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Research | Open | Published:

Solving a class of functional equations using fixed point theorems

Abstract

This paper is concerned with solvability of a class of functional equations arising in dynamic programming of multistage decision processes. Using the fixed point theorems due to Banach and Liu-Ume-Kang and iterative algorithms, some sufficient conditions which ensure the existence, uniqueness and iterative approximations of solutions for the functional equation in the Banach spaces BC(S) and B(S) and the complete metric space BB(S) are provided. Four examples are constructed to illustrate the results presented in this paper.

MSC:49L20, 90C39.

1 Introduction

It is well know that the existence problems of solutions for various functional equations arising in dynamic programming are both of theoretical and of practical interest; for example, see [119] and the references cited therein. Bellman [2], Bhakta and Choudhury [5], Liu [9] and Liu et al. [11, 12, 1416, 19] studied the existence, uniqueness and iterative approximations of solutions for the following functional equations arising in dynamic programming:

f(x)= inf y D max { u ( x , y ) , v ( x , y ) f ( a ( x , y ) ) } ,xS,
(1.1)
f(x)= inf y D max { u ( x , y ) , f ( a ( x , y ) ) } ,xS,
(1.2)
f(x)= sup y D max { u ( x , y ) , f ( a ( x , y ) ) } ,xS,
(1.3)
f(x)= opt y D opt { u ( x , y ) , f ( a ( x , y ) ) } ,xS,
(1.4)
f(x)= opt y D { u ( x , y ) max { p ( x , y ) , f ( a ( x , y ) ) } } ,xS,
(1.5)
f(x)= opt y D { u ( x , y ) min { p ( x , y ) , f ( a ( x , y ) ) } } ,xS
(1.6)

in the complete metric space BB(S), where opt stands for the sup or inf.

Motivated and inspired by the results in [119], in this paper we introduce and study a new functional equation arising in dynamic programming of multistage decision processes as follows:

f ( x ) = λ opt y D { u ( x , y ) opt { p ( x , y ) , f ( a ( x , y ) ) } } + ( 1 λ ) opt y D { v ( x , y ) opt { q ( x , y ) , f ( b ( x , y ) ) } + t ( x , y ) opt { r ( x , y ) , f ( c ( x , y ) ) } } , x S ,
(1.7)

where λ[0,1] is a constant, x and y stand for the state and decision vectors, respectively, a, b and c denote the transformations of the processes, and f(x) is the optimal return function with initial state x. Obviously, functional equation (1.7) includes functional equations (1.1)-(1.6) as special cases. Utilizing the Banach fixed point theorem and Liu-Ume-Kang fixed point theorem, some techniques in nonlinear analysis and a few iterative algorithms, we get the existence, uniqueness and iterative approximations of continuous bounded solutions, bounded solutions and solutions for functional equation (1.7) in the Banach spaces BC(S) and B(S) and the complete metric space BB(S), respectively, and discuss some error estimates between the iterative sequences generated by the iterative algorithms and the solutions. Four nontrivial examples are given to show that the results presented in this paper are more general than those in [5, 6, 11, 12, 1416, 19].

2 Preliminaries

Throughout this paper, we assume that (X,) and (Y, ) are real Banach spaces, SX is the state space, DY is the decision space, denotes the set of all positive integers, N 0 ={0}N, R=(,+), R + =[0,+) and R =(,0]. Define

Φ 1 = { φ : φ : R + R +  is nondecreasing and  φ ( t ) < t Φ 1 =  for each  t > 0 } , Φ 2 = { ( φ , ψ ) : φ Φ 1 , ψ : R + R +  is nondecreasing and  n = 0 ψ ( φ n ( t ) ) < + Φ 2 =  for each  t > 0 } , B ( S ) = { g : g : S R  is bounded } , B C ( S ) = { g : g B ( S )  is continuous } , B B ( S ) = { g : g : S R  is bounded on each bounded subsets of  S } .

Clearly, (B(S), 1 ) and (BC(S), 1 ) are Banach spaces with the norm g 1 = sup x S |g(x)|. For each kN and h,gBB(S), put

d k ( h , g ) = sup { | h ( x ) g ( x ) | : x B ¯ ( 0 , k ) } , d ( h , g ) = k = 1 1 2 k d k ( h , g ) 1 + d k ( h , g ) ,

where

B ¯ (0,k)= { x : x S  and  x k } .

Obviously, { d k } k N is a countable family of pseudometrics in BB(S). A sequence { x k } k N in BB(S) is said to converge to a point xBB(S) if d k ( x n ,x)0 as n and { x n } n N is a Cauchy sequence if d k ( x n , x m )0 as n,m for each kN. It is clear that (BB(S),d) is a complete metric space.

Lemma 2.1 ([12])

Let E be a set, p and q:ER be mappings. If opt y E p(y) and opt y E q(y) are bounded, then

| opt y E p ( y ) opt y E q ( y ) | sup y E | p ( y ) q ( y ) | .

Lemma 2.2 ([14])

Let α, β, γ and δ be in . Then

| opt { α , β } opt { γ , δ } | max { | α γ | , | β δ | } .

Lemma 2.3 (Liu-Ume-Kang fixed point theorem [17])

Let (G,ρ) be a complete metric space, { ρ k } k N be a countable family of pseudometrics on G such that for any different points x,yG, ρ k (x,y)>0 for some kN, and ρ be defined by

ρ(x,y)= k = 1 1 2 k ρ k ( x , y ) 1 + ρ k ( x , y ) ,x,yG.

Assume that T:GG satisfies that

ρ k (Tx,Ty)φ ( ρ k ( x , y ) ) ,(x,y,k) G 2 ×N,

where φ: R + R + is upper semicontinuous from the right on R + and φ(t)<t for each t>0. Then T has a unique fixed point wG and lim n T n (x)=w for each xG.

3 Main results

Now we investigate the existence, uniqueness and iterative approximations of continuous bounded solutions and bounded solutions for functional equation (1.7) in the Banach spaces BC(S) and B(S), respectively, by using the Banach fixed point theorem and iterative algorithms.

Theorem 3.1 Let S be compact, λ[0,1] and α[0,1). Let p,q,r,u,v,t:S×DR and a,b,c:S×DS satisfy that

(C1) p, q and r are bounded in S×D;

(C2) sup ( x , y ) S × D max{|u(x,y)|,|v(x,y)|+|t(x,y)|}α;

(C3) for each ( x 0 ,g)S×{p,q,r,u,v,t,a,b,c},

lim x x 0 g(x,y)=g( x 0 ,y)uniformly foryD.

Then functional equation (1.7) possesses a unique solution wBC(S) such that

(C4) for each w 0 BC(S), the iterative sequence { w n } n N 0 defined by

w n ( x ) = λ opt y D { u ( x , y ) opt { p ( x , y ) , w n 1 ( a ( x , y ) ) } } + ( 1 λ ) opt y D { v ( x , y ) opt { q ( x , y ) , w n 1 ( b ( x , y ) ) } + t ( x , y ) opt { r ( x , y ) , w n 1 ( c ( x , y ) ) } } , ( x , n ) S × N
(3.1)

converges to w and has the error estimate:

w n w 1 α n w 0 w 1 and w n w 1 α n 1 α w 0 w 1 1 , n N .
(3.2)

Proof Define a mapping H:BC(S)BC(S) by

H h ( x ) = λ opt y D { u ( x , y ) opt { p ( x , y ) , h ( a ( x , y ) ) } } + ( 1 λ ) opt y D { v ( x , y ) opt { q ( x , y ) , h ( b ( x , y ) ) } + t ( x , y ) opt { r ( x , y ) , h ( c ( x , y ) ) } } , ( x , h ) S × B C ( S ) .
(3.3)

Firstly, we show that H is a self-mapping in BC(S). Let ( x 0 ,h)S×BC(S) and ε>0. It follows from (C1), (C3) and the compactness of S that there exist constants M>0, δ>0 and δ 1 >0 such that

sup ( x , y ) S × D max { | h ( x ) | , | h ( a ( x , y ) ) | , | h ( b ( x , y ) ) | , | h ( c ( x , y ) ) | , | p ( x , y ) | , | q ( x , y ) | , | r ( x , y ) | } M ;
(3.4)
max { | u ( x , y ) u ( x 0 , y ) | , | v ( x , y ) v ( x 0 , y ) | + | t ( x , y ) t ( x 0 , y ) | } < ε 2 M , ( x , y ) S × D  with  x x 0 < δ ;
(3.5)
max { | p ( x , y ) p ( x 0 , y ) | , | q ( x , y ) q ( x 0 , y ) | , | r ( x , y ) r ( x 0 , y ) | } < ε 2 , ( x , y ) S × D  with  x x 0 < δ ;
(3.6)
| h ( x 1 ) h ( x 2 ) | < ε 2 , x 1 , x 2 S with  x 1 x 2 < δ 1 ;
(3.7)
max { a ( x , y ) a ( x 0 , y ) , b ( x , y ) b ( x 0 , y ) , c ( x , y ) c ( x 0 , y ) } < δ 1 , ( x , y ) S × D  with  x x 0 < δ .
(3.8)

On account of (C2), (3.3)-(3.8), Lemmas 2.1 and 2.2, we infer that

| H h ( x ) H h ( x 0 ) | = | λ opt y D { u ( x , y ) opt { p ( x , y ) , h ( a ( x , y ) ) } } + ( 1 λ ) opt y D { v ( x , y ) opt { q ( x , y ) , h ( b ( x , y ) ) } + t ( x , y ) opt { r ( x , y ) , h ( c ( x , y ) ) } } λ opt y D { u ( x 0 , y ) opt { p ( x 0 , y ) , h ( a ( x 0 , y ) ) } } ( 1 λ ) opt y D { v ( x 0 , y ) opt { q ( x 0 , y ) , h ( b ( x 0 , y ) ) } + t ( x 0 , y ) opt { r ( x 0 , y ) , h ( c ( x 0 , y ) ) } } | λ | opt y D { u ( x , y ) opt { p ( x , y ) , h ( a ( x , y ) ) } } opt y D { u ( x 0 , y ) opt { p ( x , y ) , h ( a ( x , y ) ) } } | + λ | opt y D { u ( x 0 , y ) opt { p ( x , y ) , h ( a ( x , y ) ) } } opt y D { u ( x 0 , y ) opt { p ( x 0 , y ) , h ( a ( x 0 , y ) ) } } | + ( 1 λ ) | opt y D { v ( x , y ) opt { q ( x , y ) , h ( b ( x , y ) ) } + t ( x , y ) opt { r ( x , y ) , h ( c ( x , y ) ) } } opt y D { v ( x 0 , y ) opt { q ( x , y ) , h ( b ( x , y ) ) } + t ( x 0 , y ) opt { r ( x , y ) , h ( c ( x , y ) ) } } | + ( 1 λ ) | opt y D { v ( x 0 , y ) opt { q ( x , y ) , h ( b ( x , y ) ) } + t ( x 0 , y ) opt { r ( x , y ) , h ( c ( x , y ) ) } } opt y D { v ( x 0 , y ) opt { q ( x 0 , y ) , h ( b ( x 0 , y ) ) } + t ( x 0 , y ) opt { r ( x 0 , y ) , h ( c ( x 0 , y ) ) } } | λ sup y D { | u ( x , y ) u ( x 0 , y ) | max { | p ( x , y ) | , | h ( a ( x , y ) ) | } } + λ sup y D { | u ( x 0 , y ) | | opt { p ( x , y ) , h ( a ( x , y ) ) } opt { p ( x 0 , y ) , h ( a ( x 0 , y ) ) } | } + ( 1 λ ) sup y D { | v ( x , y ) v ( x 0 , y ) | max { | q ( x , y ) | , | h ( b ( x , y ) ) | } + | t ( x , y ) t ( x 0 , y ) | max { | r ( x , y ) | , | h ( c ( x , y ) ) | } } + ( 1 λ ) sup y D { | v ( x 0 , y ) | | opt { q ( x , y ) , h ( b ( x , y ) ) } opt { q ( x 0 , y ) , h ( b ( x 0 , y ) ) } | + | t ( x 0 , y ) | | opt { r ( x , y ) , h ( c ( x , y ) ) } opt { r ( x 0 , y ) , h ( c ( x 0 , y ) ) } | } λ M sup y D | u ( x , y ) u ( x 0 , y ) | + λ α sup y D max { | p ( x , y ) p ( x 0 , y ) | , | h ( a ( x , y ) ) h ( a ( x 0 , y ) ) | } + ( 1 λ ) M sup y D { | v ( x , y ) v ( x 0 , y ) | + | t ( x , y ) t ( x 0 , y ) | } + ( 1 λ ) sup y D { ( | v ( x 0 , y ) | + | t ( x 0 , y ) | ) max { | q ( x , y ) q ( x 0 , y ) | , | h ( b ( x , y ) ) h ( b ( x 0 , y ) ) | , | r ( x , y ) r ( x 0 , y ) | , | h ( c ( x , y ) ) h ( c ( x 0 , y ) ) | } } < λ M ε 2 M + λ α ε 2 + ( 1 λ ) M ε 2 M + ( 1 λ ) α ε 2 < ε , x S  with  x x 0 < δ

and

| H h ( x ) | = | λ opt y D { u ( x , y ) opt { p ( x , y ) , h ( a ( x , y ) ) } } + ( 1 λ ) opt y D { v ( x , y ) opt { q ( x , y ) , h ( b ( x , y ) ) } + t ( x , y ) opt { r ( x , y ) , h ( c ( x , y ) ) } } | λ sup y D { | u ( x , y ) | | opt { p ( x , y ) , h ( a ( x , y ) ) } | } + ( 1 λ ) sup y D { | v ( x , y ) | | opt { q ( x , y ) , h ( b ( x , y ) ) } | + | t ( x , y ) | | opt { r ( x , y ) , h ( c ( x , y ) ) } | } λ α sup y D max { | p ( x , y ) | , | h ( a ( x , y ) ) | } + ( 1 λ ) sup y D { [ | v ( x , y ) | + | t ( x , y ) | ] max { | q ( x , y ) | , | r ( x , y ) | , | h ( b ( x , y ) ) | , | h ( c ( x , y ) ) | } } λ α M + ( 1 λ ) α M = α M , x S ,

which yields that Hh is bounded and continuous in S. That is, H maps BC(S) into BC(S).

Secondly, we show that H is a contraction mapping in BC(S). Given ε>0. In view of (3.3), Lemmas 2.1 and 2.2, we get that

| H h ( x ) H g ( x ) | = | λ opt y D { u ( x , y ) opt { p ( x , y ) , h ( a ( x , y ) ) } } + ( 1 λ ) opt y D { v ( x , y ) opt { q ( x , y ) , h ( b ( x , y ) ) } + t ( x , y ) opt { r ( x , y ) , h ( c ( x , y ) ) } } λ opt y D { u ( x , y ) opt { p ( x , y ) , g ( a ( x , y ) ) } } ( 1 λ ) opt y D { v ( x , y ) opt { q ( x , y ) , g ( b ( x , y ) ) } + t ( x , y ) opt { r ( x , y ) , g ( c ( x , y ) ) } } | λ sup y D { | u ( x , y ) | | opt { p ( x , y ) , h ( a ( x , y ) ) } opt { p ( x , y ) , g ( a ( x , y ) ) } | } + ( 1 λ ) sup y D { | v ( x , y ) | | opt { q ( x , y ) , h ( b ( x , y ) ) } opt { q ( x , y ) , g ( b ( x , y ) ) } | + | t ( x , y ) | | opt { r ( x , y ) , h ( c ( x , y ) ) } opt { r ( x , y ) , g ( c ( x , y ) ) } | } λ α sup y D { | h ( a ( x , y ) ) g ( a ( x , y ) ) | } + ( 1 λ ) sup y D { [ | v ( x , y ) | + | t ( x , y ) | ] × max { | h ( b ( x , y ) ) g ( b ( x , y ) ) | , | h ( c ( x , y ) ) g ( c ( x , y ) ) | } } α h g 1 , x S , h , g B C ( S ) ,

which gives that

H h H g 1 α h g 1 ,h,gBC(S),
(3.9)

that is, H is a contraction mapping in BC(S). Thus the Banach fixed point theorem yields that H has a unique fixed point wBC(S), which is a unique solution of functional equation (1.7) in BC(S).

Thirdly, we show (C4). Note that

w ( x ) = λ opt y D { u ( x , y ) opt { p ( x , y ) , w ( a ( x , y ) ) } } + ( 1 λ ) opt y D { v ( x , y ) opt { q ( x , y ) , w ( b ( x , y ) ) } + t ( x , y ) opt { r ( x , y ) , w ( c ( x , y ) ) } } , x S ,

which together with (3.1), (3.3) and (3.9) yields that

w n w 1 = sup x S | w n ( x ) w ( x ) | = sup x S | H w n 1 ( x ) H w ( x ) | = H w n 1 H w α w n 1 w 1 α n w 0 w 1 , n N ,
(3.10)

which guarantees that the sequence { w n } n N 0 converges to w. Similarly, we conclude that

w n w n + m 1 i = n n + m 1 w i w i + 1 1 = i = n n + m 1 H w i 1 H w i 1 i = n n + m 1 α w i 1 w i 1 i = n n + m 1 α i w 0 w 1 1 α n 1 α w 0 w 1 1 , ( n , m ) N × N .

Letting m in the above inequalities, we infer that (3.2) holds. This completes the proof. □

Using the proof of Theorem 3.1, we have the following.

Theorem 3.2 Let α(0,1) and λ[0,1]. Let p,q,r,u,v,t:S×DR and a,b,c:S×DS satisfy (C1) and (C2). Then functional equation (1.7) possesses a unique solution wB(S) and for each w 0 B(S), the sequence { w n } n N 0 defined by (3.1) converges to w and satisfies (C4).

Example 3.3 Consider the functional equation

f ( x ) = λ opt y R + { cos ( x 9 + y 18 ) y 2 + 2 opt { x 56 x 31 + y 5 + 1 , f ( sin 8 ( x 2 y 35 ) ) } } + ( 1 λ ) opt y R + { x 2 sin ( y 14 ) 2 ( x + 1 ) 2 + y 3 opt { x 25 ln ( 1 + x 2 ) x 2 + y 3 + 1 , f ( x 6 x 2 + y 2 + 1 ) } + x + 1 3 x + 2 + y 2 opt { x 10 cos 6 ( y 7 y ) y 3 + y 2 + 1 , f ( x 7 y x 3 + y 2 + 1 ) } } , x [ 0 , 40 ] .
(3.11)

Put X=Y=R, S=[0,40], D= R + , λ[0,1], α= 5 6 . Let p,q,r,u,v,t:S×DR and a,b,c:S×DS be defined by

p ( x , y ) = x 56 x 31 + y 5 + 1 , q ( x , y ) = x 25 ln ( 1 + x 2 ) x 2 + y 3 + 1 , r ( x , y ) = x 10 cos 6 ( y 7 y ) y 3 + y 2 + 1 , u ( x , y ) = cos ( x 9 + y 18 ) y 2 + 2 , v ( x , y ) = x 2 sin ( y 14 ) 2 ( x + 1 ) 2 + y 3 , t ( x , y ) = x + 1 3 x + 2 + y 2 , a ( x , y ) = sin 8 ( x 2 y 35 ) , b ( x , y ) = x 6 x 2 + y 2 + 1 , c ( x , y ) = x 7 y x 3 + y 2 + 1 , ( x , y ) S × D .

It is easy to see that the conditions of Theorem 3.1 are satisfied. It follows from Theorem 3.1 that functional equation (3.11) possesses a unique solution wBC(S) and (C4) holds.

Example 3.4 Consider the functional equation

f ( x ) = λ opt y R { x y 2 sin 45 ( x 9 y 7 ) x 2 + y 4 + 1 opt { sin 3 ( x y 5 ) , f ( x 59 x 4 + y 24 ) } } + ( 1 λ ) opt y R { x 3 arctan ( x 15 y 21 ) 2 π x 3 + y 2 + 1 opt { cos 9 ( x 6 y 3 ( x + y ) ) , f ( x 2 y 6 ) } + 2 x 4 sin 2 ( x y ) 3 x 4 + cos 4 ( x 8 y 6 ) opt { x 3 y 2 ( x + 1 ) 4 + y 2 , f ( x 35 y 47 x y ) } } , x R + .
(3.12)

Set X=Y=R, S= R + , D= R , λ[0,1], α= 11 12 . Let p,q,r,u,v,t:S×DR and a,b,c:S×DS be defined by

p ( x , y ) = sin 3 ( x y 5 ) , q ( x , y ) = cos 9 ( x 6 y 3 ( x + y ) ) , r ( x , y ) = x 3 y 2 ( x + 1 ) 4 + y 2 , u ( x , y ) = x y 2 sin 45 ( x 9 y 7 ) x 2 + y 4 + 1 , v ( x , y ) = x 3 arctan ( x 15 y 21 ) 2 π x 3 + y 2 + 1 , t ( x , y ) = 2 x 4 sin 2 ( x y ) 3 x 4 + cos 4 ( x 8 y 6 ) , a ( x , y ) = x 59 x 4 + y 24 , b ( x , y ) = x 2 y 6 , c ( x , y ) = x 35 y 47 x y , ( x , y ) S × D .

It is clear that the conditions of Theorem 3.2 are fulfilled. Thus Theorem 3.2 guarantees that functional equation (3.12) possesses a unique solution wB(S), which satisfies (C4).

Next we prove the existence, uniqueness and iterative approximation of solutions for functional equation (1.7) in the complete metric space BB(S) by using Liu-Ume-Kang fixed point theorem.

Theorem 3.5 Let α(0,1), λ[0,1], p,q,r,u,v,t:S×DR and a,b,c:S×DS satisfy that

(C5) p, q and r are bounded on B ¯ (0,k)×D, kN;

(C6) sup ( x , y ) B ¯ ( 0 , k ) × D max{|u(x,y)|,|v(x,y)|+|t(x,y)|}α, kN;

(C7) sup ( x , y ) B ¯ ( 0 , k ) × D max{a(x,y),b(x,y),c(x,y)}k, kN.

Then functional equation (1.7) possesses a unique solution wBB(S) such that

(C8) for each w 0 BB(S), the sequence { w n } n N 0 defined by

w n ( x ) = λ opt y D { u ( x , y ) opt { p ( x , y ) , w n 1 ( a ( x , y ) ) } } + ( 1 λ ) opt y D { v ( x , y ) opt { q ( x , y ) , w n 1 ( b ( x , y ) ) } + t ( x , y ) opt { r ( x , y ) , w n 1 ( c ( x , y ) ) } } , ( x , k , n ) B ¯ ( 0 , k ) × N × N
(3.13)

converges to w and has the following error estimates:

d k ( w n , w ) α n d k ( w 0 , w ) and d k ( w n , w ) α n 1 α d k ( w 0 , w 1 ) , ( k , n ) N × N .
(3.14)

Proof Define a mapping H:BB(S)BB(S) by

H h ( x ) = λ opt y D { u ( x , y ) opt { p ( x , y ) , h ( a ( x , y ) ) } } + ( 1 λ ) opt y D { v ( x , y ) opt { q ( x , y ) , h ( b ( x , y ) ) } + t ( x , y ) opt { r ( x , y ) , h ( c ( x , y ) ) } } , ( x , k , h ) B ¯ ( 0 , k ) × N × B B ( S ) .
(3.15)

It follows from (C5) and (C7) that for each (k,h)N×BB(S), there exist γ(k)>0 and β(k,h)>0 such that

sup ( x , y ) B ¯ ( 0 , k ) × D max { | p ( x , y ) | , | q ( x , y ) | , | r ( x , y ) | } γ ( k ) ; sup ( x , y ) B ¯ ( 0 , k ) × D max { | h ( a ( x , y ) ) | , | h ( b ( x , y ) ) | , | h ( c ( x , y ) ) | } β ( k , h ) ,

which together with (C6), (3.15) and Lemma 2.1 gives that

| H h ( x ) | = | λ opt y D { u ( x , y ) opt { p ( x , y ) , h ( a ( x , y ) ) } } + ( 1 λ ) opt y D { v ( x , y ) opt { q ( x , y ) , h ( b ( x , y ) ) } + t ( x , y ) opt { r ( x , y ) , h ( c ( x , y ) ) } } | λ sup y D { | u ( x , y ) | max { | p ( x , y ) | , | h ( a ( x , y ) ) | } } + ( 1 λ ) sup y D { | v ( x , y ) | max { | q ( x , y ) | , | h ( b ( x , y ) ) | } + | t ( x , y ) | max { | r ( x , y ) | , | h ( c ( x , y ) ) | } } λ α max { γ ( k ) , β ( k , h ) } + ( 1 λ ) max { γ ( k ) , β ( k , h ) } sup y D ( | v ( x , y ) | + | t ( x , y ) | ) λ α max { γ ( k ) , β ( k , h ) } + ( 1 λ ) α max { γ ( k ) , β ( k , h ) } = α max { γ ( k ) , β ( k , h ) } , ( x , k , h ) B ¯ ( 0 , k ) × N × B B ( S ) ,

which means that H is a self-mapping in BB(S). By virtue of (3.15), (C6), (C7), Lemmas 2.1 and 2.2, we get that

| H h ( x ) H g ( x ) | = | λ opt y D { u ( x , y ) opt { p ( x , y ) , h ( a ( x , y ) ) } } + ( 1 λ ) opt y D { v ( x , y ) opt { q ( x , y ) , h ( b ( x , y ) ) } + t ( x , y ) opt { r ( x , y ) , h ( c ( x , y ) ) } } λ opt y D { u ( x , y ) opt { p ( x , y ) , g ( a ( x , y ) ) } } ( 1 λ ) opt y D { v ( x , y ) opt { q ( x , y ) , g ( b ( x , y ) ) } + t ( x , y ) opt { r ( x , y ) , g ( c ( x , y ) ) } } | λ sup y D { | u ( x , y ) | | opt { p ( x , y ) , h ( a ( x , y ) ) } opt { p ( x , y ) , g ( a ( x , y ) ) } | } + ( 1 λ ) sup y D { | v ( x , y ) | | opt { q ( x , y ) , h ( b ( x , y ) ) } opt { q ( x , y ) , g ( b ( x , y ) ) } | + | t ( x , y ) | | opt { r ( x , y ) , h ( c ( x , y ) ) } opt { r ( x , y ) , g ( c ( x , y ) ) } | } λ α d k ( h , g ) + ( 1 λ ) sup y D { | v ( x , y ) | + | t ( x , y ) ] } d k ( h , g ) α d k ( h , g ) , ( x , k , h , g ) B ¯ ( 0 , k ) × N × B B ( S ) × B B ( S ) ,

which yields that

d k (Hh,Hg)α d k (h,g),(k,h,g)N×BB(S)×BB(S).
(3.16)

Put φ(t)=αt for all t R + . It follows from (3.16) and Lemma 2.3 that H has a unique fixed point wBB(S), which is also a unique solution of functional equation (1.7). In light of (3.13), (3.15) and (3.16), we obtain that

d k ( w n , w ) = d k ( H w n 1 , H w ) α d k ( w n 1 , w ) α n d k ( w 0 , w ) , ( k , n ) N × N
(3.17)

and

d k ( w n , w n + m ) i = n n + m 1 d k ( w i , w i + 1 ) = i = n n + m 1 d k ( H w i 1 , H w i ) i = n n + m 1 α d k ( w i 1 , w i ) i = n n + m 1 α i d k ( w 0 , w 1 ) α n 1 α d k ( w 0 , w 1 ) , ( k , n , m ) N × N × N .
(3.18)

Clearly (3.17) means that { w n } n N 0 converges to w. Thus (3.14) follows from (3.17) and (3.18) by letting m. This completes the proof. □

Remark 3.6 Theorem 3.5 extends Theorem 3.4 in [5] and Corollaries 2.2 and 2.3 in [11]. The example below shows that Theorem 3.5 extends substantially the corresponding results in [5, 11].

Example 3.7 Consider the functional equation

f ( x ) = λ opt y R + { x 3 cos 75 ( x 6 y 98 ) 3 ( x + 1 ) 3 + y 5 opt { x 69 ( x 5 y 6 ) 4 + 1 , f ( x 7 sin 4 ( x y ) x 6 + x y 3 + 1 ) } } + ( 1 λ ) opt y R + { sin 9 ( x 8 y 3 ) x y 2 + 5 opt { x 120 y 7 x 9 y 7 + 1 , f ( x 6 y 4 cos 8 ( x 3 y 9 ) x 5 y 4 + 1 ) } + x 2 y x 4 + ( y + 1 ) 2 opt { x 57 y 2 x 10 + y 4 + 1 , f ( ln ( 1 + x 15 y 23 ) ( x + 1 ) 14 ( y 23 + 1 ) ) } } , x R + .
(3.19)

Put X=Y=R, S=D= R + , λ[0,1] and α= 7 10 . Let p,q,r,u,v,t:S×DR and a,b,c:S×DS be defined by

p ( x , y ) = x 69 ( x 5 y 6 ) 4 + 1 , q ( x , y ) = x 120 y 7 x 9 y 7 + 1 , r ( x , y ) = x 57 y 2 x 10 + y 4 + 1 , u ( x , y ) = x 3 cos 75 ( x 6 y 98 ) 3 ( x + 1 ) 3 + y 5 , v ( x , y ) = sin 9 ( x 8 y 3 ) x y 2 + 5 , t ( x , y ) = x 2 y x 4 + ( y + 1 ) 2 , a ( x , y ) = x 7 sin 4 ( x y ) x 6 + x y 3 + 1 , b ( x , y ) = x 6 y 4 cos 8 ( x 3 y 9 ) x 5 y 4 + 1 , c ( x , y ) = ln ( 1 + x 15 y 23 ) ( x + 1 ) 14 ( y 23 + 1 ) , ( x , y ) S × D .

It is clear that the conditions of Theorem 3.5 are satisfied. It follows from Theorem 3.5 that functional equation (3.19) possesses a unique solution wBB(S), which satisfies (C8). But Theorem 3.4 in [5] and Corollaries 2.2 and 2.3 in [11] are unapplicable to functional equation (3.19).

Next we discuss the behaviors of solutions and iterative algorithms for functional equation (1.7) in the complete metric space BB(S).

Theorem 3.8 Let λ[0,1], (φ,ψ) Φ 2 , p,q,r,u,v,t:S×DR and a,b,c:S×DS satisfy that

(C9) sup ( x , y ) B ¯ ( 0 , k ) × D max{|p(x,y)|,|q(x,y)|,|r(x,y)|}ψ(x), kN;

(C10) sup ( x , y ) B ¯ ( 0 , k ) × D max{|u(x,y)|,|v(x,y)|+|t(x,y)|}1, kN;

(C11) sup ( x , y ) B ¯ ( 0 , k ) × D max{a(x,y),b(x,y),c(x,y)}φ(x), kN.

Then functional equation (1.7) possesses a solution wBB(S) such that

(C12) for each w 0 BB(S) with | w 0 (x)|ψ(x), (x,k) B ¯ (0,k)×N, the sequence { w n } n N 0 defined by (3.13) converges to w and d k ( w n ,w) j = n 1 ψ( φ j (k)), (k,n)N×N;

(C13) |w(x)| n = 0 ψ( φ n (x)), (x,k) B ¯ (0,k)×N;

(C14) lim n w( x n )=0 for any ( x 0 ,k) B ¯ (0,k)×N, { y n } n N D and x n {a( x n 1 , y n ),b( x n 1 , y n ),c( x n 1 , y n )}, nN;

(C15) w is a unique solution of functional equation (1.7) relative to (C14).

Proof Define a mapping H:BB(S)BB(S) by

Hh(x)=λAh(x)+(1λ)Bh(x),(x,k,h) B ¯ (0,k)×N×BB(S),
(3.20)

where

A h ( x ) = opt y D { u ( x , y ) opt { p ( x , y ) , h ( a ( x , y ) ) } } , B h ( x ) = opt y D { v ( x , y ) opt { q ( x , y ) , h ( b ( x , y ) ) } + t ( x , y ) opt { r ( x , y ) , h ( c ( x , y ) ) } } , ( x , k , h ) B ¯ ( 0 , k ) × N × B B ( S ) .
(3.21)

Note that (C9) and (C11) imply (C5) and (C7) by (φ,ψ) Φ 2 , respectively. Similar to the proof of Theorem 3.5, by (C10) we conclude that the mapping H maps BB(S) into BB(S) and satisfies that

d k (Hh,Hg) d k (h,g),(h,g,k)BB(S)×BB(S)×N,

which yields that

d ( H h , H g ) = k = 1 1 2 k d k ( H h , H g ) 1 + d k ( H h , H g ) k = 1 1 2 k d k ( h , g ) 1 + d k ( h , g ) = d ( h , g ) , ( h , g ) B B ( S ) × B B ( S ) ,
(3.22)

that is, the mapping H is nonexpansive in BB(S).

Now we show that for each n N 0 ,

| w n ( x ) | j = 0 n ψ ( φ j ( x ) ) ,(x,k) B ¯ (0,k)×N.
(3.23)

It is easy to see that (3.23) holds for n=0. Assume that (3.23) holds for some n N 0 . In terms of (C9), (C11), (C12), (3.13), (φ,ψ) Φ 2 and Lemma 2.1, we gain that

| w n + 1 ( x ) | = | λ opt y D { u ( x , y ) opt { p ( x , y ) , w n ( a ( x , y ) ) } } + ( 1 λ ) opt y D { v ( x , y ) opt { q ( x , y ) , w n ( b ( x , y ) ) } + t ( x , y ) opt { r ( x , y ) , w n ( c ( x , y ) ) } } | λ sup y D { | u ( x , y ) | max { | p ( x , y ) | , | w n ( a ( x , y ) ) | } } + ( 1 λ ) sup y D { | v ( x , y ) | max { | q ( x , y ) | , | w n ( b ( x , y ) ) | + | t ( x , y ) | max { | r ( x , y ) | , | w n ( c ( x , y ) ) | } } λ sup y D { max { ψ ( x ) , j = 0 n ψ ( φ j ( a ( x , y ) ) ) } } + ( 1 λ ) sup y D { | v ( x , y ) | max { ψ ( x ) , j = 0 n ψ ( φ j ( b ( x , y ) ) ) } + | t ( x , y ) | max { ψ ( x ) , j = 0 n ψ ( φ j ( c ( x , y ) ) ) } } λ max { ψ ( x ) , j = 0 n ψ ( φ j + 1 ( x ) ) } + ( 1 λ ) sup y D { | v ( x , y ) | + | t ( x , y ) | } max { ψ ( x ) , j = 0 n ψ ( φ j + 1 ( x ) ) } } λ ( ψ ( x ) + j = 0 n ψ ( φ j + 1 ( x ) ) ) + ( 1 λ ) ( ψ ( x ) + j = 0 n ψ ( φ j + 1 ( x ) ) ) = j = 0 n + 1 ψ ( φ j ( x ) ) , ( x , k ) B ¯ ( 0 , k ) × N ,

that is, (3.23) is true for n+1. Hence (3.23) holds for each n N 0 .

Let ε>0 and k,n,mN. Assume that opt y D = sup y D . It follows from (3.21) that for each x 0 B ¯ (0,k) there exist y, y 0 ,z, z 0 D satisfying

A w n + m 1 ( x 0 ) 2 1 ε < u ( x 0 , y ) opt { p ( x 0 , y ) , w n + m 1 ( a ( x 0 , y ) ) } , A w n 1 ( x 0 ) 2 1 ε < u ( x 0 , y 0 ) opt { p ( x 0 , y 0 ) , w n 1 ( a ( x 0 , y 0 ) ) } , A w n + m 1 ( x 0 ) u ( x 0 , y 0 ) opt { p ( x 0 , y 0 ) , w n + m 1 ( a ( x 0 , y 0 ) ) } , A w n 1 ( x 0 ) u ( x 0 , y ) opt { p ( x 0 , y ) , w n 1 ( a ( x 0 , y ) ) }
(3.24)

and

B w n + m 1 ( x 0 ) 2 1 ε < v ( x 0 , z ) opt { q ( x 0 , z ) , w n + m 1 ( b ( x 0 , z ) ) } B w n + m 1 ( x 0 ) 2 1 ε < + t ( x 0 , z ) opt { r ( x 0 , z ) , w n + m 1 ( c ( x 0 , z ) ) } , B w n 1 ( x 0 ) 2 1 ε < v ( x 0 , z 0 ) opt { q ( x 0 , z 0 ) , w n 1 ( b ( x 0 , z 0 ) ) } B w n 1 ( x 0 ) 2 1 ε < + t ( x 0 , z 0 ) opt { r ( x 0 , z 0 ) , w n 1 ( c ( x 0 , z 0 ) ) } , B w n + m 1 ( x 0 ) v ( x 0 , z 0 ) opt { q ( x 0 , z 0 ) , w n + m 1 ( b ( x 0 , z 0 ) ) } B w n + m 1 ( x 0 ) + t ( x 0 , z 0 ) opt { r ( x 0 , z 0 ) , w n + m 1 ( c ( x 0 , z 0 ) ) } , B w n 1 ( x 0 ) v ( x 0 , z ) opt { q ( x 0 , z ) , w n 1 ( b ( x 0 , z ) ) } B w n 1 ( x 0 ) + t ( x 0 , z ) opt { r ( x 0 , z ) , w n 1 ( c ( x 0 , z ) ) } .
(3.25)

On account of (3.24), (3.25) and Lemma 2.2, we obtain that

A w n + m 1 ( x 0 ) A w n 1 ( x 0 ) < u ( x 0 , y ) opt { p ( x 0 , y ) , w n + m 1 ( a ( x 0 , y ) ) } u ( x 0 , y ) opt { p ( x 0 , y ) , w n 1 ( a ( x 0 , y ) ) } + 2 1 ε | u ( x 0 , y ) | | w n + m 1 ( a ( x 0 , y ) ) w n 1 ( a ( x 0 , y ) ) | + 2 1 ε ; A w n + m 1 ( x 0 ) A w n 1 ( x 0 ) > u ( x 0 , y 0 ) opt { p ( x 0 , y 0 ) , w n + m 1 ( a ( x 0 , y 0 ) ) } u ( x 0 , y 0 ) opt { p ( x 0 , y 0 ) , w n 1 ( a ( x 0 , y 0 ) ) } 2 1 ε | u ( x 0 , y 0 ) | | w n + p 1 ( a ( x 0 , y 0 ) ) w n 1 ( a ( x 0 , y 0 ) ) | 2 1 ε ; B w n + m 1 ( x 0 ) B w n 1 ( x 0 ) < v ( x 0 , z ) opt { q ( x 0 , z ) , w n + m 1 ( b ( x 0 , z ) ) } + t ( x 0 , z ) opt { r ( x 0 , z ) , w n + m 1 ( c ( x 0 , z ) ) } v ( x 0 , z ) opt { q ( x 0 , z ) , w n 1 ( b ( x 0 , z ) ) } t ( x 0 , z ) opt { r ( x 0 , z ) , w n 1 ( c ( x 0 , z ) ) } + 2 1 ε | v ( x 0 , z ) | | w n + p 1 ( b ( x 0 , z ) ) w n 1 ( b ( x 0 , z ) ) | + | t ( x 0 , z ) | | w n + m 1 ( c ( x 0 , z ) ) w n 1 ( c ( x 0 , z ) ) | + 2 1 ε

and

B w n + m 1 ( x 0 ) B w n 1 ( x 0 ) > v ( x 0 , z 0 ) opt { q ( x 0 , z 0 ) , w n + m 1 ( b ( x 0 , z 0 ) ) } + t ( x 0 , z 0 ) opt { r ( x 0 , z 0 ) , w n + m 1 ( c ( x 0 , z 0 ) ) } v ( x 0 , z 0 ) opt { q ( x 0 , z 0 ) , w n 1 ( b ( x 0 , z 0 ) ) } t ( x 0 , z 0 ) opt { r ( x 0 , z 0 ) , w n 1 ( c ( x 0 , z 0 ) ) } 2 1 ε | v ( x 0 , z 0 ) | | w n + m 1 ( b ( x 0 , z 0 ) ) w n 1 ( b ( x 0 , z 0 ) ) | | t ( x 0 , z 0 ) | | w n + m 1 ( c ( x 0 , z 0 ) ) w n 1 ( c ( x 0 , z 0 ) ) | 2 1 ε ,

which together with (C10), (3.13), (3.20), (3.21) imply that

| w n + m ( x 0 ) w n ( x 0 ) | = | λ A w n + m 1 ( x 0 ) + ( 1 λ ) B w n + m 1 ( x 0 ) λ A w n 1 ( x 0 ) ( 1 λ ) B w n 1 ( x 0 ) | λ | A w n + m 1 ( x 0 ) A w n 1 ( x 0 ) | + ( 1 λ ) | B w n + m 1 ( x 0 ) B w n 1 ( x 0 ) | max { | A w n + m 1 ( x 0 ) A w n 1 ( x 0 ) | , | B w n + m 1 ( x 0 ) B w n 1 ( x 0 ) | } max { | u ( x 0 , y ) | | w n + m 1 ( a ( x 0 , y ) ) w n 1 ( a ( x 0 , y ) ) | , | u ( x 0 , y 0 ) | | w n + m 1 ( a ( x 0 , y 0 ) ) w n 1 ( a ( x 0 , y 0 ) ) | , | v ( x 0 , z ) | | w n + m 1 ( b ( x 0 , z ) ) w n 1 ( b ( x 0 , z ) ) | + | t ( x 0 , z ) | | w n + m 1 ( c ( x 0 , z ) ) w n 1 ( c ( x 0 , z ) ) | , | v ( x 0 , z 0 ) | | w n + m 1 ( b ( x 0 , z 0 ) ) w n 1 ( b ( x 0 , z 0 ) ) | + | t ( x 0 , z 0 ) | | w n + m 1 ( c ( x 0 , z 0 ) ) w n 1 ( c ( x 0 , z 0 ) ) | } + 2 1 ε max { | u ( x 0 , y ) | , | u ( x 0 , y 0 ) | , | v ( x 0 , z ) | + | t ( x 0 , z ) | , | v ( x 0 , z 0 ) | + | t ( x 0 , z 0 ) | } × max { | w n + m 1 ( a ( x 0 , y ) ) w n 1 ( a ( x 0 , y ) ) | , | w n + m 1 ( a ( x 0 , y 0 ) ) w n 1 ( a ( x 0 , y 0 ) ) | , | w n + m 1 ( b ( x 0 , z ) ) w n 1 ( b ( x 0 , z ) ) | , | w n + m 1 ( c ( x 0 , z ) ) w n 1 ( c ( x 0 , z ) ) | , | w n + m 1 ( b ( x 0 , z 0 ) ) w n 1 ( b ( x 0 , z 0 ) ) | , | w n + m 1 ( c ( x 0 , z 0 ) ) w n 1 ( c ( x 0 , z 0 ) ) | } + 2 1 ε max { | w n + m 1 ( a ( x 0 , y ) ) w n 1 ( a ( x 0 , y ) ) | , | w n + m 1 ( a ( x 0 , y 0 ) ) w n 1 ( a ( x 0 , y 0 ) ) | , | w n + m 1 ( b ( x 0 , z ) ) w n 1 ( b ( x 0 , z ) ) | , | w n + m 1 ( c ( x 0 , z ) ) w n 1 ( c ( x 0 , z ) ) | , | w n + m 1 ( b ( x 0 , z 0 ) ) w n 1 ( b ( x 0 , z 0 ) ) | , | w n + m 1 ( c ( x 0 , z 0 ) ) w n 1 ( c ( x 0 , z 0 ) ) | } + 2 1 ε = | w n + m 1 ( x 1 ) w n 1 ( x 1 ) | + 2 1 ε

for some x 1 {a( x 0 , y 1 ),b( x 0 , y 1 ),c( x 0 , y 1 )} and y 1 {y, y 0 ,z, z 0 }, that is,

| w n + m ( x 0 ) w n ( x 0 ) | <| w n + m 1 ( x 1 ) w n 1 ( x 1 )|+ 2 1 ε.
(3.26)

Similarly, we infer that (3.26) holds for opt y D = inf y D . Proceeding in this way, we conclude that for each nN, there exist y i D and x i {a( x i 1 , y i ),b( x i 1 , y i ),c( x i 1 , y i )} for i{1,2,,n} such that

| w n + m 1 ( x 1 ) w n 1 ( x 1 ) | | w n + m 2 ( x 2 ) w n 2 ( x 2 ) | + 2 2 ε , | w n + m 2 ( x 2 ) w n 2 ( x 2 ) | | w n + m 3 ( x 3 ) w n 3 ( x 3 ) | + 2 3 ε , | w m + 1 ( x n 1 ) w 1 ( x n 1 ) | | w m ( x n ) w 0 ( x n ) | + 2 n ε .
(3.27)

In terms of (φ,ψ) Φ 2 , (C11), (3.23) and (3.27), we deduce that

| w n + m ( x 0 ) w n ( x 0 ) | < | w m ( x n ) w 0 ( x n ) | + ε j = 0 m ψ ( φ j ( x n ) ) + ψ ( x n ) + ε j = 0 m ψ ( φ j + n ( k ) ) + ψ ( φ n ( k ) ) + ε j = n 1 m + n ψ ( φ j ( k ) ) + ε ,

which means that

d k ( w n + m , w n ) j = n 1 n + m ψ ( φ j ( k ) ) +ε.
(3.28)

Letting ε 0 + in the above inequality, we deduce that

d k ( w n + m , w n ) j = n 1 n + m ψ ( φ j ( k ) ) .
(3.29)

Notice that n = 0 ψ( φ n (t))<+ for each t>0. Thus (3.29) means that { w n } n N 0 is a Cauchy sequence in (BB(S),d) and it converges to some wBB(S). Letting m in (3.29), we conclude immediately that

d k ( w n ,w) j = n 1 ψ ( φ j ( k ) ) ,(k,n)N×N.

By virtue of (3.22), we infer that

d(Hw,w)d(Hw,H w n )+d( w n + 1 ,w)d(w, w n )+d( w n + 1 ,w)0as n,

which yields that Hw=w, that is, functional equation (1.7) possesses a solution wBB(S).

Next we show (C13). Let (x,k) B ¯ (0,k)×N. According to (C11), (3.23) and (φ,ψ) Φ 2 , we know that

| w ( x ) | | w ( x ) w n ( x ) | + | w n ( x ) | d k ( w , w n ) + j = 0 n ψ ( φ j ( x ) ) j = 0 ψ ( φ j ( x ) ) as  n ,

that is, (C13) holds.

Next we show (C14). Given ( x 0 ,k) B ¯ (0,k)×N, { y n } n N D and x n {a( x n 1 , y n ),b( x n 1 , y n ),c( x n 1 , y n )}, nN. It follows from (C11) and (φ,ψ) Φ 2 that

x n max { a ( x n 1 , y n ) , b ( x n 1 , y n ) , c ( x n 1 , y n ) } φ ( x n 1 ) φ n ( x 0 ) φ n ( k ) < k , n N ,

which together with (C11), (3.23) and (φ,ψ) Φ 2 implies that

| w ( x n ) | | w ( x n ) w n ( x n ) | + | w n ( x n ) | d k ( w , w n ) + j = 0 n ψ ( φ j ( x n ) ) d k ( w , w n ) + j = n 2 n ψ ( φ j ( k ) ) 0 as  n ,

which yields that lim n w( x n )=0.

Finally we show (C15). Suppose that functional equation (1.7) has another solution hBB(S) that satisfies (C14). Let ε>0 and x 0 S. It follows from (3.21) that there exist y, y 0 ,z, z 0 D with

A w ( x 0 ) 2 1 ε < u ( x 0 , y ) opt { p ( x 0 , y ) , w ( a ( x 0 , y ) ) } , A h ( x 0 ) 2 1 ε < u ( x 0 , y 0 ) opt { p ( x 0 , y 0 ) , h ( a ( x 0 , y 0 ) ) } , A w ( x 0 ) > u ( x 0 , y 0 ) opt { p ( x 0 , y 0 ) , w ( a ( x 0 , y 0 ) ) } 2 1 ε , A h ( x 0 ) > u ( x 0 , y ) opt { p ( x 0 , y ) , h ( a ( x 0 , y ) ) } 2 1 ε

and

B w ( x 0 ) 2 1 ε < v ( x 0 , z ) opt { q ( x 0 , z ) , w ( b ( x 0 , z ) ) } B w ( x 0 ) 2 1 ε < + t ( x 0 , z ) opt { r ( x 0 , z ) , w ( c ( x 0 , z ) ) } , B h ( x 0 ) 2 1 ε < v ( x 0 , z 0 ) opt { q ( x 0 , z 0 ) , h ( b ( x 0 , z 0 ) ) } B h ( x 0 ) 2 1 ε < + t ( x 0 , z 0 ) opt { r ( x 0 , z 0 ) , h ( c ( x 0 , z 0 ) ) } ,