Endpoint theory for set-valued nonlinear asymptotic contractions with respect to generalized pseudodistances in uniform spaces

Transcription

1 J. Math. Anal. Appl ) Endpoint theory for set-valued nonlinear asyptotic contractions with respect to generalized pseudodistances in unifor spaces Kaziierz Włodarczyk, Robert Plebaniak Departent of Nonlinear Analysis, Faculty of Matheatics, University of Łódź, Banacha 22, Łódź, Poland Received 14 February 2007 Available online 28 June 2007 Subitted by Richard M. Aron Abstract In unifor spaces, inspired by ideas of Banach, Tarafdar and Yuan, we introduce the concepts of generalized pseudodistances and generalized gauge aps, for set-valued dynaic systes we define various nonlinear asyptotic contractions and contractions with respect to these pseudodistances and gauges, provide conditions on the iterates of these set-valued dynaic systes and present a ethod which is useful for establishing conditions guaranteeing the existence and uniqueness of endpoints stationary points) of these set-valued dynaic systes and conditions that each generalized sequence of iterations in particular, each dynaic process) converges and the liit of a generalized sequence of iterations is an endpoint. The definitions, the results and the ethod are new for set-valued dynaic systes in unifor, locally convex and etric spaces and even for single-valued aps. The paper includes a nuber of various exaples which show a fundaental difference between our results and those existing in the literature Elsevier Inc. All rights reserved. Keywords: Uniqueness of endpoint; Set-valued dynaic syste; Faily of generalized pseudodistances; Faily of generalized gauge aps; Nonlinear asyptotic contraction; Nonlinear contraction; Unifor space; Locally convex space; Metric space; Closed ap; Upper seicontinuous ap; Generalized sequence of iterations 1. Introduction The faous Banach contraction principle [4] has extensive applications in any fields of atheatics and applied atheatics and because of its iportance for atheatical theory) it has been extended in different directions by any authors it is not our purpose to give a coplete list of related papers here). The investigations of the existence and uniqueness of endpoints of set-valued dynaic systes use ideas of Banach [4] and have received uch attention in recent years. Aong these generalizations, the results of Aubin and Ekeland [1], Aubin and Siegel [3], Berge [5], Justan [6], Maschler and Peleg [8], Tarafdar and Vyborny [9], Tarafdar and Yuan [10] and Yuan [11] are the ost valuable ones. * Corresponding author. E-ail addresses: wlkzxa@ath.uni.lodz.pl K. Włodarczyk), robpleb@ath.uni.lodz.pl R. Plebaniak) X/$ see front atter 2007 Elsevier Inc. All rights reserved. doi: /j.jaa

2 K. Włodarczyk, R. Plebaniak / J. Math. Anal. Appl ) In this paper, in unifor spaces, inspired by ideas of Banach [4], Tarafdar and Yuan [10] and Yuan [11], we introduce the concepts of generalized pseudodistances and generalized gauge aps, for set-valued dynaic systes we define various nonlinear asyptotic contractions and contractions with respect to these pseudodistances and gauges, provide conditions on the iterates of these set-valued dynaic systes and present a ethod which is useful for establishing conditions guaranteeing the existence and uniqueness of endpoints of these set-valued dynaic systes and conditions that each generalized sequence of iterations in particular, each dynaic process) converges and the liit of a generalized sequence of iterations is an endpoint. The definitions, the results and the ethod are new for set-valued dynaic systes in unifor, locally convex and etric spaces and even for single-valued aps. The paper includes a nuber of various exaples which show a fundaental difference between our results and those existing in the literature. 2. Definitions, notations and stateent of results Let 2 X denotes the faily of all nonepty subsets of a space X. Aset-valued dynaic syste is defined as a pair X, T ), where X is a certain space and T is a set-valued ap T : X 2 X ; in particular, a set-valued dynaic syste includes the usual dynaic syste where T is a single-valued ap. A point w X is said to be an endpoint or stationary point) oft if w is a fixed point of T i.e., w Tw)) and Tw)=w}.Adynaic process or a trajectory starting at w 0 X or a otion of the syste X, T ) at w 0 is a sequence w } defined by w Tw 1 ), N. For details, see e.g., Aubin and Siegel [3] and Aubin and Ekeland [1]. A sequence w } such that w T [] w 0 ), T [] = T T T -ties), N, is called a generalized sequence of iterations with w 0. Since the set T [] w 0 ), in general, is bigger than Tw 1 ), thus each dynaic process starting fro w 0 is a generalized sequence of iterations with respect to w 0, but the converse ay not be true. For details see Yuan [11, p. 559]. Let X be a Hausdorff unifor space with unifority defined by a saturated faily d α : α A} of pseudoetrics d α, α A, uniforly continuous on X 2.ForT : E 2 X, E X,letTE)= x E Tx). In order to present our results precisely, let us introduce soe definitions and notations. Definition 2.1. Let X be a Hausdorff unifor space. The faily V = V α :2 X [0, ],α A } is said to be a V-faily of generalized pseudodistances on X V-faily, for short) if the following four conditions hold: V1) E1,E 2 2 XE 1 E 2 V α E 1 ) V α E 2 )}; V2) x,y,z X V α x,z}) V α x,y}) + V α y,z})}; V3) for any sequence x } in X such that li sup V α x n,x }) } = 0, 2.1) n >n if there exists a sequence y } in X satisfying li V α x,y }) } = 0, 2.2) then li d α x,y ) } = 0 ; 2.3) V4) α0 AV α0 X) > 0}. The following property holds. Theore 2.1. Let X be a Hausdorff unifor space, let V =V α :2 X [0, ],α A} be a V-faily on X and let the faily F =F α : X X [0, ], α A} be defined by F α x, y) = V α x,y}), x, y) X X, α A. Ifx,y X and F α x, y) = 0}, then x = y.

3 346 K. Włodarczyk, R. Plebaniak / J. Math. Anal. Appl ) Definition 2.2. Let X be a Hausdorff unifor space. The V-faily V =V α :2 X [0, ],α A} is called a V-faily if, additionally, the following condition holds: V5) E 2 XV α E) = V α E)}. Now, we introduce various failies of gauge aps and definitions of set-valued nonlinear contractions and nonlinear asyptotic contractions with respect to V-failies. Definition 2.3. Let X be a Hausdorff unifor space and let X, T ) be a set-valued dynaic syste. Let V =V α : 2 X [0, ], α A} be a V-faily on X and let, for each α A, D α;t,v = V α E): E X TE) E V α E) > 0 } and H α;t,v = V α T [n] X) ) : V α T [n] X) ) > 0 n 0} N } where T [0] X) = X. G1) An Ω-faily of generalized gauge aps Ω-faily, for short) is by definition a faily Ω =ω ;α } α A of aps ω ;α : H α;t,v 0, ], N, α A, such that ε 0, ) η>0 N t [ε,ε+η) t Hα;T,V ω ;α t) ε }. 2.4) G2) A Π-faily of generalized gauge aps Π-faily, for short) is by definition a faily Π =π α } α A of aps π α : H α;t,v 0, ], α A, such that ε 0, ) η>0 t [ε,ε+η) t Hα;T,V π α t) ε }. 2.5) G3) A Ψ -faily of generalized gauge aps Ψ -faily, for short) is by definition a faily Ψ =ψ ;α } α A of aps ψ ;α : D α;t,v 0, ], N, α A, such that ε 0, ) η>0 N t [ε,ε+η) t Dα;T,V ψ ;α t) ε }. 2.6) G4) A Φ-faily of generalized gauge aps Φ-faily, for short) is by definition a faily Φ =ϕ α } α A of aps ϕ α : D α;t,v 0, ], α A, such that ε 0, ) η>0 t [ε,ε+η) t Dα;T,V ϕ α t) ε }. 2.7) Definition 2.4. Let X be a Hausdorff unifor space and let X, T ) be a set-valued dynaic syste. We say that T satisfies condition C) on X if one of the following conditions holds: C1) There exist a V-faily and a Ω-faily such that N n 0} N Vα T [n] X) ) > 0 V α T [] T [n] X) )) <ω ;α Vα T [n] X) ))}. 2.8) Then we say that T is a V; Ω)-asyptotic contraction on X. C2) There exist a V-faily and a Π-faily such that n 0} N Vα T [n] X) ) > 0 V α T T [n] X) )) <π α Vα T [n] X) ))}. 2.9) Then we say that T is a V; Π)-contraction on X. C3) There exist a V-faily and a Ψ -faily such that N E X TE) E Vα E) > 0 V α T [] E) ) <ψ ;α Vα E) )}. 2.10) Then we say that T is a V; Ψ)-asyptotic contraction on X. C4) There exist a V-faily and a Φ-faily such that ) E X TE) E Vα E) > 0 V α TE) <ϕα Vα E) )}. 2.11) Then we say that T is a V; Φ)-contraction on X.

4 K. Włodarczyk, R. Plebaniak / J. Math. Anal. Appl ) The relations between conditions C1) C4) are given in the following theore. Theore 2.2. Let X be a Hausdorff unifor space. Then the following are true: i) Each V; Π)-contraction T on X is a V; Ω)-asyptotic contraction on X. ii) Each V; Ψ)-asyptotic contraction T on X is a V; Π)-contraction on X. iii) Each V; Φ)-contraction T on X is a V; Ψ)-asyptotic contraction on X. Let us recall the definition of a closed ap. Definition 2.5. See Berge [5, p. 111] Klein and Thopson [7, Section 7.7], Aubin and Frankowska [2].) Let X, T ) be a set-valued dynaic syste. The ap T is called closed if for each x 0,y 0 X such that y 0 / Tx 0 ) there exist in X two neighbourhoods Nx 0 ) and Ny 0 ) of x 0 and y 0, respectively, which satisfy Tx) Ny 0 ) = for each x Nx 0 ). Our ain result is the following: Theore 2.3. Assue that: a) X is a Hausdorff coplete unifor space; b) X, T ) is a set-valued dynaic syste; c) T satisfies C) on X; and d) T [p] is closed in X for soe p N. Then: i) T has a unique endpoint w in X; and ii) Each sequence w }, where w T [] w 0 ) for N and w 0 X, converges to w. 3. Proof of Theore 2.1 Suppose that x,y X and F α x, y) = 0}. Hence, by V1), Fα x, x) = 0 }. Furtherore, denoting u = x,v = y, N,gives li sup F α u n,u ) = 0 li F α u,v ) } n >n Now, using V3), we conclude that li d α u,v ) = 0}. Consequently, we get d α x, y) = 0},i.e.x = y. 4. Proof of Theore 2.2 Note that TX) X so, by induction, we have N T [] X) T [ 1] X) X }, T [0] X) = X. 4.1) Hence, in particular, N n 0} N T [] } E n ) E n 4.2) where n 0} N En = T [n] X) }. 4.3) i) Assue that T is a V; Π)-contraction on X. Fro 4.2), 4.3) and V1) it follows that N n 0} N Vα T [] E n ) ) V α TEn ) )}. 4.4)

5 348 K. Włodarczyk, R. Plebaniak / J. Math. Anal. Appl ) In virtue of 4.4), 2.5) and 2.9), we obtain that 2.4) and 2.8) hold if we define Ω =ω ;α } α A where ω ;α = π α : H α;t,v 0, ], N, α A, sot is a V; Ω)-asyptotic contraction on X. ii) Assue that T is a V; Ψ)-asyptotic contraction onx. Thus 2.6) and 2.10) are satisfied and, by 4.1) 4.3), we see that the sets H α;t,v satisfy H α;t,v D α;t,v, α A; byv4), H α0 ;T,V for soe α 0 A. By 4.2) and 4.4) for = 1, defining Π =π α } α A where π α = ψ 1;α Hα;T,V, α A, in conclusion, we have that the conditions 2.5) and 2.9) are satisfied, so T is a V; Π)-contraction on X. iii) Assue that T is a V; Φ)-contraction on X. Since TE) E iplies N T [] E) TE)} and, consequently, by V1), N V α T [] E)) V α T E))}, therefore, defining Ψ =ψ ;α } α A by ψ ;α = ϕ α, N, α A, fro 2.7) and 2.11) we obtain 2.6) and 2.10), so T is a V; Ψ)-asyptotic contraction on X. The proof of Theore 2.2 is coplete. 5. Proof of Theore 2.3 By Theore 2.2, we ay assue without loss of generality that T is a V; Ω)-asyptotic contraction on X. The proof will be broken into several steps. Step I. For each α A, the sequence V α T [] X))} is decreasing and converges. Indeed, by 4.1) and V1), N Vα T [] X) ) ) V α TX) Vα X) }. 5.1) In conclusion, for each α A, the sequence V α T [] X))} is decreasing. Observe that, for each α A,thesetV α T [] X)): 1} is bounded. Indeed, the following two cases hold: Case I. IfV β X) is finite for soe β N, then, using 4.1) and V1), we have that the set V β T [] X)): 0} is bounded. Case II. IfV γ X) = for soe γ N, then V γ X) > 0 and by 4.1) and 2.8), one has N V γ T [] X)) V γ T X)) < ω 1;γ V α X))}. It follows fro the above that N V γ T [] X)) V γ T X)) < }. Hence V γ T [] X)): 1} is bounded. Step II. We can show that li V α T [] X) ) } 5.2) Indeed, we consider two cases: Case 1. If β A q N V β T [q] X)) = 0}, then, since T [] X) T [q] X) for each q, byv1), we get V β T [] X)) V β T [q] X)) = 0 for each q. Therefore, li V β T [] X)) = 0. Case 2. If N V α T [] X)) > 0}, then also li V α T [] X)) = 0 for each α A. Suppose that this does not hold, so, by Step I, α0 A ε0 0, ) li V α0 T [] X) ) } = ε ) First, we observe that, by Step I and 5.3), N 0} Vα0 T [] X) ) } ε 0, T [0] = I X. 5.4) Hence N Vα0 T [] X) ) } H α0 ;T,V. 5.5) Next, let us observe that, by 2.4) and in view of ε 0 > 0, η0 >0 0 N t [ε0,ε 0 +η 0 ) t Hα0 ; T,V ω } 0 ;α 0 t) ε ) Now, using Step I, 5.3) and 5.4), we obtain 1 N, ε0 V α0 T [] X) ) } <ε 0 + η )

6 K. Włodarczyk, R. Plebaniak / J. Math. Anal. Appl ) Therefore, 5.4) 5.7) and 2.8) give 1 ε0 V α0 T [ 0 ] T [] X) )) <ω 0 ;α 0 Vα0 T [] X) )) ε 0 }, which is ipossible. The proof of 5.2) is coplete. Step III. We can show that E 2 X li V α T [] E) ) } 5.8) This is deduced fro 5.2) and V1). Step IV. Let w 0 X and w }, w T [] w 0 ) for N, be arbitrary and fixed. The result is that li V α w,w +1}) } 5.9) Indeed, since w,w +1 } T [] E 1 ) for N where E 1 =w 0 } Tw 0 ), then fro V1) and 5.8) it follows that li V α w,w +1}) li V α T [] E 1 ) ) } Step V. We can show that li V α w n,w }) } 5.10) sup n >n Indeed, let α 0 A, ε 0 0, ) and η 0,ε 0 ) be arbitrary and fixed. First, we observe that, fro 5.2) and Step I, q N Vα0 T [q] X) ) <ε 0 }, and, fro 5.9), 5.11) n0 =n 0 η,q) N n n0 Vα0 w n,w n+1}) <η/q }. 5.12) Suppose to the contrary, there exist k, l N such that l>k n 0 and V α0 w k,w l )} > 2ε 0.If h = in j N: k<jand ε 0 + η V α0 w k,w j })}, 5.13) then h l and, furtherore, by 5.13), V2) and 5.12), 2η <ε 0 + η V α0 w k,w h }) h 1 j=k V α 0 w j,w j+1 })< h 1 j=k η/q = h k)η/q, which gives 2q <h k. Hence, k<h 2q <h q<h l. 5.14) Next, by V2), 5.13) and 5.12), we obtain V α0 w k,w h q }) V α0 w k,w h }) V α0 w h q,w h }) V α0 w k,w h }) q 1 j=0 V α 0 w h j 1,w h j })>ε 0 +η qη/q = ε 0,soε 0 <V α0 w k,w h q }). On the other hand, by 5.13) and 5.14), we have V α0 w k,w h q })<ε 0 + η. Therefore, ε 0 <V α0 w k,w h q })<ε 0 + η. Hence, by V1) and observing that w k,w h q } E 2 where E 2 = T [k] w 0 ) T [h q] w 0 ), we get ε 0 <V α0 E 2 ). Also, according to V1) and 5.11), since w k+q,w h } T [q] E 2 ), we note that V α0 w k+q,w h}) V α0 T [q] E 2 ) ) V α0 T [q] X) ) <ε ) In virtue of 5.13), 5.12) and 5.15), the nuber η + ε 0 satisfies ε 0 + η V α0 w k,w h}) q V α0 w k+j 1,w k+j }) + V α0 w k+q,w h}) <qη/q+ ε 0 = η + ε 0, j=1 which is ipossible. Consequently, l>k n 0 iplies V α0 w k,w l }) 2ε 0. This concludes the proof of 5.10).

7 350 K. Włodarczyk, R. Plebaniak / J. Math. Anal. Appl ) Step VI. We can show that li sup d α w n,w ) } n >n 5.16) Indeed, fro 5.10) we clai that ε>0 n1 =n 1 α,ε) N n>n1 sup V α w n,w }) : >n } } <ε and, in particular, ε>0 n1 =n 1 α,ε) N n>n1 q N Vα w n,w q+n}) <ε }. 5.17) Let i 0,j 0 N, i 0 >j 0, be arbitrary and fixed. If we define u = w i0+ and v = w j0+ for N, 5.18) then 5.17) gives li V α w,u }) = li V α w,v }) } 5.19) Therefore, by 5.10), 5.19) and V3), li d α w,u ) = li d α w,v ) } 5.20) Fro 5.18) and 5.20) we then clai that and ε>0 n2 =n 2 α,ε) N >n2 dα w,w i 0+ ) <ε/2 } 5.21) ε>0 n3 =n 3 α,ε) N >n3 dα w,w j 0+ ) <ε/2 }. 5.22) Let now α 0 A and ε 0 > 0 be arbitrary and fixed, let n 0 = axn 2 α 0,ε 0 ), n 3 α 0,ε 0 )}+1 and let k, l N be arbitrary and fixed such that k>l>n 0. Then k = i 0 + n 0 and l = j 0 + n 0 for soe i 0, j 0 N such that i 0 >j 0 and, using 5.21) and 5.22), we get d α0 w k,w l ) = d α0 w i 0+n 0,w j 0+n 0 ) d α0 w n 0,w i 0+n 0 ) + d α0 w n 0,w j 0+n 0 )< ε 0 /2 + ε 0 /2 = ε 0. Hence, we conclude that ε>0 n0 =n 0 α,ε) N k,l N, k>l>n0 d α w k,w l )<ε}. The proof of 5.16) is coplete. Step VII. There exists a unique w X such that the sequence w } converges to w. Indeed, X is a Hausdorff and coplete space and, by Step VI, w } is the Cauchy sequence. Step VIII. If u } is an arbitrary and fixed sequence such that u T [] w 0 ) for N, then u } also converges to w. Indeed, by 5.8) and V1), li W α w,u }) li W α T [] E 3 ) ) } = 0, E 3 = w 0}. 5.23) Therefore, using 5.10), 5.23) and V3), we get that the sequences w } and u } are equi-convergent, i.e., li d α w,u ) } 5.24) It reains to reark that, using 5.24) and Step VII, li d α u,w ) = li d α u,w ) + li d α w,w ) } Step IX. The point w satisfies w T [p] w),i.e.w is a fixed point of T [p].

8 K. Włodarczyk, R. Plebaniak / J. Math. Anal. Appl ) Indeed, since w T [] w 0 ) = T [p] T [ p] w 0 )) for >p, thus there exist u T [ p] w 0 ) for >psuch that w T [p] u ) for >p. 5.25) However, li w = w by Step VII, li u = w by Step VIII, and T [p] is closed by d). Therefore, fro 5.25), using [5, p. 111] and [7, Section 7.7], we obtain w T [p] w). Step X. w}=tw),i.e.w is an endpoint of T. Indeed, let v Tw)be arbitrary and fixed. By Step IX, using induction, this gives N v Tw) T [p+1] w) T [p+1] w)}. Therefore, by V1), N Vα w p+1,v }) V α T [p+1] E 4 ) )} 5.26) where E 4 =w 0,w}. Fro 5.26) and 5.8) we deduce that li V α w p+1,v }) li V α T [p+1] E 4 ) ) } 5.27) Next, assuing x = w p+1 and y = v for N, 5.28) fro 5.10) and 5.27) we have, respectively, li V α x n,x }) } = ) n >n and li V α x,y }) } 5.30) Then it follows fro 5.28) 5.30) and V3) that li d α w p+1,v ) } 5.31) We deduce fro 5.31), Step VII and Step VIII that d α w, v) li d α w p+1,w ) + li d α w p+1,v ) } So we have w = v, that is Tw)=w}. Step XI. The ap T has a unique endpoint. Indeed, if Tw)=w} and Tu)=u}, then, by 5.8), V α w,u}) li V α T [] E 5 )) = 0}, E 5 =u, w}. Consequently, by Theore 2.1, u = w. Hence, we get the clai. The proof of Theore 2.3 is now coplete. 6. Exaples and rearks First, we give soe exaples of V- and V-failies. Let X be a Hausdorff unifor space with unifority defined by a saturated faily d α : α A} of pseudoetrics d α, α A, uniforly continuous on X 2.Let δ α E) = sup d α x, y): x,y E }, E 2 X,α A. 6.1) Exaple 6.1. Let E 01 X and E 02 X be two bounded subsets of X such that each of the containts at least two points and E 01 E 02 =. Letλ α,μ α 0, ) be constants such that λ α axδ α E 01 ), δ α E 02 )} and μ α 1, α A.LetV =V α :2 X [0, ): α A} be defined by δα E) if E E 01 = E, V α E) = μ α δ α E) if E E 02 = E, E 2 X,α A. λ α if E E 01 E E E 02 E,

9 352 K. Włodarczyk, R. Plebaniak / J. Math. Anal. Appl ) We show that the faily V is a V-faily on X. Indeed, assue that A B. In the case when B E 01 = B,wehaveV α A) = δ α A) δ α B) = V α B). Next, in the case when B E 02 = B,wehaveV α A) = μ α δ α A) μ α δ α B) = V α B). Finally, in the case when B E 01 B and B E 02 B we obtain: V α A) = λ α = V α B) whenever A E 01 A and A E 02 A; V α A) = δ α A) δ α E 01 ) λ α = V α B) whenever A E 01 = A; and V α A) = μ α δ α A) μ α δ α E 02 ) δ α E 02 ) λ α = V α B) whenever A E 02 = A. Therefore, the condition V1) holds. Now we clai that V2) holds. Indeed, if x,y,z} E 01 or if x,y,z} E 02, then V α x,z}) V α x,y}) + V α y,z}).ifx,y,z} E 01 E 02 ) =, then V α x,z}) = V α x,y}) = V α y,z}) and, consequently, V α x,z}) V α x,y}) + V α y,z}). Ifx,y} E 01 and z E 02, then V α x,z}) = V α y,z}) = λ α and V α x,y}) = d α x, y) which gives V α x,z}) = λ α d α x, y) + λ α = V α x,y}) + V α y,z}). If x,z} E 01 and y E 02, then V α x,z}) = d α x, z) and V α x,y}) = V α y,z}) = λ α and hence V α x,z}) = d α x, z) δ α E 0 ) λ α < 2λ α = V α x,y}) + V α y,z}). Ify,z} E 01 and x E 02, then V α y,z}) = d α y, z) and V α x,y}) = V α x,z}) = λ α which iply V α x,z}) = λ α λ α + d α y, z) = V α x,y}) + V α y,z}). We use the sae arguent in the cases: i) x,y} E 01 and z X E 01 E 02 ); ii) x,z} E 01 and y X E 01 E 02 ); and iii) y,z} E 01 and x X E 01 E 02 ). If z E 01 and x,y} E 02, then V α x,z}) = V α y,z}) = λ α and V α x,y}) = μ α d α x, y) and, consequently, V α x,z}) = λ α λ α + μ α d α y, z) = V α x,y}) + V α y,z}). If y E 01 and x,z} E 02, then V α x,z}) = μ α d α x, z), V α x,y}) = V α y,z}) = λ α and thus V α x,z}) = μ α d α x, z) μ α δ α E 02 ) λ α < 2λ α = V α x,y}) + V α y,z}). If x E 01 and y,z} E 02, then V α y,z}) = μ α d α y, z) and V α x,y}) = V α x,z}) = λ α which iply that V α x,z}) = λ α λ α + μ α d α y, z) = V α x,y}) + V α y,z}). Weusethesae arguent in the cases: iv) z E 01 and x,y} X E 01 E 02 ); v) y E 01 and x,z} X E 01 E 02 ); and vi) x E 01 and y,z} X E 01 E 02 ). If A =x,y,z} X and each of the sets E 01 A, E 02 A and X E 01 E 02 )) A is a singleton, then V α x,z}) = V α x,y}) = V α y,z}) = λ α and, consequently, V α x,z})<v α x,y}) + V α y,z}). TheV2) is proved. Now assue that the sequences x } and y } in X satisfy conditions 2.1) and 2.2). Then 2.2) yields 0<εα <λ α 0 = 0 α,ε α ) N 0 Vα x,y }) } <ε α. 6.2) However, by definition of the faily V, 0<εα <λ α N Vα x,y }) <ε α x,y } E 01 x,y } } E ) A consequence of 6.1) 6.3) is 0<εα <λ α 0 = 0 α,ε α ) N 0 μα d α x,y ) d α x,y ) } <ε α. Hence we conclude that li d α x,y }) } Therefore, the sequences x } and y } satisfy 2.3). This shows that V3) holds. Clearly V4) also holds. Exaple 6.2. Let E 0 be a bounded subset of X which contains at least two points, let λ α 0, ) be constants such that λ α δ α E 0 ), α A, and let V =V α :2 X [0, ], α A}, where δα E) if E E V α E) = 0 = E, E 2 X,α A. 6.4) λ α if E E 0 E, Then, the following properties are satisfied:

10 K. Włodarczyk, R. Plebaniak / J. Math. Anal. Appl ) a) The faily V is a V-faily on X; b) If E 0 is closed, then the faily V is a V-faily on X; c)ife 0 E 0 and α0 λ α0 >δ α0 E 0 )}, then the faily V is not a V-faily on X. Indeed, using the sae arguent as in Exaple 6.1, we obtain the properties a) and b). In the case c), by 6.4), we have that V α0 E 0 ) = δ α0 E 0 )<λ α0 = V α0 E 0 ). Therefore, condition V5) does not hold. Now we present the exaples which illustrate Theore 2.3. Let X, d) be a etric space and let δe) = sup dx,y): x,y E} for each E X. Exaple 6.3. Let X, ) be a copact etric space where X =[0, 2] R.LetE 01 = 0, 1),letE 02 = 1, 2) and let V : 2 X [0, ] be of the for δe) if E E01 = E, VE)= 1/2)δE) if E E 02 = E, 1 ife E 01 E E E 02 E. The faily V is, of course, a V-faily. Let T : X 2 X be defined by 1/2)x + 1/4, 1/2) for 0 x<1/2, 1/2} for x = 1/2, Tx)= 1/2,1/2)x + 1/4) for 1/2 <x 1, 1/2} for 1 <x 2. First, we observe that 1/4)x + 3/8, 1/2) for 0 x<1/2, T [2] x) = 1/2} for x = 1/2 1 <x 2, 1/2,1/4)x + 3/8) for 1/2 <x 1, 1/8)x + 7/16, 1/2) for 0 x<1/2, T [3] x) = 1/2} for x = 1/2 1 <x 2, 1/2,1/8)x + 7/16) for 1/2 <x 1, and, by induction, for n>1, T [n] 2 1 n x + 2n 1, 1/2) for 0 x<1/2, 2 n+1 x) = 1/2} for x = 1/2 1 <x 2, 1/2, 2 1 n x + 2n 1 ) for 1/2 <x 1. 2 n+1 Of course, for each n N, T [n] is closed. We observe that for each n N δ T [n] X) ) ) ) 1 = 2 n 1 + 2n n+1 2 n 0 + 2n 1 2 n+1 = 1 2 n. Clearly, X E 01 X and X E 02 X. Therefore, VT [0] X)) = VX)= 1. Consequently, H T,V = V T [n] X) ) > 0: n 0} N } =1} 1/2 n : n N }. 6.5) We define the faily Ω =ω } of aps ω : H T,V [0, ] by the forulae ω t) = 1/, t H T,V, N. Now, let us observe that, for arbitary and fixed N and n 0} N, by 6.5), we have VT [] T [n] X))) = δt [] T [n] X))) = 1/2 +n < 1/+n) < 1/ = ω δt [n] X))). Therefore, the condition 2.8) holds. We see that ε 0, ) η>0 N t [ε,ε+η) t H T,V ω t) ε}. This iplies the condition 2.4). Concluding, T is a V; Ω)- asyptotic contraction. All assuptions of Theore 2.3 are satisfied. The assertions i) and ii) hold and w = 1/2 is a unique endpoint of T in X. Exaple 6.4. Let X, ) be a copact etric space where X =[0, 1] R, lete 0 =[0, 1) X and let V = V :2 X [0, )} be of the for δe) if E E0 = E, VE)= 3/2)δE 0 ) if E E 0 E.

11 354 K. Włodarczyk, R. Plebaniak / J. Math. Anal. Appl ) Fro Exaple 6.2 it follows that the faily V is a V-faily. We clai that the faily V is not a V-faily. Indeed, for E =[0, 1) we have that E E 0 = E but E E 0 E. Hence VE)= 1 3/2 = VE), so condition V5) does not hold. Let T be of the for 1/2} for 0 x<1/4, [1/2, 1) for x = 1/4, Tx)= 1/2} for 1/4 <x 1/2, 0} for 1/2 <x 1. We observe that T [2] 1/2} for x X 1/4}, x) = 0} for x = 1/4, and T [n] x) =1/2},forn>3, n N. Thus T [3] is closed. We have that X E 0 X, VX)= 3/2)δE) = 3/2, TX) E 0 = TX), VTX))= δt X)) = 1, T [2] X) E 0 = T [2] X), VT [2] X)) = δt [2] X)) = 1/2 and, for n 3, T [n] X) E 0 = T [n] X) and VT [n] X)) = δt [n] X)) = 0. Hence H T,V =VT [n] )X) > 0: n N 0}} = 1/2, 1, 3/2}.IfΠ =π} where π : H T,V 0, ] is of the for πt) = 8/9 t, t H T,V, then ε>0 η>0 t [ε,ε+η) t H T,V πt) ε}, VTX))= 1 < 8/9 3/2 = π3/2)δe 0 )) = πvx)), V T T X))) = 1/2 < 8/9 1 = πδt X))) = πv T X))), VTT [2] X))) = 0 < 8/9 1/2 = πδt [2] X))) = πvt [2] X))). Hence T is a V,Π)-contraction on X. All assuptions of Theore 2.3 are satisfied and T has a unique endpoint w = 1/2. Now we show that T is not a V,Π)-contraction on X with respect to an arbitrary faily V which is a V-faily. Indeed let V be a V-faily, and let there exists Π =π} such that G2) and C2) are satisfied. Then, for E =[0, 1], we have TE)= E. Hence, by V5), we obtain that VE)= VTE)). However, using V5), G2) and C2), we have VTE))= V T E)) < πv E)) VE)= VTE)), which is ipossible. Exaple 6.5. Let X, ) be a copact etric space, where X =[0, 1/2] R.LetT : X 2 X be a not closed thus not upper seicontinuous) set-valued ap of the for 0, 1/2] if x = 0, Tx)= 1/2} if 0 <x 1/2. Let us observe that T [2] x) =1/2} for x X. Therefore, T [2] is closed in X. Let E 0 =[1/4, 1/2] and let V : 2 X [0, ] be of the for δe) if E E0 = E, VE)= E 2 X. δe 0 ) = 1/4 if E E 0 E, By Exaple 6.2b), the faily V =V } is a V-faily. Moreover, we have X E 0 X, T X) E 0 TX) and T [n] X) E 0 = T [n] X) for n 2. Hence V T [n] X) ) δe0 ) = 1/4 ifn 0, 1}, = δt [n] X)) = δ1/2}) = 0 ifn 2, V T [] T [n] X) )) 1/4 if = 1 and n = 0, = 0 if, n N, and H T,V =VT [n] X)) > 0: n 0} N} =1/4}. Consequently, for the Ω-faily Ω =ω } of aps ω : H T,V 0, ], where ω 1/4) = 1/, N, wehave ε 0, ) η>0 N t [ε,ε+η) t H T,V ω t) ε}, VT [1] T [0] X)) = 1/4 < ω 1 V T [0] X)))} = ω 1 1/4) = 1, VT [] T [0] X)) = 0 < ω V T [0] X)))} = ω 1/4) = 1/, 2, VT [] T [1] X)) = 0 < 1/ = ω 1/4) = ω V α T [1] X)))}, 1. Fro the above, we see that T is a set-valued V; Ω)-asyptotic contraction on X. All assuptions of Theore 2.3 are satisfied. The point w = 1/2 is a unique endpoint of T in X. Exaple 6.6. Let X, ) be a copact etric space where X =[0, 1/2] R and let T : X X be of the for 0 forx = 0, Tx)= 1/2 for0<x<1/2, 0 forx = 1/2.

12 K. Włodarczyk, R. Plebaniak / J. Math. Anal. Appl ) Therefore, T [2], T [2] x) = 0forx X,isclosedinX. Let E 0 =0, 1/2} and let V : 2 X [0, ] be of the for δe) if E E0 = E, VE)= E 2 X. δe 0 ) = 1/2 if E E 0 E, By Exaple 6.2b), the faily V =V } is a V-faily. Moreover, we have X E 0 X and T [n] X) E 0 = T [n] X) for n N. Hence V T [n] X) ) δe0 ) = 1/2 if n = 0, = δt X)) = δ0, 1/2}) = 1/2 if n = 1, δt [n] X)) = δ0}) = 0 if n 2, H T,V = V T [n] X) ) > 0: n 0} N } =1/2}, V T [] T [n] X) )) 1/2 if = 1 and n = 0, = 0 if N and n = 0, 0 if, n N. Consequently, for the Ω-faily Ω =ω } of aps ω : H T,V 0, ],ω 1/2) = 1/, N, wehavethatthe following properties hold: ε 0, ) η>0 N t [ε,ε+η) t HT,V ω t) ε }, VT [1] T [0] X)) = 1/2 < ω 1 V T [0] X)))} = ω 1 1/2) = 1, VT [] T [0] X)) = 0 < 1/ = ω 1/2) = ω V T [0] X))), 2, VT [] T [1] X)) = 0 < 1/ = ω 1/2) = ω V α T [1] X)))}, 1. We see that T is a single-valued V; Ω)-asyptotic contraction on X. All assuptions of Theore 2.3 are satisfied. The point w = 0 is a unique fixed point of T in X. Exaple 6.7. Let X, ) be a copact etric space where X =[0, 3] R and let T : X 2 X be of the for 0} for 0 x<1, [0, 1) for x = 1, 1} for 1 <x<2, Tx)= [1, 2) for x = 2, 2} for 2 <x<3, [2, 3) for x = 3. We observe that 0} for 0 x 1, T [2] [0, 1) for 1 <x<2, x) = [0, 1] for x = 2, [1, 2) for 2 <x 3, 0} for 0 x 2, T [3] x) = [0, 1) for x = 2, [0, 1] for 2 <x 3, T [4] 0} for 0 x 2, x) = [0, 1) for 2 <x 3, and T [5], T [5] x) =0} for x X,isclosedinX.LetE 0 =[0, 1] and let V : 2 X [0, ] be of the for δe) if E E0 = E, VE)= E 2 X. δe 0 ) = 1 ife E 0 E, By Exaple 6.2b), the faily V =V } is a V-faily. Moreover, we have T [n] X) E 0 T [n] X) for n 0, 1, 2} and T [n] X) E 0 = T [n] X) for n 3. Hence V T [n] X) ) δe 0 ) = 1 if n 0, 1, 2}, δt = [n] X)) = δ[0, 1]) = 1 if n = 3, δt [n] X)) = δ[0, 1)) = 1 if n = 4, δt [n] X)) = δ0}) = 0 if n 5,

13 356 K. Włodarczyk, R. Plebaniak / J. Math. Anal. Appl ) V T [] T [n] X) )) 1 if N n 0, 1, 2, 3, 4} + n 5, = 0 if N n 5} + n>5, and H T,V =VT [n] X)) > 0: n 0} N}=1}. Consequently, for the Ω-faily Ω =ω } of aps ω : H T,V 0, ], N, where 2 for 0, 1, 2, 3, 4, 5}, ω 1) = 1/ for >5, we have: ε 0, ) η>0 N t [ε,ε+η) t H T,V ω t) ε}; VT [] T [n] X)) = 1 <ω V T [n] X))) = ω 1) = 2if N, n 0, 1, 2, 3, 4} and + n 5; VT [] T [n] X))) = 0 < 1/ = ω 1) = ω V T [n] X))) if N, n 5} and + n>5. Therefore, T is a set-valued V; Ω)-asyptotic contraction on X. All assuptions of Theore 2.3 are satisfied. The point w = 0 is a unique endpoint of T in X. Exaple 6.8. The V; Ω)-asyptotic contraction T : X 2 X, defined in Exaple 6.7, is not a V; Φ)-contraction. Indeed, assue that there exist a V-faily V =V } and a Φ-faily defined by Φ =ϕ : D T,V 0, ]} such that 2.7) and 2.11) hold. Then, in particular, for the set E = X =[0, 3], we have that E CX), TE)=[0, 3) E, VE)>0byV4) and, by V5), VTE))= VTE))= VE). Also, by 2.7) and 2.11), we have VTE))< ϕv E)) VE). Hence, we infer a contradiction. Exaple 6.9. The assuption c) in Theore 2.3 cannot be oitted. Indeed, let X, ) be a copact etric space where X =[0, 2] R,letV =V :2 X [0, )} be a V-faily on X and let T : X 2 X be defined by 1} for x [0, 1/2) 1/2, 1), [1, 3/2] for x = 1/2, Tx)= [1/2, 1] for x = 1, 1/2} for x 1, 2]. Then [1/2, 1) for x [0, 1), T [2] x) = [1/2, 3/2] for x = 1, [1, 3/2] for x 1, 2], T [3] [1/2, 3/2] for x [0, 1], x) = [1/2, 1] for x 1, 2] and T [n] x) =[1/2, 3/2],forx X and n 4, n N. Thus T [4] is closed. Therefore, the assuptions a), b) and d) are satisfied, while c) is not. Indeed, assue that T is a V; Ω)- asyptotic contraction on X. Then there exist the failies V =V } and Ω =ω }, ω : H T,V 0, ], N, satisfying conditions 2.4) and 2.8). Consider n 0 0} N such that VT [n0] X)) > 0, so VT [n0] X)) H T,V ;byv4) such n 0 exists. According to 2.4), for ε = VT [n0] X)), then there exists 0 N such that ω 0 V T [n0] X))) VT [n0] X)). Next, using 2.8), we get VT [0] T [n0] X))) < ω 0 V T [n0] X)). Consequently, since, for each n N, T [n] X) =[1/2, 3/2], therefore VT [n0] X)) = VT [0] T [n0] X))) < ω 0 V T [n0] X))) VT [n0] X)), which is ipossible. This proves that T is not an V; Ω)-asyptotic contraction on X. Of course, w = 1 is a unique fixed point of T but T does not have an endpoint. Exaple The assuption d) in Theore 2.3 cannot be oitted. Indeed, let X, ) be a copact etric space where X =[0, 1] R,letaV-faily V =V :2 X [0, ]} be defined by VE)= δe) = sup x,y E dx,y)}, E X, and let T : X 2 X be defined by 1/2)x + 1/4, 1/2) for 0 x<1/2, Tx)= 3/8} for x = 1/2, 1/2,1/2)x + 1/4) for 1/2 <x 1. First, we observe that 1/4)x + 3/8, 1/2) for 0 x<1/2, T [2] x) = 7/16, 1/2) for x = 1/2, 1/2,1/4)x + 3/8) for 1/2 <x 1,

14 K. Włodarczyk, R. Plebaniak / J. Math. Anal. Appl ) /8)x + 7/16, 1/2) for 0 x<1/2, T [3] x) = 15/32, 1/2) for x = 1/2, 1/2,1/8)x + 7/16) for 1/2 <x 1. Consequently, by induction, for n>1, T [n] x) = 1 2 n x + 2n 1 2 n+1, 1/2) for 0 x<1/2, 2n n+2, 1/2) for x = 1/2, 1/2, 1 2 n x + 2n 1 2 n+1 ) for 1/2 <x 1. Of course, for each n N, T [n] is not closed. We observe that, for each n N, δ T [n] X) ) ) ) 1 = 2 n 1 + 2n n+1 2 n 0 + 2n 1 2 n+1 = 1 2 n. Consequently, H T,V = V T [n] X) ) > 0: n 0} N } =1} 1/2 n : n N }, 6.6) where T [0] X) = X. We define the Ω-faily Ω =ω } of aps ω : H T,V 0, ] by the forulae ω t) = 1/, t H T,V, N. Now, for arbitrary and fixed N and n 0} N, by 6.6) we have VT [] T [n] X))) = δt [] T [n] X))) = 1/2 +n < 1/+n) < 1/ = ω δt [n] X))). This gives that T satisfies condition 2.8). Moreover, ε 0, ) η>0 N t [ε,ε+η) t H T,V ω t) ε}, sot satisfies also condition 2.4). Concluding, T is an V; Ω)-asyptotic contraction on X. We obtain that assuptions a), b), c) hold, but d) does not hold. The ap T does not have an endpoint. It is worth noticing that each sequence w }, where w T [] w 0 ) for N and w 0 X, converges to w = 1/2. Definition 6.1. See Berge [5, p. 111].) Let X, T ) be a set-valued dynaic syste. The ap T is called upper seicontinuous at x 0 X if for each open set G containing Tx 0 ) there exists a neighbourhood Ux 0 ) of x 0 such that Tx) G for each x Ux 0 ) and upper seicontinuous in X if it is upper seicontinuous at each point x of X and Tx)is copact for each x X. Reark 6.1. i) It is well known that every upper seicontinuous ap is closed [5, Theore 6, p. 112] and, if X is a copact space, then the ap is closed if and only if it is upper seicontinuous [5, Corollary, p. 112]. ii) By Reark 6.1, Theore 2.3 holds if assuption d) is replaced by assuption d ) of the for: d ) T [p] is upper seicontinuous in X for soe p N. iii) It is known that every topological vector space is copletely regular and therefore uniforisable. If X is a locally convex space with a saturated faily of seinors p α : α A}, then we can define a faily of pseudoetrics d α x, y) = p α x y). The unifor topology obtained coincides with the original topology of the space X. Therefore, Theore 2.3 also holds in Hausdorff coplete locally convex spaces and coplete etric spaces. iv) Our definitions and results are new for set-valued aps in unifor, locally convex and etric spaces. They are new even for single-valued aps. References [1] J.P. Aubin, I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, New York, [2] J.P. Aubin, H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, [3] J.P. Aubin, J. Siegel, Fixed points and stationary points of dissipative ultivalued aps, Proc. Aer. Math. Soc ) [4] S. Banach, Sur les opérations dans les ensebles abstraits et leurs applications aux équations intégrales, Fund. Math ) [5] C. Berge, Topological Spaces, Oliver and Boyd, Edinburgh, [6] M. Justan, Iterative processes with nucleolar restrictions, Int. J. Gae Theory ) [7] E. Klein, A.C. Thopson, Theory of Correspondences, John Wiley & Sons, New York, 1984.

15 358 K. Włodarczyk, R. Plebaniak / J. Math. Anal. Appl ) [8] M. Maschler, B. Peleg, Stable sets and stable points of set-valued dynaic systes with applications to gae theory, SIAM J. Control Opti ) [9] E. Tarafdar, R. Vyborny, A Generalized Multivalued) Contraction Mapping Principle, Res. Rep. Pure Math., vol. 54, The University of Queensland, Australia, [10] E. Tarafdar, G.X.-Z. Yuan, Set-valued topological contractions, Appl. Math. Lett ) [11] G.X.-Z. Yuan, KKM Theory and Applications in Nonlinear Analysis, Marcel Dekker, New York, 1999.