J. Fixed Point Theory Appl. 38:02)1( https://doi.org/10.1007/s11784-018-0562-1 c Springer International Publishing AG,  part of Springer Nature 2018

Journal of Fixed Point Theory and Applications

Some fixed point theorems for s-convex subsets in p-normed spaces based on measures of noncompactness Jian-Zhong Xiao and Ying Lu Abstract. The Kurotowskii and Hausdorff measures of noncompactness in p-normed spaces are studied, where 0 < p ≤ 1. By metric retraction and calculation of measures of noncompactness, some fixed point theorems for s-convex subsets are proved, with respect to single-valued and set-valued condensing operators, where 0 < s ≤ p. The results herein extend and generalize some of the well-known fixed point theorems, such as the types of Darbo, Sadovski, Leray-Schauder, and Martelli. Mathematics Subject Classification. 47H10, 46A16. Keywords. Kurotowskii measures of noncompactness, Hausdorff measures of noncompactness, p-normed space, s-convex set, fixed point theorem, condensing operators.

1. Introduction Let X be a linear space over K with the origin θ, where K is the field of real numbers or complex numbers. A p-norm on X is a nonnegative real-valued functional  · p on X with 0 < p ≤ 1, satisfying the following conditions: (a) xp = 0 if and only if x = θ; (b) λxp = |λ|p xp , for all x ∈ X, λ ∈ K; (c) x + yp ≤ xp + yp , for all x, y ∈ X. For p-normed space (X,  · p ), if p = 1, then it is an usual normed space. A p-normed space is also a metric linear space with a translation invariant metric d, where d is defined by d(x, y) = x − yp for x, y ∈ X. On the other hand, if (X,  · ) is an usual normed space, 0 < p < 1 and  · p =  · p , then (X,  · p ) is a p-normed space. It is a basic fact that the topology of every Hausdorff locally bounded topological linear space is given by some p-norm. This work is supported by the Natural Science Foundation of Nanjing University of Information Science and Technology (Grant no. 2014x024).

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The space Lp (μ) is a p-normed space based on the complete measure space (Ω, M, μ) with the p-norm given by  f (t)p = |f (t)|p dμ, for f ∈ Lp (μ), Ω

where Ω is a nonempty set, M is a σ-algebra in Ω, μ : M → [0, +∞) is a positive measure, and    Lp (μ) = f | f : Ω → K is measurable, and |f (t)|p dμ < +∞ . Ω

If μ is either the Lebesgue measure on Ω = [0, 1] or the counting measure on Ω = {1, . . . , n}, then the corresponding space is denoted by Lp [0, 1] and lp (n), respectively. The class of p-normed spaces (0 < p ≤ 1) is an important generalization of usual normed spaces, and it has a rich topological and geometrical structure. The related study has received a lot of attention (see [8,9,12,15,32,33,36] and the references cited therein). If 0 < p < 1, then the above p-norm is nonhomogeneous. The space 1/p p L (μ) can be equipped by a homogeneous quasi-norm  ·  =  · p and hance become a quasi-Banach space. The quasi-Banach spaces arise also in a natural way as a generalization of Banach spaces, where the triangular inequality of the norm is changed by a weaker condition. For the study of geometrical and topological aspects concerning these spaces, very interesting and important results were obtained (see [1–5,19,26,29]). In general, the p-normed space (0 < p < 1) is non-locally convex, the unit ball with center θ is not a convex set. This determines that there is much difference between a p-normed space (0 < p < 1) and an usual normed space. It is pointed out by Ding [12] and Rudin [27] that there is not open convex subset in Lp [0, 1] (0 < p < 1) except Lp [0, 1] itself. As we know, most of fixed point theorems are concerned with convex sets, but there exist some fixed point problems concerning non-convex sets in applications. It is a quite natural question whether the well-known fixed point theorems could be extended to non-convex sets. Recently, several researchers started the work for these problems. For examples, Gholizadeh et al. [17] extended the KKM theory on p-convex subsets in locally p-convex spaces; Xiao and Zhu [38] established some fixed point theorems, such as Schaudertype and Kakutani-type, on p-convex subsets in p-normed spaces; Alghamdi et al. [6] proved several Krasnosel’skii type fixed point theorems in nonconvex situations and gave an application for perturbed integral equation in p-normed spaces. We would like to focus on this direction of research. In this paper, we investigate the existence of fixed points of condensing operators on s-convex sets in p-normed spaces over the field R of real numbers, where 0 < p ≤ 1, 0 < s ≤ p. Motivated by the famous work in classical Banach spaces and locally convex spaces, due to Darbo [11], Sadovski [28], Su and Sehgal [31], O’Regan [23,24] and Martelli [22], in Sect. 3 we study the Kurotowskii and Hausdorff measures of noncompactness in p-normed spaces (0 < p ≤ 1); in Sect. 4 we prove some fixed point theorems for s-convex subsets (0 < s ≤ p) with respect to single-valued and set-valued condensing

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operators. The results herein extend and generalize some of the well-known fixed point theorems. Each of them include the case of non-convex sets in usual normed spaces as their special cases. As an application of our results, in Sect. 5, the existence of solutions for games on non-convex sets is considered in p-normed spaces.

2. Preliminaries In the sequel we denote the interior, the closure, the boundary of a subset A of X by Ao , A, and ∂A, respectively. Z+ stands for the set of positive integer. By B(x, r) we mean the open ball of X with center x ∈ X and radius r > 0. For the sake of convenience, we recall some definitions (see [8,25,39– 41]). Regarding a p-normed space as a metric space, we have the notions of completeness, boundedness, relatively compactness and compactness etc. Let (X, d), (Y, d) be two metric spaces and T : X → Y an operator. T is said to be bounded if for each bounded set A ⊂ X, T (A) is a bounded set of Y . T is said to be continuous if for every x ∈ X, limn→∞ xn = x implies limn→∞ T xn = T x. T is said to be completely continuous if T is continuous and T (A) is relatively compact for each bounded subset A of X. Let A1 , A2 ⊂ X be bounded. Then the Hausdorff metric dH (A1 , A2 ) between A1 and A2 is defined by   dH (A1 , A2 ) = max sup inf d(x, y), sup inf d(x, y) . x∈A1 y∈A2

y∈A2 x∈A1

Let T : X ⇒ Y be a set-valued operator acting from X to {A ⊂ Y : A = ∅}. T is said to be compact (quasicompact) if T (A) is relatively compact for each bounded (relatively compact) subset A of X. T is said to be upper semicontinuous, if for each nonempty open set U ⊂ Y , the set T − (U ) = {x ∈ X : T (x) ⊂ U } is open in X. Equivalently, for each nonempty closed set V ⊂ Y , the set T + (V ) = {x ∈ X : T (x) ∩ V = ∅} is closed in X. The graph of T is the set GT = {(x, y) ∈ X × Y : y ∈ T x}. T is said to have closed graph if its graph GT is a closed subset of X × Y . The Hausdorff and Kurotowskii measures βH and βK of noncompactness of nonempty, bounded subset D in X are the nonnegative real numbers βH (D) and βK (D) defined by βH (D) = inf{ε > 0 : D has a finite ε -net};  βK (D) = inf

ε>0:D⊂

n 

 Di with Di is bounded and diam Di ≤ ε; n ∈ Z

+

.

i=1

It is known that βH ≤ βK ≤ 2βH . Let T be an operator from D ⊂ X to X. T is called a k-set contraction with respect to βK (or βH ) if there is a number k ∈ [0, 1) such that βK (T A) ≤ kβK (A) (or βH (T A) ≤ kβH (A)) for all bounded sets A in D. T is said to be βK -condensing (or βH -condensing) if βK (T A) < βK (A) (or βH (T A) < βH (A)) for all bounded sets A in D with βK (A) = 0 (or βH (A) = 0). It is evident that, if T is a compact operator, then T is a k-set contraction; if T is a k-set contraction, then T is condensing. The

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following relation, between the closed graph property and the upper semicontinuity of an operator, holds: Lemma 2.1. ([13,25]) Let X, Y be two metric spaces and T : X ⇒ Y a setvalued operator. If T is upper semi-continuous and T x is closed for x ∈ X, then T has closed graph. Conversely, if T is quasicompact and has closed graph, then T is upper semi-continuous. Let X be a real linear space and 0 < s ≤ 1. We now state some notion concerning s-convex set and s-convex function (see [8,10,14,21,36]). A set C ⊂ X is said to be s-convex if the following condition is satisfied: t1 x + t2 y ∈ C, whenever x, y ∈ C, t1 , t2 ∈ [0, 1], ts1 + ts2 = 1. n n A point x ∈ X such that x = i=1 ti xi with xi ∈ X, ti ≥ 0, i=1 tsi = 1, is called a s-convex combination of x1 , x2 , . . . , xn . C is s-convex if and only if it contains all the s-convex combinations of its elements. Let C be a s-convex subset of X, the functional f : C → (−∞, +∞) is said to be s-convex if f (t1 x + t2 y) ≤ ts1 f (x) + ts2 f (y), for x, y ∈ C, t1 , t2 ∈ [0, 1], ts1 + ts2 = 1. Let A ⊂ X. The s-convex hull of A denoted by cos A is the smallest s-convex set containing A. It is given by  n  n s cos A = ti xi : ti ≥ 0, ti = 1, xi ∈ A, n ≥ 2 . i=1

i=1

For s = 1 one obtains the usual definition of convex sets and convex functionals. The following theorem for convex hulls will be needed: Lemma 2.2 (Carath´eodory Theorem, [34]). Let X be a real linear space and A ⊂ X. Let the affine hull affA be an n-dimensional subset. Then any x ∈ coA can be represented as a convex combination of n + 1 elements of A. Lemma 2.3 ([34]). Let X be a real linear space and A = {x0 , x1 , x2 , . . . , xn } ⊂ X. Let the affine hull affA be an n-dimensional subset. Then any x ∈ coA can be uniquely represented as a convex combination of x0 , x1 , x2 , . . . , xn (in this case coA is called a simplex). For the p-convex set C with θ ∈ C, the functional qC : X → [0, +∞] defined by qC (x) = inf{t > 0 | x ∈ t1/p C} is called the Minkowski p -functional of C. If A ⊂ X, then by qC (A) we mean that qC (A) = {qC (x) : x ∈ A}. But in a p-normed space space, there is much difference between a convex set and a s-convex set. A s-convex set is not translation-invariant in the case 0 < s < 1. If C is s-convex (0 < s < 1) and x0 = θ, then C − x0 is not s-convex in general. For the one-point set {x} and 0 < s < 1, we have 

cos {x} =

{tx : 0 < t ≤ 1}, if x = θ, and cos {x} = θ, if x = θ;



{tx : 0 ≤ t ≤ 1}, if x = θ, θ, if x = θ.

It is evident that if C is a closed s-convex set (0 < s < 1), then θ ∈ C.

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Lemma 2.4 ([36]). Let (X,  · p ) (0 < p ≤ 1) be a p-normed space and B a subset of X. Let f be a continuous and p-convex functional on X, where 0 < p ≤ 1. Then sup{f (x) : x ∈ cop B} = sup{f (x) : x ∈ B}. Lemma 2.5 ([8,10,21]). Let (X,  · p ) (0 < p ≤ 1) be a p-normed space and 0 < s ≤ p. (1) The ball B(θ, r) is s-convex, where r > 0. (2) If C ⊂ X is s-convex and α ∈ R, then αC is s-convex. (3) If C1 , C2 ⊂ X are s-convex, then C1 + C2 is s-convex.

(4) If {Ci : i ∈ I} is a family of s-convex sets of X, then i∈I Ci is sconvex. (5) If A ⊂ X and θ ∈ A, then cos A ⊂ coA, where coA is the convex hull of A. (6) If C is a closed s-convex set and 0 < r < s, then C is a closed r-convex set. Lemma 2.6 ([21,35,36]). Let (X, ·p ) be a p-normed space and C a s-convex subset of X with θ ∈ C, where 0 < p ≤ 1, 0 < s ≤ p. Let qC be the Minkowski s-functional of C. Then (1) qC (θ) = 0; (2) qC is positively s-homogeneous, i.e., qC (tx) = ts qC (x), for every x ∈ X and t > 0; (3) qC is sub-additive, i.e., qC (x + y) ≤ qC (x) + qC (y), for every x, y ∈ X; (4) if C is bounded, then qC (x) > 0 for every non-zero point x ∈ X; (5) if C is closed, then qC (x) is lower semi-continuous and C = {x ∈ X : qC (x) ≤ 1}; (f ) if C is absorbing, then qC (x) < +∞ for every x ∈ X; (6) if θ ∈ C o , then qC is continuous, and C o = {x ∈ X : qC (x) < 1}, and C = {x ∈ X : qC (x) ≤ 1}. Lemma 2.7 ([38]). (Mazur-type) Let (X,  · p ) (0 < p ≤ 1) be a complete pnormed space and A a relatively compact subset of X. Then cos A (0 < s ≤ p) is compact. Lemma 2.8 ([38]) . Let (X, ·p ) be a p-normed space and C a bounded closed s-convex subset of X with θ ∈ C o , where 0 < p ≤ 1, 0 < s ≤ p. For every x ∈ X define an operator by x r(x) = , max{1, (qC (x))1/s } where qC is the Minkowski s-functional of C. Then C is a retract of X and r : X → C is a continuous operator such that (1) if x ∈ C, then r(x) ≡ x; (2) if x ∈ C, then r(x) ∈ ∂C. Our results concerning condensing operators are based on the following fixed point theorems:

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Figure 1. The diametral points in closed unit ball of l1/2 (2) Theorem 2.9. ([38]) (Schauder-type) Let (X,  · p ) be a complete p-normed space and C a compact s-convex subset of X, where 0 < p ≤ 1, 0 < s ≤ p. If T : C → C is continuous, then there exists z ∈ C such that T z = z. Theorem 2.10. ([38]) (Kakutani-type) Let (X,  · p ) be a complete p-normed space and C a compact s-convex subset of X, where 0 < p ≤ 1, 0 < s ≤ p. If T : C ⇒ C is upper semi-continuous, and T x is closed s-convex for every x ∈ X , then there exists z ∈ C such that z ∈ T z.

3. The measures of noncompactness in p-normed spaces In this section we present some properties of the Kurotowskii and Hausdorff measures of noncompactness in p-normed spaces. In the case of 0 < p < 1, the measure of noncompactness is nonhomogeneous, and it is different from the ones introduced by Silva et al. [30] and Alghamdi et al. [6]. From the following examples we see that some usual properties which are true in the locally convex setting may be false in the non-locally convex setting: Example 3.1. We take the closed unit ball B of l1/2 (2) (see Fig. 1). Let x = (1, 0), y = (0, 1), z = (−1, 0). Then √ x − z1/2 = 2 < diam(B); x − y1/2 = 2 = diam(B). This means that x, y are diametral points, but x, z are not. Example 3.2. For a subset A in p-normed space (0 < p ≤ 1), it is possible that diam(coA) > diamA. Let A = {θ, a, b} is a subset of l1/2 (2) (see Fig. 2), where a = (1, 0), b = (9/16, 1/16). Then a1/2 = b1/2 = 1, a − b1/2 = √ ( 7 + 1)/4 < 1, h = (a + b)/2 ∈ coA, and so √ diam(coA) ≥ h1/2 = 3 2/4 > 1 = diamA. From Lemmas 2.2 and 2.3 we obtain the following Lemmas 3.1 and 3.2 immediately. Lemma 3.1. (Carath´eodory Type) Let X be a real linear space, A ⊂ X and θ ∈ A. Let spanA be an n-dimensional subspace. Then any x ∈ cop A can be

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Figure 2. The diameter of convex hull in l1/2 (2) represented as a p-convex combination of n + 1 elements of A which contains θ. Lemma 3.2. Let X be a real linear space, 0 < p ≤ 1 and A = {θ, x1 , x2 , . . . , xn } ⊂ X. Let spanA be an n-dimensional subspace. Then any x ∈ cop A can be uniquely represented as a p-convex combination of θ, x1 , x2 , . . . , xn (in this case cop A is called a p-simplex). Lemma 3.3. Let (X,  · p ) be an n-dimensional p-normed space (0 < p ≤ 1) and A a finite subset of X. If θ ∈ A, then diam(cop A) = diamA. Proof. Let x, y be two convex combinations of elements of A, i.e., x=

k

ti ai , y =

i=1

m

rj bj , where ai , bj ∈ A, ti , rj ≥ 0,

j=1

k

ti = 1,

i=1

m

rj = 1.

j=1

Then we have x−y =

k m

ti rj (ai − bj ) and

i=1 j=1

k m

ti rj = 1.

i=1 j=1

This shows that coA − coA ⊂ co(A − A). By Lemma 2.4, it is evident that, k 1/p if xi ∈ {x : xp ≤ 1} ∩ {x : x1 ≤ 1} (i = 1, 2, . . . , k), zp = i=1 ti xi is k a p-convex combination with ti ≥ 0 and i=1 ti = 1 (0 < p ≤ 1), then zp p ≤ 1 (0 < p < 1) ⇔ z1 1 ≤ 1.

(3.1)

Let diamA = supz∈A−A zp = r and B = r−1/p A. Then diamB = supz∈B−B zp = 1 for all p ∈ (0, 1]. Since θ ∈ B, by Lemma 2.5 we have B − B ⊂ cop (B − B) ⊂ cop B − cop B ⊂ coB − coB = co(B − B). (3.2) By Lemma 3.1, we can suppose that cop B is a p-simplex, B = {b0 = θ, b1 , b2 , . . . , bn }, without loss of generality. By Lemma 3.2, we can define a mapping T : coB − coB → cop B − cop B by  n n n n 1/p 1/p T t i bi − ri bi = t i bi − ri bi , i=0

i=0

where ti , ri ≥ 0,

n i=0

i=0 n

ti = 1,

i=0

i=0

ri = 1.

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Since the representation is unique, T is a bijective mapping and is identity in B − B. Since (X,  · 1 ) is a normed space, for each z ∈ cop B − cop B, by (3.2) and Lemma 2.4 we have T −1 z1 ≤ 1. This implies that zp ≤ 1 by (3.1). Hence, diam(cop B) = supz∈cop B−cop B zp ≤ 1 = diamB, and so diam(cop A) = r = diamA. This completes the proof.  Theorem 3.4. Let (X,  · p ) be a complete p-normed space and A a bounded subset of X. If θ ∈ A, then diam(cos A) = diamA, where 0 < p ≤ 1, 0 < s ≤ p. Proof. It is evident that diam(cos A) = diam(cos A). Since A ⊂ cos A ⊂ cop A, we only need to show that diam(cop A) ≤ diamA. Assume that diam(cop A) > diamA. Then there exist x0 , y0 ∈ cop A such that x0 − y0 p > diamA. Thus, there exist a1 , a2 , . . . , an , b1 , b2 , . . . , bm ∈ A such that n m n m x0 = ti ai , y0 = rj bj , where ti , rj ≥ 0, tpi = 1, rjp = 1. i=1

j=1

i=1

j=1

Set A0 = {a1 , a2 , . . . , an , b1 , b2 , . . . , bm }. Then A0 is a finite set, spanA0 is a finite dimensional sbspace, A0 ⊂ A and x0 , y0 ∈ cop A0 . From Lemma 3.3 it follows that diamA ≥ diamA0 = diam(cop A0 ) ≥ x0 − y0 p > diamA, which is a contradiction. This completes the proof.



Remark 3.1. In general if θ ∈ A and 0 < p < 1, it is not necessary diam (cop A) = diamA. If we take A as one-point set {x0 } with x0 = θ, then diam{x0 } = 0 and diam(cop {x0 }) = x0 p . Corollary 3.5. Let (X,  · p ) be a complete p-normed space (0 < p < 1) and A a subset of X. Then diam(cop A) = diam(A ∪ {θ}). Lemma 3.6. Let (X,  · p ) be a complete p-normed space (0 < p ≤ 1) and x0 ∈ X. Then B(cop {x0 }, r) = {x ∈ X : d(x, cop {x0 }) < r} is p-convex, cop B(x0 , r) ⊂ B(cop {x0 }, r) and βH (B(cop {x0 }, r)) = r. Proof. Let x, y ∈ B(cop {x0 }, r) and λ, μ ∈ [0, 1] with λp + μp = 1. Then there exist a, b ∈ cop {x0 } such that x − ap < r and y − bp < r. Since λa + μb ∈ cop {x0 } and (λx + μy) − (λa + μb)p ≤ λp x − ap + μp y − bp < r, we have λx + μy ∈ B(cop {x0 }, r), and so B(cop {x0 }, r) is p-convex. Clearly, B(x0 , r) ⊂ B(cop {x0 }, r). This implies that cop B(x0 , r) ⊂ B(cop {x0 }, r). Since cop {x0 } is compact, for every ε > 0, cop {x0 } has a finite ε-net {ai }m i=1 . We claim that {ai }m i=1 is a finite r + ε-net of B(cop {x0 }, r). In fact, for each z ∈ B(cop {x0 }, r), there exists a ∈ cop {x0 } such that z − ap < r; from a ∈ cop {x0 } we see that there exists ai (1 ≤ i ≤ m) such that a − ai p < ε. Hence z − ai p ≤ z − ap + a − ai p < r + ε. Since ε is arbitrary, it follows  that βH (B(cop {x0 }, r)) = r. This completes the proof. Lemma 3.7. Let (X,  · p ) be a complete p-normed space (0 < p ≤ 1) and {An }∞ n=1 a sequence of nonempty bounded closed sets in X. If An+1 ⊂ An for all n ∈ Z+ and limn→∞ βK (An ) = 0 (or limn→∞ βH (An ) = 0), then ∞ A0 = n=1 An is a nonempty compact set.

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Proof. It follows directly from the definitions of βK and βH .

83



Theorem 3.8. Let (X,  · p ) be a complete p-normed space (0 < p ≤ 1), x0 ∈ X and b ∈ R. Let A and B be bounded subsets of X. Then the following assertions hold: (1) 0 ≤ βH (A), βK (A) ≤ diamA; (2) βH ({x0 }) = βK ({x0 }) = 0; (3) if A ⊂ B, then βK (A) ≤ βK (B), βH (A) ≤ βH (B); (4) βK (A) = βK (A), βH (A) = βH (A); (5) βK (A) = 0 (or βH (A) = 0) if and only if A is relatively compact; (6) βK (A ∪ B) = max{βK (A), βK (B)}, βH (A ∪ B) = max{βH (A), βH (B)}; (7) βK (A + x0 ) = βK (A), βH (A + x0 ) = βH (A); (8) βK (bA) = |b|p βK (A), βH (bA) = |b|p βH (A); (9) βK (A + B) ≤ βK (A) + βK (B), βH (A + B) ≤ βH (A) + βH (B); (10) βK (A) = βK (cop A), βH (A) = βH (cop A); (11) |βK (A) − βK (B)| ≤ 2dH (A, B), |βH (A) − βH (B)| ≤ dH (A, B). Proof. The assertions (1)–(7) follow directly from the definitions of βK and βH . (8) It is clair for b = 0. Suppose that b = 0. For every ε > 0 there exists n n a family of sets

n {Bi }i=1 such that A ⊂ pi=1 Bi and diamBi < βK (A) + ε. Since bA ⊂ i=1 bBi and diam(bBi ) < |b| (βK (A) + ε), we have βK (bA) ≤ |b|p βK (A). Also, from this we have βK (A) = βK (b−1 bA) ≤ |b|−p βK (bA), i.e., βK (bA) ≥ |b|p βK (A). Now we prove another equality. For every ε > 0, A has a finite βH (A) + ε -net {ai }ni=1 . We consider the set {bai }ni=1 . For each x ∈ bA, we have a ∈ A such that x = ba, and so there exists i such that x − bai p = ba − bai p = |b|p a − ai p ≤ |b|p (βH (A) + ε), which shows that {bai }ni=1 is a finite |b|p (βH (A)+ε)-net of bA. Hence βH (bA) ≤ |b|p βH (A). Also, from this we have βH (A) = βH (b−1 bA) ≤ |b|−p βH (bA), i.e., βH (bA) ≥ |b|p βH (A). (9) It follows directly from (c) and the definitions of βK and βH . (10) From the assertion (3) we see that βK (A) ≤ βK (cop A) and βH (A) ≤ βH (cop A). Next we show that βK (cop A) ≤ βK (A). This will be subdivided into three steps. Step 1. We prove inequality (3.5) below. For every ε > 0 there exists

n {Bi }ni=1 such that A = i=1 Bi , βK (A) ≤ diamBi < βK (A) + ε. By taking bi ∈ B

nwe see that θ ∈ B − bi . Set Ai = cop (Bi − bi ) + cop bi . Then Bi ⊂ Ai , A ⊂ i=1 Ai and Ai is p-convex. Using Theorem 3.4 and the assertions (5) and (9) we have βK (Ai ) = βK [cop (Bi − bi )] ≤ diam[cop (Bi − bi )] = diam(Bi − bi ) = diamBi < βK (A) + ε. Let Λ = {λ = (λ1 , . . . , λn ) ∈ Rn :

n i=1

λpi = 1; λi ∈ [0, 1], i = 1, . . . , n} (3.3)

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and n

λi Ai , ∀λ = (λ1 , . . . , λn ) ∈ Λ.

(3.4)

Thus from the assertions (8) and (9) we have n βK (Γ(λ)) ≤ λpi βK (Ai ) ≤ βK (A) + ε.

(3.5)

Γ(λ) =

i=1

i=1

Step 2. We show that the union x=

n

λi xi , y =

i=1

n

λ∈Λ

Γ(λ) is a p-convex set. Let

μi yi , t ∈ [0, 1] and z = t1/p x + (1 − t)1/p y,

i=1

where λ = (λ1 , . . . , λn ), μ = (μ1 , . . . , μn ) ∈  Λ, and xi , yi ∈ Ai for all i. The n point z can be represented in the form z = i=1 γi zi , where  p p tλi /γi , if γi > 0, 1/p p p p 1/p γi = tλi + (1 − t)μi , zi = ρi xi + (1 − ρi ) yi , ρi = 0, if γi = 0. By definition of γ = (γ1 , . . . , γn ) we have ρi ∈ [0, 1]. The set Ai is p-convex, so zi ∈ Ai . Moreover, γ ∈ Λ, by the p-convexity of Λ. Hence z ∈ Γ(γ). Step 3. We prove that βK (cop A) ≤ βK (A) + 3ε. Since Ai , i = 1, . . . , n are

n all bounded, there exists M = M (α) > 0 such that xp ≤ M for all x ∈ i=1 Ai . Since the set Λ is compact, for a given ε > 0 n we can find finitely many points λ(1) , . . . , λ(k) ∈ Λ, for any x = i=1 λi xi ∈ n (j) (j) {Γ(λ) : λ ∈ Λ}, there exists λ(j) = (λ1 , . . . , λn ) ∈ Λ such that i=1 |λi − (j) λi |p < ε/M , and and so x −

n i=1

(j)

λi xi p ≤

n i=1

|λi − λi |p max xi p < (ε/M )M = ε. (j)

1≤i≤n

Hence, 

Γ(λ) ⊂

k 

Γ(λ(j) ) + B(θ, ε).

j=1

λ∈Λ

Also, since A ⊂ λ∈Λ Γ(λ), from Step 2 it follows that cop A ⊂ λ∈Λ Γ(λ). Therefore, by (3.5) we have     βK (cop A) ≤ βK Γ(λ) ≤ max diam Γ(λ(j) ) + B(θ, ε) λ∈Λ

1≤j≤k



 ≤ max diam Γ(λ(j) ) + 2ε ≤ βK (A) + 3ε. 1≤j≤k

Since ε is arbitrary, we obtain βK (cop A) ≤ βK (A). Now we have to show that βH (cop A) ≤ βH (A). This will be subdivided into three steps.

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Step 4. We prove inequality (3.6) below. Let ε > 0 be given. Then A n has a finite βH (A) + ε-net {ai } i=1 . Let Ai = {x : d(x, cop {ai }) < βH (A) + ε} n for i = 1, 2, . . . , n. Then A ⊂ i=1 Ai . By Lemma 3.6, each Ai is p-convex, and βH (Ai ) ≤ βH (A) + ε. We define Λ as (3.3) and Γ(λ) as (3.4). It follows from the assertions (8) and (9) that βH (Γ(λ) ≤

n i=1

λpi βH (Ai ) ≤ βH (A) + ε.

(3.6)

Step

5. Applying the same argument as in Step 2, we can show that the union λ∈Λ Γ(λ) is a p-convex set. Step 6. We prove that βH (cop A) ≤ βH (A) + 2ε. In fact, applying the same argument as in Step 3, we can show that there exists λ(j) ∈ Λ such that 

Γ(λ) ⊂

k 

Γ(λ(j) ) + B(θ, ε).

j=1

λ∈Λ



Since A ⊂ λ∈Λ Γ(λ), from Step 4 it follows that cop A ⊂ λ∈Λ Γ(λ). Therefore, by (3.6) we have     Γ(λ) ≤ max βH Γ(λ(j) ) + B(θ, ε) βH (cop A) ≤ βH λ∈Λ

≤ max βH 1≤j≤k

1≤j≤k



 Γ(λ(j) ) + ε ≤ βH (A) + 2ε.

Since ε was chosen arbitrarily, it follows that βH (cop A) ≤ βH (A). (11) It follows directly from (c) and the definitions of dH , βK and βH . This completes the proof.  Remark 3.2. Using Lemma 2.7, we can prove the second equality in Theorem 3.8 (10) directly from the definitions of βH . In fact, for every ε > 0, A has a finite βH (A) + ε-net V . By Lemma 2.7, cop V is compact, and so cop V has a finite ε-net W . Next we show that W is a finite βH (A) + 2ε-net of m m each x ∈ cop A, there co m exist {xi }i=1 ⊂ A and {λi }i=1 ⊂ [0, 1] with pmA. For p λ = 1 such that x = λ x . For each x ∈ A, there exists yi ∈ V i=1 i i=1 i i m i such that xi − yi p < βH (A) + ε. Let y = i=1 λi yi . Then y ∈ cop V , and so there exists z ∈ W such that y − zp < ε. Thus we have x − zp ≤ x − yp + y − zp ≤

m i=1

λpi xi − yi p + y − zp < βH (A) + 2ε.

Since ε is arbitrary, it follows that βH (cop A) ≤ βH (A), which is the desired result.

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4. Some fixed point theorems In this section we extend the fundamental fixed point theorems for condensing operators to s-convex subsets of p-normed spaces. Theorem 4.1. (Darbo-type, see [11]) Let (X,  · p ) be a complete p-normed space and C a bounded closed s-convex subset of X, where 0 < p ≤ 1, 0 < s ≤ p. If T : C → C is a continuous k-set contraction (with respect to βK or βH ), then T has a fixed point in C and the set of fixed points of T in C is compact. Proof. We only consider the case of βK . By using the same method we can prove the assertion for the case of βH . From the Darbo theorem we see that the assertion holds in p = 1, s = 1. Now we suppose 0 < s < 1. It is evident that θ ∈ C due to the closed s-convexity of C. Let {An } be a sequence defined by A1 = cos T (C), An = cos T (An−1 ), for n ≥ 2. Since C is a bounded closed s-convex subset, we have A1 ⊂ C and An+1 ⊂ An ∞ for all n ∈ Z+ . Set A0 = n=1 An . Then A0 is a bounded closed s-convex subset of C by Lemma 2.5. From A1 ⊂ C and the definition of An we see that T (A1 ) ⊂ T (C) ⊂ A1 and T (An ) ⊂ T (An−1 ) ⊂ An , and so T (A0 ) ⊂ A0 . Since T is a k-set contraction, we have βK (A1 ) ≤ βK (T (C)) ≤ kβK (C); βK (An ) = βK (T (An−1 )) ≤ kβK (An−1 ) ≤ k n βK (C). Thus limn→∞ βK (An ) = 0 due to k ∈ [0, 1). By Lemma 3.7, A0 is a nonempty compact set. Hence, as a consequence of Theorem 2.9, we deduce that there exists z ∈ A0 such that T z = z. Let C0 be the set of fixed points of T in C. Then we have C0 = T (C0 ), and so it follows that C0 is closed, from the continuity of T . Since C0 ⊂ C and T is a k-set contraction, we have βK (C0 ) = βK (T (C0 )) ≤ kβK (C0 ). This implies that βK (C0 ) = 0 from k ∈ [0, 1), and so  C0 is compact. This completes the proof. Corollary 4.2. (Krasnosel’skii-type, see [38]) Let (X,  · p ) be a complete pnormed space and C a bounded closed s-convex subset of X, where 0 < p ≤ 1, 0 < s ≤ p. Assume that T1 : C → X is a k-contractive (with respect to βK or βH ) and T2 : C → X is completely continuous. Define T by T x = T1 x + T2 x and assume that T x ∈ C for all x ∈ C. Then T has a fixed point in C and the set of fixed points of T in C is compact. Proof. It is evident that T is a k-set contraction (with respect to βK or βH ). Hence the assertion follows from Theorem 4.1.  Theorem 4.3. (Sadovski-type, see [28]) Let (X,  · p ) be a complete p-normed space and C a bounded closed s-convex subset of X, where 0 < p ≤ 1, 0 < s ≤ p. If T : C → C is a continuous βH -condensing (or βK -condensing) operator, then T has a fixed point in C and the set of fixed points of T in C is compact.

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Proof. We only consider the case of βH . By using the same method we can prove the assertion for the case of βK . Without loss of generality we suppose 0 < s < 1. Let F denote the family of all closed, s-convex subsets A of C

for which T (A) ⊂ A. Since C ∈ F, we have F = ∅. Set B = A∈F A and E = cos T (B). Then B is a closed, s-convex subset of C by Lemma 2.5. We claim that B = E. In fact, since T (B) ⊂ B, it follows that E ⊂ B. Thus T (E) ⊂ T (B) ⊂ E, so that E ∈ F, and hence B ⊂ E. By Theorem 3.8, we have βH (B) = βH (E) = βH (T (B)). Since T is βH -condensing, it follows that βH (B) = 0, and so B is compact. Thus, by Theorem 2.9, there exists a fixed point for the operator T : B → B. Let C0 be the set of fixed points of T in C. Then we have C0 = T (C0 ), and so it follows that C0 is closed, from the continuity of T . Since βH (C0 ) = βH (T (C0 )) and T is βH -condensing, it follows that βH (C0 ) = 0, and hence C0 is compact. This completes the proof.  Remark 4.1. To prove Theorem 4.3 in the case of p = 1 and s = 1, we can choose a point q ∈ C, and let F be the family of all closed, convex subsets A of C for which q ∈ A and T (A) ⊂ A, and the assertion can be verified by proceeding in a similar way (see [40]). Corollary 4.4. (Krasnosel’skii-type) Let (X,  · p ) be a complete p-normed space and C a bounded closed s-convex subset of X, where 0 < p ≤ 1, 0 < s ≤ p. Assume that T1 : C → X is a continuous βH -condensing (or βK condensing) operator and T2 : C → X is completely continuous. Define T by T x = T1 x + T2 x and assume that T x ∈ C for all x ∈ C. Then T has a fixed point in C and the set of fixed points of T in C is compact. Proof. It is evident that T is a continuous βH -condensing (or βK -condensing) operator. Hence the assertion follows from Theorem 4.3.  Theorem 4.5. Let (X,  · p ) be a complete p-normed space and C a bounded closed s-convex subset of X, where 0 < p ≤ 1, 0 < s ≤ p. If T : C ⇒ C is an upper semi-continuous βK -condensing (or βH -condensing) set-valued operator, and T x is closed s-convex for every x ∈ C, then T has a fixed point in C and the set of fixed points of T in C is compact. Proof. We only consider the case of βK . Without loss of generality we suppose 0 < s < 1. Let F denote the family

of all closed, s-convex subsets A of C for which T (A) ⊂ A, and set B = A∈F A. Applying the same argument as the proof of Theorem 4.3, we can prove that B is compact and T (B) ⊂ B. Thus, by Theorem 2.10, there exists a fixed point for the operator T : B ⇒ B. Let C0 be the set of fixed points of T in C. Then we have C0 ⊂ T (C0 ). Since T is upper semi-continuous and has closed value, from Lemma 2.1 it follows that T has closed graph, and so C0 is closed. Since T is βH -condensing and βH (C0 ) ≤ βH (T (C0 )), we have βH (C0 ) = 0, and hence C0 is compact. This completes the proof. 

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Remark 4.2. The result of Theorem 4.5 in the case of p = 1 and s = 1 can be seen in [20], and it is similar to the one in [18]. Since each compact operator is a βK -condensing (or βH -condensing) operator, Theorem 4.5 is a generalization of the Kakutani-type fixed point theorems in [38]. Corollary 4.6. (Krasnosel’skii-type) Let (X,  · p ) be a complete p-normed space and C a bounded closed s-convex subset of X, where 0 < p ≤ 1, 0 < s ≤ p. Assume that T1 : C ⇒ X is an upper semi-continuous βK -condensing (or βH -condensing) set-valued operator and T2 : C ⇒ X is an upper semicontinuous compact set-valued operator. Define T by T x = T1 x + T2 x and assume that T x ⊂ C, T1 x and T2 x are closed s-convex for every x ∈ C. Then T has a fixed point in C and the set of fixed points of T in C is compact. Proof. It is evident that T is a βH -condensing (or βK -condensing) set-valued operator. By Lemma 2.1, T1 and T2 have closed graph. Since T2 is compact, T has closed graph. Since T is a βH -condensing (or βK -condensing) operator, T is quasicompact. By Lemma 2.1, T is upper semi-continuous. Hence the assertion follows from Theorem 4.5. The approaches used in Theorems 4.7 and 4.8 were motivated by ideas in [23,24].  Theorem 4.7. Let (X,  · p ) be a complete p-normed space and C a bounded closed s-convex subset of X, where 0 < p ≤ 1, 0 < s ≤ p. Let U be an open subset of C and θ ∈ U . Suppose that T : U → C is a continuous, βK -condensing (or βH -condensing) operator and assume that x = λT x, x ∈ ∂U, λ ∈ (0, 1)

(4.1)

holds. Then T has a fixed point in U . Proof. We only consider the case of βK . Suppose T has no fixed points on ∂U (otherwise the proof is finished). Since θ ∈ U , from (4.1) we have x = λT x, x ∈ ∂U, λ ∈ [0, 1].

(4.2)

Consider B = {x ∈ U : x = tT x, t ∈ [0, 1]}. Since θ ∈ U , we see that θ ∈ B and so B = ∅. In addition, the continuity of T implies that B is closed, and from (4.2) it follows that B ∩ ∂U = ∅. By Urysohn’s lemma there exists a continuous function μ : U → [0, 1] with μ(B) = 1 and μ(∂U ) = 0. Let  μ(x)T x, x ∈ U ; (4.3) T0 x = θ, x ∈ C\U . It is evident that T0 : C → C is a continuous operator. Since μ(x) ∈ [0, 1] for x ∈ U , we have μ(x)T (x) ∈ cos T (x). Hence T0 (C) ⊂ cos T (U ). Since T is βK -condensing, for each bounded set A ⊂ C with βK (A) = 0, by Theorem 3.8 we have βK (T0 (A)) = βK (T0 (C ∩ A)) ≤ βK (cos T (U ∩ A)) ≤ βK (T (A)) < βK (A).

(4.4)

This shows that T0 is βK -condensing. By Theorem 4.3, there exists x0 ∈ C such that x0 = T0 x0 . If x0 ∈ C\U , then from (4.3) we have x0 = T0 x0 = θ. It

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is a contradiction with θ ∈ U . Hence x0 ∈ U . By (4.3) we have x0 = μ(x0 )T x0 . It follows that x0 ∈ B and so μ(x0 ) = 1, x0 is a fixed point of T . This completes the proof.  Remark 4.3. The result of Theorem 4.7 can be seen in [25], in the case of compact operator, p = 1 and s = 1. It is similar to the one in [37]. Hence Theorem 4.7 is a generalization of the Leray-Schauder alternative theorems in [25]. Theorem 4.8. Let (X,  · p ) be a complete p-normed space and C a bounded closed s-convex subset of X, where 0 < p ≤ 1, 0 < s ≤ p. Let U be an open subset of C and θ ∈ U . Suppose that T : U ⇒ C is an upper semi-continuous, βK -condensing (or βH -condensing) set-valued operator, T x is closed and sconvex for every x ∈ X, and assume that x ∈ λT x, x ∈ ∂U, λ ∈ (0, 1)

(4.5)

holds. Then T has a fixed point in U . Proof. We only consider the case of βK . Suppose T has no fixed points on ∂U (otherwise the proof is finished). Since θ ∈ U , from (4.5) we have x ∈ λT x, x ∈ ∂U, λ ∈ [0, 1].

(4.6)

Consider B = {x ∈ U : x ∈ tT x, t ∈ [0, 1]}. Since θ ∈ U , we see that θ ∈ B and so B = ∅. Let {xn } ⊂ B with limn→∞ xn = x0 . Then there exists {tn } ⊂ [0, 1] such that xn ∈ tn T xn for each n ∈ Z+ . Since [0, 1] is compact, without loss of generality we suppose limn→∞ tn = t0 > 0 and x0 = θ. Thus, Lemma 2.1 and the upper semi-continuity of T imply that T has closed graph. From this it follows that x0 ∈ t0 T x0 , i.e., x0 ∈ B, and so B is closed. In addition, from (4.6) we see that B ∩ ∂U = ∅. By Urysohn’s lemma there exists a continuous function μ : U → [0, 1] with μ(B) = 1 and μ(∂U ) = 0. Let T0 be defined as (4.3). Then T0 : C ⇒ C is an upper semi-continuous operator and T0 (C) ⊂ cos T (U ). Since T is βK -condensing, for each bounded set A ⊂ C with βK (A) = 0, we obtain the inequality (4.4). Hence T0 is βK condensing. By Theorem 4.5, there exists x0 ∈ C with x0 ∈ T0 x0 . Notice that x0 ∈ U since θ ∈ U , we have x0 ∈ μ(x0 )T x0 . It follows that x0 ∈ B and so  μ(x0 ) = 1, x0 is a fixed point of T . This completes the proof. By using metric retraction instead of topological degree, we obtain the following Theorems 4.9 and 4.10. Theorem 4.9. (Leray–Schauder-type, see [25]) Let (X,  · p ) be a complete p-normed space, where 0 < p ≤ 1. Let T : X → X be a continuous, βK condensing (or βH -condensing) operator. If Ω = {u ∈ X : ∃λ > 1, λu = T u} is bounded, then the set of fixed points of T is a nonempty and compact set. Proof. We only consider the case of βH . Since Ω is bounded, there exists M > 0 such that xp < M for all x ∈ Ω. Let D = B(θ, M ). Then Ω ⊂ D. Define a retraction r : X → D by x r(x) = , for all x ∈ X, max{1, (qD (x))1/p }

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where qD is the Minkowski p-functional of D. Since D is a open p-convex subset with θ ∈ D, by Lemma 2.8, r : X → D is continuous. Since T is continuous and βH -condensing, it implies that rT : D → D is continuous and βH -condensing. By Theorem 4.3, there exists z ∈ D such that r(T z) = z. We claim that qD (T z) ≤ 1. In fact, if α = qD (T z) > 1, then T z ∈ D, and so z = r(T z) ∈ ∂D by Lemma 2.8 (2). Thus, from r(T z) = z it implies that α1/p z = T z, i.e., z ∈ Ω, a contradiction with Ω ⊂ D. From qD (T z) ≤ 1 we see that z = r(T z) = T z. Using the same argument as the proof of Theorem 4.3, we can prove that the set of fixed points of T is compact. This completes the proof.  Theorem 4.10. (Martelli-type, see [22]) Let (X, ·p ) be a complete p-normed space, where 0 < p ≤ 1. Let T : X ⇒ X be an upper semi-continuous βK condensing (or βH -condensing) set-valued operator, and T x be closed and sconvex for every x ∈ X, where 0 < s ≤ p. If Ω = {u ∈ X : ∃λ > 1, λu ∈ T u} is bounded, then the set of fixed points of T is a nonempty compact set. Proof. We only consider the case of βH . Take D as the proof of Theorem 4.9. Then D is s-convex, open and Ω ⊂ D. Define a retraction r : X → D by x r(x) = , for all x ∈ X, (4.7) max{1, (qD (x))1/s } where qD is the Minkowski s-functional of D. By Lemma 2.8, r : X → D is continuous. Since T is upper semi-continuous and (rT )− (U ) = T − (r−1 (U )), for each open set U ⊂ D, it implies that rT : D ⇒ D is upper semi-continuous. Since T is βH condensing, for each bounded set A ⊂ D with βH (A) = 0, from (4.7) we have βH (rT (A)) ≤ βH (T (A)) < βH (A). This shows that rT is βH -condensing. Let x ∈ D be fixed. Since r is continuous and T x is closed, it is easy to see that rT x is closed. Let y1 , y2 ∈ rT x and λ1 , λ2 ∈ [0, 1] with λs1 + λs2 = 1. Then there exist z1 , z2 ∈ T x such that r(z1 ) = y1 and r(z2 ) = y2 . Set λ 1 z1 λ 2 z2 z0 = + . 1/s max{1, (qD (z1 )) } max{1, (qD (z2 ))1/s } Using Lemma 2.6 we have λs2 qD (z2 ) λs1 qD (z1 ) qD (z0 ) ≤ + ≤ λs1 + λs2 = 1 and max{1, qD (z1 )} max{1, qD (z2 )} λs2 qT x (z2 ) λs1 qT x (z1 ) + ≤ λs1 + λs2 = 1. qT x (z0 ) ≤ max{1, qD (z1 )} max{1, qD (z2 )} This means that z0 ∈ D and z0 ∈ T x = T x, and so λ1 y1 + λ2 y2 = z0 = r(z0 ) ∈ rT x. Hence T x is s-convex. By Theorem 4.5, there exists z ∈ D such that z ∈ r(T z), and so there exists y ∈ T z such that z = r(y). We will show that qD (y) ≤ 1. In fact, if α = qD (y) > 1, then y ∈ D, and so z = r(y) ∈ ∂D by Lemma 2.8 (2). Thus, from r(y) = z it follows that α1/s z = y ∈ T z, i.e.,

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z ∈ Ω, a contradiction with Ω ⊂ D. Therefore, from qD (y) ≤ 1 we see that z = r(y) = y ∈ T z, i.e., z is a fixed point of T . Using the same argument as the proof of Theorem 4.5, we can prove that the set of fixed points of T is compact. This completes the proof. 

5. Applications In this section we give an application of Theorem 4.5 to game theory. Modern game theory began with the idea regarding the existence of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Von Neumann’s original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. We now consider the existence of solutions for games on non-convex non-compact sets in p-normed spaces. For some basic notions we refer to [7,25,40]. A game is a triplet (A, B, F ), where A, B are two nonempty sets, whose elements are called strategies, and F : A×B → (−∞, +∞) is the gain (or loss) functional. The quantity F (x, y) represents the gain of the player P when he chooses the strategy x ∈ A and the player Q chooses y ∈ B; and −F (x, y) represents the gain of the player Q in the same situation. The target of the player P is to maximize his gain when Q chooses the worst strategy, i.e., to chooses x0 ∈ A such that inf y∈B F (x0 , y) = supx∈A inf y∈B F (x, y). Similarly, the player Q chooses y0 ∈ B such that supx∈A F (x, y0 ) = inf y∈B supx∈A F (x, y). Thus, we have sup inf F (x, y) = inf F (x0 , y) ≤ F (x0 , y0 ) ≤ sup F (x, y0 )

x∈A y∈B

y∈B

x∈A

= inf sup F (x, y). y∈B x∈A

(5.1)

If the equality holds in (5.1), then (x0 , y0 ) ∈ A × B is called a solution of the game (A, B, F ). We discuss the existence of solutions for a game in a product p-normed space. Let (X, ·p ), (Y, ·p ) be complete p-normed spaces, where 0 < p ≤ 1. Define product p-norm by (x, y)p = xp + yp , for all x ∈ X, y ∈ Y. It is easy to verify that (X × Y,  · p ) is a complete p-normed space. Lemma 5.1. Let (X, ·p ),(Y, ·p ) be complete p-normed spaces and A ⊂ X, B ⊂ Y nonempty bounded closed sets, where 0 < p ≤ 1, 0 < s ≤ p. If F : A × B → (−∞, +∞) is a continuous bounded functional and F −1 is βK -condensing (or βH -condensing), then the functionals ϕ : A → (−∞, +∞) and ψ : B → (−∞, +∞) are continuous, where ϕ, ψ are defined by ϕ(x) = inf F (x, y), for all x ∈ A; ψ(y) y∈B

= sup F (x, y), for all y ∈ B. x∈A

(5.2)

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Proof. We will prove that ψ is continuous. The continuity of ϕ can be proved in a similar way. We only consider the case of βH . Let {yn } ⊂ B be a sequence converging to y0 ∈ B. Suppose n is fixed. For each k ∈ Z+ , there exists xk ∈ A such that 1 ψ(yn ) − < F (xk , yn ) ≤ ψ(yn ). (5.3) k Since F −1 is βH -condensing and βH {F (xk , yn ) : k ∈ Z+ } = 0 by (5.3), we have βH {xk : k ∈ Z+ } = βH {(xk , yn ) : k ∈ Z+ } = 0. Then {xk } has a convergent subsequence. Without loss of generality, we suppose limk→∞ xk = x∗n . Since A is closed, we have x∗n ∈ A. Since F is continuous, from (5.3) we have ψ(yn ) = F (x∗n , yn ).

(5.4)

Since F is bounded and F −1 is βH -condensing, we have βH {x∗n : n ∈ Z+ } = βH {(x∗n , yn ) : n ∈ Z+ } = 0. Without loss of generality, we suppose limn→∞ x∗n = x0 ∈ A. Hence, from (5.4) and the continuity of F we have lim ψ(yn ) = F (x0 , y0 ).

(5.5)

n→∞

But for each z ∈ A, F (z, yn ) ≤ ψ(yn ) = F (x∗n , yn ), this follows that F (z, y0 ) ≤ F (x0 , y0 ). Therefore, combining (5.5) we have ψ(y0 ) = sup F (z, y0 ) = F (x0 , y0 ) = lim ψ(yn ). z∈A

n→∞

which is the desired conclusion.



By Theorem 4.5, we give the following existence result: Theorem 5.2. Let (X,  · p ),(Y,  · p ) be complete p-normed spaces and A ⊂ X, B ⊂ Y nonempty bounded closed s-convex sets, where 0 < p ≤ 1, 0 < s ≤ p. If F : A × B → (−∞, +∞) is a continuous bounded functional and F −1 is βK -condensing (or βH -condensing), and (H-1) for every x ∈ A, F (x, ·) is s-convex, and (H-2) for every y ∈ B, F (·, y) is s-concave, Then inf y∈B supx∈A F (x, y) = supx∈A inf y∈B F (x, y), and the solution set of the game (A, B, F ) is nonempty compact. Proof. Let ϕ, ψ be the functionals defined by (5.2), and let Bx = {y ∈ B : F (x, y) = ϕ(x)}, for all x ∈ A; Ay = {x ∈ A : F (x, y) = ψ(y)}, for all y ∈ B. Since the sets A, B are nonempty bounded closed, and the functionals F is continuous bounded and F −1 is βK -condensing (or βH -condensing), by Lemma 5.1, ϕ, ψ are continuous. Thus, Ax and Bx are nonempty and closed, for each (x, y) ∈ A × B.

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Now we prove that Ax and Bx are s-convex. Let t ∈ (0, 1), y1 , y2 ∈ Bx , y¯ = t1/s y1 + (1 − t)1/s y2 , x1 , x2 ∈ Ay and x ¯ = t1/s x1 + (1 − t)1/s x2 . Then y¯ ∈ B, x ¯ ∈ A. By (H-1) and (H-2) we have ϕ(x) ≤ F (x, y¯) ≤ tF (x, y1 ) + (1 − t)F (x, y2 ) = tϕ(x) + (1 − t)ϕ(x) = ϕ(x);

(5.6)

ψ(y) ≥ F (¯ x, y) ≥ tF (x1 , y) + (1 − t)F (x2 , y) = tψ(y) + (1 − t)ψ(y) = ψ(y).

(5.7)

From (5.6) and (5.7) it implies that F (x, y¯) = ϕ(x) and F (¯ x, y) = ψ(y), i.e., y¯ ∈ Bx and x ¯ ∈ Ay respectively. Hence, Ax and Bx are s-convex. Let C = A × B. Then C is s-convex. Define T : C ⇒ C by T (x, y) = Ay × Bx , for all (x, y) ∈ C. Then T (x, y) is a nonempty s-convex closed subset of C for every (x, y) ∈ C. Now we show that the graph GT of T , given by GF = {(x, y, u, v) ∈ C × C : (u, v) ∈ T (x, y)}, is closed in C × C. Suppose that {(xn , yn )}∞ n=1 ⊂ C converges to (x, y) ∈ C, converges to (u, v) ∈ C, as n → ∞. and (un , vn ) ∈ T (xn , yn ), {(un , vn )}∞ n=1 Since (un , vn ) ∈ T (xn , yn ) if and only if F (un , yn ) = ψ(yn ), F (xn , vn ) = ϕ(xn ), by the continuity of F, ϕ and ψ, we have F (u, y) = ψ(y), F (x, v) = ϕ(x), i.e., (u, v) ∈ Ay × Bx = T (x, y). Hence, the graph GT of T is closed. Let C0 be an arbitrary subset of C. Then C0 is bounded. Since F is bounded and F −1 is βK -condensing (or βH -condensing), we have βK (C0 ) = 0 (or βH (C0 ) = 0). Thus we have βK (T C0 ) = 0 (or βH (T C0 ) = 0). Hence T is βK -condensing (or βH -condensing), and is also quasicompact. By Lemma 2.1, T is upper semi-continuous. By Theorem 4.5, the set of fixed points of T in C is compact. Let (x0 , y0 ) ∈ C is a fixed point of T . Then (x0 , y0 ) ∈ T (x0 , y0 ), i.e., x0 ∈ Ay0 and y0 ∈ Bx0 . Note that x0 ∈ Ay0 and y0 ∈ Bx0 if and only if F (x0 , y0 ) = sup F (x, y0 ) ≥ inf sup F (x, y) and

(5.8)

F (x0 , y0 ) = inf F (x0 , y) ≤ sup inf F (x, y);

(5.9)

x∈A y∈B

y∈B x∈A

x∈A y∈B

and in general sup inf F (x, y) ≤ inf sup F (x, y).

x∈A y∈B

y∈B x∈A

(5.10)

From (5.8), (5.9) and (5.10) we have inf sup F (x, y) = F (x0 , y0 ) = sup inf F (x, y),

y∈B x∈A

x∈A y∈B

This shows that (x0 , y0 ) is a solution of the game (A, B, F ). Hence the solution set of the game (A, B, F ) is nonempty compact. This completes the proof. 

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Acknowledgements The authors are grateful to the referees for their valuable comments and helpful suggestions.

References [1] Albiac, F.: Nonlinear structure of some classical quasi-Banach spaces and Fspaces. J. Math. Anal. Appl. 340, 1312–1325 (2008) [2] Albiac, F., Ler´ anoz, C.: Drops in quasi-Banach spaces. J. Geom. Anal. 20, 525–537 (2010) [3] Albiac, F., Ansorena, J.L.: Integration in quasi-Banach spaces and the fundamental theorem of calculus. J. Funct. Anal. 264, 2059–2076 (2013) [4] Albiac, F., Ansorena, J.L.: Optimal average approximations for functions mapping in quasi-Banach spaces. J. Funct. Anal. 266, 3894–3905 (2014) [5] Albiac, F., Baudier, F.: Embeddability of snowflaked metrics with applications to the nonlinear geometry of the spaces Lp and lp for 0
Some fixed point theorems for s-convex subsets

Page 21 of 22

83

[19] Kalton, N.J.: Quasi-Banach Spaces, Handbook of the Geometry of Banach spaces, pp. 1101–1139. Elsevier, Amsterdam (2003) [20] Kamenskii, M., Obukhovskii, V., Zecca, P.: Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, pp. 1–96. Walter de Gruyter, Berlin (2001) [21] K¨ othe, G.: Topological Vector Spaces I, pp. 1–294. Springer, Berlin (1969) [22] Martelli, M.: A Rothe’s type theorem for non-compact acyclic-valued map. Boll. Unione Mat. Ital. 4, 70–76 (1975) [23] O’Regan, D.: Fixed point theory for closed multifunctions. Arch. Math. (Brno) 34, 191–197 (1998) [24] O’Regan, D.: Nonlinear alternatives for multivalued maps with applications to operator inclusions in abstract spaces. Proc. Am. Math. Soc. 127, 3557–3564 (1999) [25] Papageorgiou, N.S., Kyritsi-Yiallourou, STh: Handbook of Applied Analysis, pp. 1–650. Springer, Berlin (2009) [26] Rolewicz, S.: Metric Linear Spaces, pp. 1–287. PWN-Polich Scientific Publishers, Warszawa (1985) [27] Rudin, W.: Functional Analysis, 2nd edn, pp. 1–91. McGraw-Hill, New York (1991) [28] Sadovski, B.N.: On a fixed point principle. Funct. Anal. Appl. 1, 74–76 (1967) [29] S´ anchez, F.C., Garbuli´ nska-W¸egrzyn, J., Kubi´s, W.: Quasi-Banach spaces of almost universal disposition. J. Funct. Anal. 267, 744–771 (2014) [30] Silva, E.B., Fernandez, D.L., Nikolova, L.: Generalized quasi-Banach sequence spaces and measures of noncompactness. An. Acad. Bras. Ciˆe. 85(2), 443–456 (2013) [31] Su, C.H., Sehgal, V.M.: Some fixed point theorems for condensing multifonctions in locally convex spaces. Proc. Am. Math. Soc. 50, 150–153 (1975) ´ Zo ˙ ldak, M.: Stability of isometries in p-Banach spaces. [32] Tabor, J.A., Tabor, J.O., Funct. Approx. 38, 109–119 (2008) [33] Tan, D.-N.: On extension of isometries on the unit spheres of Lp -spaces for 0

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Jian-Zhong Xiao and Ying Lu School of Mathematics and Statistics Nanjing University of Information Science and Technology Nanjing 210044 People’s Republic of China e-mail: [email protected]