Fixed point theorems and the KreinS̆mulian property in locally convex spaces
 Fuli Wang^{1}Email author and
 Hua Zhou^{1}
https://doi.org/10.1186/s1366301504008
© Wang and Zhou 2015
Received: 16 June 2015
Accepted: 10 August 2015
Published: 25 August 2015
Abstract
The aim of this paper is to establish some new fixed point theorems for the superposition operators in locally convex spaces which satisfy the KreinS̆mulian property. We employ a family of measures of noncompactness in conjunction with the SchauderTychonoff fixed point theorem. As an application, the existence of solutions to a quite general nonlinear Volterra type integral equation is considered in locally integrable spaces.
Keywords
MSC
1 Introduction
As an example of algebraic settings, the captivating Krasnosel’skii fixed point theorem leads to the consideration of fixed points for the sum of two operators. It asserts that, if M is a nonempty, bounded, closed, and convex subset of a Banach space X and A, B are two maps from M into X such that \(A(M)+B(M)\subseteq M\), A is compact and B is a contraction, then \(A+B\) has at least one fixed point in M (see [1] or [2], p.31). Since then, there has been a vast literature dealing with the improvement of such a result. In the previous decade, several papers have given generalizations of this theorem involving the weak topology of Banach spaces by using the De Blasi measure of weak noncompactness (see [3–12]). The novelty of their results is that the involved operators need not to be weakly continuous. Some weak and strong compactness are addressed instead of the weak continuity since the condition of weak continuity is usually not easy to verify.
On the other hand, the Krasnosel’skii fixed point theorem has been generalized to locally convex spaces or Fréchet spaces by some authors (see [13–18]), and a family of measures of noncompactness has been introduced to the concrete spaces by Olszowy [19, 20]. All of these results naturally cause us to consider the Krasnosel’skii type fixed point theorems in locally convex spaces by means of a family of measures of weak noncompactness. This problem will be followed with interest in some special situations in the present paper.
Our goals in this paper are to establish new fixed point theorems for the solvability of (1.1) in locally convex spaces, and to study under what conditions (1.2) is solvable in \(\mathcal{L}^{1}_{\mathrm {loc}}\) by applying our new theorems.
This paper is organized as follows. In Section 2, we present the relevant definitions and results needed in our work. In Section 3, we establish some new fixed point theorems for (1.1) in the framework of locally convex spaces. In Section 4, we prove the existence of locally integrable solutions for (1.2) by virtue of our fixed point theorems and a family of measures of weak noncompactness.
2 Definitions and preliminaries
Throughout this paper, X will denote a Hausdorff locally convex topological vector space, and \(\{\vert \cdot \vert _{\rho}\}_{\rho\in\Lambda}\) a family of seminorms which generates the topology of X.
Recall that the weak topology on X is the weakest topology (the topology with the fewest open sets) such that all elements of \(X'\) (the topological dual of X) remain continuous. Explicitly, the neighborhood system of the origin for the weak topology is the collection of sets of the form \(\phi^{1}(U)\) where \(\phi\in X'\) and U is a neighborhood of the origin.
Definition 2.1
(see [5], Remark 2.1)
We say that X satisfies the KreinS̆mulian property if the closed convex hull of each weakly compact set is weakly compact.
Remark 2.2
Each Fréchet space satisfies the KreinS̆mulian property (see [22], p.233), particularly for each Banach space (see [23], p.434).
In our considerations, a family of measures of weakly noncompactness in locally convex spaces will play an important role. A single measure of weak noncompactness in Banach spaces may refer to the definition of Banaś and Rivero [24]. Next, we will use \(\mathfrak{B}(X)\) denoting the collection of all nonempty bounded subsets of X, and \(\mathfrak {W}(X)\) a subset of \(\mathfrak{B}(X)\) consisting of all weakly compact subsets of X. For \(\rho\in\Lambda\) and \(r>0\), the set \(\{ x:\vert xx_{0}\vert _{\rho}< r\}\) is denoted by \(V_{\rho}(x_{0},r)\). The closure of this set is denoted by \(\mathrm{B}_{\rho}(x_{0},r)\). We shall also sometimes use \(V(\rho)\) to stand for \(V_{\rho}(0,1)\).
Definition 2.3
 (1)
The family \(\ker(\omega_{\rho}):=\{M\in\mathfrak{B}(X):\omega _{\rho}(M)=0 \mbox{ for all } \rho\in\Lambda\}\) is nonempty and \(\ker(\omega_{\rho})\) is contained in the subfamily consisting of all relatively weakly compact sets of X;
 (2)
\(N\subseteq M \Rightarrow \omega_{\rho}(N)\leq\omega_{\rho}(M)\) for each \(\rho\in\Lambda\), where \(M,N\in\mathfrak{B}(X)\);
 (3)
\(\omega_{\rho}(\overline{\operatorname {co}}(M))=\omega_{\rho}(M)\) for each \(\rho \in\Lambda\), where \(\overline{\operatorname {co}}(M)\) is the closed convex hull of \(M\in\mathfrak{B}(X)\);
 (4)
\(\omega_{\rho}(\lambda M+(1\lambda)N)\leq\lambda\omega_{\rho}(M)+(1\lambda)\omega_{\rho}(N)\) for each \(\rho\in\Lambda\), \(\lambda\in [0,1]\) and \(M,N\in\mathfrak{B}(X)\);
 (5)
if \((M_{n})_{n=1}^{\infty}\) is a decreasing sequence of nonempty, bounded, and weakly closed subsets of X with \(\lim_{n\rightarrow\infty }\omega_{\rho}(M_{n})=0\) for each \(\rho\in\Lambda\), then \(M_{\infty}:=\bigcap_{n=1}^{\infty}M_{n}\) is nonempty.
The family \(\ker(\omega_{\rho})\) described in (1) is called the kernel of the measure of weak noncompactness \(\omega_{\rho}\). Note that the intersection \(M_{\infty}\) from (5) belongs to \(\ker(\omega_{\rho})\) since we have \(\omega_{\rho}(M_{\infty})\leq\omega_{\rho}(M_{n})\) for each \(\rho\in \Lambda\) and all \(n\in\mathbb{N}\), and \(\lim_{n\rightarrow\infty}\omega _{\rho}(M_{n})=0\).
Following the definitions given by [27, 28] in Banach spaces, we introduce the concepts of wscompactness and wwcompactness in locally convex spaces as follows.
Definition 2.4
 (i)
wscompact if it is continuous and maps relatively weakly compact sets of \(\mathcal{D}\) into relatively strongly compact ones of Y;
 (ii)
wwcompact if it is continuous and maps relatively weakly compact sets of \(\mathcal{D}\) into relatively weakly compact ones of Y.
Next, we collect a few auxiliary facts concerning the superposition operator required in the sequel.
Consider a function \(\psi(t,x)=\psi:I\times\mathbb{R}^{d}\rightarrow \mathbb{R}\), where I is an interval of \(\mathbb{R}\), and \(\mathbb {R}^{d}\) a real Euclidean space of ddimensions. We say that ψ is a Carathéodory function if it is measurable in t for each x in \(\mathbb{R}^{d}\) and continuous in x for almost everywhere (a.e., for short) given \(t\in I\).
Let \(m(I)\) be the set of all measurable functions \(x:I\rightarrow\mathbb {R}^{d}\). If ψ is a Carathéodory function, then ψ defines a mapping \(\mathcal{N}_{\psi}:m(I)\rightarrow m(I)\) by \((\mathcal{N}_{\psi}x)(t)=\psi(t,x(t))\). This mapping is called the superposition operator (or Nemytskii operator) associated to ψ. For a given measurable function \(\varphi:I\rightarrow\mathbb{R}^{d}\), the composite operator \(\mathcal{N}_{\psi}\varphi(\cdot):=\psi(\cdot,\varphi(\cdot))\) which maps I into \(\mathbb{R}\) is said to be a nonautonomous type superposition operator.
By generalizing the above concept, the solvability of (1.1) may be thought of as the existence of fixed points for the nonautonomous type superposition operator \(\mathcal{N}_{F}A\), where \(\mathcal {N}_{F}A(\cdot):=F(\cdot,A(\cdot))\) for simplicity.
The following theorem was proved by Krasnosel’skii [29] (see also [30]) in the case when I is a bounded interval and has been extended to an unbounded interval by Appell and Zabrejko [31].
Theorem 2.5
([31], Theorem 3.1, p.93)
In this case, the operator \(\mathcal{N}_{\psi}\) is continuous and bounded in the sense that it maps bounded sets into bounded ones.
3 Fixed point theorems
Let \(\mathcal{U}\) be the neighborhood system of the origin obtained from Λ. Thus if \(U\in\mathcal{U}\), there is a finite number of seminorms \(\rho_{1}, \rho_{2}, \ldots, \rho_{n}\) in Λ and real numbers \(r_{1}, r_{2}, \ldots, r_{n}\) such that \(U=\bigcap_{i=1}^{n}r_{i}V(\rho_{i})\).
A mapping \(T:X\rightarrow X\) is said to be a \(\vert \cdot \vert _{\rho}\)contraction for \(\rho\in\Lambda\) if there exists \(\alpha_{\rho}\in[0,1)\) such that \(\vert Tx_{1}Tx_{2}\vert _{\rho}\leq\alpha_{\rho} \vert x_{1}x_{2}\vert _{\rho}\) for all \(x_{1},x_{2}\in X\); if \(\alpha_{\rho}=1\) then the mapping T is said to be a \(\vert \cdot \vert _{\rho}\)nonexpansion for \(\rho\in\Lambda\).
Theorem 3.1
Let M be a nonempty, closed, and convex subset of X, and let the KreinS̆mulian property be satisfied. Suppose that \(T:M\rightarrow M\) is wscompact such that \(T(M)\) is relatively weakly compact, then T has at least one fixed point.
Proof
Let \(\mathcal{N}:=\overline{\operatorname {co}}(T(M))\). Since M is closed and convex satisfying \(T(M)\subseteq M\), then \(\mathcal{N}\subseteq M\) and therefore \(T(\mathcal{N})\subseteq T(M)\subseteq\mathcal{N}\).
It is clear that \(\mathcal{N}\) is weakly compact according to the relatively weak compactness of \(T(M)\) and the KreinS̆mulian property. Moreover, \(T(\mathcal{N})\) is relatively compact since T is wscompact. Now applying the SchauderTychonoff fixed point theorem (see [13], Theorem 2.1(b) or [2], p.32), we conclude that T has at least one fixed point \(x\in\mathcal{N}\subseteq M\) such that \(Tx=x\). □
Theorem 3.2
 (i)
\(A(M)\) is relatively weakly compact and A is wscompact;
 (ii)for each \(\rho\in\Lambda\) there exists \(\alpha_{\rho}\in[0,1)\) such thatand F is wwcompact;$$\bigl\vert F(x_{1},y)F(x_{2},y)\bigr\vert _{\rho}\leq\alpha_{\rho} \vert x_{1}x_{2} \vert _{\rho}, \quad \forall x_{1},x_{2}\in X \textit{ and } y \in Y, $$
 (iii)
\([x=F(x,Az), z\in M] \Rightarrow x\in M\).
Then there is a point x in M such that \(x=F(x,Ax)\).
Proof
For a given \(y\in A(M)\) the mapping \(F(\cdot,y)\) is a \(\vert \cdot \vert _{\rho}\)contraction for each \(\rho\in\Lambda\), so by CainNashed theorem ([13], Theorem 2.2) it has a unique fixed point in X. Let us denote by \(J:A(M)\rightarrow X\) the mapping which assigns each \(y\in A(M)\) to the unique point in X such that \(Jy=F(Jy,y)\). Thus, J is well defined.
In the framework of locally convex spaces, the following result can be thought of as an extension of Latrach et al. [9], Theorem 2.3, and also a variant of Cain and Nashed [13], Theorem 3.1, under the weak topology. This implies that we are to establish a new version of Krasnosel’skii’s fixed point theorem in locally convex spaces.
Corollary 3.3
 (i)
\(A(M)\) is relatively weakly compact, and A is wscompact;
 (ii)
B is a \(\vert \cdot \vert _{\rho}\)contraction for each \(\rho\in\Lambda\), and B is wwcompact;
 (iii)
\([x=Bx+Az, z\in M] \Rightarrow x\in M\).
Then there is a point x in M such that \(Ax+Bx=x\).
Proof
Let us take \(F(x,y):=Bx+y\) and \(Y=X\). All assumptions of Theorem 3.2 are easily verified, and then the proof immediately is achieved. □
Since assumption (iii) of Theorem 3.2 is hard to verify in real applications, we next establish a Schaefer type fixed point theorem for (1.1).
Theorem 3.4
 (i)
A maps bounded sets of X into relatively weakly compact ones of Y, and A is wscompact;
 (ii)for each \(\rho\in\Lambda\) there exists \(\alpha_{\rho}\in[0,1)\) such thatand F is wwcompact.$$\bigl\vert F(x_{1},y)F(x_{2},y)\bigr\vert _{\rho}\leq\alpha_{\rho} \vert x_{1}x_{2} \vert _{\rho}, \quad \forall x_{1},x_{2}\in X \textit{ and } y \in Y, $$
 (a)
there is a point x in X such that \(x=F(x,Ax)\), or
 (b)
the set \(\{x\in X:x=\lambda F(x/\lambda,Ax)\}\) is unbounded for \(\lambda\in(0,1)\).
Proof
As in the proof of Theorem 3.2, let us denote by \(J:A(X)\rightarrow X\) the mapping which assigns each \(y\in A(X)\) to the unique point in X such that \(Jy=F(Jy,y)\). We know that J is well defined and continuous.
Corollary 3.5
 (i)
A maps bounded sets into relatively weakly compact ones, and A is wscompact;
 (ii)
B is a \(\vert \cdot \vert _{\rho}\)contraction for all \(\rho\in\Lambda\), and B is wwcompact.
 (a)
there is a point x in X such that \(x=Ax+Bx\), or
 (b)
the set \(\{x\in X:x=\lambda Ax+\lambda B(x/\lambda)\}\) is unbounded for \(\lambda\in(0,1)\).
Recalling that a mapping \(T:M\rightarrow X\) is said to be demiclosed at 0 if for every net \((x_{\delta})\) in M converges weakly to x and \((Tx_{\delta})\) converges to 0, then we have \(Tx=0\).
Theorem 3.6
 (i)
\(A(M)\) is relatively weakly compact, and A is wscompact;
 (ii)for each \(\rho\in\Lambda\) and all \(x_{1},x_{2}\in X\), \(y\in Y\) we haveand F is wwcompact;$$\bigl\vert F(x_{1},y)F(x_{2},y)\bigr\vert _{\rho}\leq \vert x_{1}x_{2}\vert _{\rho}, $$
 (iii)
if \((x_{n})_{n\in\mathbb{N}}\) is a sequence such that \((x_{n}F(x_{n},Ax_{n}))_{n\in\mathbb{N}}\) is convergent, then \((x_{n})_{n\in \mathbb{N}}\) has a weakly convergent subsequence;
 (iv)
\(I\mathcal{N}_{F}A\) is demiclosed at 0;
 (v)
if \(\lambda\in(0,1)\) and \(x=\lambda F(x,Az)\) for some \(z\in M\), then \(x\in M\).
Then there is a point x in M such that \(x=F(x,Ax)\).
Proof
By condition (iii) the sequence \((x_{n})_{n\in\mathbb{N}}\) has a subsequence \((x_{n_{k}})_{k\in\mathbb{N}}\) which converges weakly to some \(x\in M\) (M is weakly closed because of its closeness and convexity). The demiclosedness of \(I\mathcal{N}_{F}A\) at 0 yields \((I\mathcal {N}_{F}A)x=0\), that is, \(x=\mathcal{N}_{F}Ax=F(x,Ax)\). □
The following corollary extends [4], Theorem 2.1, and [11], Theorem 3.8, to locally convex spaces.
Corollary 3.7
 (i)
\(A(M)\) is relatively weakly compact, and A is wscompact;
 (ii)
B is a \(\vert \cdot \vert _{\rho}\)nonexpansion for all \(\rho\in\Lambda\), and B is wwcompact;
 (iii)
if \((x_{n})_{n\in\mathbb{N}}\) is a sequence such that \((x_{n}Bx_{n}Ax_{n})_{n\in\mathbb{N}}\) is convergent, then \((x_{n})_{n\in \mathbb{N}}\) has a weakly convergent subsequence;
 (iv)
\(IAB\) is demiclosed at 0;
 (v)
if \(\lambda\in(0,1)\) and \(x=\lambda Bx+\lambda Ay\) for some \(y\in M\), then \(x\in M\).
Then there is a point x in M such that \(Ax+Bx=x\).
4 Application to the existence of locally integrable solutions for a general nonlinear integral equation
Let \(L^{1}[0,T]\) denote the Banach space consisting of all real functions defined and Lebesgue integrable on \([0,T]\), and let \(\Pi_{T}:\mathcal {L}^{1}_{\mathrm {loc}}\rightarrow L^{1}[0,T]\) denote the restrictive mapping.
Remark 4.1
 (1)
M is bounded if there exists a \(L_{T}>0\) for each \(T>0\) such that \(\vert x\vert _{T}\leq L_{T}\) for all \(x\in M\);
 (2)
M is relatively (strongly) compact if and only if \(\Pi_{T}(M)\) is relatively (strongly) compact in Banach space \(L^{1}[0,T]\) for each \(T>0\);
 (3)
M is relatively weakly compact if and only if \(\Pi_{T}(M)\) is relatively weakly compact in Banach space \(L^{1}[0,T]\) for each \(T>0\);
 (4)
a sequence \((x_{n})_{n\in\mathbb{N}}\) in \(\mathcal{L}^{1}_{\mathrm {loc}}\) is convergent to \(x\in\mathcal{L}^{1}_{\mathrm {loc}}\) if and only if \(\vert x_{n}x\vert _{T}\rightarrow0\) for each \(T>0\).
 (\(\mathcal{H}1\)):

\(\kappa:\Delta\rightarrow\mathbb{R}\) is measurable, where \(\Delta=\{(t,s)\in\mathbb{R}^{2}:0\leq s\leq t\}\);
 (\(\mathcal{H}2\)):

\(v:\mathbb{R}_{+}\times\mathbb{R}\rightarrow\mathbb{R}\) is a Carathéodory function, and there exist a function \(a\in\mathcal {L}_{+}^{1}\) and a constant \(b>0\) such that \(\vert v(t,x)\vert \leq a(t)+b\vert x\vert \);
 (\(\mathcal{H}3\)):

\(f:\mathbb{R}_{+}\times\mathbb{R}^{2}\rightarrow\mathbb {R}\) is a Carathéodory function, and there exist two positive functions \(\alpha,\beta\in\mathcal{L}^{\infty}_{\mathrm {loc}}\) such that$$\bigl\vert f(t,x_{1},y_{1})f(t,x_{2},y_{2}) \bigr\vert \leq\alpha(t)\vert x_{1}x_{2}\vert +\beta(t) \vert y_{1}y_{2}\vert ; $$
 (\(\mathcal{H}4\)):

\(\alpha_{T}+b\beta_{T}\vert K\vert _{T}<1\) for all \(T>0\), where$$\alpha_{T}:=\operatorname {ess}\sup_{t\in[0,T]}\alpha(t),\qquad \beta_{T}:=\operatorname {ess}\sup_{t\in [0,T]}\beta(t),\qquad \vert K\vert _{T}:=\operatorname {ess}\sup_{s\in[0,T]}\int_{s}^{T} \bigl\vert \kappa(t,s)\bigr\vert \,dt. $$
Remark 4.2
Theorem 4.3
Assume that the assumptions (\(\mathcal{H}\)1)(\(\mathcal{H}\)4) are satisfied, then (1.2) has at least one solution in \(\mathcal{L}^{1}_{\mathrm {loc}}\).
Proof
We will apply Theorem 3.4 to prove the present theorem. Let us take the spaces X and Y of Theorem 3.4 to \(\mathcal {L}^{1}_{\mathrm {loc}}\). Our proving is divided into several steps.
(1) Assumption (\(\mathcal{H}_{2}\)) shows that the superposition operator \(\mathcal{N}_{v}\) is continuous and maps bounded sets of \(\mathcal {L}^{1}_{\mathrm {loc}}\) into bounded sets of \(\mathcal{L}^{1}_{\mathrm {loc}}\) by Theorem 2.5. It follows that the operator \(\Pi_{T}K\mathcal{N}_{v}\) is continuous and maps \(L^{1}[0,T]\) into itself since Remark 4.2 shows K is continuous. The arbitrariness of \(T>0\) implies that \(A=K\mathcal {N}_{v}\) is continuous on \(\mathcal{L}^{1}_{\mathrm {loc}}\).
Now we check that \(A=K\mathcal{N}_{v}\) maps relatively weakly compact sets into relatively strongly compact ones. To this end, let \((x_{n})_{n\in\mathbb{N}}\) be a weakly convergent sequence of \(\mathcal {L}^{1}_{\mathrm {loc}}\). By Remark 4.1(3), \((\Pi_{T}x_{n})_{n\in\mathbb {N}}\) is weakly convergent for an arbitrarily given \(T>0\). Since \(\mathcal{N}_{v}\) is wwcompact on \(L^{1}[0,T]\) by [10], Lemma 3.2, the sequence \((\Pi_{T}\mathcal{N}_{v}x_{n})_{n\in\mathbb{N}}\) has a weakly convergent subsequence, say \((\Pi_{T}\mathcal{N}_{v}x_{n_{k}})_{k\in\mathbb{N}}\). Moreover, since the linear operator K is weakly continuous on \(L^{1}[0,T]\) by Lemma 4.2, the sequence \((\Pi _{T}K\mathcal{N}_{v}x_{n_{k}})_{k\in\mathbb{N}}\), i.e., \((\Pi_{T}Ax_{n_{k}})_{k\in\mathbb{N}}\), converges pointwise almost everywhere on \([0,T]\). Using Vitali’s convergence theorem [32], p.94, we conclude that \((\Pi _{T}Ax_{n_{k}})_{k\in\mathbb{N}}\) is strongly convergent in \(L^{1}[0,T]\). The arbitrariness of \(T>0\) implies that \((Ax_{n_{k}})_{k\in \mathbb{N}}\) is strongly convergent in \(\mathcal{L}^{1}_{\mathrm {loc}}\). Accordingly, A is wscompact and the condition (i) of Theorem 3.4 is fulfilled.
Example 4.4
Since the assumptions (\(\mathcal{H}_{1}\))(\(\mathcal{H}_{4}\)) are all satisfied, we apply Theorem 4.3 to derive the existence of solutions to the equation of this example.
Declarations
Acknowledgements
The authors would like to express their thanks to the referees for the careful reading and helpful remarks on the manuscript. This research was supported by Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 15KJB110001).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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