2016 International Conference on Mathematical, Computational and Statistical Sciences and Engineering (MCSSE 2016) ISBN: 978-1-60595-396-0
Fixed Point for t−φ(t,u,v) Mixed Monotone Model Operator
Yue-xiang Wu1*, Yan-mei Huo2 1Department of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, 030006, China. 2College of Economics, Shanxi University of Finance and Economics, Taiyuan, 030006, China
Abstract. This paper studies the uniqueness and existence of fixed point of t−φ(t,u,v) mixed monotone model operator in the partial order Banach space. Our conclusions essentially improve the relevant results. Moreover, as an application of our results, we prove the existence and uniqueness of a positive solution for a class of integral equations which can not be solved by using previously available methods.
Keywords. Cone and semiorder; t−φ(t,u,v) mixed monotone model operator; Fixed point
1 Introduction and Preliminaries
In this paper, we will study the existence and uniqueness of fixed point of a kind of mixed monotone operator. The resulting conclusion essentially improves the relevant results obtained by [3,8]. As an application of our results, we prove the existence and uniqueness of a positive solution for a class of integral equation which can not be solved by using previously available methods. Let the real Banach space E (or, more general case, topological linear space) be partially ordered by a cone P of E, i.e., x≤y (or alternatively denoted as y≥x) if and only if y−x∈P. 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 x∈P, λ≥0⇒λx∈P and x, −x∈P⇒x=Θ. We denote by P the interior set of P. A cone P is said to be solid cone if P≠φ. P is said to be normal if there exists a positive constant N such that Θ≤x≤y implies |x|≤N|y|. N is called the normal constant of P. We write x≪y if and only if y−x∈P. Definition 1.1 (see [2,6,8]) Let D⊂E. Operator A:D×D→E is said to be mixed monotone if A(x,y) is nondecreasing in x and nonincreasing in y, i.e., u1≤u2, v2≤v1, ui, vi∈D (i=1,2) * * * * implies A(u1,v1)≤A(u2,v2). Element x ∈D is called a fixed point of A if A(x ,x )=x . For all x, y∈E, the notation x∼y means that there exist 0<λ≤μ such that λx≤y≤μx. Clearly, ∼ is an equivalence relation. Given h>Θ(i.e.h≥Θ and h≠Θ), we denote by Ph the set Ph={x∈E, there exist λ(x), μ(x)>0, such that λ(x)h≤x≤μ(x)h}. It is easy to see that Ph⊂P. Definition 1.2 Let P be a cone of a real topological linear space E, and A:P×P→E a mixed monotone operator. Assume that for all 0 321 2. t−φ(t,u,v) mixed monotone model operator We give the following Theorem on t−φ(t,u,v) mixed monotone model operator in ordered Banach spaces. Theorem 2.1 Let P be a normal cone of a real Banach space E, h>Θ. A:Ph×Ph→Ph is a t−φ(t,u,v) mixed monotone model operator. Suppose (i) φ(t,u,v) is nonincreasing in u; (ii) φ(t,u,v) is nondecreasing in v; (iii) lim φ(t,t−1h,th)=l>0 . t→0+ * * * * Then there exists unique fixed point x ∈Ph such that A(x ,x )=x . Moreover, for any initial point x0, y0∈Ph , constructing successively the sequences * xn=A(xn−1, yn−1),yn=A(yn−1, xn−1), n=1,2,⋅⋅⋅ , we have |xn−x |→0 and * |yn−x |→0 as n→∞. Proof Firstly, choose t0∈(0,1) such that 1 l t h≤A(h,h)≤ h.(2.1) 0 t0 2 According to condition (iii), we can also adjust above 0 322 k −k k−1 −k+1 k−1 −k+1 u1=A(u0,v0)=A(t0h,t0 h)≥φ(t0,t0 h,t0 h)A(t0 h,t0 h) k−1 −k+1 k−2 −k+2 k−2 −k+2 ≥φ(t0,t0 h,t0 h)φ(t0,t0 h,t0 h)A(t0 h,t0 h) k−1 −k+1 −1 −1 ≥⋅⋅⋅≥φ(t0,t0 h,t0 h)⋅⋅⋅φ(t0,t0h,t0 h)A(t0h,t0 h) k−1 −1 k−1 ≥φ(t0,h,h) A(t0h,t0 h)≥φ(t0,h,h) φ(t0,h,h)A(h,h) k k k−1 k =φ(t0,h,h) A(h,h)≥φ(t0,h,h) t0h≥t0 t0h=t0h=u0. i.e. u1≥u0.(2.9) On the other hand, since 1 1 u,tv)≤φ(t, u,tv)−1A(u,v),(2.10) A( t t similarly, we can get −k k −1 −k+1 k−1 −k k −1 −k+1 k−1 v1=A(t0 h,t0h)=A(t0 t0 h,t0t0 h)≤φ(t0,t0 h,t0h) A(t0 h,t0 h) −k k −1 −k+1 k−1 −1 −k+2 k−2 ≤φ(t0,t0 h,t0h) φ(t0,t0 h,t0 h) A(t0 h,t0 h)≤⋅⋅⋅ −k k −1 −k+1 k−1 −1 −1 −1 ≤φ(t0,t0 h,t0h) φ(t0,t0 h,t0 h) ⋅⋅⋅φ(t0,t0 h,t0h) A(h,h) 1 A(h,h) 1 1 −k ≤ ≤ h=t h=v , −k k k−1 −1 k−1 t0 0 0 φ(t0,t0 h,t0h) φ(t0,t0 h,t0h) t0 i.e. v1≤v0.(2.11) Hence we have u0≤u1≤v1≤v0.(2.12) By using (2.8),(2.12) and the mixed monotone properties of operator A, we easily have u0≤u1≤⋅⋅⋅≤un≤⋅⋅⋅≤vn≤⋅⋅⋅≤v1≤v0.(2.13) Let tn=sup{t>0:un≥tvn}, n=0,1,2,⋅⋅⋅.(2.14) Then un≥tnvn.(2.15) * It is easy to obtain that tn is nondecreasing in n and there exists t such that lim * n→∞tn=t .(2.16) We prove that t*=1. In fact, suppose 0 323 * * Therefore by the definition of tn+1 , we have tn+1≥φ(t ,vn,un)>t , n≥N. This is a contradiction. * (ii) For all integer n, tn * * tn tn φ(t ,vn,un) φ(t ,vn,un) ≥φ( ,t*v ,(t*)−1u )×φ(t*,v ,u )A(v ,u )> ×t*[ ]v =t [ ]v . t* n n n n n n t* t* n+1 n t* n+1 Thus, by the definition of tn+1 and conditions (i) and (ii), we have * * φ(t ,vn,un) φ(t ,v0,u0) t ≥t [ ]≥tn[ ]. n+1 n t* t* * φ(t ,v0,u0) Let n→∞, we have t*≥t*[ ]>t*, this is a contradiction. t* * lim Hence t =1. i.e. n→∞tn=1. * Furthermore, similar the proof of Theorem 2.1, there exist x ∈Ph such that lim lim * n→∞un=n→∞vn=x , and x* is the fixed point of operator A. Next, the fixed point of operator A is uniqueness. In fact, suppose u*∈Ph is other fixed point of operator A. Let * −1 t*=sup{0 324 Theorem 2.2 Let P be a normal cone of a real Banach space E, h>Θ. A:Ph×Ph→Ph is a t−φ(t,u,v) mixed monotone model operator. Suppose (i) φ(t,u,v) is nonincreasing in u; (ii) φ(t,u,v) is nondecreasing in v; (iii) A(h,h)≤h. * * * * Then there exists unique fixed point x ∈Ph such that A(x ,x )=x . Moreover, for any initial point x0,y0∈Ph , constructing successively the sequences * xn=A(xn−1,yn−1), yn=A(yn−1,xn−1), n=1,2,⋅⋅⋅ , we have |xn−x |→0 and * |yn−x |→0 as n→∞. Remark 2.1 Compared with corresponding results in the literature [3, Theorem 3.1], it shows that our hypotheses are greatly weaker than that conditions. We delete the conditions in [3] that “there exist u0,v0∈Ph,u0≤v0 such that u0≤A(u0,v0),A(v0,u0)≤v0 ; there exist an adjoint sequence lim 1 {η } of A with respect to λ ,u ,v such that nη =ln ." n 0 0 0 n→∞ n λ0 Remark 2.2 In general, Theorem 2.2 generalize and extend some known results in [8], the result in [8,Theorem 3.2] is only a special case of our result. In fact, let φ(t,u,v)=tα(t,u,v) , then our result will coincide with [8,Theorem 3.2]. 3 Example As an application of our results, we give only an example. Example 3.1 Let E=C(G) denote the set of all real continuous functions on bounded closed subset G⊂RN. Equipped with the norm |x|=sup{|x(t)|:t∈G}, then E is a real Banach space. The set P={x∈C(G)|x(t)≥0, t∈G} is a normal and solid cone in E. Consider the following nonlinear integral equation: α α −1 x(t)= k1(t,s)x(s) +[k2(t,s)+x(s) ] ds,(3.1) G where α:(0,1)→(0,1) is real function, k1(t,s) and k2(t,s)∈C(G×G). 1 Conclusion 3.1 Suppose that k1(t,s), k2(t,s):G×G⟶R is nonnegative and continuous. Then Eq.(3.1) has a unique positive solution x*(t)∈P. Moreover, constructing successively the sequences xn(t) and yn(t) ( n=1, 2,⋅⋅⋅) with α α −1 xn(t)= k1(t,s)xn−1(s) +[k2(t,s)+yn−1(s) ] ds,(3.2) G and α α −1 yn(t)= k1(t,s)yn−1(s) +[k2(t,s)+xn−1(s) ] ds(3.3) G * * for any initial point (x0, y0)∈P, we have |xn−x |→0 and |yn−x |→0 as n→∞. Proof Obviously, Eq.(3.1) can be written in the form x=A(x,x), where A(x,y)=A1(x)+A2(y). α (A1x)(t)= k1(t,s)x(s) ds,(3.4) G 325 α −1 (A2y)(t)= [k(t,s)+y(s) ] ds.(3.5) G It is easy to obtain that A:P×P→P is mixed monotone operator. On the other hand, for all 0<λ<1, u, v∈P, we have 1 vα(s) A(λu, v)(t)= k (t,s)λαuα(s)ds+ [k (t,s)+ ]−1ds λ 1 2 α G G λ α α α α α −1 =λ k1(t,s)u (s)ds+λ [λ k2(t,s)+v (s)] ds G G α α α α −1 ≥λ k1(t,s)u (s)ds+λ [k2(t,s)+v (s)] ds G G α α α =λ A1(u)(t)+λ A2(v)(t)=λ A(u,v)(t).(3.6) 1 α i.e. A(λu, λv)≥λ A(u,v). Hence all conditions of Theorem 2.1 are satisfied. Therefore, conclusion 3.1 holds. Acknowledgements Supported by the innovation foundation for college teaching team of Shanxi University of Finance and Economics, 2015 education and teaching reform project of Shanxi University of Finance and Economics. References [1] Da-Jun Guo, Jin-Xian Sun, Zhao-Li Liu. Functional methods of nonlinear ordinary differential equations[M]. 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