A Generalized Fixed Point Theorem In An Extended Cone b-Metric Space Over Banach Algebra With Its Application To

A Generalized Fixed Point Theorem In An Extended Cone b-Metric Space Over Banach Algebra With Its Application To

Dynamical Programming

Kushal Roy

, Sayantan Panja

, Mantu Saha

Received 10 April 2020

Abstract

In this manuscript, we introduce the notion of an extended coneb-metric space over Banach algebra with underlying non-normal cone. A …xed point theorem has been proved for a generalized Lipschitzian mapping on such spaces. Several corollaries have been obtained from our given …xed point theorem.

Some supporting examples have been cited to examine the validity of our established result. Moreover, our …xed point result is applied to obtain solutions for some functional equations.

1 Introduction and Preliminaries

Nowadays …xed point theory is one of the prominent area of research. In metric …xed point theory our main objective is to know whether a self-mapping in a metric space or metric-type space, has …xed point or not.

The concept ofb-metric spaces had been initiated by Bakhtin [2] in 1989 which generalizes usual metric spaces and thereby he had been succeeded to generalize the Banach contraction principle over it. This concept was further improved by Czerwik [5] in his research article. Recently Kamran et al. [11] have generalized the concept of b-metric space in a totally new way. They …rst show that the coe¢ cient of a b-metric space can be a function instead of just a constant.

Huang and Zhang [9] in 2007 had been able to introduce cone metric spaces as a generalization of metric spaces in which the set of real numbers are replaced by the elements of an ordered Banach space. In view of this notion they had been succeeded to de…ne Cauchy sequence, convergence of a sequence in such spaces and obtained some …xed point theorems over it. After that several researchers have published many papers on such type of spaces (See [1], [6], [12], [15], [16]). Very recently, Liu and Xu [14] have generalized the concept of cone metric space by changing the underlying Banach space to a Banach algebra and introduced cone metric space over Banach algebra. They de…ned cone over a Banach algebra in a generalized way and investigated some partial orderings in terms of interior points of the underlying cone. They have also de…ned generalized contractive mapping in this setting where the Lipschitzian constant is a vector instead of real constant and proved some …xed point theorems. Moreover, they have given an example to show that the …xed point theorems in cone metric spaces over Banach algebras are usually not equivalent to those in case of a metric space. Though Liu and Xu have used normal cones in their article but Xu and Radenovic [19] avoided the normality of cones and de…ned solid cones to de…nec-sequences on cone metric spaces over Banach algebras.

In light of the same spirit Huang and Radenovic [8] have introduced the concept of coneb-metric spaces over Banach algebras by bundling together the concept of cone metric space over Banach algebra and b- metric space. Since b-metric spaces have various interesting properties than usual metric spaces it follows that cone b-metric spaces over Banach algebras have those interesting properties too. The authors proved several …xed point theorems on such spaces with the help of underlying ordered Banach algebra.

Mathematics Sub ject Classi…cations: 47H10, 54H25.

yDepartment of Mathematics, The University of Burdwan, Purba Bardhaman-713104, West Bengal, India

zDepartment of Mathematics, The University of Burdwan, Purba Bardhaman-713104, West Bengal, India

xDepartment of Mathematics, The University of Burdwan, Purba Bardhaman-713104, West Bengal, India

209

Before going to the next sections we need some preliminaries which help us to obtain our main results.

De…nition 1 ([18]) A vector spaceA over a …eld K(Ror C) is said to be an algebra if it is closed under multiplication (i.e., for all ; 2 A, 2 A) and

(i) ( ) = ( ),8 ; ; 2 A;

(ii) ( + ) = + and( + ) = + ,8 ; ; 2 A; (iii) r( ) = (r ) = (r ),8 ; 2 A;8r2K.

A Banach spaceA over the …eldK (RorC) is said to be a Banach algebra if (i) A is an algebra and (ii)8 ; 2 A,k k k k:k k.

Here we shall always assume that the Banach algebraAis unital, that is it has a unity elemente_Asuch thate_A = e_A= , for all 2 A:Note that the unity element of a Banach algebraA, if it exists, is unique.

A non-zero element 2 Ais said to be invertible if its inverse exists i.e. if there exists a non-zero element such that = =e_A, we write = ¹ and we call is the inverse of . One can show that in a Banach algebraA, with the unity elemente_A the inverse of an element is unique. Also for all ; 2 A, we have( ) ¹= ^{1 1}and ( ¹) ¹= .

Proposition 1 ([18]) Let A be a Banach algebra with unite_A; the spectral radius of an element 2 Ais denoted by ( )and de…ned by ( ) = sup_{r2 ( )}jrj= lim_n!1k ⁿkⁿ¹;where ( )is the spectrum of 2 A: If

( )<1then e_A is invertible and(e_A ) ¹=e_A+P₁

i=1 i:

Remark 1 The spectral radius ( )of 2 Asatis…es ( ) k k, where Ais a Banach algebra with unity e_A:

Remark 2 If ( )<1, then we getk kⁿ!0 asn! 1:

De…nition 2 ([7]) A subsetP of a unital Banach algebraAis called a cone if 1. P is non empty, closed and _A; e_A2P.

2. If ; 2P andr; s 0;then r +s 2P. 3. ; 2P implies 2P.

4. If ; 2P for some 2 Athen = _A, where _A is the zero element ofA:

A cone Pis called a solid cone ifint(P)6=;:Each conePinduces a partial ordering-onAby - i¤

2P. We write if - and 6= :When the cone is solid will stand for 2int(P):

The cone Pis said to be normal if there exists a numberL >0such that _A- - ) k k Lk k:The least positive numberL, which satis…es the normality condition is called the normal constant ofP.

Lemma 1 ([17]) If B be a real Banach space with a solid cone P and if _B - c for all c _B, then

= _B:

De…nition 3 ([19]) Let Pbe a solid cone in a Banach spaceB: A sequencef ng P is called ac-sequence if for each _B cthere exists a natural number N such that _n c whenevern N:

Lemma 2 ([19]) IfBis a real Banach space with a solid conePandf ng Pis a sequence withk nk !0 asn! 1, thenf ng is ac-sequence.

Lemma 3 ([19]) Let Abe a Banach algebra with unitye_A: Let ; 2 A such that and commute, then (i) ( ) ( ) ( ):

(ii) ( + ) ( ) + ( ):

(iii) j ( ) ( )j ( ).

Lemma 4 ([8]) If B be a real Banach space andP be a solid cone of Bthen for ; ; c2 B with - c implies c:

Lemma 5 ([19]) Let P be a solid cone of a Banach algebra A: Suppose that k 2P is an arbitrary vector andf ng P is ac-sequence, then fk _ng is also a c-sequence.

Lemma 6 ([8]) Let Abe a Banach algebra with unitye_A and 2 A:Let zbe a complex constant such that ( )<jzj. Then, (ze_A ) ¹ _jzj¹_{( )}:

Lemma 7 ([8]) Let Bbe a Banach space andPbe a solid cone ofB. Let ; ; 2P, - and - with ( )<1. Then = _B:

Lemma 8 ([19]) Let A be a Banach algebra with unity e_A and f ng A: Suppose that f ng converges 2 A and that _n and commute for alln: Then we have ( _n)! ( )asn! 1:

Lemma 9 ([8]) Let A be a Banach algebra with unity e_A and P be a solid cone of A. Let 2 A and

n = ⁿ for alln2N. If ( )<1; thenf ⁿg is ac-sequence.

Lemma 10 ([10]) For a cone Pin the Banach space(B;k:k), the followings are equivalent:

1. P is normal.

2. for arbitrary sequencef ng,f ng,f ng inB, with _n- n- n 8n2N, if limn!1 n= limn!1 n=

;thenlimn!1 n = .

3. there exists a normk:k1 onB equivalent tok:ksuch that the cone is monotone with respect tok:k1: De…nition 4 ([2]) Let X be a nonempty set and b be a real number satisfying b 1: A function Db : X X ![0;1)is said to be a b-metric onX if for all ; ; 2X, the following conditions hold:

1. D_b( ; ) = 0 i¤ = . 2. Db( ; ) =Db( ; ):

3. D_b( ; ) b[D_b( ; ) +D_b( ; )]:

The space(X; Db)is called ab-metric space.

De…nition 5 ([11]) Let X be a nonempty set and :X X ![1;1). A function D_eb :X² ![0;1)is called an extendedb-metric if for all ; ; 2X it satis…es:

1. Deb( ; ) = 0 if and only if = . 2. D_eb( ; ) =D_eb( ; ).

3. Deb( ; ) ( ; )[Deb( ; ) +Deb( ; )]:

The pair(X; D_eb)is called an extendedb-metric space.

De…nition 6 ([14]) Let X be a nonempty set and A be a real unital Banach algebra with a solid coneP. The mapping Dc : X X ! A is said to be a cone metric over Banach algebra on X if it satis…es the following conditions:

1. Dc( ; )% Afor all ; 2X andDc( ; ) = _A i¤ = . 2. D_c( ; ) =D_c( ; )for all ; 2X.

3. D_c( ; )-D_c( ; ) +D_c( ; )for all ; ; 2X:

The space(X; Dc)is called a cone metric space over the Banach algebraA.

De…nition 7 ([8]) Let X be a nonempty set, A be a real unital Banach algebra with a solid cone P and s 1 be a constant. The mappingD_cb:X X ! A is called a cone b-metric over Banach algebra on X if the following conditions hold:

1. Dcb( ; )% A for all ; 2X andDcb( ; ) = _A i¤ = . 2. Dcb( ; ) =Dcb( ; )for all ; 2X:

3. D_cb( ; )-s[D_cb( ; ) +D_cb( ; )] for all ; ; 2X:

The space(X; Dcb) is called a coneb-metric space over the Banach algebraA.

2 Main Results

In this section we introduce generalized coneb-metric space over Banach algebra without the assumption of normality of cones.

In the following de…nitions and theorems we always suppose thatAis a real Banach algebra with a unity, Pis a solid cone inA, not necessarily normal and-, and are partial orderings with respect toP: De…nition 8 Let X be a nonempty set, :X²![1;1) be a mapping. A mappingD^ecb :X X ! Ais said to be an extended coneb-metric over Banach algebra if it satis…es the following conditions:

(D^ecb1) D^ecb( ; )% A andD^ecb( ; ) = _Ai¤ = ; (Decb2) Decb( ; ) =Decb( ; );

(Decb3) Decb( ; )- ( ; )[Decb( ; ) +Decb( ; )].

The triplet(X;D^ecb;A)is called an extended cone b-metric space over Banach algebra.

Remark 3 An extended cone b-metric space over Banach algebra generalizes several known metric struc- tures, such as:

(i) If ( ; ) = 1 for all ; 2X then an extended cone b-metric space over Banach algebra reduces to a cone metric space over Banach algebra.

(ii) A coneb-metric space over Banach algebra is an extended coneb-metric space over Banach algebra for ( ; ) =s >1 for all ; 2X.

(iii) If ( ; ) = 1 for all ; 2 X andA =R with the cone P = [0;1) then an extended cone b-metric space over Banach algebra reduces to an usual metric space.

(iv) A b-metric space is an extended cone b-metric space over Banach algebra for ( ; ) = s > 1 for all

; 2X andA=Rwith the cone P= [0;1).

(v) IfA=Rwith the coneP= [0;1) then an extended coneb-metric space over Banach algebra reduces to an extendedb-metric space.

Example 1 Let A=C[0;1]be the usual unital Banach algebra with the sup norm. Let P=ff 2C[0;1] : f(t) 0for allt2[0;1]g andX=R. De…ne a mappingDecb:X²! AbyDecb( ; )(t) = (1 +j j+j j)j

jexp(t)for any ; 2X and for all t2[0;1]: Conditions(Decb1) and(Decb2) are clearly satis…ed byDecb. Now we check the condition(Decb3). For this we takeu; v; w2X as arbitrary. Then we see that

Decb(u; w)(t) = (1 +juj+jwj)ju wjexp(t)

(1 +juj+jwj)(ju vj+jv wj) exp(t) (1 +juj+jwj)(Decb(u; v)(t) +Decb(v; w)(t))

(1 +juj+jwj)(D^ecb(u; v) +D^ecb(v; w))(t)for allt2[0;1]:

Therefore D^ecb(u; w)-(1 +juj+jwj)(D^ecb(u; v) +D^ecb(v; w)). Sinceu; v; w are chosen arbitrarily, we have D^ecb is an extended coneb-metric space over Banach algebra with ( ; ) = (1 +j j+j j)for all ; 2X. Example 2 Let A=C[0;1]be the usual unital Banach algebra with the sup norm. Let P=ff 2C[0;1] : f(t) 0 for allt2[0;1]g andX=R. De…ne a mappingD^ecb:X²! AbyD^ecb( ; )(t) = (1 +j j)j

jexp(t)for any ; 2X and for all t 2[0;1]: Then clearly D^ecb satis…es conditions (D^ecb1) and (D^ecb2).

To check the condition(D^ecb3)let us choose u; v; w2X as arbitrary. Then we have Decb(u; w)(t) = (1 +ju wj)ju wjexp(t)

(1 +ju wj)(ju vj+jv wj) exp(t) (1 +ju wj)(D^ecb(u; v)(t) +D^ecb(v; w)(t))

(1 +ju wj)(Decb(u; v) +Decb(v; w))(t) for allt2[0;1]:

Therefore D^ecb(u; w)-(1 +ju wj)(D^ecb(u; v) +D^ecb(v; w)). Sinceu; v; w are chosen arbitrarily it is seen thatD^ecb satis…es condition(D^ecb3)also. HenceD^ecbis an extended coneb-metric space over Banach algebra with ( ; ) = (1 +j j)for all ; 2X but it is not a cone metric space over Banach algebra since

D^ecb(1;1:1)(t) +D^ecb(1:1;2)(t) = 1:82 exp(t)<2 exp(t) =D^ecb(1;2)(t)for all t2[0;1]:

Lemma 11 Let A be a real Banach algebra and l(6= _A) 2 A. If limn!1klⁿ⁺¹k

klⁿk exists then this limit is equal to (l):

Proof. The proof can be done easily. So we omit the proof.

Now we prove some …xed point theorems in the setting of an extended cone b-metric space over Banach algebra. Let us de…ne a subset ofAas follows:

P := l(6= _A)2P: lim

n!1

klⁿ⁺¹k

klⁿk exists :

De…nition 9 Let (X;D^ecb;A) be an extended cone b-metric space over Banach algebra andf ng be a se- quence in X:Then

(i) f ng is called D^ecb-convergent sequence if there exists some 2 X such that for every l 2 A with l _Athere exists some N 2Nsuch thatD^ecb( _n; ) l for alln N:

(ii) f ng is said to be D^ecb-Cauchy sequence if for everyl 2 Awith l _A there exists N 2Nsuch that D^ecb( _n; _n+p) l for alln N and for everyp= 1;2; .

(iii) (X;Decb;A)is calledDecb-complete if everyDecb-Cauchy sequence in X isDecb-convergent.

De…nition 10 Let (X;Decb;A) be an extended cone b-metric space over Banach algebra. A mapping G : (X;D^ecb) ! (X;D^ecb) is said to be D^ecb-orbitally continuous at a point u 2 X if for some 2 X, limi!1Gⁿⁱ =uimplies limi!1Gⁿⁱ⁺¹ =Gu. IfG isD^ecb-orbitally continuous at each point of X;then we say thatGisD^ecb-orbitally continuous inX.

Theorem 1 Let (X;D^ecb;A) be a D^ecb-complete extended cone b-metric space over Banach algebra. Let P be a solid cone in A not necessarily normal inA: Suppose that T :X !X be a mapping such that for all

2X

Decb(T ; T² )-lDecb( ; T ); (1)

where l2P , (l)<1 withlimn;m!1 ( _n; _m)< ¹_(l) and f ng =fTⁿ ₀g is the Picard iterating sequence generated by ₀2X:ThenT has a …xed point inX provided that T isD^ecb-orbitally continuous inX:

Proof. From the contractive condition (1) we have

D^ecb( _n; _n+1) =D^ecb(T _{n 1}; T² _{n 1}) -lDecb( _{n 1}; T _{n 1})

=lD^ecb( _{n 1}; _n) ...

-lⁿD^ecb( ₀; ₁): (2) Now, for alln2Nand for anyp= 1;2; ,

D^ecb( _n; _n+p)- ( _n; _n+p)[D^ecb( _n; _n+1) +D^ecb( _n+1; _n+p)]

- ( _n; _n+p)D^ecb( _n; _n+1)

+ ( _n; _n+p) ( _n+1; _n+p)[Decb( _n+1; _n+2) +Decb( _n+2; _n+p)]

- ( _n; _n+p)Decb( _n; _n+1) + ( _n; _n+p) ( _n+1; _n+p)Decb( _n+1; _n+2) + + ( _n; _n+p) ( _n+1; _n+p)::: ( _{n+p 2}; _n+p)[D^ecb( _{n+p 2}; _{n+p 1})

+D^ecb( _{n+p 1}; _n+p)]

- ( _n; _n+p)Decb( _n; _n+1) + ( _n; _n+p) ( _n+1; _n+p)Decb( _n+1; _n+2) + + ( _n; _n+p) ( _n+1; _n+p) ( _{n+p 2}; _n+p)Decb( _{n+p 2}; _{n+p 1})

+ ( _n; _n+p) ( _n+1; _n+p) ( _{n+p 1}; _n+p)D^ecb( _{n+p 1}; _n+p)]

-[ ( _n; _n+p)lⁿ+ ( _n; _n+p) ( _n+1; _n+p)lⁿ⁺¹+

+ ( _n; _n+p) ( _n+1; _n+p) ( _{n+p 2}; _n+p)l^{n+p 2} + ( _n; _n+p) ( _n+1; _n+p) ( _{n+p 1}; _n+p)l^{n+p 1}]Decb( ₀; ₁)

-

"_{n+p 1} X

r=n

l^r Yr s=1

( _s; _n+p)

#

D^ecb( ₀; ₁): (3)

Again we have,

n+pX1 r=n

l^r Yr s=1

( _s; _n+p)

n+pX1 r=n

Yr s=1

( _s; _n+p)kl^rk:

Forr2N, let us de…neu^(n+p)r =Qr

s=1 ( _s; _n+p)kl^rk. Then for anyp= 1;2; ,

nlim!1

u^(n+p)_n+1 u^(n+p)n

= lim

n!1 ( _n+1; _n+p)klⁿ⁺¹k klⁿk = lim

n!1 ( _n+1; _n+p) lim

n!1

klⁿ⁺¹k klⁿk :

Sincel2P we havelimn!1klⁿ⁺¹k

klⁿk = (l)and therefore for anyp= 1;2; ,

nlim!1

u^(n+p)_n+1 u^(n+p)n

= lim

n!1 ( _n+1; _n+p) (l)<1:

So by Ratio test we have for anyp= 1;2; ,

nlim!1 n+pX1

r=n

l^r Yr s=1

( _s; _n+p) lim

n!1 n+pX1

r=n

Yr s=1

( _s; _n+p)kl^rk= 0:

Therefore from (3) we conclude that f ng is D^ecb-Cauchy sequence in X. Since X is D^ecb-complete we have f ng is D^ecb-convergent sequence inX and let it be convergent to 2 X:Now if T is D^ecb-orbitally continuous inX then ⁿ₀ = _n ! impliesT ⁿ₀ =T _n = _n+1 !T . Hence T = and is a …xed point ofT inX:

Now here we give some supporting examples in respect of Theorem1.

Example 3 (Banach algebra with non-normal cone)

Let us consider the Banach algebra A=C_R¹[0;1]with the norm kxkA=kxk1+kx⁰k1 and usual pointwise multiplication. ObviouslyAis a Banach algebra with unitye_A= 1:Let us takeP=fx2 A:x(t) 0for all t2[0;1]g: Then it can be veri…ed thatPis a non-normal cone.

Now letX =f1;2;3g. De…ne :X²![1;1)andD^ecb :X X! A as:

( ; ) = 1 + + for all ; 2 X and D^ecb(1;1)(t) = D^ecb(2;2)(t) = D^ecb(3;3)(t) = 0, D^ecb(1;2)(t) = D^ecb(2;1)(t) = 10e^t,D^ecb(2;3)(t) =D^ecb(3;2)(t) = 40e^t,D^ecb(1;3)(t) =D^ecb(3;1)(t) = 80e^t for allt2[0;1]:

Then(X;D^ecb;A)is aD^ecb-complete extended coneb-metric space over Banach algebra but not a usual cone metric space over Banach algebra.

Let T :X !X be de…ned byT1 =T2 = 1 andT3 = 2: Then

Decb(T ; T² )-lDecb( ; T )for all 2X, with l(t) =¹₄+₄₈¹tfor all t2[0;1]: We show thatl2P . Here it is seen thatlⁿ(t) = ¹₄+₄₈¹t ⁿ for allt2[0;1]and for all n2N:Therefore(lⁿ)⁰(t) =₄₈^{n 1}₄+₄₈¹t ^{n 1}for allt2[0;1]and for all n2N:Thus klⁿkA= ¹₄+₄₈^{1 n 1 1}₄+₄₈¹ +₄₈ⁿ for all n 1: Hence we get

klⁿ⁺¹kA

klⁿkA

=

1

4+₄₈^{1 n 1}₄+₄₈¹ +ⁿ⁺¹₄₈

1

4+₄₈^{1 n 1 1}₄+₄₈¹ +₄₈ⁿ

= 1

4 + 1 48

1 + ¹⁴_n

1 + ¹³_n for alln 1: (4)

So from equation(4)it follows thatl2P . Also one can check thatT satis…es all the conditions of Theorem 1 and12X is the unique …xed point of T:

Example 4 (Banach algebra with normal cone)

Let us consider the Banach algebra A = l¹ = fu= (un)n2N:P₁

n=1junj<1g with the norm kukA = P₁

n=1junj and the multiplication de…ned by

uv= (un)n2N(vn)n2N= 0

@ X

i+j=n+1

uivj

1 A

n2N

:

ThenAis a Banach algebra with unitye_A= (1;0;0; ):Also putP=fu= (un)n2N:un 0for alln 1g. Now let us take X =l^p =f = ( _n)n2N:P₁

n=1j nj^p<1g, 0:4 < p < 1. Let : X² !R⁺ be de…ned by ( ; ) = (P₁

n=1j n nj^p)^p1 for all = ( _n)n2N, = ( _n)n2N in X: Let D^ecb : X² ! A be given by Decb( ; ) = ⁽₂n^{; )}

n2N for all ; 2X:Then(X;Decb;A)is a Decb-complete extended coneb-metric space

over Banach algebra with ( ; ) = 2^{1p 1} for all ; 2X but not a usual cone metric space over Banach algebra (See [8]).

De…neT :X !X byT( _n)_{n 1}= ₄ⁿ +₃¹n

n 1for all( _n)_{n 1}2X:ThenDecb(T ; T² )-lDecb( ; T ) for all 2 X, with l = (¹₃;0;0; ). Then one can verify that T satis…es all the conditions of Theorem 1 and ₃n+1⁴ n 1 is the unique …xed point of T inX:

Our contractive condition generalizes several contractive operators with di¤erent Lipschitz constants on an extended coneb-metric space over Banach algebra. Moreover it can be easily veri…ed that each of these contractive mappings are Decb-orbitally continuous and thus Theorem 1 holds good for such contractive mappings also.

Corollary 1 Consider the following contractive mappings over (X;Decb;A).

(i) If a mapping T : X ! X satis…es D^ecb(T ; T ) - kD^ecb( ; ) for all ; 2 X, where k 2 P with (k)<1, i.e., T is a Banach contraction then it satis…es the contractive condition(1)withl=k:

(ii) A mapping T : X !X, which satis…es D^ecb(T ; T )-k[D^ecb( ; T ) +D^ecb( ; T )] for all ; 2X and for k 2P with (k)< ¹₂, i.e., T is a Kannan type contractive map, also satis…es the contractive condition (1)with l= (e_A k) ¹k:

(iii) IfT :X !X satis…esDecb(T ; T )-k[Decb( ; T ) +Decb( ; T )] for all ; 2X, where k2P with (k) < ¹₂, i.e., T is a Chatterjea type contractive map then it satis…es the contractive condition (1) with l= (e_A k) ¹k:

(iv) A mappingT :X !X satisfying

D^ecb(T ; T )-k1D^ecb( ; ) +k2D^ecb( ; T ) +k3D^ecb( ; T ), for all ; 2X

and for k₁; k₂; k₃ 2 P, which commute with each other, with (k₁) + (k₂) + (k₃) < 1, i.e., T is a Reich type contractive map, also satis…es the contractive condition (1)with l= (e_A k₃) ¹(k₁+k₂).

(v) A mappingT :X !X satisfying

Decb(T ; T )-k₁Decb( ; ) +k₂Decb( ; T ) +Decb( ; T )

2 +k₃Decb( ; T ) +Decb( ; T ) 2

for all ; 2X and for k₁; k₂; k₃2P, which commute with each other, with (k₁) + (k₂) + (k₃)<1, i.e., T is a Hardy-Rogers type contractive map, also satis…es the contractive condition (1) with l = (e_A ^k₂^{2 k}₂³) ¹(k1+^k₂² +^k₂³):

(vi) If a mapping T :X !X satis…es D^ecb(T ; T )-kD^ecb( ; ) +LD^ecb( ; T )for all ; 2X, where k2Pwith (k)<1andL% A, i.e.,T is a weak contraction then it satis…es the contractive condition (1)withl=k.

3 An Application of Fixed Point Theorem to Dynamical Program- ming

Bellman [3] was the …rst who introduced the existence and successive approximations of solutions for several functional equations arising in dynamical programming, one of them is as follows:

f(t) = sup

t⁰2 fp(t; t⁰) +G(t; t⁰; f(q(t; t⁰)))g (5)

where X andY are Banach spaces over the …eldR. X denotes the state space (i.e., the set of initial state, action and transition of the process) and Y denotes the decision space (i.e., the set of all possible actions of the process) andt; t⁰ denotes the state and decision vectors respectively.

Here f(t) denotes the optimal return function with initial statet and p: !R, q : ! , G : R!R. There after several authors have studied the properties of solutions for various functional equations arising in dynamical programming and have tried to solve them by using various …xed point theorems, for these we may refer to [4], [13].

Let Bd( ) be the set of all real-valued bounded functions on . We de…ne a norm on Bd( ) by k k= sup_t2 j (t)jfor all 2Bd( ). Then(Bd( );k:k)forms a Banach space.

Let A = R² with k(a₁; a₂)kA = ja₁j+ja₂j for all (a₁; a₂) 2 R² and the multiplication is de…ned by ab = (a₁; a₂)(b₁; b₂) = (a₁b₁; a₁b₂+b₁a₂) for all a = (a₁; a₂); b = (b₁; b₂)2 R². Then A will be a unital Banach algebra with unity (1;0):Let us takeP=f(a₁; a₂)2R²:a₁; a₂ 0g. ThenPis a normal cone on A:

Let us de…ne a metricDecb:B_d( ) B_d( )! AbyDecb( ; ) = (k k²;k k²) = (fsup_t2 j (t) (t)jg²;fsup_t2 j (t) (t)jg²)for all ; 2B_d( ). Then (B_d( );Decb) is aDecb-complete extended cone b-metric space over Banach algebra with ( ; ) = 2for all ; 2B_d( ).

Now we are going to discuss the existence of the solution of the functional equation(5)by applying our established …xed point theorem. For this …rst we de…ne a mapping :Bd( )!Bd( )by

( f)(t) = sup

t⁰2 fp(t; t⁰) +G(t; t⁰; f(q(t; t⁰))g: (6) for allf 2Bd( ). Clearly if the functionspandGare bounded then becomes well-de…ned.

Theorem 2 Let :Bd( )!Bd( )be a mapping de…ned by(6)and suppose the following conditions hold.

(D1) p: !RandG: R!Rare continuous and bounded.

(D2) For all 2B_d( ),Gsatis…es

jG(t; t⁰; (t⁰⁰)) G(t; t⁰; ( )(t⁰⁰))j p

kk ( )k (7)

for all (t; t⁰; (t⁰⁰))2 R, where k 2(0;¹₂). Then the functional equation (7) has a bounded solution.

Proof. Let >0 be chosen arbitrarily. Then there existst⁰₁; t⁰₂2 such that for allt2 and 2Bd( ), p(t; t⁰₁) +G(t; t⁰₁; (q(t; t⁰₁)))> ( )(t) ; (8)

p(t; t⁰₂) +G(t; t⁰₂; ( )(q(t; t⁰₂²( )(t) ; (9) ( )(t) p(t; t⁰₂) +G(t; t⁰₂; (q(t; t⁰₂))) (10) and

2( )(t) p(t; t⁰₁) +G(t; t⁰₁; ( )(q(t; t⁰₁))): (11) From inequality(8)and(11)we have for allt2 ;

( )(t) ²( )(t)< G(t; t⁰₁; (q(t; t⁰₁))) G(t; t⁰₁; ( )(q(t; t⁰₁))) + jG(t; t⁰₁; (q(t; t⁰₁))) G(t; t⁰₁; ( )(q(t; t⁰₁)))j+

pkk ( )k+ : (12)

In a similar way, from equation(9)and(10)we can show that

2( )(t) ( )(t)<p

kk ( )k+ for allt2 : (13)

Since >0 is arbitrary it follows from equations (12) and (13) thatj ( )(t) ²( )(t)j p

kk ( )k for allt2 and for all 2Bd( ):Therefore for all 2Bd( ), we havek ( ) ²( )k p

kk ( )k. Hence

D^ecb( ( ); ²( )) = (k ( ) ²( )k²;k ( ) ²( )k²) - (k;0)(k ( )k²;k ( )k²)

= (k;0)D^ecb( ; ( ))

i.e. D^ecb( ( ); ²( ))-(k;0)D^ecb( ; ( ))for all 2Bd( ):Thus by Theorem1, has a …xed point in Bd( )and consequently the functional equation (5)has a bounded solution.

Acknowledgment. The authors are grateful to the honorable reviewer for his valuable suggestions and comments for the improvement of this manuscript. Also …rst and second authors acknowledge …nancial support awarded by the Council of Scienti…c and Industrial Research, New Delhi, India, through research fellowship for carrying out research work leading to the preparation of this manuscript.

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