# Fixed Point Theorems of Superlinear Operators with Applications

Authors: Shaoyuan Xu; Yan Han
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, by using the partial order method and monotone iterative techniques, the existence and uniqueness of fixed points for a class of superlinear operators are studied, without requiring any compactness or continuity. As corollaries, the new fixed point theorems forα-convex operators α>1, e-convex operators, positive α homogeneous operator α>1, generalized e-convex operator, and convex operators are obtained. The results are applied to nonlinear integral equations and partial differential equations.


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## Body

## 1. Introduction

Linear operators are a kind of operators with good properties and rich theoretical results, which have formed a classical branch in functional analysis. However, in order to solve the fixed point problems involving operators or equations in practical applications, we need a large number of nonlinear operators, including two classes of significant operators, namely, superlinear operators and sublinear operators. Since some of these operators have concavity or convexity, they bring convenience to the related research. The concepts of concave operators and convex operators were proposed in 1960s, which attracted people’s great interest. Many authors obtained a lot of meaningful results, see [1–27]. Among them, α-convex operators α>1 [12, 17], e-convex operators [13], and generalized e-convex operators [16] are a very important class of convex operators. It has important applications in many fields. However, it was difficult to study the α-convex operators α>1 (including positive α-homogeneous operators) and e-convex operators because they had strong superlinear properties [13] and described nonlinear problems [12]. Until now, the results are still very few and not very ideal (see [7], P457). Therefore, under what conditions, these operators have a unique fixed point remains a very important and meaningful problem.In [7], a fixed point theorem for a class of superlinear operators was obtained by topological degree method under the condition that there are inverse upward and downward solutions. In [17], using some results of δ-concave operator, the author transformed the positive α-homogeneous superlinear operator into δ-concave operator and studied the existence and uniqueness of the solutions of positive α-homogeneous superlinear operator equations. In [13], the existence of fixed points was investigated when the α-convex operators α>1 was a strict set contraction. In [16], Zhao and Du obtained the existence of fixed points of generalized e-concave operators and generalized e-convex operators. As an application, the singular boundary value problems for second order differential equations were discussed. In [10], according to the properties of totally ordered sets, the existence and uniqueness of new positive fixed points for a class of superlinear homogeneous operators were studied in abstract spaces. The results were applied to a class of superlinear Hammerstein-type integral equations.In this paper, we study a class of superlinear operators without requiring any compactness or continuity and obtain some new fixed point theorems for superlinear operators by using the partial order and the monotone iteration which are different from those mentioned above in the literature. As corollaries, new fixed point theorems forα-convex operators α>1, e-convex operators, positive α homogeneous operator α>1, generalized e-convex operator, and convex operators are obtained. The results are applied to nonlinear integral equations and partial differential equations.


## 2. Preliminaries

LetE be a real Banach space and P be a subset of E, θ denotes the zero element of E and intP denotes the interior of P. The subset P is called a cone if:
(i)
x∈P and λ≥0, then λx∈P(ii)
x∈P and −x∈P, then x=θ.Given a coneP⊂E, we define a partial ordering ≤ with respect to P by x≤y if and only if y−x∈P. We shall write x<y if x<y and x≠y, while x≪y will stand for y−x∈intP. A cone P is called normal if there is a number K>0 such that for all x,y∈P,
(1)θ≤x≤yimpliesx≤Ky.The least positive number satisfying the above inequality is called the normal constant ofP.LetD⊂E, A:D⟶E be an operator. If there exists a point x∈D such that Ax=x, then x is called a fixed point of A in D. Let u0,v0∈E, and u0≤v0, then
(2)u0,v0=x∈E∣u0≤x≤v0,is said to be an ordering interval. The operatorA:D⟶E is said to be increasing; if for any x,y∈D, x≤y implies Ax≤Ay.Throughout this paper, we always assume thatE is a real Banach space and ≤ is a partial ordering with respect to P; θ denotes the null element of E.Definition 1 (see [19]).
LetD⊂E. D is called a star-shaped subset of the real Banach space E; if for any x∈D and 0<t<1, it holds that tx∈D.
Note that a convex setD in the real Banach space E with the null element θ∈D is a star-shaped subset of E. Especially, any cone P in the real Banach space E is a star-shaped subset of E.Definition 2 (see [7]).
LetD be a star-shaped subset of the real Banach space E and A:D⟶D be an operator, then
(1)
A is said to be sublinear, if for all x∈D and 0<t<1, Atx≥tAx;(2)
A is said to be superlinear, if for all x∈D and 0<t<1, Atx≤tAx.Definition 3 (see [4, 7]).
Lete>θ. A:P⟶P is called an e-concave operator, if
(i)A is e-positive, that is, AP−θ⊂Pe, where
(3)Pe=x∈Ethereexistλ,μ>0,suchthatλe≤x≤μe.
(ii) For allx∈Pe and 0<t<1, there exists η=ηt,x>0 such that
(4)Atx≥1+ηtAx,
whereη=ηt,x is called the characteristic function of A.
Similarly, if in the above definition, (ii) is replaced by the following (ii′):
(ii′) For all x∈Pe and 0<t<1, there exists η=ηt,x>0 such that
(5)Atx≤1−ηtAx,where η=ηt,x is called the characteristic function of A; then, A:P⟶P is called an e-convex operator.Definition 4 (see [16]).
Lete>θ. A:P⟶P is called a generalized e-concave operator, if
(i)Ae∈Pe, where
(6)Pe=x∈Ethereexistλ,μ>0,suchthatλe≤x≤μe.
(ii) For allx∈Pe and 0<t<1, there exists η=ηt,x>0 such that
(7)Atx≥1+ηtAx,
whereη=ηt,x is called the characteristic function of A.
Similarly, if in the above definition, we replace (ii) by the following (ii′):
(ii′) For all x∈Pe and 0<t<1, there exists η=ηt,x>0 such that
(8)Atx≤1+ηt−1Ax,
whereη=ηt,x is called the characteristic function of A; then, A:P⟶P is called a generalized e-convex operator.Definition 5 (see [4, 17]).
LetA:P⟶P be an operator, α>0.
(1)
A is said to be an α-concave operator, if for any x∈P and 0<t<1, Atx≥tαAx(2)
A is said to be an α-convex operator, if for any x∈P and 0<t<1, Atx≤tαAx(3)
A is said to be a positive α-homogeneous operator, if for any x∈P and t>0, Atx=tαAx.Remark 6 (see [9]).
Anyα-convex operator α>1 must be an e-convex operator, where the characteristic function ηt,x=1−tα−1.Remark 7.
Clearly, anye-convex operator must be a superlinear operator. Thus, α-convex operators α>1 and e-convex operators are special superlinear operators.Remark 8.
Any generalizede-convex operator A must be a superlinear operator if ηt,x≥1/t2 for any x∈Pe and 0<t<1 where η=ηt,x is the characteristic function of A. Thus, generalized e-convex operators are special superlinear operators under suitable conditions.Remark 9.
NotingA:P⟶P is called a convex operator if Atx+1−ty≤tAx+1−tAy for all x,y∈P and 0<t<1; we can easily see that any convex operator A:P⟶P satisfying Aθ=θ must be a superlinear operator.


## 3. Main Results

In [18], the author proved that there was no operator which was decreasing and e-convex, where e>θ. Now, we give some important theorems of increasing superlinear operators, which generalize increasing e-convex operators.Theorem 10.
LetP be a normal cone in E and A:P⟶P be an increasing superlinear operator. If there exist a∈0,1 and u0,v0∈P, u0<v0 such that u0≤Au0,Av0≤av0, then the operator A has a unique fixed point x∗ in u0,v0. For any x0∈u0,v0 and iterated sequence xn=Axn−1n=1,2,⋯, we have xn−x∗⟶0n⟶∞.Proof.
We firstly prove the existence of the fixed point. Letun=Aun−1,vn=Avn−1. Since A is increasing, we have
(9)u0≤u1≤⋯≤un≤⋯≤vn≤⋯≤v1≤v0.
Takev0′=v0,vn′=a−1Avn−1′n=1,2,⋯, then
(10)u0≤vn′≤v0n=1,2,⋯,(11)vn≤anvn′n=1,2,⋯.
Equation (10) can be proved by iteration. Indeed, for n=1, we get
(12)u0≤Au0≤a−1Au0≤a−1Av0′=a−1Av0′=v1′≤a−1av0=v0,
which means equation (10) holds when n=1. Suppose that equation (10) holds for n=k, that is
(13)u0≤vk′≤v0.
By the fact thatA is increasing, we obtain Au0≤Avk′≤Av0, then
(14)u0≤Au0≤a−1Au0≤a−1Av0≤a−1Avk′≤a−1Av0=v1′≤v0,
which impliesu0≤vk+1′≤v0. Thus, equation (10) holds for all n∈ℕ. Now, we prove that equation (11) is also true. Indeed, if n=1, then
(15)v1=Av0=aa−1Av0=aa−1Av0′=av1′,
that is, (11) holds when n=1. Suppose (11) holds for n=k, i.e.,
(16)vk≤akvk′.
It follows thatAvk≤akAvk′ since A is an increasing superlinear operator. Hence, we see that
(17)vk+1=Avk≤Aakvk′≤akAvk′=ak+1a−1Avk′=ak+1vk+1′,
which givesvk+1≤ak+1vk+1′. So, equation (11) holds for all n∈ℕ.Combining equations (9), (10), and (11), for any p≥1, we know
(18)θ≤vn−un≤anvn′−un≤anv0−anu0=anv0−u0,(19)θ≤un+p−un≤vn−un,θ≤vn−vn+p≤vn−un.By equations (18) and (19) and the normality of P, we can check that vn−un⟶0n⟶∞, which implies that {un} and {vn} are Cauchy sequences in E. Then, there exist u∗, v∗∈u0,v0 such that un⟶u∗, vn⟶v∗n⟶∞, and u∗=v∗. Denote x∗=u∗=v∗. We have un≤u∗≤v∗≤vn by (9). Therefore,
(20)un+1=Aun≤Au∗≤Av∗≤Avn=vn+1.Letn⟶∞ in (13), then u∗≤Au∗≤Av∗≤v∗. This gives u∗=Au∗=Av∗=v∗; that is, the operator A has a fixed point x∗ in u0,v0.Next, we prove the uniqueness of the fixed point. If there existsx¯∈ [u0, v0] such that Ax¯=x¯, then u0≤x¯≤v0. By the monotonicity of A, we see Au0≤Ax¯≤Av0, i.e., u1≤x¯≤v1. It is easy to deduce that un≤x¯≤vn, for any n≥1. So x¯=x∗ as n⟶∞.At last, for anyx0∈ [u0, v0], the sequence xn=Axn−1n=1,2,⋯ satisfies
(21)un≤xn≤vnn=1,2,⋯,by iteration. Lettingn⟶∞, we know xn⟶x∗ (n⟶∞).Similarly, if the superlinear operator has an upward solution, we have the following result.Theorem 11.
LetP be a normal cone in E and A:P⟶P be an increasing superlinear operator. If there exist a∈0,1 and u0,v0∈P, u0<v0 such that au0≤Au0,Av0≤v0, then the equation Ax=ax has a unique fixed point x∗ in u0,v0. For any x0∈u0,v0 and the iterated sequence xn=Axn−1n=1,2,⋯, we have xn−x∗⟶0n⟶∞.Proof.
LetB=a−1A, then
(22)Bu0=a−1Au0≥a−1au0=u0,Bv0=a−1Av0≤a−1v0.
For anyx∈P and 0<t<1, we obtain
(23)Btx=a−1Atx≤a−1tAx=tBx.Thus,B is a superlinear operator which satisfies all conditions of Theorem 10. The conclusions are true by Theorem 10.Similar to Theorem10, we immediately get the following result.Theorem 12.
LetP be a normal cone in E and A:P⟶P be an increasing superlinear operator. If there exists ε∈0,1 such that Aθ>θ, A3θ≤εA2θ, then the operator A has a unique fixed point x∗ in Aθ,A2θ. For any x0∈Aθ,A2θ and iterated sequence xn=Axn−1n=1,2,⋯, we have xn−x∗⟶0n⟶∞.Proof.
We use Theorem10 to give the proof of Theorem 12. Set u0=θ,v0=A2θ. Then, u0,v0∈P. Since the operator A is increasing and Aθ>θ, we have A2θ≥Aθ. Obviously, we have A2θ>Aθ (otherwise if A2θ=Aθ, then A3θ=A2θ≤εA2θ0<ε<1, which implies that A2θ=θ, so Aθ=θ. This is a contradiction since Aθ>θ.
Now lettinga=ε∈0,1, we see that
(24)u0=Aθ≤A2θ=Au0,Av0=A3θ≤εA2θ=av0.
So, all conditions of Theorem10 are satisfied. By Theorem 10, we know that the conclusions of Theorem 12 hold true.Remark 13.
Compared with ([7], Theorem 3.1), in order to obtain the existence and uniqueness of positive fixed points, the superlinear operator A:P⟶P in Theorem 10 and Theorem 11 does not need any compactness or continuity. It is quite different from [7] (Theorem 3.1), which required that A:P⟶P is a condensing operator.Remark 14.
Since superlinear operators include three classes of operators: generalizede-convex operators, e-convex operators, and α-convex operators, Theorem 10 and Theorem 11 improve or generalize lots of famous results in [5, 7, 9, 12–17].Corollary 15.
LetP be a normal cone in E and A:P⟶P be an increasing e-convex operator. If there exist a∈0,1 and w0,v0∈P, w0<v0 such that w0≤Aw0,Av0≤av0, then the operator A has a unique fixed point x∗ in w0,v0. For any x0∈w0,v0 and iterated sequence xn=Axn−1n=1,2,⋯, we have xn−x∗⟶0n⟶∞.Corollary 16.
LetP be a normal cone in E and A:P⟶P be an increasing e-convex operator. If there exist a∈1,∞ and w0,v0∈P, w0<v0 such that aw0≤Aw0,Av0≤v0, then the equation Ax=ax has a unique fixed point x∗ in w0,v0. For any x0∈w0,v0 and the iterated sequence xn=Axn−1n=1,2,⋯, we have xn−x∗⟶0n⟶∞.Corollary 17.
LetP be a normal cone in E and A:P⟶P be an increasing α-convex (α>1) operator. If there exist a∈0,1 and u0,v0∈P, u0<v0 such that u0≤Au0,Av0≤av0, then the operator A has a unique fixed point x∗ in u0,v0. For any x0∈u0,v0 and the iterated sequence xn=Axn−1n=1,2,⋯, we have xn−x∗⟶0n⟶∞.Corollary 18.
LetP be a normal cone in E and A:P⟶P be an increasing α-convex (α>1) operator. If there exist a∈1,∞ and u0,v0∈P, u0<v0 such that au0≤Au0,Av0≤v0, then the equation Ax=ax has a unique fixed point x∗ in u0,v0. For any x0∈u0,v0 and the iterated sequence xn=Axn−1n=1,2,⋯, we have xn−x∗⟶0n⟶∞.Corollary 19.
LetP be a normal cone in E and A:P⟶P be an increasing positive αα>1 homogeneous operator. If there exist a∈0,1 and u0,v0∈P, u0<v0 such that u0≤Au0,Av0≤av0, then the operator A has a unique fixed point x∗ in u0,v0. For any x0∈u0,v0 and the iterated sequence xn=Axn−1n=1,2,⋯, we have xn−x∗⟶0n⟶∞.Corollary 20.
LetP be a normal cone in E and A:P⟶P be an increasing positive αα>1 homogeneous operator. If there exist a∈1,∞ and u0,v0∈P, u0<v0 such that au0≤Au0,Av0≤v0, then the equation Ax=ax has a unique fixed point x∗ in u0,v0. For any x0∈u0,v0 and the iterated sequence xn=Axn−1n=1,2,⋯, we have xn−x∗⟶0n⟶∞.Corollary 21.
LetP be a normal cone in E and A:P⟶P be an increasing generalized e-convex operator satisfying ηt,x≥1/t2 for any x∈Pe and 0<t<1 where η=ηt,x is the characteristic function of A. If there exist a∈0,1 and w0,v0∈P, w0<v0 such that w0≤Aw0,Av0≤av0, then the operator A has a unique fixed point x∗ in w0,v0. For any x0∈w0,v0 and iterated sequence xn=Axn−1n=1,2,⋯, we have xn−x∗⟶0n⟶∞.Corollary 22.
LetP be a normal cone in E and A:P⟶P be an increasing generalized e-convex operator satisfying ηt,x≥1/t2 for any x∈Pe and 0<t<1 where η=ηt,x is the characteristic function of A. If there exist a∈1,∞ and w0,v0∈P, w0<v0 such that aw0≤Aw0,Av0≤v0, then the equation Ax=ax has a unique fixed point x∗ in w0,v0. For any x0∈w0,v0 and the iterated sequence xn=Axn−1n=1,2,⋯, we have xn−x∗⟶0n⟶∞.Corollary 23.
LetP be a normal cone in E and A:P⟶P be an increasing convex operator satisfying Aθ=θ. If there exist a∈0,1 and w0,v0∈P, w0<v0 such that w0≤Aw0,Av0≤av0, then the operator A has a unique fixed point x∗ in w0,v0. For any x0∈w0,v0 and iterated sequence xn=Axn−1n=1,2,⋯, we have xn−x∗⟶0n⟶∞.Corollary 24.
LetP be a normal cone in E and A:P⟶P be an increasing convex operator satisfying Aθ=θ. If there exist a∈1,∞ and w0,v0∈P, w0<v0 such that aw0≤Aw0,Av0≤v0, then the equation Ax=ax has a unique fixed point x∗ in w0,v0. For any x0∈w0,v0 and the iterated sequence xn=Axn−1n=1,2,⋯, we have xn−x∗⟶0n⟶∞.Remark 25.
In Corollary15 and Corollary 16, the existence and uniqueness of positive fixed points are proved, without appealing to the monotonicity or any compactness and continuity of the e-convex operator A:P⟶P. This is very different from [9] (Theorem 9), which required that there existed M>1 homogeneous increasing functional F:Pe⟶0,+∞. In addition, Corollary 15 and Corollary 16 in the paper are quite different from [14] (Corollary 2.4), which only obtained the existence of positive fixed points while the condition required the strong condition of that there existed ε0>0 such that
(25)Ax≥ε0Axe,∀x∈P+,lim¯t⟶0+ηx,t>max1−ε0Aε0eeN2,1−1MN,uniformlyforx∈Ce,
withM=supAx∣x∈P,x=1.Remark 26.
In Corollary17 and Corollary 18, the existence and uniqueness of positive fixed points are proved, without appealing to the monotonicity of α-convex operator α>1 or any compactness and continuity of the operator A:P⟶P. This is very different from [12] (Theorem 9), [8] (Theorem 2), and [15] (Theorem 1.3), which required that there existed a linear operator L:E⟶E which satisfied certain conditions, and the increasing α-convex operator (α>1) was completely continuous, respectively.Remark 27.
In Corollary19 and Corollary 20, the positive α-homogeneous operator α>1A:P⟶P does not need to have any compactness or continuity, but Theorem 1 in [17] requested that the α-homogeneous operator α>1A:P⟶P can be decomposed into A=FC, where F:Pe⟶0,+∞ was an increasing positive β functional and C:Pe⟶Pe was an increasing operator in Pee>θ. Therefore, the methods and techniques of Corollary 19 and Corollary 20 are different from those of [17] (Theorem 1).Remark 28.
In this paper, we use the partial order and the monotone iteration to study the fixed point theorems of superlinear operators in Banach spaces. The methods and techniques are different from those used in the literature [7–10, 12, 14, 15, 17], but the existence and uniqueness of the fixed points and the convergence of the iterative sequences of superlinear operators are obtained.


## 4. Applications

Now, we give some examples to show the applications of our main results in nonlinear integral equations and partial differential equations.Example 1.
Letα>1. Consider Hammerstein integral equation
(26)xt=Axt=∫−∞+∞Kt,sxsαds.Conclusion 29.
LetK:R×R⟶R be a nonnegative continuous function. If there exists a constant 0<c<1 and two continuous functions u=u0t,v=v0t satisfying 0<u0t≤v0t, −∞<t<+∞, and
(27)u0t≤∫−∞+∞Kt,su0sαds,∫−∞+∞Kt,sv0sαds≤cv0t.
Then, equation (26) has a unique solution x∗t satisfying u0≤x∗≤v0. For any x0t which satisfies u0t≤x0t≤v0t, the iterated sequence
(28)xnt=Axn−1t=∫−∞+∞Kt,sxn−1sαds,uniformly converges to x∗t in −∞,+∞.Proof.
LetE=CBR be a bounded continuous function space in Rn. Define x=supt∈Rxt, then E is a Banach space. Let P=CB+R denote all nonnegative continuous functions in E, then P is a normal cone in E. We claim that A:P⟶P is a homogeneous operator. In fact, by equation (26), we have
(29)Aλxt=∫−∞+∞Kt,sλxsαds=λα∫−∞+∞Kt,sλxsαds≤λAxt,
which meansA:P⟶P is a homogeneous operator. It is clear that A satisfies all conditions of Theorem 10. The conclusion is true.Similarly, we also have the following.Example 2.
Letα>1. Consider Hammerstein integral equation (see the equation (9) in [10])
(30)xt=Axt=∫01Kt,sxsαds.Conclusion 30.
LetK:0,1×0,1⟶0,1 be a nonnegative continuous function. If there exists a constant 0<c<1 and two continuous functions u=u0t,v=v0t satisfying 0<u0t≤v0t, 0<t<1, and
(31)u0t≤∫01Kt,su0sαds,∫01Kt,sv0sαds≤cv0t.
Then, equation (30) has a unique solution x∗t satisfying u0≤x∗≤v0. For any x0t which satisfies u0t≤x0t≤v0t, the iterated sequence
(32)xnt=Axn−1t=∫01Kt,sxn−1sαds,
uniformly converges tox∗t in −∞,+∞.Remark 31.
In Example2, we obtain the existence of positive solutions of the integral equation (30), without requiring that the integral kernel Kt,s can be decomposed into Kt,s=htgs (see condition C1 in [10]). The methods and techniques used in this paper are different from those in [10].Example 3.
LetΩ be a bounded convex domain in Rnn≥2 whose boundary ∂Ω belongs to C2+μ for some 0<μ<1. Consider the Dirichlet problem
(33)Lu=fx,u,u∂Ω=0,where fx,u is nonnegative and continuous on x∈Ω¯ and u≥0 and
(34)Lu=−∑i,j=1naijx∂2u∂xi∂xj+∑i=1nbix∂u∂xi+cxu,
i.e., there exists a positive constantμ0 such that
(35)∑i,j=1naijxξiξj≥μ0ξ2,
foranyx∈Ω¯andξ=ξ1,ξ2,⋯,ξn∈Rn,andaijx=ajix,cx≥0. Here, all functions aijx,bix, and cx belong to CμΩ¯ (see [3]).
Finding the solution of the above problem is equivalent to finding the fixed point of the integral operatorA:
(36)Aux=∫Ω¯Gx,yfy,uydy,where Gx,y is the corresponding Green function, which satisfies
(37)0<Gx,y<K0x−y2−n,n>2K0lnx−y,n=2x,y∈Ω,x≠y.
Hence (see Guo and Lakshmikantham [4]), the linear integral operator
(38)Gvx=∫ΩGx,yvydy,
is a completely continuous operator fromCΩ¯ into CΩ¯, and therefore, operator A maps P into P and is completely continuous, where P=ux∈CΩ¯∣ux≥0,∀x∈Ω¯ is a normal cone of space CΩ¯.Conclusion 32.
Let the functionfx,ux be increasing and satisfy
(39)fx,tu<tfx,u,∀u>0,x∈Ω,0<t<1.
If there exista∈0,1 and θ<v0=vx0∈P, such that ∫Ω¯Gx0,yfy,vydy≤avx0 for some x0∈Ω, then the Dirichlet problem has a unique fixed point x∗ in θ,vx0.Proof.
Firstly, we prove that the operatorA is e-convex, where
(40)ex=∫Ω¯Gx,ydy,∀x∈Ω¯.
Here, we need to use a conclusion about integral operator (17), which can be found in Amann [2]: linear integral operator (17) is e-positive, i.e., for any v>θ, there exist α=αv>0 and β=βv>0 such that αe≤Gv≤βe, i.e.,
(41)α∫Ω¯Gx,ydy≤∫Ω¯Gx,yvydy≤β∫Ω¯Gx,ydy,∀x∈Ω¯.
Now, letu>θ. Then, there exists an x1∈Ω such that ux1>0, and it follows from (39) that
(42)0≤fx1,2ux1<2fx1,ux1.
Consequently,fu>θ, where f denotes the Nemitskyi operator:
(43)fux=fx,ux.
Thus, from (41), we know that there exist α>0 and β>0 such that
(44)αe≤Au=Gfu≤βe,
i.e.,A satisfies condition (i) of Definition 4.
Next, supposeu∈P satisfying α1e≤u≤β1eα1=α1u>0,β1=β1u>0 and 0<t<1. Since ex>0 for any x∈Ω, we have by (39)
(45)tfx,ux−fx,tux>0,∀x∈Ω,and hence, by (41), there exists α2>0 such that
(46)∫Ω¯Gx,ytfy,uy−fy,tuydy≥α2ex,∀x∈Ω¯.
On the other hand, it is clear that(47)∫Ω¯Gx,yfy,uydy≤Mex,∀x∈Ω¯,where
(48)M=maxx∈Ω¯fx,ux.
It follows therefore from (46) and (47) that
(49)∫Ω¯Gx,yfy,tuydy≤t1−α2Mt∫Ω¯Gx,yfy,uydy,∀x∈Ω¯,
i.e.,Atu≤t1−ηAu, where η=α2/Mt>0. Thus, the operator A satisfies condition (ii) of Definition 4, and therefore, A is e-convex.
Takew0=θ, then w0<v0 and w0≤Aw0. Therefore, all conditions of Corollary 15 are satisfied. By Corollary 15, we see that the conclusion is true.



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