1 Introduction and preliminaries

Since the notion of mixed monotone operator was established in 1987 (see [1]), a number of scholars have explored the existence and uniqueness of different types of fixed points for such operators (see [1–18]-20]). In order to solve the fixed point problems, two common methods are usually utilized in the study of mixed monotone operators. One is to use the compactness or continuity of the operators whenever such condition is satisfied (see [1–4]), the other is to take the advantage of certain concavity or convexity by means of cone theory and monotone iterative techniques once such concavity or convexity property lies in the operators (see [5–21]). These methods have been widely applied in various fields, including integral equations (see [22, 23]), higher-order mixed integro-differential equations (see [24–33]), fractional integro- differential equations (see [34–37]) and some other related applications (see [38–44]). These works demonstrate the versatility and applicability of mixed monotone operators in various mathematical problems.

Among them, the authors in [11] presented a new concept, named concave- convex mixed monotone operators. They gained several theorems of this type of operators with certain concavity and convexity. The advantage of introducing concave- convex mixed monotone operators is that it can unify a large number of mixed monotone operators with certain concave-convex properties, and thus cope with the fixed point problems together in a general way. However, in the existing literature, such as [11], a crucial question arises about the fixed point theorem of concave- convex operators, which is as following:

Question A: If we delete the condition (C) :

(C)

in Theorem 2.1 in [11], then do the fixed point results in Theorem 2.1 in [11] remain true?

In this paper, we positively answer this question. Following [11], we continue to investigate the fixed point problems regarding concave- convex mixed monotone operators. Without requiring any compactness or continuity of the operators, we also get the existence and uniqueness conclusions for these operators, as well as the convergence of the iterative sequences. The novelty of this work is that we utilize a new iterative technique to establish some new fixed point theorems of mixed monotone operators in which the existence and uniqueness of the fixed points, as well as the convergence of iterative sequences are proven by only requiring weaker conditions than that in [11] and [20], thus improve the main results in [11] and [20] greatly. Furthermore, we also obtain a number of new fixed point results about mixed monotone operators with certain concavity and convexity without requiring coupled upper or lower solutions. In addition, the main results are applied to two types of nonlinear integral equations. Our work extends and improves the previous related works.

Now we review some basic definitions, notations and known facts in the theory of cone and partial order, which can be found in [1, 19].

A real Banach space S is commonly defined as a real normed vector space where all Cauchy sequences must converge in S.

Let S be a real Banach space and U be a subset of S. Denote by the null element of S and by intU the interior of U. The subset U is called a cone if:

(i) and , then

(ii) i.e., , i.e., and , then .

Remark 1.1   Let the cone . Then we have since for any (where ), .

Given a cone , we define a partial ordering with respect to U by if and only if . We shall write if and , while will stand for when . A cone U is called normal if there exists such that for all ,

The least positive number of k is said to be the normal constant of U.

A cone U is called solid, if is not empty, i.e., .

Let . An operator is said to be mixed monotone if is nondecreasing in and nonincreasing in , i.e., and imply . An element is named a fixed point of if is said to be convex if for with and each we have

is said to be concave if is convex.

Let , write

If for some , then is named a fixed point of in M. The operator is said to be increasing or nondecreasing, if for any , implies .

Let and , then

is said to be an ordering interval.

Definition 1.1 ([11])  Let M = U or and with . Let be an operator.

(1) is said to be -concave, if for any and , ;

(2) is said to be -convex, if for any and , .

Remark 1.2 ([11])  Any -convex operator must be an e-convex operator, where the characteristic function .

Definition 1.3 ([11])  Let . An operator is said to be concave- convex, if there exists a function and a function such that implies and also satisfies the following two conditions:

;

.

Definition 1.4 ([20])  Let be an operator and . Suppose that

(i)  ,

(ii)  there is a real number satisfying

(1)

Then is said to be a generalized e-concave operator, and is named its characteristic function.

Similarly, in the above-mentioned definition, if the condition (ii) is replaced by the following

(ii’)

then is called a generalized e-convex operator, and is called its characteristic function.

2 Some basic results on abstract cones in order real Banach spaces

In this section, we will show some useful and interesting results about abstract cone and partial order in real ordered Banach spaces.

In what follows, the set S always represents a real Banach space.

Lemma 2.1  Assume the cone is solid and . Then for any , there exists a sufficient small number such that .

Proof.  For any , without loss of generality, assume that . Since , it follows that there is a sufficient small number r>0 such that

(2)

Obviously, we see

(3)

By 2.1 and 2.2 we have . Set , then there exists sufficient small such that , which completes the proof of Lemma 2.1.

By Lemma 2.1, we can deduce a number of useful results about cone and partial order.

Corollary 2.1  Assume the cone is solid and . If , then there exists a real number such that .

Proof.  Since , set , then . By Lemma 2.1, it follows that for any , there exists a real number such that

Take , then we have

which implies , a desired result.

Remark 2.1  In Corollary 2.1 above, we see the conclusion implies , where and so . Hence Corollary 2.1 can deduce [45], Lemma 3.1], that is to say, Corollary 2.1 generalizes [45], Lemma 3.1].

Proposition 2.1 ([46])  Assume the cone and .

(i) ;

(ii) .

Furthermore, suppose U is solid, we have

(iii) , so ;

(iv) ;

(v) ;

(vi) ;

(vii) ; .

Proof.  (i) Suppose , then , so we see and . Hence since . That is, . Now let us show . Otherwise if , then by we see , i.e., . On the other hand, since , i.e., , we have . So which implies , i.e., , a contradiction to . Thus . Therefore, .

(ii) similar to (i).

(iii) Suppose , by Lemma 2.1, for any , there exists such that . Taking , then we get . Hence, . Besides, if , then , so by the arguments above we have , i.e., .

(iv) Suppose . Noting that a basic fact as follows: for any and r>0, the ball in S can be written as

where Then we see there exists r>0 such that

hence

(4)

Since , it follows that

(5)

By 2.3 and 2.4 we have

which implies . That is, . So, .

(v) Similar to (iv).

(vi) Suppose , we need to prove , i.e., there exists r₁>0 such that

(6)

Since , there exists r>0 such that

So, for any , we get . Hence, it follows from that

(7)

So, . Now, we prove

(8)

i.e.,

In fact, for any , we have . In order to prove 2.7, it suffices to show . Indeed, setting , then we see , i.e., So by 2.6 we have , i.e., . Thus, 2.7 holds, which implies 2.5 holds by taking .

(vii) It is obvious that (vi) implies (vii). We only prove the second conclusion in (vii), namely, . The proof of the first conclusion in (vii) is obviously seen when one takes in the second one in (vii). Set , then implies , i.e., So by (vi) we have

since . Thus i.e., . So .

Lemma 2.2  Assume the cone is solid and . Then if and only if there is a point and a real number such that .

Proof.  (): Assume Taking , then we see . So .

(): Assume there is a point and a real number such that . Then by Proposition 2.1 (vii) we see and . So

Thus it follows from Proposition 2.1(v) that , i.e., Hence, Lemma 2.2 is true.

Lemma 2.3  Assume the cone is solid and . Then .

Proof.  (1) Firstly, let us show In fact, for any , there exist such that . Take . Then . So by Lemma 2.2, Hence, .

(2) Now we prove . For any , since , by Lemma 2.2, there exist such that and . Thus, , which implies that . Hence, .

That is to say,

3 Fixed points for concave-(-) convex mixed monotone operators

In this section, using some of the basic results on the abstract cones obtained in the last section, we will present a number of new fixed point theorems of concave- convex mixed monotone operator by virtue of cone theory as well as monotone iterative techniques in the setting of real ordered Banach spaces.

In what follows, we assume U is a normal cone in the real Banach space S.

Theorem 3.1  Let with and be a concave- convex mixed monotone operator, where and are two functions such that implies . Suppose that

(i) there exists such that and

(9)

(ii) there is a point such that

Then the operator admits the unique fixed point in , and for any , the iterated sequences

(10)

always converge to . Namely, and as .

Proof.  Now we prove the existence and the uniqueness of the fixed point. Set

(11)

Since is mixed monotone and , by 3.1 we have By induction, we see

Obviously, since from (i). Without loss of generality, we assume 0<r₀<1. Put , then we have . In general, we set

(12)

Then it follows that and

(13)

By induction, we can prove that

(14)

Actually, without loss of generality, let 0<t_m<1. From 3.5 and the fact that is concave- convex mixed monotone, we get

(15)(16)

By 3.7 and 3.8, we have

(17)

From 3.4 and 3.9, it follows that

(18)

which implies that {t_m} is increasing and so 3.6 holds. Hence, exists and . We now check

(19)

Otherwise, we have . Thus by 3.5 and the fact that is concave- convex mixed monotone, we see

(20)

and

(21)

From (ii), 3.12 and 3.13, we get

(22)

It follows from 3.4 and 3.14 that

(23)

Letting in 3.15 we have

a contradiction. This yields t^* = 1. Now, for all , we see

and

So, on account of the normality of the cone U, we get as and hence and are both Cauchy in S. Hence, by the fact that S is complete, there are two points such that and . Set . Since is mixed monotone, it follows from 3.2 that

(24)

Taking in 3.16 we know and

Using standard method (see [11]) we easily show and such fixed point of is unique in .

Next, we prove the convergence of the iterated sequences. For any initial , i.e., by 3.2 and the fact that is mixed monotone, we see

and

i.e., and . It is easily seen by induction that and Hence, it follows that

since . This shows that the convergence of the iterated sequence defined as 3.2 holds. Therefore, all the conclusions of Theorem 3.1 hold.

Remark 3.1  Compared with Theorem 2.1 in [11], Theorem 3.1 deletes the condition

(25)

in the condition (iii) of Theorem 2.1 in [11] and the other conditions are unchangeable, but the conclusions remain hold, so Theorem 3.1 improves Theorem 2.1 in [11]. More- over, the technique in the proof of Theorem 3.1 is different from that in the proof of Theorem 2.1 in [11] since the latter depends strongly on the condition 3.17.

Lemma 3.1  Let be a concave- convex mixed monotone operator with and for all and . Assume that there is a point with such that . Then , and moreover, there exist and such that

Proof.  The method and technique to prove Lemma 3.1 are similar to that in Lemma 2.1 in [15]. We omit the proof to avoid the repeatedness.

Theorem 3.2  Let and be a concave- convex mixed monotone operator with and for all and . Then the operator admits the unique fixed point in U_e. Moreover, for any initial value , the iterated sequences

always converge to . Namely, and as .

Proof.  In order to prove Theorem 3.2, we need to use Theorem 3.1. Now we will verify the operator satisfies all the conditions in Theorem 3.1.

In fact, since and the operator maps to U_e, it follows that By Lemma 3.1, there exist and such that

i.e., the condition (i) in Theorem 3.1 is satisfied.

On the other hand, since is concave- convex with for all and , it is obvious that the condition (ii) in Theorem 3.1 is also satisfied. Hence, the conclusions of Theorem 3.2 follow from Theorem 3.1.

4 Fixed points for mixed monotone operators with certain concavity and convexity

In this section, we will use the main results obtained in last section to deduce some new fixed point results for mixed monotone operators with certain concavity and convexity in the setting of ordered real Banach spaces.

In what follows, U always represents a normal cone of the real Banach space S.

Theorem 4.1  Let and with . Suppose that the operator is mixed monotone and satisfies the following conditions:

(i) there exists such that and

(ii) for fixed , is generalized e-concave (see Definition 1.4) with characteristic function ; for fixed , is -convex (see Definition 1.1). Furthermore, the number and the function satisfy

(H₁) is monotone in , and

(H₂) implies

Then admits the unique fixed point in , and for any , the iterated sequences

always converge to . Namely, and as .

Proof.  According to Theorem 3.1, it suffices to check is concave- convex with for all , where

(26)

Actually, since is generalized e-concave with regard to the first variable , and is -convex with respect to the second variable , it follows that for all and ,

and moreover, we know holds by means of (H₂) and (4.1). That is to say, is concave- convex with and satisfying 4.1. In addition, by (H₁), is monotone in , then it follows that the condition (ii) in Theorem 3.1 is satisfied. Indeed, for all we have and so it follows that

(27)

since is monotone in and we suppose is nondecreasing in without loss of generality. Take , then by 4.2 we get

(28)

Considering

(29)

by 4.3 and 4.4 we have

which means the condition (ii) in Theorem 3.1 is satisfied. Thus, the operator meets all the conditions of Theorem 3.1. Hence, the conclusions of Theorem 4.1 hold on the basis of Theorem 3.1.

Remark 4.1  Compared to Theorem 2.4 in [11], Theorem 4.1 does not acquire the characteristic function should satisfy the continuity condition “ is continuous in t from left", which is the crucial condition in (H₁) of Theorem 2.4 in [11], and the conclusions stay unchanged. So Theorem 4.1 improves Theorem 2.4 in [11].

Lemma 4.1  Let and the operator . Then the following are equivalent:

(a) For any 0<t<1 there is such that

(b) For any 0<t<1 there is such that

where .

Proof.  In Lemma 4.1, the relation between and is such that

Corollary 4.1  Let and be the same as in Theorem 4.1. Suppose that

(i) there exists such that and

(ii) there exist and a nonnegative function with such that for any , it holds that

(30)

and

(31)

Then admits the unique fixed point in , and for any initial value , the iterated sequences

always converge to . Namely, and as .

Proof.  Let us use Theorem 4.1 to prove it. We merely need to verify that condition (ii) from Theorem 4.1 is satisfied since the condition (i) has been met in the assumption. In fact, by 4.5 and Lemma 4.1, we see

(32)

where . That is to say, is generalized e-concave with its characteristic function with respect to the first variable . At the same time, 4.6 implies that is -convex with regard to the second variable . In addition, the condition (H₁) is satisfied since for any and . Besides, note that , then it follows that

so for any , we have

hence, we get

Then it is obviously seen that

which implies that

thus (H₂) is also satisfied. Based on the arguments above, we know that all the conditions of Theorem 4.1 are satisfied. Thus, the conclusions hold by Theorem 4.1.

Theorem 4.2  Let and with . Suppose that the operator is mixed monotone and satisfies the following conditions:

(i)

(ii) for fixed , is generalized e-concave with characteristic function ; for fixed , is convex. Furthermore, the function satisfies

(H₁) is monotone in , and

(H₂) there exists such that and implies

(33)

Then admits the unique fixed point in , and for any initial value , the iterated sequences

always converge to . Namely, and as .

Proof.  According to Theorem 3.1, it suffices to check is concave- convex with for all , where

(34)

In fact, since is generalized e-concave with respect to the first variable , and is convex with respect to the second variable , it follows from the fact the operator is mixed monotone that for all and , we see

and

Moreover, we know holds by virtue of 4.8 and 4.9. That is to say, is concave- convex. In addition, by (H₁), is monotone in , then it follows that the condition (ii) in Theorem 3.1 is satisfied. In fact, for all we see and

Take Since is monotone in , without loss of generality, suppose is nondecreasing in , then we have

so it follows that

i.e., Considering

we have

which yields the condition (ii) in Theorem 3.1 is satisfied. Thus all the conditions of Theorem 3.1 are met for the operator . Hence, the conclusions of Theorem 4.2 hold on the basis of Theorem 3.1.

Remark 4.2  Compared to Theorem 2.5 in [11], Theorem 4.2 does not require the characteristic function to satisfy the continuity condition “ is continuous in t from left", which is the crucial condition in (H₁) of Theorem 2.5 in [11], and the conclusions remain the same. So Theorem 4.2 improves Theorem 2.5 in [11].

Theorem 4.3  Let and with . Suppose that the operator is mixed monotone and satisfies the following conditions:

(i) there exists such that ;

(ii)

(iii) for fixed , is generalized e-concave with characteristic function ; for fixed , is generalized e-convex with characteristic function . Furthermore, these functions and satisfy the following

(H₁) the function is monotone in , and

(H₂) for all , it holds that

(35)

Then admits the unique fixed point in , and for any , the iterated sequences

always converge to . Namely, and as .

Proof.  According to Theorem 3.1, it suffices to check that is concave- convex, where

In fact, for all and , we have

and moreover, holds by virtue of (4.10). So, is concave- convex. Hence the conclusions of Theorem 4.3 follow from Theorem 3.1.

Theorem 4.4  Let and with . Suppose that the operator is mixed monotone and satisfies the following conditions:

(36)

(ii) for fixed , is concave; for fixed , is generalized e-convex with characteristic function . Moreover, the function satisfies

(H₁) is monotone in , and

(H₂) there exists such that and implies

(37)

Then admits the unique fixed point in , and for any initial value , the iterated sequences

always converge to . Namely, and as .

Proof.  Let

By 4.11 and the fact that is mixed monotone, we have

(38)

Since is mixed monotone, by (H₂), it follows that , thus . Now, we begin to show is concave- convex. It is sufficient to prove is concave- convex, where

(39)

In fact, for all , we have

By 4.12 and 4.14, we obtain holds for all . Hence, based on the arguments above we see is concave- convex, and so all the conditions of Theorem 3.1 are satisfied, which means that Theorem 4.4 holds on the basis of Theorem 3.1.

Remark 4.3  Compared to Corollary 3.3 in [20], Theorem 4.4 deletes the continuity condition “ is continuous in t from left", which is the crucial condition the characteristic function should satisfy in the assumption (iii) of Corollary 3.3 in [20], and the conclusions remain the same. So Theorem 4.4 improves [20], Corollary 3.3].

Theorem 4.5  Let and with . Suppose that the operator is mixed monotone and satisfies the following conditions:

(i) there exists such that ;

(ii)

(iii) for fixed , is -concave; for fixed , is generalized e-convex with characteristic function . Moreover, the function satisfies

(H₁) the function is monotone in , and

(H₂) for all , it holds that

(40)

Then admits the unique fixed point in , and for any , the iterated sequences

always converge to . Namely, and as .

Proof.  According to Theorem 3.1, it suffices to check that is concave- convex, where

In fact, for all and , we have

and moreover, holds by virtue of 4.15. Hence, is concave- convex. The conclusions of Theorem 4.5 hold by virtue of Theorem 3.1.

Theorem 4.6  Let and be a mixed monotone operator. If there exist such that and

(H) for fixed , is -concave; for fixed , is -convex, then the operator admits the unique fixed point in U_e. Moreover, for any initial value , the iterated sequences

always converge to . Namely, and as .

Proof.  Obviously, it can be shown that the operator is concave- convex mixed monotone, where

Note that for all , since , it follows that

Namely, So, is concave- convex mixed monotone operator with and indeed. Hence, the conclusions of Theorem 4.6 hold by virtue of Theorem 3.2.

Corollary 4.2  Let U be solid and be a mixed monotone operator. If there exist such that and

(H) for fixed , is -concave; for fixed , is -convex,

then the operator admits the unique fixed point in . Moreover, for any initial value , the iterated sequences

always converge to . Namely, and as .

Proof.  Since U is solid, . Hence there exists By Lemma 2.3, we have , which finishes the proof of Corollary 4.2 by Theorem 4.6.

Obviously, the following simple result is derived from Corollary 4.2; the proof is omitted.

Corollary 4.3  Let be the same as in Corollary 4.2. Assume that there exists such that

(H) for fixed , is -concave; for fixed , is -convex.

Then the operator admits the unique fixed point in . Moreover, for any , the iterated sequences

always converge to . Namely, and as .

Remark 4.4  Compared with Corollary 2.1 and Corollary 2.2 in [11], Corollary 4.2 and Corollary 4.3 delete the following condition of coupled upper and lower solutions

(ii) there exist two points with such that

in Corollary 2.1 and Corollary 2.2 in [11], and the other conditions remain unchangeable, while the conclusions are the same. So Corollary 4.2 and Corollary 4.3 improve Corollary 2.1 and Corollary 2.2 in [11], respectively.

5 Applications

In this section, we will give applications of some main results gained above to nonlinear integral equations. As a result, it shows that the fixed point theorems of concave- convex mixed monotone operator obtained in the previous section are powerful to study the existence and uniqueness of the positive solutions to two classes of nonlinear integral equations.

Theorem 5.1  Let G be a closed subset of . Suppose is a continuous function and are two positive real numbers with . Then the following nonlinear nonlinear integral equation

(41)

admits a unique positive solution . Moreover, the iterated sequences and with

and

for any initial , we have

as .

Proof.  Let , the space of continuous bounded functions on G. We define , then S is a Banach space. Let

It is not difficult to observe that U is a normal solid cone in S and . Obviously, can be written as , where

We now confirm that the operator fulfills all the requirements specified in Corollary 4.2. In fact, put and , then is a mixed monotone operator and for fixed , is -concave; for fixed , is -convex, where . That is to say, all the conditions in Corollary 4.2 are satisfied. As a results, the conclusions of Theorem 5.1 hold by Corollary 4.2.

Remark 5.1  If one takes in Theorem 5.1, then Theorem 5.1 is reduced to the case of [11, Example 3.1]. Obviously, Theorem 5.1 extends and improves [11, Example 3.1], since Theorem 5.1 concludes that has a unique positive solution in , the set of all positive continuous bounded functions on G, while Example 3.1 in [11] can only state that has a unique positive solution in a given interval such as [10⁻²,1], but cannot in the whole .

In what follows, we will use to represent the family of all bounded continuous functions of . Define the norm . Then S is a real Banach space. Denote by all the nonnegative functions in . It is obvious that S is ordered by the normal cone U with the order relation as if and only if for all .

Theorem 5.2  Let be a nonnegative and continuous function. Consider the following Hammerstein integral equation:

(42)

where is a real constant and Further, suppose the following conditions are satisfied:

(i) f is nondecreasing, i.e.,

(ii) for any , there exists a real function with and such that

(iii) there exist two functions and a real number r₀>0 with such that

Then admits a unique continuous positive solution with . Furthermore, for any initial value with and , the sequences and with

and

always converge to . Namely,

Proof.  It is easily seen that Eq. (5.2) can be written as where Here and are defined as

(43)

We will prove Theorem 5.2 by means of Corollary 4.1. It is sufficient to check that the operator satisfies all the conditions of Corollary 4.1.

At first, we can demonstrate that is a mixed monotone operator. Actually, for any , suppose that with (that is, , then by (i) and we see

which means is mixed monotone.

Next, by (iii) and 5.3, we see that there exist and r₀>0 with such that

so the condition (i) in Corollary 4.1 is satisfied.

Thirdly, let us check the condition (ii) in Corollary 4.1 is also satisfied. In fact, by (ii), for any , there exists with and such that

so it follows that for given and for any and , we have

(44)

noting that in 5.4 since and we see .

On the other hand, for given and for any and , we get

(45)

noting that in 5.5 since and we see and so .

By 5.4 and 5.5 we hold that the condition (ii) in Corollary 4.1 is satisfied. In a word, all the conditions of Corollary 4.1 are satisfied. Therefore, the conclusions of Theorem 5.2 hold by Corollary 4.1.

Remark 5.2  In Corollary 4.1, if we take as a constant with and , then Corollary 4.1 is reduced to [11], Theorem 2.6]. So Corollary 4.1 generalizes [11], Theorem 2.6] and also Theorem 5.2 generalizes [11], Example 3.1].

6 Conclusions

In this paper, we prove some useful results about abstract cone and partial order in real ordered Banach spaces. Utilizing these results, we obtain a number of new fixed point theorems of concave- convex mixed monotone operators by means of monotone iterative techniques in real ordered Banach spaces with the normal cones. Then, these results continue to deduce other new fixed point results for mixed monotone operators with certain concavity and convexity. Our main results improve and extend the previous findings. In order to explain these results, we show that the results are powerful to study the existence and uniqueness of the positive solutions to two classes of nonlinear integral equations. The main novelty of this paper is that much deeper fixed point results about mixed monotone operators with certain concavity and convexity are presented by virtue of a number of new monotone iterative techniques in the setting of ordered Banach spaces. Future research will be focused on the applications of the obtained new fixed point results regarding mixed monotone operators to nonlinear mixed monotone systems and discuss their global dynamical properties.