Fixed point theorems of φ convex ψ concave mixed monotone operators and applications

J. Math. Anal. Appl. 339 (2008) 970 98 www.elsevier.com/locate/jmaa Fixed point theorems of φ convex ψ concave mixed monotone operators and applications Zhang Meiyu a,b a School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, PR China b Foundation Department, Taiyuan Institute of Technology, Taiyuan, Shanxi 030008, PR China Received 6 October 2006 Available online August 2007 Submitted by Richard M. Aron Abstract In this paper, φ convex ψ concave mixed monotone operators are introduced and some new existence and uniqueness theorems of fixed points for mixed monotone operators with such convexity concavity are obtained. As an application, we give one example to illustrate our results. 2007 Elsevier Inc. All rights reserved. Keywords: Cone; φ Convex ψ concave operator; Mixed monotone operator; Fixed point. Introduction The study of mixed monotone operators have been discussed by many authors since it was introduced in 987, because it has not only important theoretical meaning but also wide applications. Recently, some authors focused on mixed monotone operators with certain concavity and convexity (see [3,5 9]). In this paper, we introduce φ convex ψ concave mixed monotone operator which is different from previous ones. The questions discussed in literatures are concavity in essence. However, the questions discussed in this paper are convexity in essence, of which results are seldom seen up to now. Our methods are different from previous ones, that is to say we seek pre-images instead of images for mappings. The aim of this paper is to obtain the existence and uniqueness of positive fixed points for φ convex ψ concave mixed monotone operators without assuming the operators to be continuous or compact. To demonstrate the applicability of our results, in the final section of the paper, we give an application to certain integral equations. 2. Preliminaries Here we present some concepts. For details see [ 4]. This work was supported by Fund of the National Natural Science of China (037068) and Fund of the National Natural Science of Shanxi Province (2004003). E-mail address: zmy.03@63.com. 0022-247X/$ see front matter 2007 Elsevier Inc. All rights reserved. doi:0.06/j.jmaa.2007.07.06

M. Zhang / J. Math. Anal. Appl. 339 (2008) 970 98 97 Let E be a real Banach space. We denote the zero element of E by θ. A non-empty convex closed set P E is called a cone in E if it satisfies (i) x P, λ 0 λx P ; (ii) x P, x P x = θ. E can be partially ordered by the cone P E, that is, x y (or y x) if and only if y x P.Ifx y and x y, then we denote x<y(or y>x). Denote [x,y]={z E x z y}, where x,y E and x y. A cone P is said to be solid if P ={x P x is an interior point of P }. We write x y (or y x) if y x P. It is easy to see that we can choose t 0 (0, ) sufficiently small such that x t 0 y for x E and y P. Hence, we can also choose t 0 (0, ) sufficiently small such that t 0 y x t 0 y for x,y P. A cone P is called normal if there exists a constant M>0such that θ x y implies x M y. Definition 2.. Let P be a cone in a real Banach space E, D D 0 E and λd 0 D 0 for λ (0, ). A : D 0 D 0 E is said to be φ convex ψ concave in D, if there exist functions φ : (0, ) D (0, + ) and ψ : (0, ) D (0, + ) such that (t, x) (0, ) D implies 0 < φ(t, x)ψ(t, x) < t, and also A satisfies the following two conditions: (H ) A(tx,y) φ(t,x)a(x,y) for (t,x,y) (0, ) D D; (H 2 ) A(x,ty) ψ(t,y) A(x,y) for (t,x,y) (0, ) D D. A : D 0 D 0 E is called a φ convex ψ concave in D mixed monotone operator, if and only if A is both φ convex ψ concave in D and mixed monotone. Definition 2.2. (See [7].) Let P be a cone of a real Banach space E, e>θ, write P e ={x E λ,μ > 0 such that λe x μe}. An operator A : P P is said to be e-convex (e-concave), if it satisfies the following two conditions: (i) A is e-positive, i.e., A(P \{θ}) P e ; (ii) there exists a function η : (0, ) P e (0, + ) such that A(tx) ( + η(t, x))tax (A(tx) ( + η(t, x))tax) for (t, x) (0, ) P e, where η(t,x) is called the characteristic function of A. Definition 2.3. Let P be a cone of a real Banach space E. Suppose that D P and α (, + ). An operator A : D D is said to be α-convex (α-concave) if it satisfies A(tx) t α Ax (A(tx) t α Ax)for (t, x) (0, ) D. Definition 2.4. Let P be a cone of a real Banach space E. Suppose that D P. An operator A : D D is said to be convex (concave) if it satisfies A(tx + ( t)y) tax + ( t)ay (A(tx + ( t)y) tax + ( t)ay)for (t,x,y) (0, ) D D. 3. Main results Lemma 3.. Let E be a real Banach space, P be a normal and solid cone of E, S be a complete ordered closed subset of E with S 0 = S \{θ} P and λs S for λ [0, ]. Let u 0,v 0 S 0, A : P P E be a mixed monotone operator with A(([θ,v 0 ] S) ([θ,v 0 ] S)) S and A(u 0,v 0 ) u 0 v 0 A(v 0,u 0 ). Assume that there exists a function η : (0, ) ([u 0,v 0 ] S) (0, + ) such that A(tx, x) η(t,x)a(x,tx) for (t, x) (0, ) ([u 0,v 0 ] S), and (t, x) (0, ) ([u 0,v 0 ] S) implies 0 <η(t,x)<t. Suppose that (i) for u, v [u 0,v 0 ] S with A(u, v) u v, there exists u S such that u A(u,v) u v; similarly, for u, v [u 0,v 0 ] S with u v A(v, u), there exists v S such that u v A(v,u) v; (ii) there exists an element w 0 [u 0,v 0 ] S such that η(t,x) η(t,w 0 ) for (t, x) (0, ) ([u 0,v 0 ] S), and lim s t η(s,w 0 )<t for t (0, ).

972 M. Zhang / J. Math. Anal. Appl. 339 (2008) 970 98 Proof. By the condition (i), there exists u S such that u 0 A(u,v 0 ) u v 0, and then there exists v S such that u v A(v,u ) v 0. Likewise, there exists u 2 S such that u A(u 2,v ) u 2 v, then there exists v 2 S such that u 2 v 2 A(v 2,u 2 ) v. In general, there exists u n S such that u n A(u n,v n ) u n v n, then there exists v n S such that u n v n A(v n,u n ) v n (n =, 2,...). Taking r n = sup{r (0, ) u n rv n }, one has 0 <r 0 <r < <r n <r n+ < <. Therefore, lim n r n = sup{r n n = 0,, 2,...}=r (0, ]. Since r n+ >r n = sup{r (0, ) u n rv n }, one has u n r n+ v n. Noticing that S is complete ordered and λs S for λ [0, ],wehaveu n <r n+ v n. Hence, Therefore, u n A(u n,v n ) η(r n+,v n )A(v n,r n+ v n ) η(r n+,v n )A(v n,u n ) η(r n+,v n )v n. r n = sup { r (0, ) u n rv n } η(rn+,v n ) η(r n+,w 0 ) (3.) (n =, 2,...). We can prove that r. Otherwise we have r (0, ). Letn + in (3.), by the condition (ii), we get lim r n = r lim η(r n+,w 0 ) = lim η(s,w 0)<r, n n s r which is a contradiction. Since θ u n+p u n v n u n ( r n )v n ( r n )v 0 for n, p =, 2,..., by the normality of P, we know that u n+p u n θ (n )for p =, 2,..., i.e., {u n } is a Cauchy sequence in E. Hence, there exists u E such that u n u (n ), and lim n v n = lim n u n = u. It is easy to see that u 0 u n u v n v 0 for n =, 2,... Noticing that S is closed, we get u [u n,v n ] S [u 0,v 0 ] S (n = 0,, 2,...). Finally, we show that u is a fixed point of A in [u 0,v 0 ] S. It follows from the mixed monotone properties of A that u n A(u n,v n ) A ( u,u ) A(v n,u n ) u n. Letting n,wehavea(u,u ) = u, which completes the proof. Theorem 3.2. Let E be a real Banach space, P be a normal and solid cone of E, S be a complete ordered closed and λs S for λ [0, ]. Let u 0,v 0 S 0, A : P P E be a mixed monotone operator with A(([θ,v 0 ] S) ([θ,v 0 ] S)) S and A(u 0,v 0 ) u 0 v 0 A(v 0,u 0 ). Assume that there exists a function ϕ : (0, ) ([u 0,v 0 ] S) ([u 0,v 0 ] S) (0, + ) such that A(tx, y) ϕ(t,x,y)a(x,ty) for (t,x,y) (0, ) ([u 0,v 0 ] S) ([u 0,v 0 ] S), and (t, x) (0, ) ([u 0,v 0 ] S) implies 0 <ϕ(t,x,x)<t. Suppose (i) for u, v [u 0,v 0 ] S with A(u, v) u v, there exists u S such that u A(u,v) u v; similarly, for u, v [u 0,v 0 ] S with u v A(v, u), there exists v S such that u v A(v,u) v; (ii) there exists an element w 0 [u 0,v 0 ] S such that ϕ(t,x,x) ϕ(t,w 0,w 0 ) for (t, x) (0, ) ([u 0,v 0 ] S), and lim s t ϕ(s,w 0,w 0 )<t for t (0, ). Proof. Set η : (0, ) ([u 0,v 0 ] S) (0, + ) be η(t,x) = ϕ(t,x,x) for (t, x) (0, ) ( [u 0,v 0 ] S ). Then we have A(tx, x) η(t,x)a(x,tx),0<η(t,x)<t and η(t,x) η(t,w 0 ) for (t, x) (0, ) ([u 0,v 0 ] S); and lim s t η(s,w 0 ) = lim s t ϕ(s,w 0,w 0 )<tfor t (0, ). Hence, all the conditions of Lemma 3. are satisfied, by Lemma 3., we get the conclusion of Theorem 3.2 hold.

M. Zhang / J. Math. Anal. Appl. 339 (2008) 970 98 973 Theorem 3.3. Let E be a real Banach space, P be a normal and solid cone of E, S be a complete ordered closed and λs S for λ [0, ]. Let A : P P E be a φ convex ψ concave in [u 0,v 0 ] S mixed monotone operator with A(([θ,v 0 ] S) ([θ,v 0 ] S)) S, where u 0,v 0 S 0 with A(u 0,v 0 ) u 0 v 0 A(v 0,u 0 ). Suppose (i) for u, v [u 0,v 0 ] S with A(u, v) u v, there exists u S such that u A(u,v) u v; similarly, for u, v [u 0,v 0 ] S with u v A(v, u), there exists v S such that u v A(v,u) v; (ii) there exists an element w 0 [u 0,v 0 ] S such that φ(t,x)ψ(t,x) φ(t,w 0 )ψ(t,w 0 ) for (t, x) (0, ) ([u 0,v 0 ] S), and lim s t φ(s,w 0 )ψ(s, w 0 )<t for t (0, ). Proof. Set ϕ : (0, ) ([u 0,v 0 ] S) ([u 0,v 0 ] S) (0, + ) be ϕ(t,x,y) = φ(t,x)ψ(t,y) for (t,x,y) (0, ) ( [u 0,v 0 ] S ) ( [u 0,v 0 ] S ). Then we have A(tx, y) φ(t,x)a(x,y) φ(t,x)ψ(t,y)a(x,ty) = ϕ(t,x,y)a(x,ty) for (t,x,y) (0, ) ( [u 0,v 0 ] S ) ( [u 0,v 0 ] S ) ; 0 <ϕ(t,x,x)<t, ϕ(t,x,x) ϕ(t,w 0,w 0 ) for (t, x) (0, ) ( [u 0,v 0 ] S ) ; and lim s t ϕ(s,w 0,w 0 ) = lim s t φ(s,w 0 )ψ(s, w 0 )<t for t (0, ). Hence, all the conditions of Theorem 3.2 are satisfied, by Theorem 3.2, we get the conclusion of Theorem 3.3 hold. Theorem 3.4. Let E be a real Banach space, P be a normal and solid cone of E, S be a complete ordered closed and λs S for λ [0, ]. Let A : P P E be a φ convex ψ concave in [u 0,v 0 ] S mixed monotone operator with A(([θ,v 0 ] S) ([θ,v 0 ] S)) S, where u 0,v 0 S 0 with A(u 0,v 0 ) u 0 v 0 A(v 0,u 0 ). Suppose (i) writing S ={(u, v) ([u 0,v 0 ] S) ([u 0,v 0 ] S) A(u, v) u v} and S 2 ={(u, v) ([u 0,v 0 ] S) ([u 0,v 0 ] S) u v A(v, u)}, assume that S {(A(a, b), b) (a, b) S } and S 2 {(a, A(b, a)) (a, b) S 2 }; (ii) there exists an element w 0 [u 0,v 0 ] S such that φ(t,x)ψ(t,x) φ(t,w 0 )ψ(t,w 0 ) for (t, x) (0, ) ([u 0,v 0 ] S); and lim s t φ(s,w 0 )ψ(s, w 0 )<t for t (0, ); (iii) for x,y [u 0,v 0 ] S and x y, assume that x = A(a,b ), y = A(a 2,b 2 ), x a i,b i y, a i,b i S(i=, 2) and b 2 b imply a a 2. Then A has one unique fixed point in [u 0,v 0 ] S. Proof. Clearly, (u 0,v 0 ) S S 2, S S 2. By the condition (i), we can easily get: () for u, v [u 0,v 0 ] S with A(u, v) u v, one has (u, v) S {(A(a, b), b) (a, b) ([u 0,v 0 ] S) ([u 0,v 0 ] S),A(a,b) a b}, and hence, there exists u [u 0,v 0 ] S such that u = A(u,v) u v; (2) for u, v [u 0,v 0 ] S with u v A(v, u), one has (u, v) S 2 {(a, A(b, a)) (a, b) ([u 0,v 0 ] S) ([u 0,v 0 ] S), a b A(b, a)}, and hence, there exists v [u 0,v 0 ] S such that u v A(v,u)= v. That is to say that the condition (i) of Theorem 3.3 is satisfied, hence, all the conditions of Theorem 3.3 are satisfied. So we can get two sequences {u n } and {v n } with u i,v i S 0 (i = 0,, 2,...), u n = A(u n+,v n ) u n+ v n+ A(v n+,u n+ ) = v n (n = 0,, 2,...); u n r n v n, where 0 <r n < (n = 0,, 2,...) and lim n r n = ; and hence, lim n u n = lim n v n = u [u 0,v 0 ] S is a fixed point of A. Now we prove that u is the unique fixed point of A in [u 0,v 0 ] S. Suppose that x [u 0,v 0 ] S isalsoafixed point of A. Noticing u 0 = A(u,v 0 ) x = A(x,x ) v 0 = A(v,u ), by the condition (iii), we have A(u 2,v ) =

974 M. Zhang / J. Math. Anal. Appl. 339 (2008) 970 98 u x v = A(v 2,u 2 ). Likewise, u 2 x v 2. In general, u n x v n (n = 0,, 2,...). Hence, it follows from the normality of the cone P that x = lim n u n = lim n v n = u, which completes the proof. Remark 3.5. Clearly, in Lemma 3., Theorems 3.2 3.4, A(([θ,v 0 ] S) ([θ,v 0 ] S)) S can be replaced by A(([θ,v 0 ] S 0 ) ([θ,v 0 ] S 0 )) S 0. Theorem 3.6. Let E be a real Banach space, P be a normal and solid cone of E, S be a complete ordered closed and λs S for λ [0, + ). Let A : P P E be a φ convex ψ concave in S 0 mixed monotone operator with A(S 0 S 0 ) S 0. Suppose (i) for u, v S 0 with A(u, v) u v, there exists u S 0 such that u A(u,v) u v; similarly, for u, v S 0 with u v A(v, u), there exists v S 0 such that u v A(v,u) v; (ii) there exists an element w 0 S 0 such that φ(t,x)ψ(t,x) φ(t,w 0 )ψ(t, w 0 ) for (t, x) (0, ) S 0, and lim s t φ(s,w 0 ) ψ(s,w 0 )<t for t (0, ); moreover, there exists an element h S 0 such that 0 < φ(t,h),ψ(t,h) t 2 for t (0,t ), where t (0, ] is a constant. Then A has at least one fixed point in S 0. Proof. Since w 0,h,A(h,h) S 0 P, we can choose t 0 (0,t ) (0, ) sufficiently small such that t 0 h w 0, A(h, h) t 0 h. Taking u 0 = t 0 h and v 0 = t 0 h,wehavew 0,u 0,v 0 S 0 [u 0,v 0 ], u 0 v 0, ( A(u 0,v 0 ) φ(t 0,h)A h, ) h φ(t 0, h)a(h, h) φ(t 0,h) h t 0 h = u 0, t 0 t 0 ( ) A(v 0,u 0 ) ψ(t 0,h) A h, h t 0 ψ(t 0,h) A(h, h) t 0 ψ(t 0,h) h h = v 0. t 0 Hence, A : ([θ,v 0 ] S 0 ) ([θ,v 0 ] S 0 ) S 0 be a φ convex ψ concave in [u 0,v 0 ] S 0 mixed monotone operator. Therefore, by Theorem 3.3, we get A has at least one fixed point in S 0 [u 0,v 0 ]. Theorem 3.7. Let E be a real Banach space, P be a normal and solid cone of E, S be a complete ordered closed and λs S for λ [0, + ). Let A : P P E be a φ convex ψ concave in S 0 mixed monotone operator with A(S 0 S 0 ) S 0. Suppose (i) writing S ={(u, v) S 0 S 0 A(u, v) u v} and S 2 ={(u, v) S 0 S 0 u v A(v, u)}, assume that S {(A(a, b), b) (a, b) S } and S 2 {(a, A(b, a)) (a, b) S 2 }; (ii) there exists an element w 0 S 0 such that φ(t,x)ψ(t,x) φ(t,w 0 )ψ(t, w 0 ) for (t, x) (0, ) S 0, and lim s t φ(s,w 0 ) ψ(s,w 0 )<t for t (0, ); moreover, there exists an element h S 0 such that 0 < φ(t,h),ψ(t,h) t 2 for t (0,t ), where t (0, ] is a constant; (iii) for x,y S 0 and x y, assume that x = A(a,b ), y = A(a 2,b 2 ), x a i,b i y, a i,b i S 0 (i =, 2) and b 2 b imply a a 2. Then A has one unique fixed point in S 0. Proof. Just as the proof of Theorem 3.5, we can choose t 0 (0,t ) (0, ) sufficiently small such that t 0 h A(h, h), w 0 t 0 h. Taking u 0 = t 0 h and v 0 = t 0 h,wehavew 0,u 0,v 0 S 0 [u 0,v 0 ] and A(u 0,v 0 ) u 0 v 0 A(v 0,u 0 ). For (u, v) S ={(u, v) ([u 0,v 0 ] S) ([u 0,v 0 ] S) A(u, v) u v} ( S {(A(a, b), b) a,b S 0, A(a, b) a b}), one has there exists u S 0, such that u 0 u = A(u,v) u v v 0, and hence u S 0 [u 0,v 0 ], (u, v) {(A(a, b), b) (a, b) S }. Therefore S {(A(a, b), b) (a, b) S }. Similarly, we have S 2 = {(u, v) ([u 0,v 0 ] S) ([u 0,v 0 ] S) u v A(v, u)} {(a, A(b, a)) (a, b) S 2 }. That is to say that the condition (i) of Theorem 3.4 is satisfied, hence, all the conditions of Theorem 3.4 are satisfied. By Theorem 3.4, we know that A has one unique fixed point u in [u 0,v 0 ] S.

M. Zhang / J. Math. Anal. Appl. 339 (2008) 970 98 975 Now we show that the fixed point of A is unique in S 0. In fact, suppose that x,y S 0 are both fixed points of A. We can choose the above t 0 (0,t ) sufficiently small such that u 0 = t 0 h x,y t 0 h = v 0 also. Hence, x and y are both fixed points of A in [u 0,v 0 ] S. Noticing that the fixed point of A in [u 0,v 0 ] S is unique, we get x = y, which complete the proof. Remark 3.8. () In Theorems 3.6 and 3.7 the condition (ii) can be changed into (ii ) there exists an element w 0 S 0 such that φ(t,x)ψ(t,x) φ(t,w 0 )ψ(t, w 0 ) for (t, x) (0, ) S 0, and there exists the limit lim s t φ(s,w 0 )ψ(s, w 0 ) for t (0, ); moreover, 0 <φ(t,w 0 ) t 2,0< ψ(t,w 0 ) t 2 for t (0, ). (2) The condition (ii) of Lemma 3. (or Theorem 3.2, or Theorems 3.3 and 3.4) should be changed into separately: (ii ) η(t,x) (or ϕ(t,x,x), orφ(t,x)ψ(t,x)) is non-increasing (non-decreasing) in x, and is continuous from left in t. The condition (ii) of Theorems 3.6 and 3.7 should be changed into separately: (ii ) φ(t, x)ψ(t, x) is non-increasing (non-decreasing) in x, and is continuous from left in t; moreover, there exists an element h S 0 such that 0 <φ(t,h) t 2,0<ψ(t,h) t 2 for t (0,t ), where t (0, ] is a constant. (3) In Lemma 3., Theorems 3.2 3.4, 3.6, 3.7 assume that the functions η : (0, ) D (0, + ), ϕ : (0, ) D D (0, + ), φ : (0, ) D (0, + ) and ψ : (0, ) D (0, + ) are single-valued functions η : (0, ) (0, + ), ϕ : (0, ) (0, + ), φ : (0, ) (0, + ) and ψ : (0, ) (0, + ) separately. Then the condition (ii) of Lemma 3. (or Theorem 3.2, or Theorems 3.3 and 3.4) should be changed into separately: (ii ) there exists the limit lim s t η(s) (or lim s t ϕ(s), or lim s t φ(s)ψ(s)) and lim s t η(s) < t (or lim s t ϕ(s) < t, or lim s t φ(s)ψ(s) < t)for t (0, ). The condition (ii) of Theorems 3.6, 3.7, should be changed into separately: (ii 2 ) there exists the limit lim s t η(s) (or lim s t ϕ(s), or lim s t φ(s)ψ(s)) and lim s t η(s) < t (or lim s t ϕ(s) < t, or lim s t φ(s)ψ(s) < t) for t (0, ); moreover, 0 <φ(t) t 2,0<ψ(t) t 2 for t (0,t ), where t (0, ) is a constant. Now we use Theorems 3.3, 3.4, 3.6, 3.7 to obtain a series of new fixed point theorems for mixed monotone operators with certain convexity and concavity. Theorem 3.9. Let E be a real Banach space, P be a normal and solid cone of E, S be a complete ordered closed subset of E with λs S for λ [0, ] and S 0 = S \{θ} P. Let A : P P P be a mixed monotone operator, (i) for fixed y P, A(,y): P P is α-convex; for fixed x P, A(x, ) : P P is β-concave, where α, β (, + ), α + β>; (ii) for u, v [u 0,v 0 ] S with A(u, v) u v, there exists u S such that u A(u,v) u v; similarly, for u, v [u 0,v 0 ] S with u v A(v, u), there exists v S such that u v A(v,u) v. Proof. Set φ : (0, ) P (0, + ) and ψ : (0, ) P (0, + ) be φ(t,x) = t α and ψ(t,x) = t β for (t, x) (0, ) P, separately. Then A : P P P is a φ convex ψ concave in [u 0,v 0 ] S mixed monotone operator. Taking w 0 [u 0,v 0 ] S arbitrarily, we have φ(t,x)ψ(t,x) = φ(t,w 0 )ψ(t, w 0 ) = t α+β for (t, x) (0, ) ([u 0,v 0 ] S), and lim s t φ(s,w 0 )ψ(s, w 0 ) = t α+β <t for t (0, ). Hence, all the conditions of Theorem 3.3 are satisfied. Therefore, by Theorem 3.3, the proof is completed. Similar to Theorem 3.9, the following three theorems can be also obtained by using Theorems 3.6, 3,4, 3.7 separately, so we omit the proof.

976 M. Zhang / J. Math. Anal. Appl. 339 (2008) 970 98 Theorem 3.0. Let E be a real Banach space, P be a normal and solid cone of E, S be a complete ordered closed and λs S for λ [0, + ). Let A : P P P be a mixed monotone operator with A(S 0 S 0 ) S 0. Suppose (i) for fixed y P, A(,y): P P is α-convex; for fixed x P, A(x, ) : P P is β-concave, where α, β 2; (ii) for u, v S 0 with A(u, v) u v, there exists u S 0 such that u A(u,v) u v; similarly, for u, v S 0 with u v A(v, u), there exists v S 0 such that u v A(v,u) v. Then A has at least one fixed point in S 0. Theorem 3.. Let E be a real Banach space, P be a normal and solid cone of E, S be a complete ordered closed subset of E with λs S for λ [0, ] and S 0 = S \{θ} P. Let A : P P P be a mixed monotone operator, (i) for fixed y P, A(,y): P P is α-convex; for fixed x P, A(x, ) : P P is β-concave, where α, β (, + ), α + β>; (ii) writing S ={(u, v) ([u 0,v 0 ] S) ([u 0,v 0 ] S) A(u, v) u v} and S 2 ={(u, v) ([u 0,v 0 ] S) ([u 0,v 0 ] S) u v A(v, u)}, assume that S {(A(a, b), b) (a, b) S } and S 2 {(a, A(b, a)) (a, b) S 2 }; (iii) for x,y [u 0,v 0 ] S and x y, assume that x = A(a,b ), y = A(a 2,b 2 ), x a i,b i y, a i,b i S(i=, 2) and b 2 b imply a a 2. Then A has one unique fixed point in [u 0,v 0 ] S. Theorem 3.2. Let E be a real Banach space, P be a normal and solid cone of E, S be a complete ordered closed and λs S for λ [0, + ). Let A : P P P be a mixed monotone operator with A(S 0 S 0 ) S 0. Suppose (i) for fixed y P, A(,y): P P is α-convex; for fixed x P, A(x, ) : P P is β-concave, where α, β 2; (ii) writing S ={(u, v) S 0 S 0 A(u, v) u v} and S 2 ={(u, v) S 0 S 0 u v A(v, u)}, assume that S {(A(a, b), b) (a, b) S } and S 2 {(a, A(b, a)) (a, b) S 2 }; (iii) for x,y S 0 and x y, assume that x = A(a,b ), y = A(a 2,b 2 ), x a i,b i y, a i,b i S 0 (i =, 2) and b 2 b imply a a 2. Then A has one unique fixed point in S 0. Theorem 3.3. Let E be a real Banach space, P be a normal and solid cone of E, S be a complete ordered closed subset of E with λs S for λ [0, ] and S 0 = S \{θ} P. Let A : P P P be a mixed monotone operator, (i) for fixed y P, A(,y): P P is convex; for fixed x P, A(x, ) : P P is α-concave, α< ; (ii) for u, v [u 0,v 0 ] S with A(u, v) u v, there exists u S such that u A(u,v) u v; similarly, for u, v [u 0,v 0 ] S with u v A(v, u), there exists v S such that u v A(v,u) v. Proof. Set φ : (0, ) P (0, + ) and ψ : (0, ) P (0, + ) be φ(t,x) = and ψ(t,x) = t α for (t, x) (0, ) P, separately. Then A(tx, y) ta(x,y)+ ( t)a(θ,y) φ(t,x)a(x,y), A(x, ty) t α A(x,y) = ψ(t,y) A(x,y) for (t,x,y) (0, ) P P ; and 0 < φ(t, x)ψ(t, x) = t α <t for (t, x) (0, ) P. Hence, A : P P P is a φ convex ψ concave in P mixed monotone operator. We have φ(t,x)ψ(t,x) = t α and lim s t φ(s,x)ψ(s,x) = t α <tfor (t, x) (0, ) ([u 0,v 0 ] S). So by Theorem 3.3, we complete the proof.

M. Zhang / J. Math. Anal. Appl. 339 (2008) 970 98 977 Similar to Theorem 3.3, the following theorem can be also obtained by using Theorem 3.3. Theorem 3.4. Let E be a real Banach space, P be a normal and solid cone of E, S be a complete ordered closed subset of E with λs S for λ [0, ] and S 0 = S \{θ} P. Let A : P P P be a mixed monotone operator, (i) for fixed y P, A(,y): P P is α-convex, α>; for fixed x P, A(x, ) : P P is concave; (ii) for u, v [u 0,v 0 ] S with A(u, v) u v, there exists u S such that u A(u,v) u v; similarly, for u, v [u 0,v 0 ] S with u v A(v, u), there exists v S such that u v A(v,u) v. Theorem 3.5. Let E be a real Banach space, P be a normal and solid cone of E, S be a complete ordered closed subset of E with λs S for λ [0, ] and S 0 = S \{θ} P. Let A : P P P be a mixed monotone operator, (i) for fixed x P, A(x, ) : P P is α-concave, α<0; for fixed y P, A(,y): P P is e-convex (e θ) with its characteristic function η : (0, ) P e (0, + ), where η(t,x) is monotone in x and continuous in t from left, and (t, x) (0, ) ([u 0,v 0 ] S) implies 0 <η(t,x)<t α ; (ii) for u, v [u 0,v 0 ] S with A(u, v) u v, there exists u S such that u A(u,v) u v; similarly, for u, v [u 0,v 0 ] S with u v A(v, u), there exists v S such that u v A(v,u) v. Proof. Clearly, A((P \{θ}) P) P e = P and [u 0,v 0 ] S P. Set φ : (0, ) P e (0, + ) and ψ : (0, ) P e (0, + ) be φ(t,x) = t( + η(t,x)) and ψ(t,x) = t α for (t, x) (0, ) P e, separately. Then A(tx, y) φ(t,x)a(x,y), A(x, ty) ψ(t,y) A(x,y) for (t,x,y) (0, ) P e P e ; and 0 < φ(t, x)ψ(t, x) = t α ( + η(t,x)) < t α ( + t α ) = t for (t, x) (0, ) ([u 0,v 0 ] S). Hence, A : P P P is a φ convex ψ concave in [u 0,v 0 ] S mixed monotone operator. If η(t,x) is increasing in x, then we take w 0 = v 0 ;ifη(t,x) is decreasing in x, then we take w 0 = u 0. Then we have φ(t,x)ψ(t,x) = t α ( + η(t,x)) t α ( + η(t,w 0 )) = φ(t,w 0 )ψ(t, w 0 ) for (t, x) (0, ) ([u 0,v 0 ] S); and lim s t φ(s,w 0 )ψ(s, w 0 ) = t α ( + η(t,w 0 )) < t for t (0, ). So by Theorem 3.3, we complete the proof. Theorem 3.6. Let E be a real Banach space, P be a normal and solid cone of E, S be a complete ordered closed subset of E with λs S for λ [0, ] and S 0 = S \{θ} P. Let A : P P P be a mixed monotone operator, (i) for fixed y P, A(,y): P P is α-convex, α (, + ); for fixed x P, A(x, ) : P P is e-concave (e θ)with its characteristic function η : (0, ) P e (0, + ), where η(t,x) is monotone in x and continuous in t from left, and (t, x) (0, ) ([u 0,v 0 ] S) implies η(t,x) > max{0,t α 2 }; (ii) for u, v [u 0,v 0 ] S with A(u, v) u v, there exists u S such that u A(u,v) u v; similarly, for u, v [u 0,v 0 ] S with u v A(v, u), there exists v S such that u v A(v,u) v. Proof. Clearly, A(P (P \{θ})) P e = P and [u 0,v 0 ] S P = P e. Set φ : (0, ) P e (0, + ) and ψ : (0, ) P e (0, + ) be φ(t,x)= t α and ψ(t,x)= t(+η(t,x)) for (t, x) (0, ) P e, separately. Then A(tx, y) t α A(x,y) = φ(t,x)a(x,y), A(x, ty) t( + η(t, y))a(x, y) = ψ(t,y) A(x,y) for (t,x,y) (0, ) P e P e ; and 0 < φ(t, x)ψ(t, x) = t α (+η(t,x)) < = t for (t, x) (0, ) ([u t α t α 2 0,v 0 ] S). Hence, A : P P P is a φ convex ψ concave in [u 0,v 0 ] S mixed monotone operator. If η(t,x) is increasing in x, then we take w 0 = u 0 ; if η(t,x) is decreasing in x, then we take w 0 = v 0. Then we have φ(t,x)ψ(t,x) = t α (+η(t,x)) t α (+η(t,w 0 )) =

978 M. Zhang / J. Math. Anal. Appl. 339 (2008) 970 98 φ(t,w 0 )ψ(t, w 0 ) for (t, x) (0, ) ([u 0,v 0 ] S); and lim s t φ(s,w 0 )ψ(s, w 0 ) = lim s t <t for t (0, ). Hence, by Theorem 3.3, the proof can be completed. t α (+η(t,w 0 )) s α (+η(s,w 0 )) = Theorem 3.7. Let E be a real Banach space, P be a normal and solid cone of E, S be a complete ordered closed subset of E with λs S for λ [0, ] and S 0 = S \{θ} P. Let A : P P P be a mixed monotone operator, (i) e θ, for fixed y P, A(,y): P P is e-convex with its characteristic function η : (0, ) P e (0, + ); for fixed x P, A(x, ) : P P is e-concave with its characteristic function μ : (0, ) P e (0, + ); assume that η(t,x) is increasing (decreasing) in x and continuous in t from left, μ(t, x) is decreasing (increasing) in x and continuous in t from left, and (t, x) (0, ) ([u 0,v 0 ] S) implies + η(t,x) < t( + μ(t, x)); (ii) for u, v [u 0,v 0 ] S with A(u, v) u v, there exists u S such that u A(u,v) u v; similarly, for u, v [u 0,v 0 ] S with u v A(v, u), there exists v S such that u v A(v,u) v. Proof. Clearly, A((P \{θ}) (P \{θ})) P e = P P \{θ} and [u 0,v 0 ] S P. Set φ : (0, ) P e (0, + ) and ψ : (0, ) P e (0, + ) be φ(t,x) = t( + η(t,x)) and ψ(t,x) = t(+μ(t,x)) for (t, x) (0, ) P e, separately. Then A(tx, y) φ(t,x)a(x,y), A(x, ty) ψ(t,y) A(x,y) for (t,x,y) (0, ) P e P e ; and 0 < φ(t,x)ψ(t,x) = +η(t,x) +μ(t,x) <t for (t, x) (0, ) ([u 0,v 0 ] S). Hence, A : P P P is a φ convex ψ concave in [u 0,v 0 ] S mixed monotone operator. Taking w 0 = v 0 (w 0 = u 0 ), we have φ(t,x)ψ(t,x) φ(t,w 0 )ψ(t, w 0 ) for (t, x) (0, ) ([u 0,v 0 ] S); and lim s t φ(s,w 0 )ψ(s, w 0 ) = +η(t,w 0) +μ(t,w 0 ) <t for t (0, ). Hence, by Theorem 3.3, the proof can be completed. Remark 3.8. () The condition (i) of Lemma 3., Theorems 3.2, 3.3 and the condition (ii) of Theorems 3.9, 3.3 3.7 can be changed into the condition: (j) there exist u,v S 0 such that u 0 A(u,v 0 ) u v A(v,u ) v 0 ; likewise, there exist u 2,v 2 S 0 such that u A(u 2,v ) u 2 v 2 A(v 2,u 2 ) v ; in general, there exist u n,v n S 0 such that u n A(u n,v n ) u n v n A(v n,u n ) v n (n =, 2,...). And in this case, the results of these theorems can be changed into: Then A has at least one fixed point x in [u 0,v 0 ] S, and lim n u n = lim n v n = x. (2) The condition (i) of Lemma 3., Theorems 3.2, 3.3 and the condition (ii) of Theorems 3.9, 3.3 3.7 can be changed into (I) or (I 0 ) as follows: (I) for u, v [u 0,v 0 ] S 0 with A(u, v) u v A(v, u), there exist u S 0 and v S 0 such that u A(u,v) u v and u v A(v,u) v; (I 0 ) for u, v [u 0,v 0 ] S 0 with A(u, v) u v A(v, u), there exist u S 0 and v S 0 such that u A(u,v ) u v A(v,u ) v. (3) The condition (i) of Lemma 3.4 and the condition (ii) of Theorem 3. can be changed into the condition: (k) there exist u,v S 0 such that u 0 = A(u,v 0 ) u v A(v,u ) = v 0 ; likewise, there exist u 2,v 2 S 0 such that u = A(u 2,v ) u 2 v 2 A(v 2,u 2 ) = v ; in general, there exist u n,v n S 0 such that u n = A(u n,v n ) u n v n A(v n,u n ) = v n (n =, 2,...). And in this case, the conclusions of these theorems can be changed into: Then A has one unique fixed point x in [u 0,v 0 ] S, and lim n u n = lim n v n = x. (4) The condition (i) of Lemma 3.4 and the condition (ii) of Theorem 3. can be changed into (i ) or (i ) as follows: (i ) writing S ={(u, v) ([u 0,v 0 ] S) ([u 0,v 0 ] S) u v}, assume that S {(A(a, b), b) (a, b) S,A(a,b) a} {(u, A(v, u)) (u, v) S,v A(v, u)}; (i ) writing S ={(u, v) ([u 0,v 0 ] S) ([u 0,v 0 ] S) A(u, v) u v A(v, u)}, assume that S {(A(a, b), b) (a, b) S } {(u, A(v, u)) (u, v) S }.

M. Zhang / J. Math. Anal. Appl. 339 (2008) 970 98 979 (5) The condition (i) of Theorem 3.6 and the condition (ii) of Theorem 3.0 can be changed into the condition: (j ) there exist u i,v i S 0 (i = 0, ) such that u 0 A(u,v 0 ) u v A(v,u ) v 0 ; likewise, there exist u 2,v 2 S 0 such that u A(u 2,v ) u 2 v 2 A(v 2,u 2 ) v ; in general, there exist u n,v n S 0 such that u n A(u n,v n ) u n v n A(v n,u n ) v n (n =, 2,...). And in this case, the results of these theorems can be changed into: Then A has at least one fixed point x in S 0, and lim n u n = lim n v n = x. (6) The condition (i) of Theorem 3.6 and the condition (ii) of Theorem 3.0 can be changed into (I ) or (I 2 ) as follows: (I ) for u, v S 0 with A(u, v) u v A(v, u), there exist u S 0 and v S 0 such that u A(u,v) u v and u v A(v,u) v; (I 2 ) for u, v S 0 with A(u, v) u v A(v, u), there exist u S 0 and v S 0 such that u A(u,v ) u v A(v,u ) v. (7) The condition (i) of Theorem 3.7 and the condition (ii) of Theorem 3.2 can be changed into the condition: (k ) there exist u i,v i S 0 (i = 0, ) such that u 0 = A(u,v 0 ) u v A(v,u ) = v 0 ; likewise, there exist u 2,v 2 S 0 such that u = A(u 2,v ) u 2 v 2 A(v 2,u 2 ) = v ; in general, there exist u n,v n S 0 such that u n = A(u n,v n ) u n v n A(v n,u n ) = v n (n =, 2,...). And in this case, the conclusions of these theorems can be changed into: Then A has one unique fixed point x in S 0, and lim n u n = lim n v n = x. (8) The condition (i) of Theorem 3.7 and the condition (ii) of Theorem 3.2 can be changed into (I ) or (I ) as follows: (I ) writing S ={(u, v) S 0 S 0 u v}, assume that S {(A(a, b), b) (a, b) S,A(a,b) a} {(u, A(v, u)) (u, v) S,v A(v, u)}; (I ) writing S ={(u, v) S 0 S 0 A(u, v) u v A(v, u)}, assume that S {(A(a, b), b) (a, b) S } {(u, A(v, u)) (u, v) S }. 4. Applications As an application of our results, we give only an example. Example 4.. Consider the integral equation x(t) = k(t,s)x γ (s) ds where γ R \{0, }, is a bounded closed domain in R N (N ). Suppose that k(t,s) = h(t)f (s) for t,s, where h, f {x C() x(t) > 0, t } and C() is the set of all continuous functions on R N. Then Eq. (4.) has one unique positive continuous solution x = ( f(s)hγ (s) ds) γ h. Moreover, selecting α and β such that γ = α β, and then selecting l {2, 3,...} such that α<lβ 2 ; constructing successively the sequences {x n } n= and {y n } n= with x n = a n h and y n = b n, where ( an b β ) ( α β ) n an b n α a n =, bn f(s)h α β = (s) f(s)h α β (n =, 2,...) (s) for any initial x 0 = a 0 h and y 0 = b 0 h with ( 0 <a 0 <a = ) f(s)h α β α+β (s) ds <b0 < +, a 0 b lβ 0 < ( a ) +lβ and ( a ) (l+)β+β 2 +α (l+)β <a 0 b β2 +α 0 ; we have sup t x n (t) x (t) 0, sup t x n (t) x (t) 0 (n ). Proof. Clearly, P ={x C() x(t) 0, t } is a normal and solid cone of the real Banach space E = C(), P ={x C() x(t) > 0, t }. Set A : P P P be A(x,y) = (s) k(,s)xα y β (s) ds = h (s) f(s)xα ds for y β (s) (4.)

980 M. Zhang / J. Math. Anal. Appl. 339 (2008) 970 98 x,y P, and S ={μh μ [0, + )}. Obviously, S is a complete ordered closed subset of E, S 0 = S \{θ} P and λs = S for λ [0, + ); A : P P P is a mixed monotone operator and A( P P) S 0 ; for fixed y P, A(,y): P P is α-convex; for fixed x P, A(x, ) : P P is β-concave. Clearly, x = a h S 0 is the unique fixed point of A in P, where ( a = ) ( f(s)h α β α+β (s) ds = ) f(s)h γ γ (s) ds. () For 0 <a<a <b, firstly, we have θ A(ah, bh) ah a h bh A(bh, ah), i.e., 0 < aα b β f(s)h α β (s) ds < a < a <b< bα a β f(s)h α β (s) ds; secondly, setting a = cα b β f(s)h α β (s) ds and b = dα c β f(s)h α β (s) ds (i.e., ah = A(ch, bh) and bh = A(dh, ch), c,d R ), we have ( ab β ) ( α bc c =, d = β f(s)hα β (s) ds f(s)hα β (s) ds ) α = ( a β b β2 +α ( f(s)hα β (s) ds) α+β If 0 <a<a <b, ab β <(a ) +β and (a ) β+β2 +α <a β b β2 +α, then a<c<a <d<b. In fact, clearly, we have a<c. It follows from ab β <(a ) +β that c α ab = β f(s)hα β (s) ds <(a ) α, i.e., c<a ; and a β b β2 +α < ( a ) ( ) β+β 2 +α b α a < ( a ) ( ) β+β 2 +α b α 2 a, hence, d α2 = ) α 2. a β b β2 +α ( f(s)hα β (s) ds) α+β <b α2, i.e., d<b. Since (a ) β+β2 +α <a β b β2 +α, we have (a ) α(+β) < a β b β2 +α = (bc β ) α, i.e., a <d. (a ) β( α+β) If 0 <a<a <band ab lβ <(a ) +lβ, then we have ( a ab β ) (l )β ( < a ) +β ( < a ) ( +β a, ab 2β ) (l 2)β ( < a ) +2β ( a ) +2β, b b (( ab cd lβ lβ ) α ( ab β ) lβ 2( ) b αβ ) α = 2 ( (a ) +lβ (a ) +β a a ) +lβ (( ab lβ ) α ( ab 2β ) lβ 2) α < 2 ( (a ) +lβ (a ) +2β a ) +lβ ( < a ) +lβ. If 0 <a<a <band a (l+)β b β2 +α >(a ) (l+)β+β2 +α, then we have ( a a β b β2 +α ) lβ ( > a ) β+β 2 +α ( > a ) β+β 2 +α, a ( a c (l+)β d β2 +α β b β2 +α ) β 2 +α ( α = 2 a (l+)β b β2 +α (a ) β+β2 +α (a ) (l+)β+β2 +α ( ) lβ 2 α b α ( > a ) (l+)β+β 2 +α ( > a ) (l+)β+β 2 +α. a ) ( ) lβ 2 α α b α ( a ) (l+)β+β 2 +α a

M. Zhang / J. Math. Anal. Appl. 339 (2008) 970 98 98 (2) Clearly A(x n,y n ) = A(y n,x n ) = a β n a n b β n h f(s)h α β (s) ds = x n, f(s)hα β (s) ds an β b n h f(s)h α β (s) ds = y n, n=, 2,... f(s)hα β (s) ds b β n From (), noticing that 0 <a 0 <a <b 0 < +, a 0 b lβ 0 <(a ) +lβ and (a ) (l+)β+β2 +α <a (l+)β 0 b β2 +α 0, we get a 0 <a <a <b <b 0, a b lβ <(a ) +lβ, (a ) (l+)β+β2 +α <a (l+)β b β2 +α. Likewise, a 0 <a <a 2 <a < b 2 <b <b 0, a 2 b lβ 2 <(a ) +lβ, (a ) (l+)β+β2 +α <a (l+)β 2 b β2 +α 2. In general, a n <a n <a <b n <b n, a n bn lβ <(a ) +lβ, (a ) (l+)β+β2 +α <a n (l+)β b β2 +α n, n =, 2,...Sowehaveθ x n = A(x n,y n ) x n y n y n = A(y n,x n ), n =, 2,..., where x n = a n h, y n = b n h S 0, n = 0,, 2,... Therefore, by Theorem 3.9 (in the case the condition (ii) is changed into (j), see Remark 3.8()), we get A has at least one fixed point u in [x 0,y 0 ] S, and lim n x n = lim n y n = u. However, x = ( f(s)hγ (s) ds) γ h [x 0,y 0 ] S S 0 P is the unique fixed point of A in P. Hence u = x, i.e., x is the unique positive continuous solution of Eq. (4.), and lim n x n x =lim n sup t x n (t) x (t) =lim n y n x = lim n sup t y n (t) x (t) =0. Remark 4.2. Here we only need γ 0, intheγ -homogeneous equation (4.). We used Theorem 3.9 to discuss the solution of Eq. (4.) and obtain the iterative sequences, both of which converge on the unique solution of (4.), for initial values with certain initial value conditions. The iterative sequences are obtained by seeking pre-images of mappings. References [] D.J. Guo, V. Lakshmikantham, Coupled fixed points of nonlinear operators with application, Nonlinear Appl. (987) 623 632. [2] D.J. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, 988. [3] D.J. Guo, Existence and uniqueness of positive fixed point for mixed monotone operators with applications, Appl. Anal. 46 (992) 9 00. [4] D.J. Guo, Nonlinear Functional Analysis, Shandong Sci. and Tech. Press, Jinan, 2002 (in Chinese). [5] Z.D. Liang, L.L. Zhang, S.J. Li, Fixed point theorems for a class of mixed monotone operators, Anal. Appl. 22 (2003) 529 542. [6] Y.S. Wu, G.Z. Li, On the fixed point existence and uniqueness theorems of mixed monotone operators and their applications, Acta Math. Sinica 46 () (2003) 2 26 (in Chinese). [7] S.Y. Xu, B.G. Jia, Fixed-point theorems of φ concave-( ψ) convex mixed monotone operators and applications, J. Math. Anal. Appl. 295 (2004) 645 657. [8] Z.T. Zhang, New fixed point theorems of mixed monotone operators and applications, J. Math. Anal. Appl. 204 (996) 307 39. [9] Z.T. Zhang, Fixed point theorems of mixed monotone operators and its applications, Acta Math. Sinica 4 (6) (998) 6 66 (in Chinese).