FIXED-POINT AND COINCIDENCE THEOREMS FOR SET-VALUED MAPS WITH NONCONVEX OR NONCOMPACT DOMAINS IN TOPOLOGICAL VECTOR SPACES KAZIMIERZWŁODARCZYK AND DOROTA KLIM

A technique, based on the investigations of the images of maps, for obtaining fixed-point and coincidence results in a new class of maps and domains is described. In particular, we show that the problem concerning the existence of fixed points of expansive set-valued maps and inner set-valued maps on not necessarily convex or compact sets in Hausdorff topological vector spaces has a solution. As a consequence, we prove a new intersection theorem concerning not necessarily convex or compact sets and its applications. We also give new coincidence and section theorems for maps defined on not necessarily convex sets in Hausdorff topological vector spaces. Examples and counterexamples show a fundamental difference between our results and the well-known ones.

In the past decade, there was a renewed interest in the fixed-point and coincidence theory of set-valued maps in topological vector spaces (see, e.g., [2,3,8,9,10,11,12,28,36,37,38,39,40,42,43,44,45,46,47,48]), partially due to new and powerful methods of investigations introduced into it (notably based on those introduced by Fan and Browder).Most of the work has centered around the fixed-point and coincidence theory of maps on convex compact sets, but there are also a considerable number of papers devoted to maps on nonconvex and noncompact sets (see, e.g., [8,45]).
There exist a number of introductions to and surveys of fixed-point and coincidence theory.We mention [47] among the more recent ones but also some elder ones [14,49,50].See also many references therein.
A natural question arises: whether expansive set-valued maps and inner setvalued maps on not necessarily convex or compact sets have fixed points and, as a consequence, theorems of intersection type hold and whether the maps F : C → 2 E and G : C → 2 E on not necessarily convex sets in Hausdorff topological vector spaces have coincidences.The affirmative answers are given in this paper.Using a technique based on the investigation of the images of maps, we obtain a number of new fixed-point, coincidence, intersection, and section theorems of Fan-Browder type.Examples and counterexamples show a fundamental difference between our results and the known results of the above-mentioned authors.

Fixed points and coincidences of expansive set-valued maps on not necessarily convex or compact sets in topological vector spaces
Let C be a subset of a Hausdorff topological vector space E over K (K = R or C).
A set-valued map F : C → E (which will always be denoted by capital letters) is a map which assigns a unique F(c) ∈ 2 E (here 2 E denotes the family of all subsets of E) to each c ∈ C. We say that c ∈ C is a fixed point of F : We say that a map F : Maps in the usual sense will be considered as special (single-valued) setvalued maps and these ordinary maps will always be denoted by small letters f : C → E.
We prove the following theorem.
Theorem 2.1.Let C be a nonempty subset of a Hausdorff topological vector space E over R, let F : C → 2 E , and let K be a convex subset of E. Assume that the following conditions hold: Then there exists u ∈ C such that u ∈ F(u).
Proof.By (iii), the compact set F(C) is covered by the sets F(c), c ∈ C, which are open in F(C).Clearly, there exists a finite set {c 1 ,...,c n } ⊂ C such that F(c i ) are nonempty, 1 ≤ i ≤ n, and F(C) = n i=1 F(c i ).Let {ϕ 1 ,...,ϕ n } be a partition of unity with respect to this cover, that is, a finite family of real-valued nonnegative continuous maps ϕ i on F(C) such that ϕ i vanish outside F(c i ) and are less than or equal to one everywhere, 1 ≤ i ≤ n, and n i=1 ϕ i (y) = 1 for all y ∈ F(C).Let σ be a simplex spanned by points c 1 ,...,c n and let ϕ : F(C) → σ be a continuous map defined by the formula ϕ(y If y ∈ K is arbitrary and fixed and ϕ i (y) = 0 for some i ∈ {1, ...,n}, then y ∈ F(c i ), so c i ∈ F −1 (y).As a consequence, for each y ∈ K, ϕ(y) is a convex linear combination of points of F −1 (y) and by (iv), we get for each From Brouwer's theorem, we get u = ϕ(u) for some u ∈ σ and hence, since σ ⊂ K, by (2.1), u = ϕ(u) ∈ F −1 (u) ⊂ C, and therefore, u ∈ F(u) and u ∈ C, as required.
By using various sets K, a number of variations of Theorem 2.1 can be obtained, of which the following two are typical.
Theorem 2.2.Let C be a nonempty subset of a Hausdorff topological vector space E over R and let F : C → 2 E .Assume that the following conditions hold: Then there exists u ∈ C such that u ∈ F(u).
Proof.We use Theorem 2.1 for K = F(C).
Theorem 2.3.Let C be a nonempty subset of a Hausdorff topological vector space E over R and let F : C → 2 E .Assume that the following conditions hold: Proof.Indeed, if ϕ and σ are as in the proof of Theorem 2.1 and C is convex, then σ ⊂ C ⊂ F(C), and we may use Theorem 2.1 for let T be a closed triangle with vertices (0,0), (1,0) and (0,1), and let If in Theorems 2.2 or 2.3 we omit at least one of the assumptions, then we can construct a counterexample.
We say that a single-valued map f : C → E and a set-valued map F : The following theorem is a generalization of the above one.
Theorem 2.6.Let C be a nonempty convex subset of a Hausdorff topological vector space E over R, let F : C → 2 E be an expansive map, and let f : C → E be a single-valued continuous map such that f (C) ⊂ F(C).Assume that the following conditions hold: Proof.Let ϕ and σ be as in the proof of Theorem 2.1.We have ϕ : F(C) → σ, σ ⊂ C, and y ∈ F(ϕ(y)) for each y ∈ f (C).On the other hand, ϕ • f : σ → σ and, by the theorem of Brouwer, (ϕ

Fixed points and coincidences of set-valued inner maps on not necessarily convex sets in topological vector spaces
We say that a map This section is devoted to new fixed-point and coincidence theorems for set-valued inner maps on not necessarily convex sets.
We have the following theorem.
Theorem 3.1.Let C be a nonempty compact subset of a Hausdorff topological vector space E over R and let F : C → 2 E be an inner map such that F(C) is a convex subset of E. Assume that the following conditions hold: Then there exists u ∈ C such that u ∈ F(u).
Proof.In virtue of (i) and (ii), there exists a finite set {y 1 ,..., Let {ϕ 1 ,...,ϕ n } be a partition of unity with respect to this cover, let σ be a simplex spanned by points y 1 ,..., y n , and let a continuous map ϕ : C → σ be defined by the formula On the other hand, from Brouwer's theorem, we get that u = ϕ(u) for some u ∈ σ and, by (3.1), we have u = ϕ(u) ∈ F(u), as required.
6 Fixed-point and coincidence theorems Recall that a map F : C → 2 E is called upper semicontinuous if, for each c ∈ C and any open set V containing F(c), there is an open set U containing c such that F(U ∩ C) ⊂ V (for details, see [4]).A map F : C → 2 E is called upper demicontinuous on C (after Fan [20]) if, for each c ∈ C and any open half-space H in E containing F(c), there is a neighbourhood N(c) of c in C such that F(x) ⊂ H for each x ∈ N(c).It is clear that the condition of upper semicontinuity is stronger than that of upper demicontinuity.
Let C and D be nonempty sets.The maps F : We now establish the following theorem.
Theorem 3.2.Let C be a nonempty compact subset of a Hausdorff locally convex topological vector space E over R, let F : C → 2 E be an inner map, and let G : C → 2 E be an upper demicontinuous map such that G(C) ⊂ F(C).Assume that the following conditions hold: Then there exists Proof.Let ϕ and σ be as in the proof of Theorem 3.1.Since σ ⊂ F(C) ⊂ C and ϕ : and G • ϕ is upper demicontinuous on the compact convex set F(C).By [20,Theorem 6], there exists v ∈ F(C) such that v ∈ G(ϕ(v)).Moreover, in virtue of (3.1), ϕ(v) ∈ F(v).This implies the assertion for u = ϕ(v).

Intersection theorem with applications on not necessarily convex or compact sets in topological vector spaces
Various intersection theorems concerning convex and compact sets, with their applications, are given in [6,17,18,21,22,33,35].From Theorem 2.2, we get the following new intersection theorem.
It follows from Theorem 2.2 that u ∈ F(u) for some u ∈ C.This shows that, for each i ∈ {1, ...,n}, u = (u 1 ,...,u i−1 ,u i ,u i+1 ,...,u n Hence C 1 is a noncompact and nonconvex subset of K 1 , the assumptions of Theorem 4.1 are satisfied and C ∩ 2 i=1 S i = ∅.As an application of Theorem 4.1 we obtain the following theorem. Theorem 4.3.Let E be a Hausdorff topological vector space over R and let n ≥ 2. Let C 1 ,...,C n be nonempty (not necessarily convex or compact) subsets of E, let K 1 ,...,K n be convex compact subsets of E, and let C = n j=1 C j , K = n j=1 K j .Let f 1 ,..., f n be real-valued maps defined on K, let t 1 ,...,t n be real numbers, and let the following conditions hold: and for each point (y 1 ,..., y i−1 , y i+1 ,..., y n ) of n j =i K j , the set {c i ∈ C i : f i (y 1 ,..., y i−1 ,c i , y i+1 ,..., y n ) > t i } is a nonempty convex subset of C i ; (iii) for each i, 1 ≤ i ≤ n, and for each point c i ∈ C i , the set y 1 ,..., y i−1 , y i+1 ,..., y n ∈ is an open subset of n j =i K j .Then there is a point u in C such that f i (u) > t i for each i, Proof.Define the subsets S i of K to be S i = {y : y ∈ K, f i (y) > t i }, i ∈ {1, ...,n}.Clearly, (ii) is equivalent to the condition: (ii ) for each i ∈ {1, ...,n} and for each point (y 1 ,..., y i−1 , y i+1 ,..., y n ) of n j =i K j , the section S i (y 1 ,..., y i−1 , y i+1 ,..., y n ), formed by all points c i ∈ C i such that (y 1 ,..., y i−1 ,c i , y i+1 ,..., y n ) ∈ S i , is a nonempty convex subset of K i , and (iii) is equivalent to the condition: (iii ) for each i∈{1,...,n} and for each point c i ∈ C i , the section S i (c i ), formed by all points (y 1 ,..., y i−1 , y i+1 ,..., y n ) of n j =i K j such that y 1 ,..., y i−1 ,c i , y i+1 ,..., y n ∈ S i , ( is an open subset of n j =i K j .We can apply Theorem 4.1 to obtain C ∩ n i=1 S i = ∅.Hence, by the definition of S i , the point u from this intersection satisfies f i (u) > t i for each i ∈ {1, ...,n}.

Coincidence theorems for set-valued maps and section theorems on not necessarily convex sets in topological vector spaces
Using his infinite-dimensional version of the KKM theorem as a tool, Fan [16] established a geometrical "lemma" concerning convex and compact sets.Next, Browder [6] restated it in the more convenient form of a fixed-point theorem.
A weaker form (with a relaxed compactness assumption) of this theorem was afterwards obtained by Fan [21].Finally, Lassonde [33] extended these results.He gave a proof of the following interesting coincidence theorem: Theorem 5.1.Let X be a convex space (i.e., a convex set in a vector space with any topology that induces the Euclidean topology on the convex hulls of its finite subsets), Y a topological space, and F the map of X into 2 Y for which the following conditions hold:

nonempty and convex;
(iii) for some c-compact set K ⊂ X, the set Y \ x∈K F(x) is compact.Then, for each single-valued continuous map f of X into Y , there exists an x ∈ X such that f (x) ∈ F(x).
Theorem 5.2.Let C be a nonempty compact set in a Hausdorff topological vector space E over R and let f : Then there exists a point u ∈ C such that f (u) ∈ F(u).Section theorems concerning convex compact sets in Hausdorff topological vector spaces, with various applications, are given by Fan [18,20].In the proof of Theorem 5.2, we need the following two new auxiliary section theorems of Fan type.
Theorem 5.4.Let C be a nonempty compact set (not necessarily convex) in a Hausdorff topological vector space E over K. Let f : C → E and g : C → E be continuous maps on C, and let f (C) be convex.Let K be a subset of g(C) × f (C) having the following properties: (i) for each fixed w ∈ f (C), the set {t ∈ C : (g(t),w) ∈ K} is closed in C; (ii) for each t ∈ C, (g(t), f (t)) ∈ K; (iii) for any fixed t ∈ C, the set {w ∈ f (C) : (g(t),w) / ∈ K} is convex (or empty).
Then there exists a point Proof.We use KKM set-valued maps.Define a map H : f (C) → 2 E as follows: Obviously, by (i), H(w) is a compact subset of C and thus f (H(w)) is a compact subset of f (C) for each w ∈ f (C).Let {w 1 ,...,w m } be any finite and fixed subset of f (C).We prove that conv{w 1 ,..., To this goal, we assume that f (s ∈ H(w i ) for all i = 1,...,m, that is, (g(s),w i ) / ∈ K for any i = 1,...,m.Therefore, by (iii), w i , i = 1,...,m, are contained in a convex set U = {w ∈ f (C) : (g(s),w) / ∈ K}.Consequently, conv{w 1 ,...,w m } ⊂ U and, in particular, f (s) ∈ U, that is, (g(s), f (s)) / ∈ K, which, by (ii), is impossible.We must have f (s Then there exists a point Proof.Here, B denotes a complement of the set K in g(C) × f (C) where K is defined in Theorem 5.4 Proof of Theorem 5.2.We define a set B = {(c, y) ∈ C × f (C) : y ∈ F(c)} and apply Theorem 5.5 for g = I E .

Coincidences for upper semicontinuous set-valued maps on not necessarily convex sets in locally convex spaces
Let F : (a) Then either F and G have a Φ-coincidence or there exist p ∈ Γ and λ > 0 (b) Then either F and G have a Φ-coincidence or there exists p ∈ Γ and, for any c ∈ C and any u ∈ G(c), there exists v ∈ F(c) such that K. Włodarczyk and D. Klim 11 Indeed, for an arbitrary and fixed w ∈ Φ(G(c) × F(c)), there exists p w ∈ Γ such that p w (w) = 0 and, by the continuity of p w , there exist a neighbourhood M w of w and µ w > 0 such that µ w = Inf{p w (t) : t ∈ M w }.Since the family {M w : , there exists a finite subset {w 1 ,...,w m } of Φ(G(c) × F(c)) such that the family {M wi : i = 1,2,...,m} covers Φ(G(c) × F(c)) and we may assume that Min µ wi : i = 1,...,m .( Now we prove that (ii) for each c ∈ C, there exist p c ∈ Γ, λ c > 0, and a neighbourhood W c of c, such that Indeed, let c ∈ C be arbitrary and fixed and we define open sets A c and B c as follows: where p c and λ c are as in (i).Since F(c  p(Φ(u,•)) attains its minimum on a compact set
In this section, we will give further applications of Theorem 5.4.In particular, we derive some minimax theorem (Theorem 7.1), Hartman-Stampacchia type variational inequalities (Theorem 7.2), and a theorem of Iohvidov type (Theorem 7.3(b)) for maps on not necessarily convex sets.One of them will be used later to prove new results concerning Φ-coincidences and Φ-fixed points (in particular, coincidences and fixed points) of continuous single-valued maps on not necessarily convex sets (Theorem 7.4).
A real map ψ, defined on a topological vector space E, is said to be lower semicontinuous (upper semicontinuous) on E if, for each real number µ, the set {x ∈ E : ψ(x) > µ} ({x ∈ E : ψ(x) < µ}) is open.
A real map, ψ defined on a convex set A of a vector space E, is said to be quasiconcave (quasi-convex) on A if, for each real number µ, the set {a ∈ A : ψ(a) > µ} ({a ∈ A : ψ(a) < µ}) is convex.
As a consequence of Theorem 5.4, we obtain the following theorem.
Theorem 7.1.Let C be a nonempty compact set (not necessarily convex) in a Hausdorff topological vector space E over K. Let f : C → E and g : C → E be continuous maps on C and let f (C) be convex.
If E is a locally convex Hausdorff topological vector space over K and E denotes the topological vector space of continuous linear functionals on E, let λ; x denote the pairing between λ in E and x in E. Now, we show the following theorem.
is the set of all continuous seminorms p α on E, α ∈ Z, {p α1 , p α2 ,..., p αn } is a finite subset of Γ and p α = p α1 + p α2 + ••• + p αn , then there exists at least one c ∈ C such that, for each w ∈ f (C), Let, additionally, Φ(g(s), f (s)) = 0 for all s ∈ C. Then there exists c ∈ C such that Let, additionally, Φ(g(s), f (s)) = 0 for all s ∈ C. Then there exists c ∈ C such that Then Ψ satisfies the conditions of (iv) and (v) and, consequently, there exists Our new coincidence theorem does not require convexity.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

Remark 5 . 3 .
If f = I E (the identity map) and C is convex, then Theorem 5.2 becomes the Browder theorem [6, Theorem 1].However, his method of proving this fact (based on the partition of unity) is absolutely different from ours.
By virtue of [16, Lemma 1, page 305], this yields f (c) ∈ { f (H(w)) : w ∈ f (C)} for some c ∈ C and we conclude that {g(c)} × f (C) ⊂ K for some c ∈ C. Theorem 5.5.Let C be a nonempty compact set (not necessarily convex) in a Hausdorff topological vector space E over K. Let f : C → E and g : C → E be continuous maps on C, and let f (C) be convex.Let B be a subset of g(C) × f (C) and suppose that (i) for each fixed y ∈ f (C), the set {c ∈ C : (g(c), y) ∈ B} is open in C; (ii) for any fixed c ∈ C, the set {y ∈ f (C) : (g(c), y) ∈ B} is nonempty and convex.
and G are upper semicontinuous, there exist neighbourhoods U c and V c of c, such that F(x) ⊂ A c for x ∈ U c ∩ C, and G(y) ⊂ B c for y ∈ V c ∩ C. Consequently, we may assume that W c = U c ∩ V c .Finally, for each c ∈ C, let p c , λ c , and W c be as in (ii).Since the family {W c : c ∈ C} is an open cover of a compact set of C, there exists a finite subset {c 1 ,..., c n } of C such that the family {W ci : i = 1,...,n} covers C and we may assume that p = Max p ci : i = 1,...,n , λ= Min λ ci : i = 1,...,n .(6.6) (b) If F and G do not have a Φ-coincidence in C, let p and λ be as in (a) and let c ∈ C be arbitrary and fixed.Observe that, for any u ∈ G(c), the continuous map and G = −F.Then F and G satisfy the assumptions of Theorem 6.1(b) for Φ defined byΦ(u,v) = u − v, (u,v) ∈ G(c) × F(c), c ∈ C.The sets C, F(C), G(C), F(c), and G(c) are nonconvex for all c ∈ C.Moreover, C ⊂ F(C), C ⊂ G(C), the sets F(C) and G(C) are not contained in C and any c ∈ C is a coincidence of F and G.