sequence (𝐴𝑥_𝑛)_𝑛∈ℕ has a weakly convergent subsequence in 𝑋.

Definition 6. Let 𝑋 be a Banach space. A mapping 𝑇∶𝐷(𝑇)⊆𝑋→𝑋 is called a nonlinear contraction if there exists a continuous and nondecreasing function 𝜑∶ℝ⁺→ℝ⁺ such that 𝑇𝑥−𝑇𝑦≤𝜑𝑥−𝑦, (8) for all 𝑥,𝑦∈𝐷(𝑇), where 𝜑(𝑟)<𝑟 for 𝑟>0.

In this paper we consider the semilinear equation 𝐴𝑢+𝜆𝐹(𝑢)=𝑓, (9) where 𝐴∶𝐻→𝐻 is a linear self-adjoint operator, satisfying a strong positivity condition, 𝐹∶𝐻→𝐻 is a nonlinear Lipschitz operator, and 𝜆 is a positive parameter. Using the Banach fixed point theorem, we prove an existence and uniqueness result about the considered equation. Thus, we obtain here the same type of result as in [2], replacing the maximal monotonicity of linear part 𝐴 of the semilinear equation with the hypothesis that 𝐴 is self-adjoint. So, the principal result of this paper can be applied in the study of nonlinear Lipschitz perturbations of a linear integral operator with symmetric kernel. Further a result of continuous dependence on the free term and a fixed point theorem are presented. As applications we present some common fixed point theorems and approximation results for a pair of nonlinear mappings. Finally, the existence of solution of an integral equation is provided to illustrate the usability of the obtained results.

2. Results

Theorem 7. Let 𝐴∶𝐻→𝐻 be a linear self-adjoint operator and 𝐹∶𝐻→𝐻 nonlinear, satisfying the following conditions:(i)𝐹 is a Lipschitz operator, that is, there is a constant 𝑀>0 such that 𝐹(𝑥)−𝐹𝑦≤𝑀𝑥−𝑦, (10) for all 𝑥,𝑦∈𝐻;(ii)𝐴 is a strongly positive operator, that is, there is a constant 𝑐>0 such that ⟨𝐴𝑥,𝑥⟩≥𝑐‖𝑥‖², (11) for all 𝑥∈𝐻.
Then the equation 𝐴𝑢+𝜆𝐹(𝑢)=𝑓 has a unique solution for all 𝑓∈𝐻 and 𝜆∈(0;𝑐/𝑀).

Proof. Let us choose 𝑠 in the spectrum of 𝐴. We have 𝑠≥inf{⟨𝐴𝑥,𝑥⟩∣𝑥∈𝐻,‖𝑥‖=1}≥𝑐, (12) and we obtain that every real number 𝜔∈(−∞;𝑐) is in the resolvent set of the operator 𝐴. Consequently, we have Rg(𝐴−𝜔𝐼)=𝐻, (13) for all 𝜔<𝑐, where 𝐼 is the identity of 𝐻.
Let 𝛿<0. We write (9) in the equivalent form 𝐴_𝛿𝑢+𝐹_𝛿(𝑢)=𝑓, (14) where 𝐴_𝛿=𝐴−𝛿𝐼 and 𝐹_𝛿=𝜆𝐹+𝛿𝐼. We have Rg(𝐴_𝛿)=𝐻 and 𝐹_𝛿(𝑥)−𝐹_𝛿𝑦≤(𝜆𝑀+|𝛿|)𝑥−𝑦, (15) for all 𝑥, 𝑦∈𝐻.
Also 𝐴_𝛿𝑥,𝑥=⟨𝐴𝑥,𝑥⟩−𝛿‖𝑥‖²≥(𝑐−𝛿)‖𝑥‖²=(𝑐+|𝛿|)‖𝑥‖²∀𝑥∈𝐻. (16)
From (16) we obtain 𝐴_𝛿𝑥≥(𝑐+|𝛿|)‖𝑥‖∀𝑥∈𝐻. (17)
Consequently, there exists 𝐴_𝛿⁻¹∶𝐻→𝐻 which is linear and continuous, that is 𝐴_𝛿⁻¹∈ℒ(𝐻), the Banach space of all linear and bounded operators from 𝐻 to 𝐻. Moreover, we have 𝐴_𝛿⁻¹_ℒ(𝐻)≤1|𝛿|.𝑐+ (18)
Now (14) can be equivalently written as 𝑢+𝐴_𝛿⁻¹𝐹_𝛿𝑢=𝐴_𝛿⁻¹𝑓. (19)
We consider the operator 𝑇∶𝐻→𝐻 defined by 𝑇𝑢=−𝐴_𝛿⁻¹𝐹_𝛿𝑢+𝐴_𝛿⁻¹𝑓. (20)
Therefore (19) becomes 𝑢=𝑇𝑢, (21) and so, the problem of the solvability of (9) is reduced to the study of fixed points of the operator 𝑇. We have 𝑇𝑥−𝑇𝑦=𝐴_𝛿⁻¹𝐹_𝛿𝑥−𝐴_𝛿⁻¹𝐹_𝛿𝑦=𝐴_𝛿⁻¹𝐹_𝛿𝑥−𝐹_𝛿𝑦≤𝐴_𝛿⁻¹_ℒ(𝐻)𝐹_𝛿𝑥−𝐹_𝛿𝑦≤𝜆𝑀+|𝛿|𝑐+|𝛿|𝑥−𝑦∀𝑥,𝑦∈𝐻. (22)
It results that 𝑇 is a strict contraction from  𝐻  to  𝐻  because 𝜆𝑀<𝑐. According to the Banach fixed point theorem, 𝑇 has a unique fixed point, and thus the proof of Theorem 7 is complete.

Let us consider now the dependence of solution of (9) on the data 𝑓.

Theorem 8. Under the assumptions from the hypothesis of Theorem 7, let 𝑖∈{1,2}, and let 𝑢_𝑖 be the unique solution of the equation 𝐴𝑢+𝜆𝐹(𝑢)=𝑓_𝑖,𝑖∈{1,2};𝑓₁,𝑓₂∈𝐻. (23) Then 𝑢₁−𝑢₂≤1𝑓𝑐−𝜆𝑀¹−𝑓₂. (24)

Proof. According to the equivalent form (19) of (9), we have 𝑢₁−𝑢₂=−𝐴_𝛿⁻¹𝐹_𝛿𝑢₁+𝐴_𝛿⁻¹𝑓₁+𝐴_𝛿⁻¹𝐹_𝛿𝑢₂−𝐴_𝛿⁻¹𝑓₂≤𝐴_𝛿⁻¹𝐹_𝛿𝑢₁−𝐴_𝛿⁻¹𝐹_𝛿𝑢₂+𝐴_𝛿⁻¹𝑓₁−𝐴_𝛿⁻¹𝑓₂≤𝜆𝑀+|𝛿||𝛿|𝑢𝑐+¹−𝑢₂+1|𝛿|𝑓𝑐+¹−𝑓₂. (25)
It results that (𝑐−𝜆𝑀)𝑢₁−𝑢₂≤𝑓₁−𝑓₂, (26) and thus our assertion is proved.

In fact, Theorem 8 establishes the continuous dependence of the solution of (9) on the free term and signifies the stability of the solution.

3. Consequences of Principal Result

The previous results prove that, under the considered assumptions, the operator 𝐴+𝜆𝐹 is invertible for all 𝜆∈(0;𝑐/𝑀) and the inverse 𝐶=(𝐴+𝜆𝐹)⁻¹ is a Lipschitz operator. From (24) it results that the operator 𝐶 satisfies 𝐶𝑓₁−𝐶𝑓₂≤1𝑓𝑐−𝜆𝑀¹−𝑓₂∀𝑓₁,𝑓₂∈𝐻. (27)

Suppose now that 𝑐>1 and 𝜆∈(0;(𝑐−1)/𝑀)⊂(0;𝑐/𝑀). We obtain 𝑀𝜆<𝑐−1 or 1/(𝑐−𝜆𝑀)<1, and, consequently, 𝐶 is a strict contraction. It results that there exists an unique element 𝑢^∗∈𝐻 such that 𝐶(𝑢^∗)=𝑢^∗ which is equivalent to (𝐴+𝜆𝐹)(𝑢^∗)=𝑢^∗. So we obtained the following fixed point theorem.

Theorem 9. Let 𝐴∶𝐻→𝐻 be a linear self-adjoint operator and 𝐹∶𝐻→𝐻 nonlinear, satisfying the following conditions:(i)𝐹 is a Lipschitz operator, that is, there is a constant 𝑀>0 such that 𝐹(𝑥)−𝐹𝑦≤𝑀𝑥−𝑦, (28) for all 𝑥,𝑦∈𝐻;(ii)𝐴 is a strongly positive operator, that is, there is a constant 𝑐>0 such that ⟨𝐴𝑥,𝑥⟩≥𝑐‖𝑥‖², (29) for all 𝑥∈𝐻.
If 𝑐>1 and 𝜆∈(0;(𝑐−1)/𝑀), then the operator 𝐴+𝜆𝐹 has a unique fixed point.

The way used in obtaining Theorem 9 can be applied in the study of the following problem: extracting operators which have the unique fixed point property from a family of operators {𝑉_𝜆/𝜆∈𝐑}.

Let 𝐸 be a Banach space, and ℱ={𝑉_𝜆∶𝐸→𝐸/𝜆∈𝐑}, 𝑉_𝜆 satisfying some constraints for any𝜆∈𝐑. Our intention is to extract a subfamily 𝒢⊂ℱ, so that 𝑉_𝜆 can have a unique fixed point for all 𝑉_𝜆∈𝒢. It is easy to observe, using the same method as in obtaining Theorem 9, that the following result holds.

Theorem 10. If(i) there exists 𝐽⊂𝐑, 𝐽≠𝜙, so that 𝑉_𝜆 is invertible and 𝑉_𝜆⁻¹𝑥−𝑉_𝜆⁻¹𝑦≤𝑔(𝜆)𝑥−𝑦,𝑥,𝑦∈𝐸;𝑔(𝜆)>0, (30) for all 𝜆∈𝐽;(ii) there exists 𝐼⊂𝐽, 𝐼≠𝜙, so that 𝑔(𝜆)<1 for all 𝜆∈𝐼, then 𝑉_𝜆 has a unique fixed point for all 𝜆∈𝐼.

As an application of the observations established above, we develop here the Krasnoselskii and Ky Fan type approximation results for certain pair of maps.

Theorem 11. Let 𝐴, 𝐹, 𝜆, 𝑀 and 𝑐 be as in Theorem 9, 𝑇=(𝐴+𝜆𝐹)⁻¹, and let 𝐻 be an ordered Hilbert space. Let 𝐷 be a nonempty closed convex subset of 𝐻 and 𝜓 a set additive measure of noncompactness on 𝐻. Let 𝑆∶𝐻→𝐻 be a mapping satisfying the following:(i)𝑇(𝐷)⊆𝐷 and 𝑆(𝐷)⊆𝐷,(ii)𝑆 is continuous and 𝜓-condensing,(iii)𝑇 and 𝑆 are weakly isotone.
Then there exists a unique 𝑥^∗∈𝐷 such that 𝐴𝑆𝑥^∗+𝜆𝐹𝑆𝑥^∗=𝑥^∗.

Proof. By Theorem 9, 𝑇=(𝐴+𝜆𝐹)⁻¹∶𝐻→𝐻 is a contraction with contractive constant 𝛼=1/(𝑐−𝜆𝑀)<1. Thus 𝑇 is a shrinking mapping. Now all of the conditions of Corollary 1.18 [7] are satisfied so there exists an 𝑥^∗∈𝐷 such that 𝑇𝑥^∗=𝑆𝑥^∗=𝑥^∗ which implies that (𝐴+𝜆𝐹)𝑜𝑆𝑥^∗=𝑥^∗, and hence 𝐴𝑆𝑥^∗+𝜆𝐹𝑆𝑥^∗=𝑥^∗.

Theorem 12. Let 𝐴, 𝐹, 𝜆, 𝑀, and 𝑐 be as in Theorem 9, 𝑇=(𝐴+𝜆𝐹)⁻¹, and let 𝐻 be an ordered Hilbert space. Let 𝐷 be a nonempty closed convex subset of 𝐻 and 𝜓 a set additive measure of noncompactness on 𝐻. Let 𝑆∶𝐻→𝐻 be a mapping satisfying the following:(i)𝑇(𝐷)⊆𝐷 and 𝑆(𝐷)⊆𝐷,(ii)𝑆 is a nonlinear contraction,(iii)𝑇 and 𝑆 are weakly isotone.
Then there exists a unique 𝑥^∗∈𝐷 such that 𝐴𝑆𝑥^∗+𝜆𝐹𝑆𝑥^∗=𝑥^∗.

Proof. By Theorem 9, 𝑇=(𝐴+𝜆𝐹)⁻¹∶𝐻→𝐻 is a contraction with contractive constant 𝛼=1/(𝑐−𝜆𝑀)<1. Thus 𝑇 is a shrinking mapping. Now all of the conditions of Corollary 1.19 [7] are satisfied so there exists an 𝑥^∗∈𝐷 such that 𝑇𝑥^∗=𝑆𝑥^∗=𝑥^∗ which implies 𝐴𝑆𝑥^∗+𝜆𝐹𝑆𝑥^∗=𝑥^∗.

As an application of Corollaries 1.24 or 1.25 and 1.30 or 1.31 [7], we obtain the following results, respectively

Theorem 13. Let 𝐴, 𝐹, 𝜆, 𝑀, and 𝑐 be as in Theorem 9, 𝑇=(𝐴+𝜆𝐹)⁻¹, and let 𝐻 be an ordered Hilbert space. Let 𝐷 be a nonempty closed convex subset of 𝐻 and 𝜓 a set additive measure of weak noncompactness on 𝐻. Assume that 𝑆,𝑇∶𝐻→𝐻 are sequentially weakly continuous mappings satisfying the following:(i)𝑇(𝐷)⊆𝐷 and 𝑆(𝐷)⊆𝐷,(ii)𝑆 is 𝜓-condensing or 𝑆 is a nonlinear contraction,(iii)𝑇 and 𝑆 are weakly isotone.
Then there exists a unique point 𝑥^∗∈𝐷 such that 𝐴𝑆𝑥^∗+𝜆𝐹𝑆𝑥^∗=𝑥^∗.

Theorem 14. Let 𝐴, 𝐹, 𝜆, 𝑀, and 𝑐 be as in Theorem 9, 𝑇=(𝐴+𝜆𝐹)⁻¹, and let 𝐻 be an ordered Hilbert space. Let 𝐷 be a nonempty closed convex subset of 𝐻 and 𝜓 a set additive measure of weak noncompactness on 𝐻. Assume that 𝑆, 𝑇 satisfy the following:(i)𝑇 is a ww-compact mapping,(ii)𝑆 is continuous ws-compact and 𝜓-condensing or 𝑆 is continuous ws-compact and nonlinear contraction,(iii)𝑇 and 𝑆 are weakly isotone,(iv)𝑇(𝐷)⊆𝐷 and 𝑆(𝐷)⊆𝐷.
Then there exists a unique 𝑥^∗∈𝐷 such that 𝐴𝑆𝑥^∗+𝜆𝐹𝑆𝑥^∗=𝑥^∗.

Theorem 15. Let 𝐻, 𝐴, 𝐹, 𝜆, 𝑀, and 𝑐 be as in Theorem 9 and 𝑇=(𝐴+𝜆𝐹)⁻¹ sequentially weakly continuous. Assume that 𝐷 is nonempty closed bounded convex subset of 𝐻 and 𝑆∶𝐷→𝐻 is sequentially weakly continuous mapping satisfying the following: (i)𝑆(𝐷) is relatively weakly compact,(ii)(𝑥=𝑇𝑥+𝑆𝑦;𝑦∈𝐷)⇒𝑥∈𝐷.
Then there exists a unique point 𝑥^∗∈𝐷 such that (𝐴+𝜆𝐹)𝑜(𝐼−𝑆)𝑥^∗=𝑥^∗.

Proof. By Theorem 9, 𝑇=(𝐴+𝜆𝐹)⁻¹∶𝐻→𝐻 is a strict contraction with contractive constant 𝛼=1/(𝑐−𝜆𝑀)<1. Now Theorem 2.1 [6] implies that there exists an 𝑥^∗∈𝐷 such that 𝑆𝑥^∗+𝑇𝑥^∗=𝑥^∗ which implies that (𝐴+𝜆𝐹)𝑜(𝐼−𝑆)𝑥^∗=𝑥^∗.

Theorem 16. Let 𝐴, 𝐹, 𝜆, 𝑀, and 𝑐 be as in Theorem 9, 𝑇=(𝐴+𝜆𝐹)⁻¹, and let 𝐻 be an ordered Hilbert space. Let 𝐷 be a nonempty closed convex subset of 𝐻 and 𝜓 a set additive measure of noncompactness on 𝐻. Let 𝑆∶𝐻→𝐻 be a mapping satisfying the following:(i)𝑆(𝐷)⊆𝐷, (ii)𝑆 is continuous and 𝜓-condensing or (𝑆 is a nonlinear contraction),(iii)𝑃𝑇 and 𝑆 are weakly isotone,
where 𝑃 is the proximity map on 𝐷. Then there exists 𝑥₀∈𝐷 such that 𝑥₀−𝑇𝑥₀=𝑆𝑥₀−𝑇𝑥₀=𝑑𝑇𝑥₀,𝐷=𝑑𝑇𝑥₀,𝐼_𝐷𝑥₀. (31)

More precisely, either(1)𝑆 and 𝑇 have a common fixed point 𝑥₀∈𝐷, or(2)there exists 𝑥₀∈𝜕𝐷 with 0<𝑇𝑥₀−𝑆𝑥₀=𝑇𝑥₀−𝑥₀=𝑑𝑇𝑥₀,𝐷=𝑑𝑇𝑥₀,𝐼_𝐷𝑥₀. (32)

Proof. Let 𝑃 be the proximity map on 𝐷; that is, for each 𝑥∈𝐻, we have ‖𝑃𝑥−𝑥‖=𝑑(𝑥,𝐷). It is well known that 𝑃 is nonexpansive in 𝐻. As 𝑇 is shrinking map, so 𝑃𝑇∶𝐷→𝐷 is also shrinking mapping. By Theorem 4.1 (or 4.2), there exists 𝑥₀∈𝑀 such that 𝑥₀=𝑆𝑥₀=𝑃𝑇𝑥₀. Thus we obtain, as in Theorem 4.1 [8] the desired conclusion.

Following the proof of Corollary 4.5 [8] and using Theorem 16, we obtain the following common fixed point theorem.

Theorem 17. Let 𝐴, 𝐹, 𝜆, 𝑀, and 𝑐 be as in Theorem 9, 𝑇=(𝐴+𝜆𝐹)⁻¹, and let 𝐻 be an ordered Hilbert space. Let 𝐷 be a nonempty closed convex subset of 𝐻 and 𝜓 a set additive measure of noncompactness on 𝐻. Assume that 𝑆∶𝐻→𝐻 is a mapping satisfying the following:(i)𝑆(𝐷)⊆𝐷, (ii)𝑆 is continuous and 𝜓-condensing or (𝑆 is a nonlinear contraction),(iii)𝑃𝑇 and 𝑆 are weakly isotone,
where 𝑃 is the proximity map on 𝐷. Suppose that 𝑇 satisfies one of the following conditions for each 𝑥∈𝜕𝐷, with 𝑥≠𝑇𝑥:(i)‖𝑇𝑥−𝑦‖<‖𝑇𝑥−𝑥‖ for some 𝑦 in 𝐼_𝐷(𝑥);(ii)there is a 𝛾 such that |𝛾|<1 and 𝛾𝑥+(1−𝛾)𝑇𝑥∈𝐼_𝐷(𝑥);(iii)𝑇𝑥∈𝐼_𝐷(𝑥); (iv)for each 𝛾∈(0,1), 𝑥≠𝛾𝑇𝑥;(v)there exists an 𝛼∈(1,∞) such that, ‖𝑇𝑥−𝑥‖^𝛼≥‖𝑇𝑥‖^𝛼−𝑟^𝛼;(vi)there exists 𝛽∈(0,1) such that, ‖𝑇𝑥−𝑥‖^𝛽≤‖𝑇𝑥‖^𝛽−𝑟^𝛽.
Then 𝐹(𝑆)∩𝐹(𝑇)≠∅.

4. An Application

Fixed point theorems for certain operators have found various applications in differential and integral equations (see [7–10] and references therein). In this section, we present an application of our Theorem 7 to establish a solution of a nonlinear integral equation.

Let 𝐾∶[0,1]×[0,1]→𝐑 be a continuous function, and suppose that 𝐾 is symmetric (i.e., 𝐾(𝑥,𝑦)=𝐾(𝑦,𝑥) for all 𝑥,𝑦∈[0,1]). We consider the linear operator 𝐵∶𝐿²_𝐑[0,1]→𝐿²_𝐑[0,1] defined by 𝐵𝑢(𝑡)=∫¹₀𝐾(𝑡,𝑠)𝑢(𝑠)𝑑𝑠. It is easy to observe that 𝐵 is a self-adjoint operator. Now let 𝐴∶𝐿²_𝐑[0,1]→𝐿²_𝐑[0,1] defined by 𝐴𝑢(𝑡)=𝑢(𝑡)+∬¹₀𝐾(𝑡,𝑠)𝐾(𝑠,𝑧)𝑢(𝑧)𝑑𝑧𝑑𝑠; that is, 𝐴=𝐼+𝐵² where 𝐼 is the identity of 𝐿²_𝐑[0,1]. 𝐴 is a self-adjoint strongly positive operator satisfying ⟨𝐴𝑢,𝑢⟩₂=⟨𝑢,𝑢⟩₂+⟨𝐵²𝑢,𝑢⟩₂≥‖𝑢‖²₂(⟨⋅,⋅⟩₂ and ‖⋅‖₂ signify the inner product and the norm in 𝐿²_𝐑[0,1]).

Define 𝑓∶[0,1]×𝐑→𝐑, (𝑡,𝑢)↦𝑓(𝑡,𝑢), having partial derivative of first order in the second variable 𝑢 and |(𝜕𝑓/𝜕𝑢)(𝑡,𝑢)|≤𝑀 for all (𝑡,𝑢)∈[0,1]×𝐑(𝑀>0). 𝐹∶𝐿²_𝐑[0,1]→𝐿²_𝐑[0,1] defined by 𝐹𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)) is a Lipschitz operator of constant 𝑀 from 𝐿²_𝐑[0,1] into 𝐿²_𝐑[0,1].

Define nonlinear integral equation 𝑢(𝑡)+¹₀¹₀𝐾(𝑡,𝑠)𝐾(𝑠,𝑧)𝑢(𝑧)𝑑𝑧𝑑𝑠+𝜆𝑓(𝑡,𝑢(𝑡))=𝑔(𝑡),𝑔∈𝐿²_𝐑[0,1], (33) where 𝜆∈(0;1/𝑀). All of the conditions of Theorem 7 are satisfied, so the above-mentioned integral equation has a unique solution for all 𝜆∈(0;1/𝑀). Moreover, by Theorem 8 the solution depends continuously on the free term 𝑔.

Acknowledgment

The work of first author was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0087.