# Existence of Fixed Points for a Class of Decreasing Operators with Parameter and Applications

Authors: Xingchang Li; Shiqin Tian
Journal: Journal of Mathematics
 (2022)

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## Abstract

This paper studies the existence of fixed points for a class of decreasing operators with parameter in real Banach spaces. The existence theorems of fixed point are obtained as when the parameter is increasing, there will still be a large fixed point. These results have reduced the requirements of convexity, compactness, and lattice structure of spaces. By this new method, the existence of solutions for a class of second-order differential equations with parameter in infinite intervals is studied.


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## Body

## 1. Introduction and Preliminaries

The solvability of nonlinear equations can be transformed into the existence of fixed points for operators. The existence of fixed points has been considered widely [1–11]. There are many conclusions about fixed points for increasing operators, but less for decreasing operators [6–11]. The partial-order method is subject to many constraints to verify the convergence of decreasing operators, and it is also difficult to construct iterative conditions.The existence theory of fixed points for decreasing operators has been developed rapidly in the field of microeconomics [12–16] and has been found having various applications to economic models. In [15], the existence of game equilibrium for alternative strategy is proved. However, the above results require convexity, compactness, or lattice structure in player’s strategy space.Inspired by [16], this paper studies a class of decreasing operators with parameter in real Banach spaces by cone theory, and the existence theorems of fixed points for decreasing operators with parameter are obtained. It is proved that when the parameter is increasing, there will still be a larger fixed point. This result reduces the requirements for convexity, compactness, and lattice structure of spaces. As an application of the results in this paper, we study the existence of solutions to the initial value problem of a class of second-order initial value differential equations in infinite intervals. It is proved that the solution of the problem is monotonically increasing with respect to the parameters.The result of this study can also be used as a mathematical tool to solve the comparative static analysis of the existence of Nash equilibrium in noncooperative alternative strategy games. It provides a mathematical means for us to study the phenomenon of diminishing returns, mutual substitution, incomplete competition, and various modern economic phenomena characterized by “difference.”Here some concepts and theorems are given, which are related to incomplete preference.LetE be a nonempty set. An ordering relation ≺¯ on E may satisfy the following axioms:(i)
Reflexive:x≺¯x, for any x∈E;(ii)
Symmetry: for any x,y∈E, x≺¯y and y≺¯x imply x=y(iii)
Transitive: for any x,y,z∈E, x≺¯y and y≺¯z imply x≺¯z(iv)
Complete: if x≺¯yandy≺¯x, for any x,y∈E, there is at least one inequality to be establishedDefinition 1 (see [17]).
LetE be a nonempty set. An order relation ≺¯ defined among certain elements of E is said to be partial order if the order relation satisfies reflexive, transitivity, and antisymmetry axioms. Then, E≺¯ is called a poset.Definition 2 (see [17]).
LetE be the Banach space and θ be a zero element of E. Let P be a nonempty convex closed sets of E. If the following conditions are satisfied,(1)
x∈P,λ≥0⇒λx∈P(2)
x∈P,−x∈P⇒x=θ; then, P is called a coneDefinition 3 (see [17]).
A coneP⊂E is called normal if there is a constant N>0, such that θ≺¯x≺¯y implies x≤Ny, for all x,y∈E. The least positive constant N satisfying the above inequality is called the normal constant of P. We call P a regular cone. If every increasing sequence in E has an upper bound in order, it must have a limit. If, for arbitrary x,y∈E, supx,y and infx,y exist, the cone P⊂E is called minimal.Definition 4 (see [18]).
LetE,≺¯ be a partially ordered set and D⊂E. The operator A:D⟶E is said to be a decreasing operator if, for all x,y∈D,x≺¯y implies Ay≺¯Ax.Definition 5 (see [18]).
LetD⊂E and the operator A:D⟶E be continuous and bounded. The operator A:D⟶E is said to be a condensing operator if, for any nonrelatively compact and bounded set, S⊂D implies αAS<αS, where α means noncompact measurement in E.Remark 1.
If the operatorA satisfies continuity and compactness, then it is condensing.Lemma 1 (see [11]).
LetP be a normal cone in a real Banach space E and A:u0,v0⟶E be decreasing. Suppose that there is a∈E which satisfies(1)a≺¯Aa,a≺¯A2a.
If the operatorA is condensing, then A has a fixed point in a,Aa.Lemma 2 (see [18]).
LetP be a normal cone in a real Banach space E and A:u0,v0⟶E be an increasing and condensing operator such that u0≺¯Au0;Av0≺¯v0. Then, A has a maximal fixed point u∗ and a minimal fixed point u∗ in u0,v0; moreover,(2)u∗=limn⟶+∞un=limn⟶+∞An−1u0,v∗=limn⟶+∞vn=limn⟶+∞An−1v0.Lemma 3 (Sadovskii [18]).
LetE be a real Banach space, D⊂E be nonempty, closed bounded convex, and A:D⟶D be condensing or completely continuous. Then, A has a fixed point in D.


## 2. Main Results and Proof

LetE be a Banach space and P be a cone of E. We define a partial ordering ≺¯ with respect to P by x≺¯y if y−x∈P.Theorem 1.
LetP be a normal cone in a real Banach space E, T be a nonempty partially ordered set, and ελ=x∈E|x=Fx,λ,λ∈T,xλ∈ελ. Assume that, for any λ∈T,F⋅,λ:E⟶E is a condensing operator which satisfies the following conditionsA1 For any x∈E,Fx,⋅ is increasing on λ, and for any λ∈T,F⋅,λ is decreasing on xA2 For any λ∈T, there exists a a∈E, such that a≺¯Fa,λanda≺¯F2a,λ
Then, the following is true:(1)
For anyλ∈T,F⋅,λ has a fixed point xλ in E; if F⋅,λ has another fixed point x^λ≠xλ in E, then xλ and x^λ are incomparable(2)
For anyλ1,λ2∈T,λ1≤λ2,xλ1∈ελ1, there exists a xλ2∈ελ2 such that xλ1≺¯xλ2Proof.
(1)
For anyλ∈T, Lemma 1 implies that the operator F⋅,λ has a fixed point xλ in a,Fa,λNext, it will be proved that any two unequal fixed points of operatorF⋅,λ in E cannot be compared. If xλ and x^λ are any two unequal fixed points of operator F⋅,λ in E, then xλ=Fxλ,λandx^λ=Fx^λ,λ. Suppose that xλ and x^λ can be compared. Without loss of generality, let xλ≺¯x^λ and x^λ≠xλ, that is, xλ≺x^λ. Since F⋅,λ is decreasing on x, we know that Fx^λ,λ≺¯Fxλ,λ. And because xλ=Fxλ,λ,x^λ=Fx^λ,λ, we have x^λ≺¯xλ. Yet this is in contradiction to xλ≺x^λ. So, any two unequal fixed points of operator F⋅,λ in E cannot be compared.(2)
For anyλ1,λ2∈T,λ1≤λ2 and xλ1∈ελ1, it is easy to know that xλ1≺¯Fxλ1,λ2 by the condition A1 that Fxλ1,⋅ is increasing on λ. Noticing that F⋅,λ is decreasing on x, we know that(3)F2xλ1,λ2≺¯Fxλ1,λ2,F2xλ1,λ2≺¯F3xλ1,λ2,where F2xλ,λ=FFxλ,λ,λ. F2⋅,λ is increasing on x. So, from condition A2, the following conclusion can be obtained:(4)F2xλ1,λ2≺¯F4xλ1,λ2,F3xλ1,λ2≺¯Fxλ1,λ2.
(3) and (4) show that(5)xλ1≺¯F2xλ1,λ2≺¯⋯≺¯F2n+2xλ1,λ2≺¯⋯≺¯F2n+1xλ1,λ2≺¯⋯≺¯F3xλ1,λ2≺¯Fxλ1,λ2.
Let(6)un=F2n+2xλ1,λ2,vn=F2n+1xλ1,λ2,n=0,1,2,3….
By (3) and (5), we have(7)u0≺¯v0,u0≺¯F2u0,λ2,F2v0,λ2≺¯v0,(8)u0≺¯u1≺¯⋯≺¯un≺¯⋯≺¯vn≺¯⋯≺¯v1≺¯v0.
SinceF2⋅,λ2 is an increasing and condensing operator on x, which satisfies the conditions (7) and (8), Lemma 2 implies F2⋅,λ2 has a maximal fixed point v∗ and a minimal fixed point u∗ in u0,v0. Therefore, we obtain(9)u∗=limn⟶+∞un=limn⟶+∞F2n+2x∗,λ2,v∗=limn⟶+∞vn=limn⟶+∞F2n+1x∗,λ2.
Letxλ2 be a fixed point of F2⋅,λ2. Then,(10)F2Fxλ2,λ2,λ2=F3xλ2,λ2=FF2xλ2,λ2,λ2=Fxλ2,λ2,and thus, Fxλ2,λ2 is also a fixed point of F2⋅,λ2.
We distinguish two cases to discuss:(i)
Ifu∗=v∗, that is, F2⋅,λ2 has only one fixed point, then xλ2=Fxλ2,λ2. This proves that F⋅,λ2 must have a fixed point xλ2 in the ordered interval u0,v0, and xλ1≺¯xλ2.(ii)
Ifu∗≺v∗, (10) implies that Fu∗,λ2 and Fv∗,λ2 are fixed points of F2⋅,λ2. Since F⋅,λ is decreasing on x, we have(11)u∗≺¯Fv∗,λ2≺¯Fu∗,λ2≺¯v∗.
For∀x∈u∗,v∗, since F⋅,λ2 is decreasing on x, we have Fv∗,λ2≺¯Fx,λ2≺¯Fu∗,λ2. By (11), we have Fx,λ2∈u∗,v∗. Since P is a normal cone of real E, we have that u∗,v∗ closed bounded convex ordered interval. So, F⋅,λ satisfies the conditions in Lemma 3. Thus, Lemma 3 implies that the operator F⋅,λ has a fixed point xλ2 in u∗,v∗, and xλ1≺¯xλ2.
The proof of Theorem1 is completed.Remark 2.
For operatorF, the requirements of lattice structure, continuity, and compactness are weaker.Theorem 2.
LetP be a regular cone in a real Banach space E, T be a nonempty partially ordered set, and ελ=x∈E|x=Fx,λ,λ∈T,xλ∈ελ. Assume that, for any λ∈T,F⋅,λ:E⟶E is a condensing operator which satisfies the following conditions:A1 For any x∈E,Fx,⋅ is increasing in λ, and for any λ∈T,F⋅,λ is decreasing in xA2 For any λ∈T, there exists a a∈E, such that a≺¯Fa,λanda≺¯F2a,λ
Then, the following is true.(1)
For anyλ∈T,F⋅,λ has a fixed point xλ in E; if F⋅,λ has another fixed point x^λ≠xλ in E, then xλ and x^λ are incomparable.(2)
For anyλ1,λ2∈T,λ1≤λ2,xλ1∈ελ1, there exists xλ2∈ελ2 such that xλ1≺¯xλ2.Proof.
Similar to the proof of Theorem1, by Lemma 3, we can get the result.
In order to support the main results, let us consider the following a nontrivial example that shows that, for anyx∈E,Fx,⋅ is increasing in λ, and for any λ∈T,F⋅,λ is decreasing in x.Example 1.
For anyx=x1,x2∈R2,λ∈T,T=0,1, we define Fx,λ=λ−1/2x2,λ−2/3x1. With the standard product order, it is easy to check that, for any x∈E,Fx,⋅ is increasing in λ, and for any λ∈T,F⋅,λ is decreasing in x; for any λ∈T, θ∈E, such that θ≺¯Fθ,λ,θ≺¯F2θ,λ. Then, the equation x=Fx,λ,λ∈T, has a nontrivial solution xλ=3/4λ,1/2λ, and xλ is increasing in λ.


## 3. Applications

In this section, we use Theorem1 to show the existence of unique solution for the second-order initial value problem:(12)u″t=λat+λa0t+∑i=1naitutai+∫01kt,susds−1,0≤t<+∞,u0=0,u′0=b0,where 0<λ<+∞,at,a0t,aiti=1,2,3,… are nonnegative continuous functions in 0≤t<+∞, and kt,s is nonnegative continuous function, for all 0≤s≤t<+∞, 0<a1<⋯<an=1,b0≥0. Then, problem (12) has a C2 positive solution uλ∗t, and uλ∗t is increasing on λ.Proof.
Supposing thatut is C2 function, for all 0≤t<+∞, if(13)u′t=u′0+∫0tu″sds,0≤t<+∞,is substituted into(14)ut=u0+∫0tu′sds,0≤t<+∞,then the following conclusion holds:(15)ut=u0+tu′0+∫0tt−su″sds,0≤t<+∞.
So, the nonnegative solutionut of the initial value problem (12) for all 0≤t<+∞ is equivalent to the fixed point of the integral operator so that ut is the nonnegative C2 solution of the initial value problem (12) which is equivalent to that ut which is the nonnegative continuous solution of the following integral equation:(16)ut=b0t+λ∫0tt−sasds+λ∫0tt−sa0t+∑i=1naisusai+∫0tks,s1us1ds1−1ds,0≤t<+∞.
LetJ=0,∞,CJ=u:utis continuous inJ,E=u∈CJ:supt∈Jut/t+1<∞. It is easy to know that E is a Banach space with maximum norm u=supt∈Jut/t+1. Let C+J=u∈CJ:ut≥0,∀t∈J, P=u∈E:ut≥0,∀t∈J, and P be a normal cone in E.
Consider the operator:(17)Au,λ=b0t+λ∫0tt−sasds+λ∫0tt−sa0s+∑i=1naisusai+∫0tks,s1us1ds1−1ds,0≤t<+∞.
For any fixedλ, Ax,⋅ is a completely continuous operator from P into P, and thus, Ax,⋅ is condensing. For any fixed λ, Ax,⋅ is also a decreasing operator from P into P and for any 0<λ<+∞, Au,⋅ is increasing on λ.
Lettingut=θ in (17), we have(18)Aθ,λ=b0t+∫0tt−sasds≥θ.
For∀t≥0,λ>0,(19)A2θ,λ=b0t+∫0tt−sasds+∫0tt−sa0s+∑i=1naisAθ,λai+∫0tks,s1Aθ,λds1−1ds≥θ.
So, the condition of Theorem1 holds. By Theorem 1, we get that A has a fixed point uλ∗t in P. Then, problem (12) has a C2 positive solution uλ∗t, and uλ∗t is increasing on λ.
The problem in this section is proved.



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