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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009, Article ID 892691, 14 pages doi:10.1155/2009/892691 Research Article On Series-Like Iterative Equation with a General Boundary Restriction Wei Song,1 Guo-qiu Yang,1 and Feng-chun Lei2 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China Correspondence should be addressed to Wei Song, dawenhxi@126.com Received 26 August 2008; Revised 19 November 2008; Accepted February 2009 Recommended by Tomas Dominguez Benavides By means of Schauder fixed point theorem and Banach contraction principle, we investigate the existence and uniqueness of Lipschitz solutions of the equation P f ◦ f F Moreover, we get that the solution f depends continuously on F As a corollary, we investigate the existence and uniqueness of Lipschitz solutions of the series-like iterative equation ∞ an f n x F x , x∈B n with a general boundary restriction, where F : B → A is a given Lipschitz function, and B, A are compact convex subsets of RN with nonempty interior Copyright q 2009 Wei Song et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Introduction Let f be a self-mapping on a topological space X For integer n ≥ define the nth iterate f ◦ f n−1 and f id, where id denotes the identity mapping on X, and ◦ of f by f n denotes the composition of mappings Let C X, X be the set of all continuous self-mappings on X An equation with iteration as its main operation is simply called an iterative equation It is one of the most interesting classes of functional equations 1–4 because it concludes the problem of iterative roots 2, 5, , that is, finding f ∈ C X, X such that f n is identical to a given F ∈ C X, X and the problem of invariant curves Iteration equations also appear in the study on transversal homoclinic intersection for diffeomorphisms , normal form of dynamical systems , and dynamics of a quadratic mapping 10 The well-known Feigenbaum equation f x − 1/λ f f λx , arising in the discussion of period-doubling bifurcations 11, 12 , is also an iterative equation As a natural generalization of the problem of iterative roots, a class of iterative equations which is called polynomial-like iterative equation: λ1 f x λ2 f x ··· λn f n x F x , x∈I a, b , 1.1 Fixed Point Theory and Applications always fascinates many scholars’ attentions 3, 13 It is more difficult than the analogous differential equation, where each f j is replaced with the jth derivative f j of f because differentiation is a linear operator but iteration is not In 1986, Zhang 14 constructed an interesting operator called “structural operator” L : f → Lf for 1.1 and used the fixed point theory in Banach spaces to get the solutions of 1.1 By means of this method, Zhang and Si made a series of works concerning these qualitative problems such as 15–19 Recently, Zhang et al 20, 21 developed this method and made a series of works on 1.1 Furthermore, they have got the nonmonotonic and decreasing solutions of 1.1 , and the convexity of solutions is also considered In 2002, Kulczycki and Tabor 22 investigated iterative functional equations in the ˙ class of Lipschitz functions In 2004, Tabor and Zołdak 23 studied the iterative equations in Banach spaces In the above references, the authors first gave theorems for the existence of solutions of P f ◦f F 1.2 By virtue of these theorems, in 22 , the authors considered the existence of Lipschitz solutions of the iterative functional equation: ∞ an f n x F x , x ∈ B, 1.3 n where B is a compact convex subset of RN with nonempty interior, and F : B → B is a given Lipschitz function In 23 , the existence of solutions of Ai f i x F x , i f φi x F x , x∈B 1.4 i is investigated, where B is a nonempty closed subset of a Banach Space X But they all considered the case F|∂B id|∂B It is easy to see that 1.1 is the special case of 1.3 with 0, i n 1, n 2, , and B a, b Since the left-hand side of 1.3 is a functional series, in this paper we call it series-like iterative equation In 14–20 , the authors considered the solutions of 1.1 with F : I → I, F a a, F b b In 22 , the authors considered the solutions of 1.3 with F : B → B, F|∂B id∂B In fact, the above authors had studied the solutions of ∞ an f n x F x , x ∈ B, n F : B −→ B, F|∂B id∂B , 1.5 Fixed Point Theory and Applications where B ⊂ RN is a convex compact set with nonempty interior In 21 , the authors considered the solutions of 1.1 with F : I → J, F a d, F b c, I a, b , J c, d Obviously, the more general case is ∞ an f n x x ∈ B, F x , 1.6 n F : B −→ A, F|∂B g, where B, A are convex compact subsets of RN with nonempty interior, and g : ∂B → ∂A is a continuous surjective map Since g could be any map, in this paper we call 1.6 serieslike iterative equation with a general boundary restriction It is easy to see that 14–22 all considered one special case of 1.6 The problem of differentiable solutions of iterative equations has also fascinated many scholars’ attentions In Zhang 16 and Si 19 , the C1 and C2 solutions of 1.1 are considered In Wang and Si 24 , the differentiable solutions of H x, φn1 x , , φni x x∈I F x , a, b 1.7 are considered Murugan and Subrahmanyam 25–27 offered theorems on the existence and uniqueness of differentiable solutions to the iterative equations involving iterated functional series: ∞ λi Hi f i x F x , x∈I a, b , i 1.8 ∞ λi Hi x, φ ai1 x , ,φ aini x F x , x∈I a, b i But the references above only considered the case that F a a, F b b The problem of differentiable solutions of higher dimensional iterative equations is also interesting By constructing a new operator for the structure of 1.3 , which simplifies the procedure of applying fixed point theorems in some sense, Li 28 studies the smoothness of solutions of 1.3 In 29 , C1 solutions of ∞ λn x f n x F x , x∈B 1.9 n are discussed, where B is a compact convex subset of RN and for any n ≥ 1, λn x : B → R is continuous The boundary restrictions are not considered in the two references above because they only consider the case that F B ⊆ B It should be pointed out that Mai and Liu 30 made an important contribution to Cm solutions of iterative equations Using the method of approximating fixed points by small shift of maps, choosing suitable metrics, and finding a relation between uniqueness Fixed Point Theory and Applications and stability of fixed points of maps of general spaces, Mai and Liu proved the existence, uniqueness of Cm solutions of a relatively general kind of iterative equations: G x, f x , , f n x 0, x ∈ J, 1.10 where J is a connected closed subset of R and G ∈ Cm J n , R , n ≥ Here, Cm J n , R denotes the set of all Cm mappings from J n to R Inspired and motivated by the above work as well as 14–30 , we will study 1.6 and investigate the existence and uniqueness of Lipschitz solution of this equation The rest of this paper is organized as follows In Section 2, we will give some definitions and lemmas In Section 3, we will give a main theorem concerning the existence and uniqueness of solution of P f ◦f F 1.11 In Sections and 5, we will study some special cases of 1.6 by means of the above main theorem Preliminary Let B be a compact convex subset of RN with nonempty interior Let C B, RN RN | f is continuous}, N ∈ Z In C B, RN , we use the supremum norm f B sup f x , for f ∈ C B, RN , {f : B → 2.1 x∈B where · denotes the usual metric of RN Obviously, C B, RN is a complete metric space Definition 2.1 Let A, B be two convex compact subsets of RN with nonempty interior For m ∈ 0, , M ∈ 1, ∞ , define Lip B, A, m, M : {f : B −→ A | f is continuous, f B m x−y ≤ f x −f y A, ∀x, y ∈ B, ≤ M x − y } 2.2 Let g : ∂B → ∂A be a continuous surjective map, and let C B, A, m, M, g denote the subset of Lip B, A, m, M whose elements satisfy f ∂B ∂A and f|∂B g Lemmas 2.2–2.4 can be proved by a corresponding method which is contained in the proofs of Observation 2.2, Lemma 2.3, and Lemma 2.4 of 22 Lemma 2.2 Let m ∈ 0, , M ∈ 1, ∞ , and f ∈ Lip B, A, m, M be arbitrary, then f −1 ∈ Lip A, B, 1/M, 1/m Fixed Point Theory and Applications Lemma 2.3 For every m > 0, the mapping L : f ∈ Lip B, A, m, ∞ −→ f −1 ∈ Lip A, B, 0, m 2.3 is well defined and Lipschitz with constant 1/m Lemma 2.4 For ≤ K, M < ∞ and F ∈ Lip B, A, 0, K , the mapping SF : f ∈ Lip A, B, 0, M −→ f ◦ F ∈ Lip B, B, 0, K, M 2.4 is Lipschitz with constant Lemmas 2.5 and 2.6 can be proved by a method which is contained in the proof of Proposition in 23 Lemma 2.5 If H, G are homeomorphisms from B to A with Lipschitz constant L, then H − G L H −1 − G−1 A Lemma 2.6 If f, g ∈ Lip B, B, m, M , then f k − g k B ≤ k−1 j Mj f − g B ≤ B Lemma 2.7 For any M ∈ 1, ∞ and g : ∂B → ∂B, which is a surjective map, C B, B, 0, M, g is a compact subset of C B, RN Proof It is easy to see that C B, B, 0, M, g is uniformly bounded and equicontinuous By Ascoli-Arzel´ lemma for any sequence {fn }∞ ⊂ C B, B, 0, M, g , there exists a subsequence a n {fnk }∞ of {fn }∞ which converges to a continuous map f ∈ C B, RN Without any loss of k n ∂B, and f x − generality, we suppose limn → ∞ fn f We can easily get f|∂B g, f ∂B B, so for any f y ≤ M x − y , ∀x, y ∈ B We only need to prove f B B Since fn B y By the compactness of B, we suppose limn → ∞ xn x y ∈ B there is a xn ∈ B with fn xn Noticing that y−f x we can get f x fn xn − f x ≤ fn xn − fn x fn x − f x , 2.5 y Then, C B, A, 0, M, g is compact Let DN {x | x ∈ RN , x ≤ 1} Then, ∂DN D , and ∂B is homeomorphic to SN−1 SN−1 Obviously, B is homeomorphic to N Lemma 2.8 If f : DN → DN is continuous and f SN−1 ⊂ SN−1 Let f0 denote f|SN−1 If deg f0 / 0, then f is surjective, where deg f0 denotes the degree of f0 Proof Suppose that f is not surjective Let x0 ∈ DN \f DN If x0 ∈ SN−1 , then f0 is homotopic ∈ 0, a contradiction So x0 / SN−1 , then there exists a retraction to a constant and deg f0 N N−1 Thus, r ◦ f : DN → SN−1 is a continuous mapping, and mapping r : D \ {x0 } → S r ◦ f |SN−1 This means that f0 ∗ N−1 is trivial, then deg f0 So f is surjective f0 Fixed Point Theory and Applications Lemma 2.9 Let M ∈ 1, ∞ and C B, B, 0, M, g be defined as above, where the surjective map g : ∂B → ∂B is the restriction of the elements of C B, B, 0, M, g If deg g / 0, then C B, B, 0, M, g is a convex subset of C B, RN Proof For ∀t ∈ 0, and ∀f, h ∈ C B, B, 0, M, g , tf tf − t h |∂B − t h is continuous and 1−t g tg g 2.6 It is easy to see that tf − t h x − tf 1−t h y ≤M x−y , By lemma 2.8, tf − t h is surjective Thus, tf C B, B, 0, M, g is convex ∀x, y ∈ B 2.7 − t h ∈ C B, B, 0, M, g , that is, Main Result Theorem 3.1 Give M, K ∈ 1, ∞ and A, B which are compact convex subsets of RN with nonempty interior Suppose that both g : ∂B → ∂B and T : ∂B → ∂A are continuous surjective maps and deg g / If there exist a decreasing function α : 1, ∞ → 0, and a continuous map P defined on C B, B, 0, M, g such that P f ∈ C B, A, α M , ∞, T , ∀f ∈ C B, B, 0, M, g M · α M ≥ K 3.1 Then, for any F ∈ C B, A, 0, K, T ◦ g , there exists a f ∈ C B, B, 0, M, g such that P f ◦f F 3.2 Furthermore, if P is Lipschitz with a Lipschitz constant d which satisfies d/α M < 1, then f is unique, and f depends continuously on F Proof Firstly, we prove that P f : B → A is a homeomorphism for all f ∈ C B, B, 0, M, g Since the interior of A is nonempty, α M > otherwise we would get that F is a constant, while we suppose F B A Then, by Lemma 2.2, P f is a homeomorphism We also get M ≥ K/α M By Lemmas 2.3 and 2.4, the mapping , α M K −→ f ◦ F ∈ Lip B, B, 0, α M L : f ∈ Lip B, A, α M , ∞ −→ f −1 ∈ Lip A, B, 0, SF : f ∈ Lip A, B, 0, α M are both well defined and continuous 3.3 Fixed Point Theory and Applications From the above discussions, for ∀f ∈ C B, B, 0, M, g , we can get that P f ∈ C B, A, α M , ∞, T , P f −1 , T −1 , α M L ◦ P f ∈ C A, B, 0, SF ◦ L ◦ P f ∈ C B, B, 0, K ,g α M 3.4 ⊂ C B, B, 0, M, g These mean that SF ◦ L ◦ P : C B, B, 0, M, g → C B, B, 0, M, g is well defined and continuous By Lemmas 2.7, 2.9 and Schauder’s fixed points theorem, SF ◦ L ◦ P has a fixed point f in C B, B, 0, M, g Then, SF ◦ L ◦ P f x ∀x ∈ B, f x , 3.5 which implies P f −1 ◦F f 3.6 This means that f satisfies the assertion of the theorem For f1 , f2 ∈ C B, B, 0, M, g , by Lemma 2.5, SF ◦ L ◦ P f1 − SF ◦ L ◦ P f2 P f1 ≤ P f1 ≤ α M d ≤ α M −1 −1 −1 ◦ F − P f2 −1 − P f2 B ◦F A 3.7 P f1 − P f2 f1 − f2 B B B Since d/α M < 1, the mapping SF ◦ L ◦ P is a contraction on C B, B, 0, M, g By Banach contraction principle, f is unique Suppose F1 , F2 ∈ C B, A, 0, K, T ◦ g and f1 , f2 ∈ C B, B, 0, M, g such that P f1 ◦ f1 F1 and P f2 ◦ f2 F2 , then, f1 − f2 SF1 ◦ L ◦ P f1 − SF2 ◦ L ◦ P f2 B P f1 −1 ≤ P f1 −1 ≤ d α M ◦ F1 − P f2 −1 ◦ F2 B ◦ F1 − P f2 −1 ◦ F1 B B f1 − f2 B α M P f2 F1 − F2 −1 B ◦ F1 − P f2 −1 ◦ F2 B 3.8 Fixed Point Theory and Applications This means that f1 − f2 B ≤ α M 1− −1 d α M F1 − F2 B , 3.9 then f depends continuously on F Theorem 3.2 Let the sequence {ak }∞ ⊂ R satisfy that k all M ∈ 1, ∞ the mapping P : f ∈ Lip B, B, 0, M −→ ∞ ∞ k ak is absolutely convergent, then for ak f k ∈ C B, RN 3.10 k is well defined and continuous ∞ Proof Since ∞ ak is absolutely convergent and {f k }k is a uniformly bounded sequence of k continuous maps on the compact space B to itself, by Weierstrass M-test, P f is well defined and continuous By Lemma 2.6 and the absolute convergence of ∞ ak , the continuity of the mapping k P can be easily got Iterative Equation in R Let I a, b and J c, d be two compact intervals Let h1 id|∂I and h2 r1 be the c, g1 b d and g2 a d, g2 b antipodal maps on ∂I Let g1 , g2 : ∂I → ∂J satisfy g1 a and deg h2 −1 c Obviously, g1 g1 ◦ h1 and g2 g1 ◦ h2 Obviously, deg h1 Theorem 4.1 Suppose that the sequence {ak }∞ ⊂ R satisfy a1 > and k convergent and M, K ≥ 1, if ∞ ≥ a1 − ∞ n ∞ hi−1 a < k i an Mn−1 ≥ k ak is absolutely K , M hi−1 b , k ∞ k 4.1 1, 4.2 i Then, for any F ∈ C I, J, 0, K, gk , 1.6 has a solution in C I, I, 0, K, hk , where I ∞ i−1 a , ∞1 hi−1 b , k 1, Moreover, if J i hk i k ∞ k ak α M f is unique and depends continuously on F k−2 j Mj < 1, a, b and 4.3 Fixed Point Theory and Applications Proof For t ∈ 1, ∞ define α t by α t min{max a1 − ∞2 |ai |ti−1 , , 1} Since ≤ α t ≤ 1, i we obtain that α : 1, ∞ → 0, From 4.1 , we have M · α M ≥ K By Theorem 3.2, we can define P : C I, I, 0, M, hk → C I, R , k 1, by ∞ P f x f i−1 x , ∀x ∈ I, 4.4 i 1, It is easy to see that P f a where f ∈ C I, I, 0, M, hk , k ∞ i−1 b For x, y ∈ I with y > x, one can check that P f b i hk < α M y − x ≤ P f y − P f x ≤ 2a1 y − x ∞ i hi−1 a and k 4.5 This means that P f I J, and P f is an orientation preserving homeomorphism of I The above discussions imply that P f ∈ C I, J, α M , ∞, g1 4.6 For any f, g ∈ C I, I, 0, M, hk by Lemma 2.6, we get that P f −P g ∞ sup I x∈I ≤ sup ≤ i ∞ f i−1 x − g i−1 x i · f i−1 x − g i−1 x x∈I i ∞ 4.7 · f i−1 − g i−1 i ≤ ∞ ∞ · i i−2 I Mj f − g I j By Theorem 3.1, the assertion is true Example 4.2 54 f x 55 ∞ i F1 x i−2 −1 54i−1 fi x x : I −→ I, x2 , F1 x∈I ∂I g1 0, , 4.8 10 Fixed Point Theory and Applications x2 ∈ C I, I, 0, 2, g1 Let M Obviously, F1 x α M ∞ a1 − since n ∞ k ak k−2 j Mj ∞ k 1/54 α M k−2 j k−1 ∞ 54 − 55 an Mn−1 4j α M 4i−1 54 i ∞ k ≤ 248 > , 275 i−1 2k−1 · 4k−2 /54k−1 α M 275 23 × 248 4.9 Then, by Theorem 4.1, the equation has an unique strictly increasing solution in C I, I, 0, 4, h1 For ⎧ ⎪2x, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨1 , ⎪2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪2x − 1, ⎩ F2 x , x∈ , , 4 x∈ ,1 , x ∈ 0, 4.10 it is easy to see that F2 ∈ C I, I, 0, 2, g1 Then, 54 f x 55 ∞ −1 i−2 54i−1 i fi x F2 : I −→ I, x∈I F2 x , F2 ∂I 0, , 4.11 g1 has an unique increasing solution in C I, I, 0, 4, h1 x 1/2 sin 2πx ∈ C I, I, 0, 5, g1 For a nonmonotonic example, we consider F3 x As mentioned in 20 , F3 has a local maximum at a point x1 and a local minimum at a point x2 in 0, The equation 210 f x 211 ∞ −1 210 i i−2 i−1 fi x F3 : I −→ I, F3 x , F3 ∂I x∈I 0, , 4.12 g1 has an unique nonmonotonic solution in C I, I, 0, 10, h1 Example 4.3 For convenience, we only consider {ak }∞ with a2i k − x2 ∈ C I, I, 0, 2, g2 By Theorem 4.1, for I 0, , F1 x 254 f x 255 ∞ −1 i i−2 256i−1 f 2i−1 x F1 : I −→ I, F1 F1 x , ∂I g2 0, i x ∈ 0, , 1, 2, Obviously, 4.13 Fixed Point Theory and Applications 11 has an unique strictly decreasing solution in C I, I, 0, 4, h2 For ⎧ ⎪1 − 2x, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨1 , ⎪2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩2 − 2x, F2 x , x∈ , , 4 x∈ ,1 , x ∈ 0, 4.14 it is easy to see that F2 ∈ C I, I, 0, 2, g2 Then, 254 f x 255 ∞ i−2 −1 i f 2i−1 x 256i−1 F2 : I −→ I, F2 F2 x , ∂I x ∈ 0, , 4.15 g2 has an unique decreasing solution in C I, I, 0, 4, h2 − x − 1/2 sin 2πx ∈ C I, I, 0, For a nonmonotonic example, we consider F3 x 5, g2 F3 has a local maximum at a point x1 and a local minimum at a point x2 in 0, The equation 1000 f x 1001 ∞ i −1 i−2 1000i−1 f 2i−1 x F3 : I −→ I, F3 F3 x , ∂I x ∈ 0, , 4.16 g2 has an unique nonmonotonic solution in C I, I, 0, 10, h2 Iterative Equation in RN N ≥ In 22 , Kulczycki and Tabor got the existence of solutions of the iterative 1.6 on compact convex subsets of RN , but they only discussed the case F|∂B id∂B In this section, we will continue the work of 22 and discuss the solutions for a special case of 1.6 on unit closed ball of RN Let ξ : SN−1 → SN−1 be a homeomorphism which satisfies ξ ◦ ξ idSN−1 Obviously, deg ξ ±1 Theorem 5.1 Let {ai }∞1 ⊂ 0, with i with ∞ i a1 − ∞ i 1, a1 > and there exist two constants M, K ≥ M2i−2 ≥ K M 5.1 12 Fixed Point Theory and Applications Then, for any F ∈ C DN , DN , 0, K, ξ , ∞ f 2i−1 x x ∈ DN , F x , 5.2 i F:D N −→ D , N F ∞ k has a solution f ∈ C DN , DN , 0, M, ξ Moreover, if unique and depends continuously on F ∞ i Proof For t ∈ 1, ∞ , define α t max{a0 − P : C DN , DN , 0, M, ξ → C DN , RN by ∞ P f x ξ SN−1 ak 2k−3 j Mj /α M < 1, then f is t2i−2 , 0}, then ≤ α t ≤ a0 ≤ Define f 2i−2 x 5.3 i Then, we get ∞ P f x −P f y f 2i−2 x − i ∞ f 2i−2 y i ≥ a1 x − y − ∞ f 2i−2 x − f 2i−2 y i ≥ a1 − ∞ M2i−2 x−y 5.4 i α M x−y , P f x −P f y ≤ ∞ a1 M2i−2 x−y i For all x ∈ DN , we have ∞ f 2i−2 x ∈ conv x, f x , f x , ⊂ DN 5.5 i ∞ 2i−2 N−1 Since P f |SN−1 |S id|SN−1 and deg id|SN−1 1, then P f DN DN i ξ and P f is a homeomorphism from DN to DN By the above discussion, we get that ∞ 2i−1 ξ For f, g ∈ P f ∈ C DN , DN , α M , ∞, id|SN−1 But P f ◦ f |SN−1 i ξ N N C D , D , 0, M, ξ , by Lemma 2.6, we get that P f −P g DN ≤ ∞ 2k−3 ak k Mj f − g DN 5.6 j Obviously, the maps id|SN−1 and ξ are the concrete forms of the maps T and g in Theorem 3.1 By Theorem 3.1, the assertion is true Fixed Point Theory and Applications 13 2 Example 5.2 For F x − x1 , x2 dist x, S1 1, − x1 − x1 x2 , −x2 , where x 1 x1 , x2 ∈ D , and dist x, S denotes the distance of the point x from S Obviously, F|S1 r1 , where r1 denotes the antipodal map on S1 By simple calculation, we get that for any x, y ∈ D2 , F x −F 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