In [25], C. B. Zhai and X. M. Cao introduced the concept of τ-⊎-concave operators A:P⟶P, where P is a cone in a Banach space, and proved the existence and uniqueness of fixed points for such operators. They did so without requiring upper and lower solutions, compactness or continuity conditions. Krasnoselskii [15] studied u₀-concave operators with u0≻θ. In [27], Zhang and Zhai used the fixed point theorem for increasing α-concave operators to obtain the existence and uniqueness of positive solutions for Neumann boundary value problems. The same authors in [26], proved new fixed point theorems for mixed monotone operators, and then they established some criterions for the local existence–uniqueness of positive solutions to some boundary value problems.

Indeed, there has been much attention focused on problems of positive solutions for diverse nonlinear boundary value problems (See, for instance, [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24]). However, most of these works studied the scalar case. Therefore, motivated by some papers, for example [25, 26] and the references therein, we propose in the present work to extend a fixed point theorem and its application to the vector case. In other words, we construct a fixed point theorems for a vector operator, and then we apply it to systems of nonlinear Neumman boundary value problems of the following type

(1.1)−x′′(t)+θ2x(t)=λf(t,x(t),x(t),y(t)), 0<t<1,−y′′(t)+ω2y(t)=βg(t,x(t),y(t),y(t)), 0<t<1,x′(0)=x′(1)=1, y′(0)=y′(1)=0,

in order to obtain existence and uniqueness of the positive solution.

Let E,‖.‖ be a real Banach space which is partially ordered by a cone P⊂E, i.e., x⪯y if and only if y-x∈ P. If x⪯y and x≠ y, then we denote x≾ y or y≻x. By 𝜃 we denote the zero element of E. Recall that a non-empty closed convex set P⊂E is a cone if it satisfies (i)x∈P,λ≥0⇒λx∈P, (ii)x∈P,−x∈P⇒x=θ. A cone P is said to be solid if its interior P° is non-empty. P is called normal if there exists a constant N>0 such that, for all x,y∈E,θ⪯x⪯y implies ‖x‖≤N‖y‖; in this case N is called the normality constant of P.

For all x,y∈ E, the notation x y means that there exist λ>0 and μ>0 such that λx⪯y⪯μx. Clearly ≃ is an equivalence relation. Given h≻θ (i.e., h∈ P and h≠ 𝜃), we denote by P_h the set Ph={x∈E:x~h}. It is easy to see that for h ∈ P, Ph⊂P is convex and λPh=Ph for all λ>0. If P°≠∅ and h∈P°, it is clear that Ph=P°. Let us give the definition of mixed monotone operators with three variables as it is known in the literature.

Definition 1.1.([3]) Let X,⪯ be a partially ordered set and A:X×X×X⟶X. Then the trivariate operator A is said to have the mixed monotone property if A(., u, y) and A(x, u, .) are monotone non-decreasing, and A(x, ., y) is monotone non-increasing, i.e., for any x, u, y ∈ X

x1,x2∈X, x1⪯x2⟹A(x1,u,y)⪯A(x2,u,y),u1,u2∈X, u1⪯u2⟹A(x,u1,y)≽A(x,u2,y),y1,y2∈X, y1⪯y2⟹A(x,u,y1)⪯A(x,u,y2).

The organization of this paper can be described as follows. In section 2, after introducing the definition of cooperative and competitive mixed monotone vector operators, we present two fixed point theorems corresponding to these two cases. We prove the first result and leave the second to the reader, since the steps of the proof will be analogous. In section 3, we give some applications of the results obtained in section 2 on the existence and uniqueness of solutions of system (1.1). Our results will be illustrated by concrete examples.

Inspired by the works [25] and [26], we present in this section our fixed point theorems for a system of two operators with three variables, which can be written as a vector operator. In other words, if X,⪯ is a partially ordered set and, if A1,A2:X×X×X⟶X are two operators, then we define the vector operator Φ=(A1,A2):X×X×X×X⟶X×X, noted Φ=(A1,A2), by

(2.1)Φ(x,y,u,v)=(A1(x,u,y),A2(x,v,y)), ∀x,y,u,v∈X.

Then, we introduce the following definition.

Definition 2.1. Let X,⪯ be a partially ordered set. Let A1,A2:X×X×X⟶X be two operators and Φ=(A1,A2) be given as in (2.1).

(i) We say that the operator Φ=(A1,A2) is a cooperative mixed monotone vector operator if A₁, A₂ are mixed monotone as in Definition 1.1.

(ii) We say that Φ=(A1,A2) is a competitive mixed monotone vector operator if A1(.,u,y), A2(x,u,.) are monotone non-decreasing, and A1(x,.,y), A1(x,u,.), A2(x,.,y), A2(.,u,y) are monotone non-increasing.

2.1. Cooperative mixed monotone vector operator

Lemma 2.2. Let E be a real Banach space and P be a cone in E. Consider two operators A1,A2:P×P×P⟶P such that Φ=(A1,A2) satisfies the following conditions:

(C₁) Φ=(A1,A2) is cooperative mixed monotone, and there exist h,k∈ P with h≠θ,k≠θ such that

(2.2)A1(h,h,k)∈Ph and A2(h,k,k)∈Pk;

(C₂) There exist positive-valued functions τ1,τ2 on interval (a,b), φ1,φ2 on (a,b)×(a,b)×P×P×P and ψ1,ψ2:(a,b)×(a,b)×(0,1] such that

(i) τ1,τ2:(a,b)⟶(0,1) are surjections.

(ii) For any x,u∈Ph, for any y,v∈Pk, for any t,s∈(a,b) and any ε∈(0,1]

(2.3)infx,u∈[εh,1εh],y∈[εk,1εk]φ1(t,s,x,u,y)=ψ1(t,s,ε)>min{τ1(t),τ2(s)},infx∈[εh,1εh],v,y∈[εk,1εk]φ2(t,s,x,v,y)=ψ2(t,s,ε)>min{τ1(t),τ2(s)}

and

(2.4)A1τ1(t)x,1τ1(t)u,τ2(s)y≽φ1(t,s,x,u,y)A1(x,u,y),A2τ1(t)x,1τ2(s)v,τ2(s)y≽φ2(t,s,x,v,y)A2(x,v,y).

Then A1:Ph×Ph×Pk⟶Ph, A2:Ph×Pk×Pk⟶Pk. Moreover, there exist x0,u0∈Ph, y0,v0∈Pk and r∈(0,1)such that

(2.5)ru0⪯x0⪯u0,rv0⪯y0⪯v0   and  x0⪯A1(x0,u0,y0)⪯A1(u0,x0,v0)⪯u0,y0⪯A2(x0,v0,y0)⪯A2(u0,y0,v0)⪯v0.

Proof. For any x,u∈Ph and y,v∈Pk there exists λ*∈(0,1) such that

λ*h⪯x,u⪯1λ*h  and  λ*k⪯y,v⪯1λ*k.

It follows from (C2)(i) that there exist t*,s*∈(a,b) such that τ1(t*)=λ* and τ2(s*)=λ*, which gives

τ1(t*)h⪯x,u⪯1τ1(t*)h  and  τ2(s*)k⪯y,v⪯1τ2(s*)k.

Then, by the mixed monotone properties of operators A₁, A₂ and condition (C₂)(ii), we have

A1(x,u,y)≽A1τ1(t*)h,1τ1(t*)h,τ2(s*)k≽φ1(t*,s*,h,h,k)A1(h,h,k)

and

A1(x,u,y)⪯A11τ1(t*)h,τ1(t*)h,1τ2(s*)k⪯1φ1(t*,s*,1τ1(t*)h,τ1(t*)h,1τ2(s*)k)A1(h,h,k).

From A1(h,h,k)∈Ph, we have A1(x,u,y)∈Ph and hence, A1:Ph×Ph×Pk⟶Ph. Analogously we obtain A2:Ph×Pk×Pk⟶Pk.

Now, since A1(h,h,k)∈Ph,A2(h,k,k)∈Pk, there exists λ0∈(0,1) such that

λ0h⪯A1(h,h,k)⪯1λ0h  and  λ0k⪯A2(h,k,k)⪯1λ0k.

It follows from (C2)(i) that there exist t0,s0∈(a,b) such that τ1(t0)=λ0 and τ2(s0)=λ0, which gives

(2.6)τ1(t0)h⪯A1(h,h,k)⪯1τ1(t0)h  and  τ2(s0)k⪯A2(h,k,k)1τ2(s0)k.

Set ε0=min{τ1(t0),τ2(s0)}. Then we have ψi(t0,s0,ε0n)≤ψi(t0,s0,ε0n−1), for all n=1,2,... and i=1,2. By (C₂) (ii), we can choose a positive integer m such that

(2.7)∏mi=1ψ1(t0,s0,ε0i)ε0≥1ε0 and ∏mi=1ψ2(t0,s0,ε0i)ε0≥1ε0.

Put x0=ε0mh, u0=1ε0mh, v0=1ε0mk and y0=ε0mk. It is clear that x0,u0∈Ph with x0=ε02mu0<u0 and y0,v0∈Pk with y0=ε02mv0<v0. Furtheremore, for any r∈(0,ε02m)⊂(0,1), x0≽ru0 and y0≽rv0. Also, by the mixed monotone properties, A1(x0,u0,y0)⪯A1(u0,x0,v0) and A2(x0,v0,y0)⪯A2(u0,y0,v0). Moreover, combining (C₂)(ii) with (2.6) and (2.7), we have on the one hand,

A1(x0,u0,y0)=A1τ1(t0)[τ1(t0)]m−1h,1τ1(t0)1[τ1 (t0 )]m−1h,τ2(s0)[τ2(s0)]m−1k     ≽φ1t0,s0,[τ1(t0)]m−1h,1[τ1 (t0 )]m−1h,[τ2(s0)]m−1k      A1[τ1(t0)]m−1h,1[τ1 (t0 )]m−1h,[τ2(s0)]m−1k     ≽ψ1(t0,s0,ε0m−1)A1[τ1(t0)]m−1h,1[τ1 (t0 )]m−1h,[τ2(s0)]m−1k     ≽ψ1(t0,s0,ε0m−1)...ψ1(t0,s0,1)A1(h,h,k)     ≽ψ1(t0,s0,ε0m−1)...ψ1(t0,s0,1)τ1(t0)h     ≽ψ1(t0,s0,ε0m)...ψ1(t0,s0,ε0)ε0h     ≽ε0mh=x0.

On the other hand,

A1(u0,x0,v0)=A11[τ1 (t0 )]mh,[τ1(t0)]mh,1[τ2 (s0 )]mk     ⪯1φ1t0,s0,1[τ1 (t0 )]mh,[τ1 (t0 )]mh,1[τ2 (s0 )]mk      A11[τ1 (t0 )]m−1h,[τ1(t0)]m−1h,1[τ2 (s0 )]m−1k     ⪯1ψ1(t0,s0,ε0m)A11[τ1 (t0 )]m−1h,[τ1(t0)]m−1h,1[τ2 (s0 )]m−1k     ⪯1ψ1(t0,s0,ε0m)...1ψ1(t0,s0,ε0)A1(h,h,k)     ⪯1ψ1(t0,s0,ε0m)...1ψ1(t0,s0,ε0)1τ1(t0)h     ⪯1ε0mh=u0.

in a similar way, we obtain

A2(x0,v0,y0)≽y0  and  A2(u0,y0,v0)⪯v0.

Theorem 2.3. Let P be a normal cone in a Banach space E. Consider two operators A1,A2:P×P×P⟶P such that (C1),(C2) in Lemma 2.2 hold. Then the operator Φ=(A1,A2):Ph×Pk×Ph×Pk⟶Ph×Pk,

defined by (2.1), has a unique fixed point (x*,y*)∈Ph×Pk, that is, Φ(x*,y*,x*,y*)=(x*,y*), or equivalently A1(x*,x*,y*)=x* and A2(x*,y*,y*)=y*. Moreover, for any initial x0,u0∈Ph and y0,v0∈Pk, constructing successively the sequences

(2.8)xn=A1(xn−1,un−1,yn−1),   yn=A2(xn−1,vn−1,yn−1),un=A1(un−1,xn−1,vn−1),   vn=A2(un−1,yn−1,vn−1), n=1,2,...,

we have ‖xn−x*‖⟶0, ‖un−x*‖⟶0 nd ‖yn−y*‖⟶0, ‖vn−y*‖⟶0 (as n⟶∞).

Proof. Let x0,u0∈Ph, y0,v0∈Pk and r∈ (0,1) be as obtained in Lemma 2.2. By constructing successively the sequences as in (2.8), and using Lemma 2.2 combined with the mixed monotone property of the operators A1,A2, we obtain

x0⪯x1⪯...⪯xn⪯...⪯un⪯...⪯u1⪯u0,y0⪯y1⪯...⪯yn⪯...⪯vn⪯...⪯v1⪯v0.

In addition, we have

xn≽x0≽ru0≽run,yn≽y0≽rv0≽rvn, n=1,2,...

We put

rn=sup{r>0:xn≽run and yn≽rvn}, n=1,2,...

Then, xn≽rnun and yn≽rnvn, n=1,2,..., and therefore

xn+1≽xn≽rnun≽rnun+1,yn+1≽yn≽rnvn≽rnvn+1, n=1,2,...

Hence, rn+1≥rn, that is, {rn} is an increasing convergent sequence with {rn}⊂(0,1]. Set r*=limn⟶∞rn. We claim that r^*=1, otherwise, 0<rn≤r*<1. By (C2)(i), there exist t*,s*∈(a,b) such that τ1(t*)=r* and τ2(s*)=r*. We distinguish two cases.

First case. There exists n₀ such that rn0=r*. Thus, for all n≥n0 we have rn=r* and

xn+1=A1(xn,un,yn)≽A1rnun,1rnxn,rnvn        =A1τ1(t*)un,1τ1(t*)xn,τ2(s*)vn        ≽φ1(t*,s*,un,xn,vn)A1(un,xn,vn)        ≽ψ1t*,s*,ε0mun+1,

Analogously we have

yn+1≽ψ2(t*,s*,ε0m)vn+1.

Which means that

n+1=r*≥min{ψ1(t*,s*,ε0m),ψ2(t*,s*,εm)}   >min{τ1(t*),τ2(s*)}=r*.

This is a contradiction.

Second case. For all integer n, rn<r*<1, then 0<rnr*<1. By (C2)(i), there exist αn,βn∈(a,b) such that τ1(αn)=rnr*=τ2(βn). In this case we have

xn+1=A1(xn,un,yn)  ≽A1rnun,1rnxn,rnvn  =A1rnr*r*un,r*rn1r*xn,rnr*r*vn  =A1τ1(αn)r*un,1τ1(αn)1r*xn,τ2(βn)r*vn  ≽φ1αn,βn,r*un,1r*xn,r*vnA1r*un,1r*xn,r*vn  ≽ψ1(αn,βn,(r*ε0)m)ψ1(t*,s*,ε0m)un+1.

Analogously we obtain

yn+1≽ψ2(αn,βn,(r*ε0)m)ψ2(t*,s*,ε0m)vn+1.

It follows that

rn+1≥min{ψ1(αn,βn,(r*ε0)m)ψ1(t*,s*,ε0m),ψ2(αn,βn,(r*ε0)m)ψ2(t*,s*,ε0m)}≥min{τ1(αn)ψ1(t*,s*,ε0m),τ2(βn)ψ2(t*,s*,ε0m)}=min{rnr*ψ1(t*,s*,ε0m),rnr*ψ2(t*,s*,ε0m)}.

If n⟶∞, we get

r*≥min{ψ1(t*,s*,ε0m),ψ2(t*,s*,ε0m)}>min{τ1(t*),τ2(s*)}=r*.

This is also a contradiction.

Now, by the same reasoning as in [4, Theore 2.2] and [26, Lemma 2.1] we obtain limn→∞xn=limn→∞un=x* and limn→∞yn=limn→∞vn=y*. Hence

xn+1=A1(xn,un,yn)⪯A1(x*,x*,y*)⪯A1(un,xn,vn)=un+1,yn+1=A2(xn,vn,yn)⪯A2(x*,y*,y*)⪯A2(un,yn,vn)=vn+1.

If n⟶∞, we get x*=A1(x*,x*,y*) and y*=A2(x*,y*,y*). That is, (x*,y*) is a fixed point of Φ in Ph×Pk.

Now, for any x0,u0∈Ph and y0,v0∈Pk, we can choose a small number λ1∈(0,1) such that

λ1h⪯x0,u0⪯1λ1h,  λ1k⪯y0,v0⪯1λ1k.

From (C₂)(i), there exist t1,s1∈(a,b) such that τ1(t1)=λ1=τ2(s1), and hence

τ1(t1)h⪯x0,u0⪯1τ1(t1)h,  τ2(s1)k⪯y0,v0⪯1τ2(s1)k.

Similarly to Lemma 2.2, set ε1=min{[τ1(t1)],[τ2(s1)]} and choose a sufficiently large integer m such that

∏mi=1ψ1(t1,s1,ε1i)ε1≥1ε1 and ∏mi=1ψ2(t1,s1,ε1i)ε1≥1ε1.

Put x¯0=ε1mh, u¯0=1ε1mh, v¯0=1ε1mk and y¯0=ε1mk. Then, x¯0,u¯0∈Ph and y¯0,v¯0∈Pk with x¯0<x0,u0< u¯0 and y¯0<y0,v0< v¯0. Construct the sequences

x¯n=A1(x¯n−1,u¯n−1,y¯n−1),   y¯n=A2(x¯n−1,v¯n−1,y¯n−1),u¯n=A1(u¯n−1,x¯n−1,v¯n−1),   v¯n=A2(u¯n−1,y¯n−1,v¯n−1), n=1,2,...

Therefore, there exist (u*,v*)∈Ph×Pk such that Φ(u*,v*,u*,v*)=(u*,v*) and limn→∞x¯n=limn→∞u¯n=u*, limn→∞y¯n=limn→∞v¯n=v*. By the uniqueness of fixed points of operator 𝚽 in Ph×Pk, we have x*=u* and y*=v*. Moreover, by induction, x¯n⪯xn,un⪯ u¯n and y¯n⪯yn,vn⪯ v¯n, for n=1,2,... Finally, by the normality of the cone P we get limn→∞xn=limn→∞un=x* and limn→∞yn=limn→∞vn=v*.

2.2. Competitive Mixed Monotone Vector Operator

We will give below, another result of existence and uniqueness of a fixed point concerning competitive mixed monotone vector operators. Similarly to the case of cooperative mixed monotone vector operators, we will have a lemma, then the existence theorem. The steps of the proofs are not very far from those of the previous case, for that we leave them to the reader.

Lemma 2.4. Let E be a real Banach space and P be a cone in E. Consider two operators A1,A2:P×P×P⟶P such that Φ=(A1,A2) satisfies the following conditions:

(C1)′ Φ=(A1,A2) is competitive mixed monotone, and there exist h,k ∈ P with h≠θ,k≠θ such that

A1(h,h,k)∈Ph and A2(h,k,k)∈Pk;

(C2)′ There exist positive-valued functions τ on interval (a,b), φ1,φ2 on (a,b)×P×P×P and ψ1,ψ2:(a,b)×(0,1] such that

(i) τ:(a,b)⟶(0,1) is surjection.

(ii) For any x,u∈Ph, for any y,v∈Pk, for any t∈(a,b) and any ε∈(0,1]

infx,u∈[εh,1εh],y∈[εk,1εk]φ1(t,x,u,y)=ψ1(t,ε)>τ(t),infx∈[εh,1εh],v,y∈[εk,1εk]φ2(t,x,v,y)=ψ2(t,ε)>τ(t)

and

A1τ(t)x,1τ(t)u,1τ(t)y≽φ1(t,x,u,y)A1(x,u,y),A21τ(t)x,1τ(t)v,τ(t)y≽φ2(t,x,v,y)A2(x,v,y).

Then A1:Ph×Ph×Pk⟶Ph, A2:Ph×Pk×Pk⟶Pk. Moreover, there exist x0,u0∈Ph, y0,v0∈Pk and r∈(0,1) such that

ru0⪯x0⪯u0,rv0⪯y0⪯v0   and  x0⪯A1(x0,u0,v0)⪯A1(u0,x0,y0)⪯u0,y0⪯A2(u0,v0,y0)⪯A2(x0,y0,v0)⪯v0.

Theorem 2.5. Let P be a normal cone in a Banach space E. Consider two operators A1,A2:P×P×P⟶P such that (C1)′ and (C2)′ of Lemma 2.4 hold. Then, the operator Φ=(A1,A2):Ph×Pk×Ph×Pk⟶Ph×Pk defined by (2.1) has a unique fixed point (x*,y*)∈Ph×Pk, that is, Φ(x*,y*,x*,y*)=(x*,y*), or equivalently A1(x*,x*,y*)=x* and A2(x*,y*,y*)=y*. Moreover, for any initial x0,u0∈Ph and y0,v0∈Pk, constructing successively the sequences

xn=A1(xn−1,un−1,vn−1),   yn=A2(un−1,vn−1,yn−1),un=A1(un−1,xn−1,yn−1),   vn=A2(xn−1,yn−1,vn−1), n=1,2,...,

we have ‖xn−x*‖⟶0, ‖un−x*‖⟶0 and ‖yn−y*‖⟶0, ‖vn−y*‖⟶0 (as n⟶∞).