An integral Suzuki-type fixed point theorem with application
aa r X i v : . [ m a t h . F A ] S e p AN INTEGRAL SUZUKI-TYPE FIXED POINT THEOREMWITH APPLICATION
SOKOL BUSH KALIAJ
Abstract.
In this paper, we present an integral Suzuki-type fixed point theorem for multivaluedmappings defined on a complete metric space in terms of the ´Ciri´c integral contractions. As anapplication, we will prove an existence and uniqueness theorem for a functional equation arising indynamic programming of continuous multistage decision processes. Introduction and Preliminaries
The Banach contraction principle [1] is a very famous theorem in nonlinear analysis and has manyuseful applications and generalizations. Over the years, it has been generalized in different directionsand spaces by several mathematicians. In 2008 Suzuki [18] introduced a new type of mappings whichgeneralize the well-known Banach contraction principle. The same has been extended in various ways,see e.g. papers [10]- [13], [17], [8], [9] and [19]. Using the idea of the Kikkawa-Suzuki fixed pointtheorem [10], Dori´c and Lazovi´c ( [5], Theorem 2.1) have proved the following theorem in terms ofgeneralized multivalued contractions considered in [4], Definition 3.
Theorem 1.1.
Let ( X, d ) be a complete metric space and let T : X → CB ( X ) be a multivaluedmapping. Assume that T is ( r, φ ) -Suzuki contraction with (1.1) φ ( r ) = (cid:26) if ≤ r < − r if ≤ r < i.e., there exists r ∈ [0 , such that the implication (1.2) φ ( r ) d ( x, T x ) ≤ d ( x, y ) ⇒ H ( T x, T y ) ≤ rT M ( x, y ) , holds whenever x, y ∈ X , where T M ( x, y ) = max (cid:26) d ( x, y ) , d ( x, T x ) , d ( y, T y ) , d ( x, T y ) + d ( y, T x )2 (cid:27) . Then, T has a fixed point. In this paper, we first define new Suzuki-type contractions in terms of the ´Ciri´c integral contrac-tions, ( r, φ, ψ )-Suzuki integral contractions. Then, a fixed point theorem for multivalued mappings T : X → CB ( X ) of ( r, φ, ψ )-Suzuki integral type contractions is proved. The main result heregeneralizes Theorem 1.1. In addition, using our result, we proved the existence and uniqueness ofsolutions for a functional equation arising in dynamic programming of continuous multistage decisionprocesses, see Theorem 3.3. Similar theorems are presented in the papers [3], [7], [14], [15] and [6].Throughout this paper, ( X, d ) is a metric space and R + is the set of all non-negative real numbers,i.e., R + = [0 , + ∞ ). A mapping ψ : R + → R + is said to be subadditive , if the inequality ψ ( t ′ + t ′′ ) ≤ Mathematics Subject Classification.
Primary 47H10, 54E50; Secondary 28A10.
Key words and phrases.
Complete metric space, integral Suzuki-type fixed point theorem, multivalued mappings,functional equation, dynamic programming. ψ ( t ′ ) + ψ ( t ′′ ) holds whenever t ′ , t ′′ ∈ R + . We now setΨ = { ψ : R + → R + : ψ is subadditive, non-decreasing and continuous on R + ,ψ ( t ) ≥ t for all t > ,ψ − (0) = { }} . The family Ψ is not empty, see [8]. For any ψ ∈ Ψ and [ a, b ] ⊂ R + , by Theorem VIII.2.4, p.211, [16], ψ ′ ( t ) exists at almost t ∈ [ a, b ]. Further, by Theorem VIII.2.5, p.212, [16], ψ ′ is summable on [ a, b ]and Z ba ψ ′ ( t ) dt ≤ ψ ( b ) − ψ ( a ) . By CB ( X ) the family of all nonempty closed bounded subsets of X is denoted. Let H ( · , · ) be theHausdorff metric, i.e., H ( A, B ) = max (cid:26) sup x ∈ A d ( x, B ) , sup y ∈ B d ( y, A ) (cid:27) , for all A, B ∈ CB ( X ) , where d ( x, B ) = inf { d ( x, y ) : y ∈ B } . It is well-known that if (
X, d ) is a complete metric space, then ( CB ( X ) , H ) is also a complete metricspace. We say that a multivalued mapping T : X → CB ( X ) is an ( r, φ, ψ ) -Suzuki integral contraction ,if φ defined by (1.1) and there exist r ∈ [0 ,
1) and ψ ∈ Ψ such that the implication(1.3) φ ( r ) Z d ( x,T x )0 ψ ′ ( t ) dt ≤ ψ ( d ( x, y )) ⇒ ψ ( H ( T x, T y )) ≤ rT R ( x, y )holds whenever x, y ∈ X , where T R ( x, y ) = max (Z d ( x,y )0 ψ ′ ( t ) dt, Z d ( x,T x )0 ψ ′ ( t ) dt, Z d ( y,T y )0 ψ ′ ( t ) dt, Z d ( x,Ty )+ d ( y,T x )2 ψ ′ ( t ) dt ) ; T : X → CB ( X ) is said to be ( r, φ, ψ ) -Suzuki contraction , if φ defined by (1.1) and there exist r ∈ [0 ,
1) and ψ ∈ Ψ such that the implication(1.4) φ ( r ) ψ ( d ( x, T x )) ≤ ψ ( d ( x, y )) ⇒ ψ ( H ( T x, T y )) ≤ rT ψ ( x, y )holds whenever x, y ∈ X , where T ψ ( x, y ) = max (cid:26) ψ ( d ( x, y )) , ψ ( d ( x, T x )) , ψ ( d ( y, T y )) , ψ (cid:18) d ( x, T y ) + d ( y, T x )2 (cid:19)(cid:27) . In the special case when ψ ∈ Ψ is absolutely continuous on R + (i.e. ψ is absolutely continuous overevery interval [0 , r ] , ( r > T is ( r, φ, ψ )-Suzuki integral contraction type if and only is T is ( r, φ, ψ )-Suzuki contraction type. Note that if ψ ( t ) = t , for all t >
0, then ( r, φ, ψ )-Suzuki contractions coincides with ( r, φ )-Suzuki contractions.We say that T is ( r, ψ ) - ´Ciri´c integral contraction , if there exist r ∈ [0 ,
1) and ψ ∈ Ψ such that theinequality ψ ( H ( T x, T y )) ≤ rT R ( x, y )holds whenever x, y ∈ X . Clearly, if T is ( r, ψ )- ´Ciri´c integral contraction, then T is ( r, φ, ψ )-Suzukiintegral contraction. The multivalued mapping T is said to have a fixed point, if there exists z ∈ X such that z ∈ T z .We say that a single valued mapping S : X → X is an ( r, φ, ψ ) -Suzuki integral contraction , if themultivalued mapping T : X → CB ( X ) defined as follows T x = { Sx } , for all x ∈ X, N INTEGRAL SUZUKI-TYPE FIXED POINT THEOREM 3 is an ( r, φ, ψ )-Suzuki integral contraction. We say that S is ( r, ψ ) - ´Ciri´c integral contraction , if T is( r, ψ )- ´Ciri´c integral contraction. The single valued mapping S is said to have a fixed point, if thereexists z ∈ X such that z = Sz . 2. The main result
The main result is Theorem 2.3. Let us start with the following auxiliary lemma.
Lemma 2.1.
Let ( X, d ) be a complete metric space and let T : X → CB ( X ) be a multivaluedmapping. Assume that T is an ( r, φ, ψ ) -Suzuki integral contraction. Then, for any x ∈ X , we have (2.1) ψ ( H ( T x, T y )) ≤ r Z d ( x,y )0 ψ ′ ( t ) dt, for all y ∈ T x.
Proof.
Fix x ∈ X and choose an arbitrary y ∈ T x . Then, d ( x, T x ) ≤ d ( x, y ) , and since 0 < φ ( r ) ≤
1, we obtain φ ( r ) Z d ( x,T x )0 ψ ′ ( t ) dt ≤ Z d ( x,T x )0 ψ ′ ( t ) dt ≤ Z d ( x,y )0 ψ ′ ( t ) dt. (2.2)By Theorem VIII.2.5 in [16], we have also Z d ( x,y )0 ψ ′ ( t ) dt ≤ ψ ( d ( x, y )) . The last inequality together with (2.2) yields φ ( r ) Z d ( x,T x )0 ψ ′ ( t ) dt ≤ ψ ( d ( x, y )) . Hence, by hypothesis, we obtain ψ ( H ( T x, T y )) ≤ rT R ( x, y ) = r max { Z d ( x,y )0 ψ ′ ( t ) dt, Z d ( x,T x )0 ψ ′ ( t ) dt, Z d ( y,T y )0 ψ ′ ( t ) dt, Z d ( x,Ty )+02 ψ ′ ( t ) dt } , and since R d ( x,T x )0 ψ ′ ( t ) dt ≤ R d ( x,y )0 ψ ′ ( t ) dt and d ( x, T y )2 ≤ d ( x, y )2 + d ( y, T y )2 ≤ max { d ( x, y ) , d ( y, T y ) } it follows that ψ ( H ( T x, T y )) ≤ r max (Z d ( x,y )0 ψ ′ ( t ) dt, Z d ( y,T y )0 ψ ′ ( t ) dt, Z max { d ( x,y ) ,d ( y,T y ) } ψ ′ ( t ) dt ) = r max (Z d ( x,y )0 ψ ′ ( t ) dt, Z d ( y,T y )0 ψ ′ ( t ) dt ) . (2.3)Hence, by inequality ψ ( d ( y, T y )) ≤ ψ ( H ( T x, T y )) SOKOL BUSH KALIAJ we obtain ψ ( d ( y, T y )) ≤ r max (Z d ( x,y )0 ψ ′ ( t ) dt, Z d ( y,T y )0 ψ ′ ( t ) dt ) . By Theorem VIII.2.5 in [16], we also have Z d ( y,T y )0 ψ ′ ( t ) dt ≤ r max (Z d ( x,y )0 ψ ′ ( t ) dt, Z d ( y,T y )0 ψ ′ ( t ) dt ) , and since r ∈ [0 , (Z d ( x,y )0 ψ ′ ( t ) dt, Z d ( y,T y )0 ψ ′ ( t ) dt ) = Z d ( x,y )0 ψ ′ ( t ) dt. The last result together with (2.3) implies ψ ( H ( T x, T y )) ≤ r Z d ( x,y )0 ψ ′ ( t ) dt. Since y was arbitrary, the last result means that (2.1) holds and this ends the proof. (cid:3) Lemma 2.2.
Let ( X, d ) be a complete metric space and let T : X → CB ( X ) be an ( r, φ, ψ ) -Suzukiintegral contraction. Assume that a sequence ( z n ) ⊂ X converges to a point z ∈ X and z n +1 ∈ T z n for all n ∈ N . Then, (2.4) ψ ( d ( z, T x )) ≤ r max (Z d ( z,x )0 ψ ′ ( t ) dt, Z d ( x,T x )0 ψ ′ ( t ) dt ) , for all x ∈ X \ { z } . Proof.
Fix an arbitrary x ∈ X \ { z } . Then d ( z, x ) >
0, and since ( z n ) converges to z , there exists p ∈ N such that d ( z n , z ) < d ( z,x )3 whenever n ≥ p , and since z n +1 ∈ T z n it follows that for each n ≥ p , we have d ( z n , T z n ) ≤ d ( z n , z n +1 ) ≤ d ( z n , z ) + d ( z, z n +1 ) ≤ d ( z, x ) = d ( z, x ) − d ( z, x ) ≤ d ( z, x ) − d ( z n , z ) ≤ d ( z n , x ) . Hence, φ ( r ) Z d ( z n ,T z n )0 ψ ′ ( t ) dt ≤ Z d ( z n ,T z n )0 ψ ′ ( t ) dt ≤ Z d ( z n ,x )0 ψ ′ ( t ) dt ≤ ψ ( d ( z n , x )) , whenever n ≥ p . By hypothesis the last result implies ψ ( H ( T z n , T x )) ≤ rT R ( z n , x ) = r max { Z d ( z n ,x )0 ψ ′ ( t ) dt, Z d ( z n ,T z n )0 ψ ′ ( t ) dt, Z d ( x,T x )0 ψ ′ ( t ) dt, Z d ( zn,Tx )+ d ( x,Tzn )2 ψ ′ ( t ) dt } , whenever n ≥ p . Further, by inequalities ψ ( d ( z n +1 , T x )) ≤ ψ ( H ( T z n , T x )) , Z d ( z n ,T z n )0 ψ ′ ( t ) dt ≤ ψ ( d ( z n , T z n )) ≤ ψ ( d ( z n , z n +1 )) N INTEGRAL SUZUKI-TYPE FIXED POINT THEOREM 5 and d ( z n , T x ) + d ( x, T z n )2 ≤ max { d ( z n , T x ) , d ( x, T z n ) } ≤ max { d ( z n , T x ) , d ( x, z n +1 ) } it follows that ψ ( d ( z n +1 , T x )) ≤ r max { Z d ( z n ,x )0 ψ ′ ( t ) dt, Z d ( z n ,z n +1 )0 ψ ′ ( t ) dt, Z d ( x,T x )0 ψ ′ ( t ) dt, Z d ( z n ,T x )0 ψ ′ ( t ) dt, Z d ( x,z n +1 )0 ψ ′ ( t ) dt } whenever n ≥ p . Hence, by Theorem IX.4.1 in [16], p.252, we get ψ ( d ( z, T x )) = lim n →∞ ψ ( d ( z n +1 , T x )) ≤ r max (Z d ( z,x )0 ψ ′ ( t ) dt, Z d ( x,T x )0 ψ ′ ( t ) dt, Z d ( z,T x )0 ψ ′ ( t ) dt ) , and since R d ( z,T x )0 ψ ′ ( t ) dt ≤ ψ ( d ( z, T x )) we obtain Z d ( z,T x )0 ψ ′ ( t ) dt ≤ r max (Z d ( z,x )0 ψ ′ ( t ) dt, Z d ( x,T x )0 ψ ′ ( t ) dt, Z d ( z,T x )0 ψ ′ ( t ) dt ) . The last result impliesmax (Z d ( z,x )0 ψ ′ ( t ) dt, Z d ( x,T x )0 ψ ′ ( t ) dt, Z d ( z,T x )0 ψ ′ ( t ) dt ) = max { Z d ( z,x )0 ψ ′ ( t ) dt, Z d ( x,T x )0 ψ ′ ( t ) dt } and consequently ψ ( d ( z, T x )) ≤ r max (Z d ( z,x )0 ψ ′ ( t ) dt, Z d ( x,T x )0 ψ ′ ( t ) dt ) . Since x was arbitrary, the last result yields that (2.4) holds and the proof is finished. (cid:3) We are now ready to present the main result.
Theorem 2.3.
Let ( X, d ) be a complete metric space and let T : X → CB ( X ) be a multivaluedmapping. Assume that T is an ( r, φ, ψ ) -Suzuki integral contraction. Then, T has a fixed point.Proof. Let us start with an arbitrary point z ∈ X . Choose a real number r such that r < r < d ( z , T z ) = 0, then z is a fixed point and the proof is finished. Assume that d ( z , T z ) > z ∈ T z with d ( z , z ) ≥ d ( z , T z ) >
0. Since z ∈ T z , by Lemma 2.1, we have ψ ( d ( z , T z )) ≤ ψ ( H ( T z , T z )) ≤ r Z d ( z ,z )0 ψ ′ ( t ) dt ≤ rψ ( d ( z , z )) < rψ ( d ( z , z )) . We assume that r > r = 0 ⇒ ψ ( d ( z , T z )) = 0 ⇒ d ( z , T z ) = 0 ⇒ z ∈ T z . SOKOL BUSH KALIAJ If d ( z , T z ) = 0, then z is a fixed point and the proof is finished. Assume that d ( z , T z ) = t > ψ is continuous at t , given ε = rψ ( d ( z , z )) − ψ ( t ) > δ > t ≤ t < t + δ ⇒ ψ ( t ) ≤ ψ ( t ) < rψ ( d ( z , z )) , and since there exists z ∈ T z such that t ≤ d ( z , z ) < t + δ it follows that(2.5) ψ ( t ) ≤ ψ ( d ( z , z )) < rψ ( d ( z , z )) . We now assume that z n ∈ T z n − has been chosen. Then, by Lemma 2.1, we have rψ ( d ( z n − , z n )) − ψ ( d ( z n , T z n )) > . If d ( z n , T z n ) = 0, then z n is a fixed point and the proof is finished. Assume that d ( z n , T z n ) = t n > ψ is continuous at t n , given ε n = rψ ( d ( z n − , z n )) − ψ ( t n ) > δ n > t n ≤ t < t n + δ n ⇒ ψ ( t n ) ≤ ψ ( t ) < rψ ( d ( z n − , z n )) , and since there exists z n +1 ∈ T z n such that t n ≤ d ( z n , z n +1 ) < t n + δ n it follows that(2.6) ψ ( t n ) ≤ ψ ( d ( z n , z n +1 )) < rψ ( d ( z n − , z n )) . Since z n +1 ∈ T z n , by Lemma 2.1, we have rψ ( d ( z n , z n +1 )) − ψ ( d ( z n +1 , T z n +1 )) > . By above construction we obtain a sequence ( z n ) ⊂ X such that(2.7) ψ ( d ( z n , z n +1 )) < rψ ( d ( z n − , z n )) , for all n ∈ N . Hence, we get ψ ( d ( z n , z n +1 )) < r n ψ ( d ( z , z )) , for all n ∈ N , and since ψ ( d ( z n , z n +1 )) ≥ d ( z n , z n +1 ) it follows that d ( z n , z n +1 ) < r n ψ ( d ( z , z )) , for all n ∈ N . Hence + ∞ X n =1 d ( z n , z n +1 ) ≤ + ∞ X n =1 r n ψ ( d ( z , z )) . Since r <
1, the last result yields that ( z n ) is a Cauchy sequence in X and by completeness of X itfollows that ( z n ) converges to a point z ∈ X .We are going to prove that z is a fixed point of T . To see this, we suppose that z T z . Considerthe following possible cases:(i) 0 ≤ r < ,(ii) ≤ r < N INTEGRAL SUZUKI-TYPE FIXED POINT THEOREM 7 ( i ) Fix an arbitrary w ∈ T z . Since d ( z, T z ) ≤ d ( z, T w ) + H ( T w, T z )we obtain Z d ( z,T z )0 ψ ′ ( t ) dt ≤ ψ ( d ( z, T z )) ≤ ψ ( d ( z, T w )) + ψ ( H ( T w, T z )) , (2.8)and since by Lemma 2.1 and Lemma 2.2 we have also ψ ( H ( T w, T z )) ≤ r Z d ( w,z )0 ψ ′ ( t ) dtψ ( d ( z, T w )) ≤ r max (Z d ( z,w )0 ψ ′ ( t ) dt, Z d ( w,T w )0 ψ ′ ( t ) dt, ) and Z d ( w,T w )0 ψ ′ ( t ) dt ≤ ψ ( d ( w, T w )) ≤ ψ ( H ( T z, T w )) ≤ r Z d ( z,w )0 ψ ′ ( t ) dt< Z d ( z,w )0 ψ ′ ( t ) dt it follows that Z d ( z,T z )0 ψ ′ ( t ) dt ≤ ψ ( d ( z, T z )) ≤ r Z d ( z,w )0 ψ ′ ( t ) dt. Since w was arbitrary, the last results yields Z d ( z,T z )0 ψ ′ ( t ) dt ≤ ψ ( d ( z, T z )) ≤ r Z d ( z,w )0 ψ ′ ( t ) dt, for all w ∈ T z. (2.9)From equality d ( z, T z ) = inf { d ( z, w ) : w ∈ T z } it follows that there exists a sequence ( w n ) ⊂ T z such thatlim n →∞ d ( z, w n ) = d ( z, T z ) . Then, by (2.9), we obtain Z d ( z,T z )0 ψ ′ ( t ) dt ≤ ψ ( d ( z, T z )) ≤ r Z d ( z,w n )0 ψ ′ ( t ) dt, for all n ∈ N . Hence, by Theorem IX.4.1 in [16], we get Z d ( z,T z )0 ψ ′ ( t ) dt ≤ ψ ( d ( z, T z )) ≤ r Z d ( z,T z )0 ψ ′ ( t ) dt and since 0 < r < Z d ( z,T z )0 ψ ′ ( t ) dt = 0 ⇒ ψ ( d ( z, T z )) = 0 ⇒ d ( z, T z ) = 0or Z d ( z,T z )0 ψ ′ ( t ) dt > ⇒ Z d ( z,T z )0 ψ ′ ( t ) dt < Z d ( z,T z )0 ψ ′ ( t ) dt. These contradictions show that z ∈ T z .( ii ) Fix an arbitrary x ∈ X \ { z } . Since d ( x, T x ) ≤ d ( x, z ) + d ( z, T x ) SOKOL BUSH KALIAJ we get Z d ( x,T x )0 ψ ′ ( t ) dt ≤ Z d ( x,z )+ d ( z,T x )0 ψ ′ ( t ) dt ≤ Z d ( z,T x )0 ψ ′ ( t ) dt + Z d ( x,z )+ d ( z,T x ) d ( z,T x ) ψ ′ ( t ) dt ≤ Z d ( z,T x )0 ψ ′ ( t ) dt + ψ [ d ( z, x ) + d ( z, T x )] − ψ ( d ( z, T x )) ≤ Z d ( z,T x )0 ψ ′ ( t ) dt + ψ ( d ( z, x )) . (2.10)Since x = z , by Lemma 2.2, we have Z d ( z,T x )0 ψ ′ ( t ) dt ≤ ψ ( d ( z, T x )) ≤ r max (Z d ( z,x )0 ψ ′ ( t ) dt, Z d ( x,T x )0 ψ ′ ( t ) dt ) . (2.11)If R d ( x,T x )0 ψ ′ ( t ) dt ≤ R d ( z,x )0 ψ ′ ( t ) dt , then φ ( r ) Z d ( x,T x )0 ψ ′ ( t ) dt ≤ Z d ( z,x )0 ψ ′ ( t ) dt ≤ ψ ( d ( z, x ))(2.12)Otherwise, if R d ( x,T x )0 ψ ′ ( t ) dt > R d ( z,x )0 ψ ′ ( t ) dt , then we obtain by (2.11) that Z d ( z,T x )0 ψ ′ ( t ) dt ≤ ψ ( d ( z, T x )) ≤ r Z d ( x,T x )0 ψ ′ ( t ) dt, and by (2.10) we get φ ( r ) Z d ( x,T x )0 ψ ′ ( t ) dt = (1 − r ) Z d ( x,T x )0 ψ ′ ( t ) dt ≤ ψ ( d ( z, x )) . Since x was arbitrary, the last result together with (2.12) yields φ ( r ) Z d ( x,T x )0 ψ ′ ( t ) dt ≤ ψ ( d ( z, x )) , for all x ∈ X \ { z } . Then, by hypothesis it follow that ψ ( H ( T x, T z )) ≤ r max { Z d ( z,x )0 ψ ′ ( t ) dt, Z d ( z,T z )0 ψ ′ ( t ) dt, Z d ( x,T x )0 ψ ′ ( t ) dt, Z max { d ( z,T x ) ,d ( x,T z ) } ψ ′ ( t ) dt } , N INTEGRAL SUZUKI-TYPE FIXED POINT THEOREM 9 whenever x ∈ X \ { z } . Clearly, if x = z , then the last inequality also holds. Thus, the last inequalityholds for all x ∈ X . In particular, for x = z n , we have ψ ( d ( z n +1 , T z )) ≤ ψ ( H ( T z n , T z )) ≤ r max { Z d ( z,z n )0 ψ ′ ( t ) dt, Z d ( z,T z )0 ψ ′ ( t ) dt, Z d ( z n ,z n +1 )0 ψ ′ ( t ) dt, Z max { d ( z,z n +1 ) ,d ( z n ,T z ) } ψ ′ ( t ) dt } , Hence, by Theorem IX.4.1 in [16], it follows that0 < ψ ( d ( z, T z )) = lim n →∞ ψ ( d ( z n +1 , T z )) ≤ r Z d ( z,T z )0 ψ ′ ( t ) dt ≤ rψ ( d ( z, T z )) < ψ ( d ( z, T z )) . This contradiction shows that z ∈ T z and the proof is finished. (cid:3)
Corollary 2.4.
Let ( X, d ) be a complete metric space and let T : X → CB ( X ) be a multivaluedmapping. If T is an ( r, ψ ) - ´Ciri´c integral contraction, then T has a fixed point. Corollary 2.5.
Let ( X, d ) be a complete metric space and let S : X → X be a mapping. If S is an ( r, φ, ψ ) -Suzuki integral contraction, then S has a unique fixed point.Proof. Since the multivalued mapping T : X → CB ( X ) defined as follows T x = { Sx } , for all x ∈ X, is an ( r, φ, ψ )-Suzuki integral contraction, by Theorem 2.3, T has a fixed point z and, consequently, z is a fixed point for S .It remains to prove that the fixed point z is unique. Assume that z ′ ∈ X is another fixed pointfor S and z = z ′ . Then, by Lemma 2.2, we have Z d ( z,T z ′ )0 ψ ′ ( t ) dt ≤ ψ ( d ( z, T z ′ )) ≤ r max (Z d ( z,z ′ )0 ψ ′ ( t ) dt, Z d ( z ′ ,T z ′ )0 ψ ′ ( t ) dt, ) and since R d ( z ′ ,T z ′ )0 ψ ′ ( t ) dt = 0 and d ( z, T z ′ ) = d ( z, z ′ ) it follows that0 ≤ Z d ( z,z ′ )0 ψ ′ ( t ) dt ≤ ψ ( d ( z, z ′ )) ≤ r Z d ( z,z ′ )0 ψ ′ ( t ) dt Hence, r = 0 or Z d ( z,z ′ )0 ψ ′ ( t ) dt = 0 ! ⇒ ψ ( d ( z, z ′ )) = 0 ⇒ d ( z, z ′ ) = 0 , or < r < Z d ( z,z ′ )0 ψ ′ ( t ) dt > ! ⇒ Z d ( z,z ′ )0 ψ ′ ( t ) dt < Z d ( z,z ′ )0 ψ ′ ( t ) dt. These contradicts show that z = z ′ and this ends the proof. (cid:3) Corollary 2.6.
Let ( X, d ) be a complete metric space and let S : X → X be a mapping. If S is an ( r, ψ ) - ´Ciri´c integral contraction, then S has a unique fixed point. Corollary 2.7.
Let ( X, d ) be a complete metric space and let T : X → CB ( X ) be a multivaluedmapping. If T is an ( r, φ ) -Suzuki multivalued contraction, then T has a fixed point. Proof.
Clearly, T is an ( r, φ, ψ )-Suzuki integral contraction with ψ ( t ) = t for all t ∈ R + . Then, byTheorem 2.3, T has a fixed point and the proof is finished. (cid:3) An Application to Dynamic Programming
We will now prove an existence and uniqueness theorem for a functional equation arising in dynamicprograming of continuous multistage decision processes. From now on X and Y will be the Banachspaces. Let S ⊂ X be the state space and let D ⊂ Y be the decision space . In the paper [2], Bellmanand Lee gave the following basic form of the functional equation of dynamic programming: f ( x ) = sup y ∈ D H ( x, y, f [ T ( x, y )]) , where x and y represent the state and decision vectors respectively, T : S × D → S represents thetransformation of the process and f ( x ) represent the optimal return function with initial state x .We will study the existence and the uniqueness of the solution of the following functional equation:(3.1) f ( x ) = sup y ∈ D [ g ( x, y ) + G ( x, y, f [ T ( x, y )]) ] , for all x ∈ S, where g : S × D → R and g : S × D × R → R are bounded functions.Let B ( S ) be the Banach space of real valued bounded functions defined on S with norm || h || = sup {| h ( x ) | : x ∈ S } , for all h ∈ B ( S ) . Since g and G are bounded functions we can define a mapping A : B ( S ) → B ( S ) as follows(3.2) ( Ah )( x ) = sup y ∈ D [ g ( x, y ) + G ( x, y, h [ T ( x, y )]) ] , for each h ∈ B ( S ) and x ∈ S . It is easy to see that any fixed point of A is a solution of the functionalequation (3.1) and, conversely, any bounded solution of (3.1) is a fixed point of A .Clearly, if A is an ( r, φ, ψ )- Suzuki integral contraction, then there exists r ∈ [0 ,
1) and ψ ∈ Ψ suchthat the implication φ ( r ) Z || h − Ah || ψ ′ ( t ) dt ≤ ψ ( || h − ℓ || ) ⇒ ψ ( || Ah − Aℓ || ) ≤ rA R ( h, ℓ ) , holds whenever h, ℓ ∈ B ( S ), where φ is defined by (1.1) and A R ( h, ℓ ) = max { Z || h − ℓ || ψ ′ ( t ) dt, Z || h − Ah || ψ ′ ( t ) dt, Z || ℓ − Aℓ || ψ ′ ( t ) dt, Z || h − Aℓ || )+ || ℓ − Ah || ψ ′ ( t ) dt } . Since Ψ is a subfamily of F defined in [7], we obtain by Lemmas 3.1 and 3.2 in [7] the followingauxiliary lemmas. Lemma 3.1.
Let R be a subset of R + such that sup R < + ∞ , and let ψ ∈ Ψ . Then, ψ (sup R ) = sup ψ ( R ) . Lemma 3.2.
Assume that the functions g and G are bounded and let ε > , x ∈ S , h, ℓ ∈ B ( S ) andlet ψ ∈ Ψ . Then, there exists y , y ∈ D such that (3.3) ψ ( | ( Ah )( x ) − ( Aℓ )( x ) | ) ≤ max { ψ ( | a ( x, y ) | ) , ψ ( | b ( x, y ) | ) } + ε, where a ( x, y ) = G ( x, y , h [ T ( x, y )]) − G ( x, y , ℓ [ T ( x, y )]) ,b ( x, y ) = G ( x, y , h [ T ( x, y )]) − G ( x, y , ℓ [ T ( x, y )]) . We are now ready to prove the existence and uniqueness of the solution of the functional equation(3.1) under certain conditions.
N INTEGRAL SUZUKI-TYPE FIXED POINT THEOREM 11
Theorem 3.3.
Assume that the following conditions are satisfied: (i) g and G are bounded functions, (ii) there exist r ∈ [0 , and ψ ∈ Ψ such that for each h, ℓ ∈ B ( S ) , the inequality φ ( r ) Z || h − Ah || ψ ′ ( t ) dt ≤ ψ ( || h − ℓ || ) implies that for each ( x, y ) ∈ S × D , we have ψ ( | G ( x, y, h [ T ( x, y )]) − G ( x, y, ℓ [ T ( x, y )]) | ) ≤ rA R ( h, ℓ ) . Then, the functional equation (3.1) possesses unique bounded solution on S .Proof. Assume that an arbitrary ε > h, ℓ ∈ B ( S ) such that(3.4) φ ( r ) Z || h − Ah || ψ ′ ( t ) dt ≤ ψ ( || h − ℓ || )are given. Fix an arbitrary x ∈ S . Then, by Lemma 3.2, there exists y , y ∈ D such that ψ ( | ( Ah )( x ) − ( Aℓ )( x ) | ) ≤ max { ψ ( | a ( x, y ) | ) , ψ ( | b ( x, y ) | ) } + ε. By hypothesis, we also have ψ ( | a ( x, y ) | ) ≤ rA R ( h, ℓ ) and ψ ( | b ( x, y ) | ) ≤ rA R ( h, ℓ ) . It follows that ψ ( | ( Ah )( x ) − ( Aℓ )( x ) | ) ≤ rA R ( h, ℓ ) + ε. Since x was arbitrary, the last inequality holds for all x ∈ S . Hence,sup x ∈ S ψ ( | ( Ah )( x ) − ( Aℓ )( x ) | ) ≤ rA R ( h, ℓ ) + ε, and since by Lemma 3.1 we also havesup x ∈ S ψ ( | ( Ah )( x ) − ( Aℓ )( x ) | ) = ψ (sup x ∈ S | ( Ah )( x ) − ( Aℓ )( x ) | ) = ψ ( || Ah − Aℓ || ) , it follows that ψ ( || Ah − Aℓ || ) ≤ rA R ( h, ℓ ) + ε. Since ε > ψ ( || Ah − Aℓ || ) ≤ rA R ( h, ℓ ) . Thus, inequality (3.4) implies (3.5), whenever h, ℓ ∈ B ( S ). This means that A is an ( r, φ, ψ )-Suzukiintegral contraction. Therefore, by Corollary 2.5, A has a unique fixed point and, therefore, thefunctional equation (3.1) possesses unique bounded solution on S , and this ends the proof. (cid:3) References [1] S. Banach,
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