Abstract
In this article we study several classes of sum operator equations on ordered Banach spaces and present some new existence and uniqueness results of positive solutions, which extend the existing corresponding results. Moreover, we establish some pleasant properties of nonlinear eigenvalue problems for several classes of sum operator equations. As applications, we utilize the main results obtained in this paper to study two classes nonlinear problems; one is the integral equation , where f and G are both nonnegative, is a parameter; the other is the elliptic boundary value problem for the Lane-Emden-Fowler equation , in Ω, on ∂ Ω, where Ω is a bounded domain with smooth boundary in (), and is allowed to be singular on ∂ Ω. The new results on the existence and uniqueness of positive solutions for these problems are given, which complement the existing results of positive solutions for these problems in the literature.
MSC:47H10, 47H07, 45G15, 35J60, 35J65.
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1 Introduction and preliminaries
With the development of nonlinear sciences, nonlinear functional analysis has been an active area of research over the past several decades. As an important branch of nonlinear functional analysis, nonlinear operator theory has attracted much attention and has been widely studied, especially nonlinear operators which arise in the connection with nonlinear differential and integral equations have been extensively studied (see for instance [1–12]). It is well known that the existence and uniqueness of positive solutions to nonlinear operator equations is very important in theory and applications. Many authors have studied this problem; for a small sample of such work, we refer the reader to [7, 10, 13–23]. The operator equation considered in this papers is always of the following form:
In [23], Zhao considered the existence of solutions for the sum operator equation
where A is increasing e-concave, B is increasing e-convex and is a strict set contraction. Motivated by the works [22, 23], Sang et al. considered the operator equation (1.1), where A is -concave, B is -convex and is also a strict set contraction. However, we can see that the conditions of the main results in [23, 24] are strong and of utmost convenience.
Recently, we considered successively the operator equation (1.1) and the following operator equation:
the operators A, B in (1.1) are increasing, α-concave and sub-homogeneous, respectively; the operators A, B in (1.2) are mixed monotone and increasing α-concave (or sub-homogeneous), respectively. In [7], by using the properties of cones and a fixed point theorem for increasing general α-concave operators, we established the existence and uniqueness of positive solutions for the operator equation (1.1), and we utilized the main results to present the existence and uniqueness of positive solutions for the following two problems; one is a fourth-order two-point boundary value problem for elastic beam equations,
where and are real functions; and the other is an elliptic value problem for Lane-Emden-Fowler equations
where , are allowed to be singular on ∂ Ω. In [21], by using the properties of cones and a fixed point theorem for mixed monotone operators, we established the existence and uniqueness of positive solutions for the operator equation (1.2), and we utilized the results obtained to study the existence and uniqueness of positive solutions for a nonlinear fractional differential equation boundary value problem,
where is the Riemann-Liouville fractional derivative of order . These results are useful and interesting. For completeness, in this paper we will further consider the following several classes of sum operators:
-
(i)
the sum of increasing operators and decreasing operators;
-
(ii)
the sum of increasing operators and mixed monotone operators;
-
(iii)
the sum of decreasing operators and mixed monotone operators;
-
(iv)
the sum of increasing operators, decreasing operators and mixed monotone operators.
Motivated by our works [7, 10, 21], we will study the above cases (i)-(iv). So this article is a continuation of our papers [7, 10, 21], and we will present some interesting results on the existence and uniqueness of positive solutions for the above several classes of sum operator equations. To demonstrate the applicability of our abstract results, we give, in the last section of the paper, some applications to nonlinear integral equations and elliptic boundary value problems for the Lane-Emden-Fowler equations.
In the following two subsections, we state some definitions, notations, and known results. For convenience of the readers, we refer to [7–13, 20–22, 25–27] for details.
1.1 Some basic definitions and notations
Suppose that E is a real Banach space which is partially ordered by a cone , i.e., if and only if . If and , then we denote or . By θ we denote the zero element of E. Recall that a non-empty closed convex set is a cone if it satisfies (i) ; (ii) .
Putting , a cone P is said to be solid if is non-empty. Moreover, P is called normal if there exists a constant such that, for all , implies ; in this case N is called the normality constant of P. If , the set is called the order interval between and . We say that an operator is increasing (decreasing) if implies ().
For all , the notation means that there exist and such that . Clearly, ∼ is an equivalence relation. Given (i.e., and ), we denote by the set . It is easy to see that .
Definition 1.1 Let or and α be a real number with . An operator is said to be α-concave if it satisfies
Notice that the definition of an α-concave operator mentioned above is different from that in [26], because we need not require the cone to be solid in general.
Definition 1.2 An operator is said to be sub-homogeneous if it satisfies
Definition 1.3 (See [10, 21, 27])
is said to be a mixed monotone operator if is increasing in x and decreasing in y, i.e., , , imply . An element is called a fixed point of A if .
1.2 Some fixed point theorems and properties
In this subsection, we assume that E is a real Banach space with a partial order introduced by a cone P of E. Take , , is given as in Section 1.1.
In the paper [7], we considered the existence and uniqueness of positive solutions to the operator equation (1.1) on ordered Banach spaces and established the following conclusion.
Theorem 1.1 (See Theorem 2.2 in [7])
Let P be a normal cone in E, be an increasing α-concave operator and be an increasing sub-homogeneous operator. Assume that
-
(i)
there is such that and ;
-
(ii)
there exists a constant such that , .
Then the operator equation (1.1) has a unique solution in . Moreover, constructing successively the sequence , for any initial value , we have as .
In the paper [10], we present the following fixed point theorem for a class of general mixed monotone operators and established some pleasant properties of nonlinear eigenvalue problems for mixed monotone operators.
Theorem 1.2 (See Lemma 2.1 and Theorem 2.1 in [10])
Let P be a normal cone in E. Assume that is a mixed monotone operator and satisfies:
-
(i)
there exists with such that ;
-
(ii)
for any and , there exists such that .
Then:
-
(1)
;
-
(2)
there exist and such that , ;
-
(3)
the operator equation (1.2) has a unique solution in ;
-
(4)
for any initial values , constructing successively the sequences
we have and as .
Theorem 1.3 (See Theorem 2.3 in [10])
Assume that the operator A satisfies the conditions of Theorem 1.2. Let () denote the unique solution of nonlinear eigenvalue equation in . Then we have the following conclusions:
(R1) If for , then is strictly decreasing in λ, that is, implies ;
(R2) If there exists such that for , then is continuous in λ, that is, () implies ;
(R3) If there exists such that for , then , .
Based on Theorem 1.2, in [21] we considered the operator equation (1.2) and established the following conclusions.
Theorem 1.4 (See Theorem 2.1 in [21])
Let P be a normal cone in E, . is a mixed monotone operator and satisfies
is an increasing sub-homogeneous operator. Assume that
-
(i)
there is such that and ;
-
(ii)
there exists a constant such that , .
Then:
-
(1)
, ;
-
(2)
there exist and such that
-
(3)
the operator equation (1.2) has a unique solution in ;
-
(4)
for any initial values , constructing successively the sequences
we have and as .
Theorem 1.5 (See Theorem 2.4 in [21])
Let P be a normal cone in E, . is a mixed monotone operator and satisfies
is an increasing α-concave operator. Assume that
-
(i)
there is such that and ;
-
(ii)
there exists a constant such that , .
Then:
-
(1)
, ;
-
(2)
there exist and such that
-
(3)
the operator equation (1.2) has a unique solution in ;
-
(4)
for any initial values , constructing successively the sequences
we have and as .
2 Main results
In this section we consider the existence and uniqueness of positive solutions for several classes of sum operator equations. We always assume that E is a real Banach space with a partial order induced by a cone P of E. Take , and as given in the Introduction.
2.1 The sum of increasing operators and decreasing operators
Now we first consider the following sum operator equations:
Theorem 2.1 Let P be a normal cone, be an increasing operator and be a decreasing operator. Assume that:
(H11) for any and , there exist () such that
(H12) there exists such that .
Then:
-
(i)
there exist and such that
-
(ii)
the operator equation (2.1) has a unique solution in ;
-
(iii)
for any initial values , constructing successively the sequences
we have , as .
Proof Firstly, from (2.3), we have
Since , there exist constants such that
Also from , there exists a constant such that
Let , . Then . From (2.3) and (2.4), we obtain
Note that , we can get .
Next we define an operator by . Then is a mixed monotone operator and . Moreover, for any and , we have
Hence, the operator T satisfies the condition (ii) in Theorem 1.2. An application of Theorem 1.2 implies that: there are and such that , ; operator equation has a unique positive ; for any initial values , constructing successively the sequences
we have and as . That is:
-
(i)
there exist and such that
-
(ii)
the operator equation (2.1) has a unique solution in ;
-
(iii)
for any initial values , constructing successively the sequences
we have , as .
□
Note that , and we can easily obtain the following conclusions.
Corollary 2.2 Let P be a normal cone, be an increasing operator and be a decreasing operator. Assume that:
(H13) for any and , there exist () such that
Then:
-
(i)
there exist and such that
-
(ii)
the operator equation (2.1) has a unique solution in ;
-
(iii)
for any initial values , constructing successively the sequences
we have , as .
Corollary 2.3 Let . Let P be a normal cone, be an increasing -concave operator and be a decreasing -convex operator. Assume that (H12) holds. Then:
-
(i)
there exist and such that
-
(ii)
the operator equation (2.1) has a unique solution in ;
-
(iii)
for any initial values , constructing successively the sequences
we have , as .
Proof Let , , . Then for and
Hence, the conclusions follow from Theorem 2.1. □
Corollary 2.4 Let and P be a normal cone. Let be an increasing α-concave operator and be an increasing sub-homogeneous operator, be a decreasing operator which satisfies (2.3). Assume that:
(H14) there exists such that ;
(H15) there exists such that , ;
(H16) there exists such that .
Then:
-
(i)
there exist and such that
-
(ii)
the following operator equation:
(2.5)
has a unique solution in ;
-
(iii)
for any initial values , constructing successively the sequences
we have , as .
Proof Define an operator by . Then is an increasing operator and . Since , there exist such that
Then , , and thus
Note that and , we can get . Hence, .
From the proof of Theorem 1.1, there exists with respect to t, such that
Let , . Then and , .
Therefore, operators A, B satisfy all the conditions of Theorem 2.1. So we easily obtain the following conclusions:
-
(i)
there exist , and such that
-
(ii)
the operator equation (2.5) has a unique solution in ;
-
(iii)
for any initial values , constructing successively the sequences
we have , as .
□
Corollary 2.5 Assume that all the conditions of Theorem 2.1 hold. Let () denote the unique solution of operator equation (2.2). Then we have the following conclusions:
-
(i)
if () for , then is strictly decreasing in λ, that is, implies ;
-
(ii)
if there exists such that () for , then is continuous in λ, that is, () implies ;
-
(iii)
if there exists such that () for , then , .
Proof Define an operator by . Then is a mixed monotone operator. From the proof of Theorem 2.1, we have , and
where . Evidently, for . Hence, the conclusions follow from Theorem 1.3. □
Similarly, we can easily obtain the following result.
Corollary 2.6 Assume that all the conditions of Corollary 2.3 hold. Let () denote the unique solution of operator equation (2.2). Then we have the following conclusions:
-
(i)
if , then is strictly decreasing in λ, that is, implies ;
-
(ii)
is continuous in λ, that is, () implies ;
-
(iii)
if , then , .
2.2 The sum of increasing operators and mixed monotone operators
Next, we consider the following sum operator equations:
Theorem 2.7 Let P be a normal cone, be an increasing operator and be a mixed monotone operator. Assume that:
(H21) for any , , there exists such that
(H22) for any , , there exists such that
(H23) there exists such that .
Then:
-
(i)
there exist and such that
-
(ii)
the operator equation (2.6) has a unique solution in ;
-
(iii)
for any initial values , constructing successively the sequences
we have , as .
Proof From (2.9), we obtain
Since , there exist constants such that
Also, from , there exists a small constant such that
Let . Then for . From (2.8)-(2.10),
Note that , we can get .
Next, we define an operator by . Then is a mixed monotone operator and . Moreover, for any and , we have
Hence, all the conditions of Theorem 1.2 are satisfied. An application of Theorem 1.2 implies that: there are and such that , ; operator equation has a unique solution ; for any initial values , constructing successively the sequences
we have and as . That is:
-
(i)
there exist and such that
-
(ii)
the operator equation (2.6) has a unique solution in ;
-
(iii)
for any initial values , constructing successively the sequences
we have , as .
□
Corollary 2.8 Let P be a normal cone, be an increasing operator and be a mixed monotone operator. Assume that:
(H24) for any , , there exists such that
(H25) for any , , there exists such that
Then:
-
(i)
there exist and such that
-
(ii)
the operator equation (2.6) has a unique solution in ;
-
(iii)
for any initial values , constructing successively the sequences
we have , as .
Corollary 2.9 Let and P be a normal cone. Let be an increasing operator which satisfies (H21), be an increasing sub-homogeneous operator and be a mixed monotone operator which satisfies
Assume that:
(H26) there exist such that
(H27) there exists a constant such that
Then:
-
(i)
there exist and such that
-
(ii)
the following operator equation:
(2.12)
has a unique solution in ;
-
(iii)
for any initial values , constructing successively the sequences
we have , as .
Proof Define an operator by . Then is a mixed monotone operator and . Since , there exist such that
Then , , and thus
Note that and , we can get . Hence, .
Note that (H27) and from the proof of Theorem 1.4, there exists with respect to t such that
Let , . Then and
Therefore, the operators A, B satisfy all the conditions of Theorem 2.7. So we easily obtain the following conclusions:
-
(i)
there exist and such that
-
(ii)
the operator equation (2.12) has a unique solution in ;
-
(iii)
for any initial values , constructing successively the sequences
we have , as .
□
Corollary 2.10 Let and P be a normal cone. Let be an increasing operator which satisfies (H21), be an increasing α-concave operator and be a mixed monotone operator which satisfies
Assume that (H26) holds and
(H28) there exists a constant such that , .
Then:
-
(i)
there exist and such that
-
(ii)
the operator equation (2.12) has a unique solution in ;
-
(iii)
for any initial values , constructing successively the sequences
we have , as .
Proof Consider the same operator B defined by the proof of Corollary 2.9, we have is a mixed monotone operator and . From Definition 1.1, we have , , . Since , there exist such that
Then , and thus
Note that and , we can get . Hence, .
Note that (H28) and from the proof of Theorem 1.5, we know that there exists with respect to t such that
Let , . Then and , .
Therefore, the operators A, B satisfy all the conditions of Theorem 2.7. So we easily obtain the following conclusions:
-
(i)
there exist and such that
-
(ii)
the operator equation (2.12) has a unique solution in ;
-
(iii)
for any initial values , constructing successively the sequences
we have , as .
□
Similar to Corollary 2.5, we have the following result.
Corollary 2.11 Assume that all the conditions of Theorem 2.7 hold. Let () denote the unique solution of operator equation (2.7). Then we have the following conclusions:
-
(i)
if () for , then is strictly decreasing in λ, that is, implies ;
-
(ii)
if there exists such that () for , then is continuous in λ, that is, () implies ;
-
(iii)
if there exists such that () for , then , .
2.3 The sum of decreasing operators and mixed monotone operators
In the following we also consider the operator equations (2.6) and (2.7).
Theorem 2.12 Let P be a normal cone, be a decreasing operator and be a mixed monotone operator. Assume that (H22) and (H23) hold and
(H31) for any and , there exists such that
Then:
-
(i)
there exist and such that
-
(ii)
the operator equation (2.6) has a unique solution in ;
-
(iii)
for any initial values , constructing successively the sequences
we have , as .
Proof From (2.14), we have
Since , there exist constants such that
Also from , there exists a small constant such that
Let . Then for . From (H22) and (2.14), (2.15),
Note that , we can get .
Next, we define an operator by . Then is a mixed monotone operator and .
Moreover, for any and , we have
Hence, all the conditions of Theorem 1.2 are satisfied. Application of Theorem 1.2 implies that: there are and such that , ; operator equation has a unique solution ; for any initial values , constructing successively the sequences
we have and as . That is,
-
(i)
there exist and such that
-
(ii)
the operator equation (2.6) has a unique solution in ;
-
(iii)
for any initial values , constructing successively the sequences
we have , as .
□
Corollary 2.13 Let P be a normal cone, be a decreasing operator and be a mixed monotone operator. Assume that:
(H32) for any and , there exists such that
(H33) for any , , there exists such that
Then:
-
(i)
there exist and such that
-
(ii)
the operator equation (2.6) has a unique solution in ;
-
(iii)
for any initial values , constructing successively the sequences
we have , as .
Corollary 2.14 Assume that all the conditions of Theorem 2.12 hold. Let () denote the unique solution of operator equation (2.7). Then we have the following conclusions:
-
(i)
if () for , then is strictly decreasing in λ, that is, implies ;
-
(ii)
if there exists such that () for , then is continuous in λ, that is, () implies ;
-
(iii)
if there exists such that () for , then , .
2.4 The sum of increasing operators, decreasing operators, and mixed monotone operators
From the above results, we can easily obtain the following results on operator equations:
By Theorem 2.12 and Corollary 2.9, Corollary 2.10, we have the following conclusions.
Theorem 2.15 Let and P be a normal cone. Let be a decreasing operator which satisfies (H31), operators , be the same as for Corollary 2.9. Assume that (H26), (H27) hold. Then:
-
(i)
there exist and such that
-
(ii)
the operator equation (2.16) has a unique solution in ;
-
(iii)
for any initial values , constructing successively the sequences
we have , as .
Theorem 2.16 Let and P be a normal cone. Let be a decreasing operator which satisfies (H31), operators , be the same as for Corollary 2.10. Assume that (H26), (H28) hold. Then:
-
(i)
there exist and such that
-
(ii)
the operator equation (2.16) has a unique solution in ;
-
(iii)
for any initial values , constructing successively the sequences
we have , as .
From Corollary 2.9, Corollary 2.10, and Corollary 2.4, we can easily obtain the following results.
Theorem 2.17 Let and P be a normal cone, operators , satisfy the conditions of Corollary 2.4, where is -concave, operators , satisfy the conditions of Corollary 2.9, where satisfies (2.11) with α replaced by . Then:
-
(i)
there exist and such that
-
(ii)
the operator equation (2.17) has a unique solution in ;
-
(iii)
for any initial values , constructing successively the sequences
where , we have , as .
Theorem 2.18 Let and P be a normal cone, operator is -concave, operators , satisfy the conditions of Corollary 2.10, where is -concave. Then:
-
(i)
there exist and such that
-
(ii)
the operator equation (2.17) has a unique solution in ;
-
(iii)
for any initial values , constructing successively the sequences
where , we have , as .
3 Some applications
In this section, we will apply the main results to study nonlinear problems which include nonlinear integral equations and nonlinear elliptic boundary value problems for the Lane-Emden-Fowler equations. And then we will obtain new results on the existence and uniqueness of positive solutions for these problems, which are not consequences of the corresponding fixed point theorems in the literature.
3.1 Applications to nonlinear integral equations
A standard approach, in studying the existence of positive solutions of boundary value problems (BVPs for short) for ordinary differential equations, is to rewrite the problem as an equivalent positive-solution problem for a Hammerstein integral equation of the form
in the space , where the nonlinearity f and the kernel G (the Green function of the problem) are both nonnegative, is a parameter. One seeks fixed points of a Hammerstein integral operator in a suitable cone of positive functions.
Set , the standard cone. It is easy to see that P is a normal cone of which the normality constant is 1. Then . Assume that is continuous with and there exist with , such that
Theorem 3.1 Assume that and
(H31) is continuous (), is increasing in for fixed and is decreasing in for fixed ;
(H32) for , there exist () such that
Then, for any given , the integral equation (3.1) has a unique positive solution in . Moreover, for any initial values , constructing successively the sequences:
we have , as . Further, (i) if () for , then is strictly increasing in λ, that is, implies ; (ii) if there exists such that () for , then is continuous in λ, that is, () implies ; (iii) if there exists such that () for , then , .
Proof Define two operators and by
It is easy to see that u is the solution of (3.1) if and only if . From (H31), we know that is increasing and is decreasing. Further, from (H32), we can prove that A, B satisfy (H11). Next we prove that . Set , . Then .
For any , from (H31) and (3.2), we have
Let , . Note that is continuous with and from (3.2), we get and in consequence, . That is, . Hence, all the conditions of Theorem 2.1 are satisfied. It follows from Theorem 2.1 and Corollary 2.5 that the operator equation has a unique solution in , that is, . So is a unique positive solution of the integral equation (3.1) in for given . From Corollary 2.5, we have (i) if () for , then is strictly increasing in λ, that is, implies ; (ii) if there exists such that () for , then is continuous in λ, that is, () implies ; (iii) if there exists such that () for , then , .
Let , . Then , also satisfy the conditions of Theorem 2.1. By Theorem 2.1, for any initial values , constructing successively the sequences
we have , as . That is,
as . □
Theorem 3.2 Assume that with satisfies (H31) and
(H33) is continuous, increasing in for fixed , , decreasing in for fixed , ;
(H34) for , there exist () such that
Then, for any given , the integral equation (3.1) has a unique positive solution in . Moreover, for any initial values , constructing successively the sequences:
we have , as . Further, the conclusions (i), (ii), and (iii) in Theorem 3.1 also hold.
Proof Define two operators and by
It is easy to see that u is the solution of (3.1) if and only if . From (H31) and (H33), we know that is increasing and is mixed monotone. Further, from (H34), we can prove that A, B satisfy (H21) and (H22). Next we prove that .
For any , from (H31), (H33), and (3.2), we have
Let , . Note that is nonnegative and continuous with and from (3.2), we get and in consequence, . That is, . Hence, all the conditions of Theorem 2.7 are satisfied. It follows from Theorem 2.7 and Corollary 2.11 that the operator equation has a unique solution in , that is, . So is a unique positive solution of the integral equation (3.1) in for given . From Corollary 2.11, we have (i) if () for , then is strictly increasing in λ, that is, implies ; (ii) if there exists such that () for , then is continuous in λ, that is, () implies ; (iii) if there exists such that () for , then , .
Let , . Then , also satisfy the conditions of Theorem 2.7. By Theorem 2.7, for any initial values , constructing successively the sequences , , , we have , as . That is,
as . □
Theorem 3.3 Assume that with satisfies all the conditions of in Theorem 3.1 and satisfies (H33) and (H34). Then, for any given , the integral equation (3.1) has a unique positive solution in . Moreover, for any initial values , constructing successively the sequences:
we have , as . Further, the conclusions (i), (ii), and (iii) in Theorem 3.1 also hold.
Proof Similar to the proofs of Theorem 3.1 and Theorem 3.2, the conclusions follow from Theorem 2.12 and Corollary 2.14. □
3.2 Applications to nonlinear elliptic BVPs for the Lane-Emden-Fowler equations
Let Ω be a bounded domain with smooth boundary in (). Consider the following singular Dirichlet problem for the Lane-Emden-Fowler equation:
where and the nonlinear term is allowed to be singular on ∂ Ω.
The problem (3.3) arises in the study of non-Newtonian fluids, boundary layer phenomena for viscous fluids, chemical heterogeneous catalysts, as well as in the theory of heat conduction in electrically materials (see [28–32]). The theory of singular elliptic boundary value problems for partial differential equations has become an important area of investigation in the past three decades, see [28–40] and references therein. By means of sub-supersolutions and various techniques related to the maximum principle for elliptic equations, some existence and nonexistence results, a unique positive solution are established. In [3, 7], we investigated the existence and uniqueness of positive solutions to the singular Dirichlet problem for the Lane-Emden-Fowler equation (3.3), where is increasing in for each in [3]; with is increasing in for each , in [7] and . However, to our knowledge, the results on the existence-uniqueness of positive solutions for singular elliptic equation are still few. The purpose here is to establish the existence-uniqueness of positive solutions to the singular Dirichlet problem for the Lane-Emden-Fowler equation (3.3) by using some fixed point results in Section 2.
Throughout this subsection, denote the Sobolev space (see [41]), where and k is a nonnegative integer. And denote the eigenfunction corresponding to the smallest eigenvalue of the problem in Ω, and . For convenience, we can assume that in . Moreover, it is well known that (see for instance [42]) there exist two positive constants , such that the first eigenvalue function satisfies
where .
Lemma 3.4 (See [[43], Theorem 3, p.468])
Let Ω be a bounded domain in with smooth boundary ∂ Ω. Let and assume that, for some , u satisfies
Then either , or there exists such that in Ω.
The proof of this result is due to Brezis and Nirenberg and the result is inspired by the work of Stampachia. Brezis and Nirenberg obtained this result in order to solve a similar eigenvalue problem as considered here. Actually, the result was extended to more general operators, such as , under some suitable restrictions in order to solve a large class of problems (see for example the problems considered recently in the work of Covei [44]). Here we recall the result since can be used to prove the following simple but useful lemma.
Lemma 3.5 (See [[3], Theorem 3.1, p.1278])
Let Ω be a bounded domain with smooth boundary in (). If and for , then there exists a constant such that
where depends only upon N and Ω.
Theorem 3.6 Assume that and
(H35) , is nonnegative on , Hölder continuous in the variable x with the Hölder exponent for each and continuous in the variable u for each ;
(H36) is increasing in u for each x and is decreasing in u for each x, and for any , there exists a constant , , such that
(H37) , satisfy the conditions of integrability, i.e.,
Then the problem (3.3) has a unique positive solution with respect to , where . Moreover, (i) if () for , then is strictly increasing in λ, that is, implies ; (ii) if there exists such that () for , then is continuous in λ, that is, () implies ; (iii) if there exists such that () for , then , .
Proof For the sake of convenience, set , the Banach space of continuous functions on with the norm . Set , the standard cone. It is clear that P is a normal cone in E and the normality constant is 1, is given as in the Section 1.1. We divide the proof into several steps.
Step 1. We consider the following linear elliptic boundary value problem:
where . Since , we can choose a sufficiently small number such that
Then from (H36),
Thus we get by applying the integrability condition (H37) that says that
namely, . By the classical theory of linear elliptic equations (see [45]), the problem (3.5) admits a unique strong solution . Recall that . Using the Sobolev imbedding theory, with . Now we define an operator by
where is the unique strong solution of (3.5) for . Evidently, . Suppose that ϕ is the solution of (3.5) with , then . Then from Lemma 3.5, there exists a positive constant such that
Note that . By the maximal principle, . Since for , an application of Lemma 3.4 implies that
Combining (3.4) and (3.9), there exists a positive constant such that
Let is the unique strong solution of (3.5) for . From (3.6) and (3.7), and applying the comparison principle, we conclude that
and, from (3.8) and (3.10), we get for any . So we find that is well defined. Further, from (H36) and the comparison principle, we can easily prove that is increasing. In the following we prove that for any and . For any and , we have
and
From (H36) we also get for any . Therefore,
Using the comparison principle again, we can obtain immediately. So we have for , .
Step 2. We consider the following linear elliptic boundary value problem:
where . Since , we can choose a sufficiently small number such that
Then from (H36),
Thus we get by applying the integrability condition (H37) that says that , namely, . By the classical theory of linear elliptic equations, the problem (3.11) admits a unique strong solution . Recall that . Using the Sobolev imbedding theory, with . Now we define an operator by , , where is the unique strong solution of (3.11) for . Similar to Step 1,we can prove that is well defined. Using the comparison principle again, we can easily see that is decreasing and for any and .
Step 3. Now all the conditions of Corollary 2.2 are satisfied. It follows from Corollary 2.2 and Corollary 2.5 that the operator equation has a unique solution in , that is, . So is a unique positive solution of the problem (3.3) in for given . By the theory of the linear elliptic equation, the problem
admits a unique solution , and hence . Recalling the uniqueness of the solution of (3.3), one can see easily that . Thus the problem (3.3) has a unique classical solution . Moreover, by using Corollary 2.5 and the theory of the linear elliptic equation, we can easily prove that (i) if () for , then is strictly increasing in λ, that is, implies ; (ii) if there exists such that () for , then is continuous in λ, that is, () implies ; (iii) if there exists such that () for , then , . □
Similar to the proofs of Theorem 3.6 and Theorems 3.2, 3.3, we can easily obtain the following conclusions.
Theorem 3.7 Assume that and satisfies all the conditions of Theorem 3.6, satisfies
(H38) is nonnegative on , Hölder continuous in the variable x with the Hölder exponent for each and is continuous in the variables u, v for each ;
(H39) for any , there exists a constant such that
(H310) satisfies the condition of integrability, i.e.,
Then the problem (3.3) has a unique positive solution with respect to , where . Further, the conclusions (i), (ii), and (iii) in Theorem 3.6 also hold.
Theorem 3.8 Assume that with satisfying all the conditions of in Theorem 3.6 and satisfying (H38), (H39), and (H310). Then the problem (3.3) has a unique positive solution with respect to , where . Further, the conclusions (i), (ii), and (iii) in Theorem 3.6 also hold.
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The research was supported by the Youth Science Foundation of China (11201272), the Science Foundations of Shanxi Province (2010021002-1; 2013011003-3) and the Science Foundation of Business College of Shanxi University (2012050).
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Yang, C., Zhai, C. & Hao, M. Uniqueness of positive solutions for several classes of sum operator equations and applications. J Inequal Appl 2014, 58 (2014). https://doi.org/10.1186/1029-242X-2014-58
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DOI: https://doi.org/10.1186/1029-242X-2014-58