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Fixed point theorems in locally convex spaces and a nonlinear integral equation of mixed type
 Fuli Wang^{1}Email author and
 Hua Zhou^{1}
https://doi.org/10.1186/s1366301504770
© Wang and Zhou 2015
 Received: 21 May 2015
 Accepted: 30 November 2015
 Published: 14 December 2015
Abstract
In this paper, we provide a new approach for discussing the solvability of a class of operator equations by establishing fixed point theorems in locally convex spaces. Our results are obtained extend some Krasnosel’skii type fixed point theorems. As an application, we investigate the existence and global attractivity of solutions for a general nonlinear integral equation of mixed type of Urysohn and Volterra.
Keywords
 fixed point theorem
 locally convex space
 nonlinear integral equation
MSC
 47H10
 45G10
 45D05
1 Introduction
As an example of algebraic settings, the captivating Krasnosel’skii’s fixed point theorem (see [1] or [2], p.31) leads to the consideration of fixed points for the sum of two operators. It asserts that, if M is a bounded, closed, and convex subset of a Banach space X and A, B are two mappings from M into X such that A is compact and B is a contraction, \(A(M)+B(M)\subseteq M\), then \(A+B\) has at least one fixed point in M. Since then, there has been a vast literature dealing with the improvements of such a result. For instance, in locally convex spaces or in Fréchet spaces all kinds of extensions and generalizations of Krasnosel’skii’s fixed point theorem have been obtained by some authors (see e.g. [3–11] etc.).
Our results may also be applied to find the existence and uniqueness of solutions of differential equations of fractional order with integral boundary conditions [14].
Definition 1.1
(see [15], Definition 1)
The solutions of equation (1.2) are said to be globally attractive, if for arbitrary solutions \(x_{1}(t)\) and \(x_{2}(t)\) of equation (1.2) it follows that \(\lim_{t\rightarrow\infty} \vert x_{1}(t)x_{2}(t)\vert =0\).
On the other hand, owing to the deficiency of a single measure of noncompactness in the space \(C(\mathbb{R}_{+},\mathbb{R}^{d})\), the technique of a family of measures of noncompactness in conjunction with the SchauderTychonoff’s fixed point theorem was applied by Olszowy [11, 18].
Since the calculating of measures of noncompactness is very hard in the majority of locally convex spaces, in the present paper we will show that by establishing some new fixed point results the use of the technique of measures of noncompactness may be avoided. Our goals in this paper are to establish new fixed point theorems for the solvability of equation (1.1) in locally convex spaces, and to study under what conditions equation (1.2) is solvable in \(C(\mathbb{R}_{+},\mathbb{R}^{d})\) by applying our new theorems, and under what conditions the solutions of equation (1.2) are globally attractive. The results obtained will provide a new approach for discussing a class of operator equations in locally convex spaces.
The organization of this paper is as follows. In Section 2, we establish the fixed point theorems for equation (1.1) in locally convex spaces. In Section 3, we prove the existence and the global attractivity of solutions for equation (1.2).
2 Fixed point theorems in locally convex spaces
Throughout this section, 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.
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 \(U=\bigcap_{i=1}^{n}r_{i}V(\rho_{i})\), where \(V(\rho)=\{x:\vert x\vert _{\rho}<1\}\).
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 \(x_{1},x_{2}\in X\).
Theorem 2.1
 (i)
A is continuous, and \(A(M)\) is a relatively compact subset of Y;
 (ii)F is continuous, and for each \(\rho\in\Lambda\) there exists \(\alpha_{\rho}\in[0,1)\) such thatfor all \(x_{1},x_{2}\in X\) and \(y\in Y\);$$ \bigl\vert F(x_{1},y)F(x_{2},y)\bigr\vert _{\rho}\leq\alpha_{\rho} \vert x_{1}x_{2} \vert _{\rho}, $$
 (iii)
\(z=F(z,Ax)\), \(x\in M \Rightarrow z\in M\).
Proof
For a given \(y\in A(M)\) by assumption (ii) the mapping \(F(\cdot,y)\) defined by \(z\mapsto F(z,y)\) is a \(\vert \cdot \vert _{\rho}\)contraction for each \(\rho\in\Lambda\), so it has a unique fixed point in X according to [3], Theorem 2.2. Now, 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 \(Jy=F(Jy,y)\). Accordingly, the mapping J is well defined.
The following corollary shows that our above result extends the Krasnosel’skii fixed point theorem in locally convex spaces.
Corollary 2.2
 (i)
A is continuous, and \(A(M)\) is relatively compact;
 (ii)
B is a \(\vert \cdot \vert _{\rho}\)contraction for each \(\rho\in\Lambda\);
 (iii)
\(y=By+Ax\), \(x\in M \Rightarrow y\in M\).
Since the assumption (iii) of Theorem 2.1 sometimes is hard to verify in actual applications, we next establish a Schaefer type fixed point theorem for equation (1.2).
Theorem 2.3
 (i)
A maps bounded sets of X into relatively compact ones of Y;
 (ii)for each \(\rho\in\Lambda\) there exists \(\alpha_{\rho}\in[0,1)\) such thatfor all \(x_{1},x_{2}\in X\) and \(y\in A(X)\).$$ \bigl\vert F(x_{1},y)F(x_{2},y)\bigr\vert _{\rho}\leq\alpha_{\rho} \vert x_{1}x_{2} \vert _{\rho}, $$
 (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 some \(\lambda\in(0,1)\).
Proof
As in the proof of Theorem 2.1, 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 2.4
 (i)
A is continuous and maps bounded sets into relatively weakly compact ones;
 (ii)
B is a \(\vert \cdot \vert _{\rho}\)contraction for each \(\rho\in\Lambda\).
 (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 some \(\lambda\in(0,1)\).
3 The existence and global attractivity of solutions for a nonlinear integral equation of mixed type
Remark 3.1
 (a)
a nonempty subset M of \(C(\mathbb{R}_{+},\mathbb{R}^{d})\) is said to be bounded if for each \(T>0\) there exists a \(L_{T}>0\) such that \(\vert x\vert _{T}\leq L_{T}\) for all \(x\in M\);
 (b)
the set \(\{x\in C(\mathbb{R}_{+},\mathbb{R}^{d}):\vert x(t)\vert \leq h(t)\}\) is a nonempty, bounded, closed, and convex subset of \(C(\mathbb {R}_{+},\mathbb{R}^{d})\), where \(h:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}\) is a continuous function;
 (c)
a sequence \((x_{n})_{n\in\mathbb{N}}\) is convergent to x in \(C(\mathbb{R}_{+},\mathbb{R}^{d})\) if and only if \(\vert x_{n}x\vert _{T}\rightarrow0\) for each \(T>0\), i.e. \((x_{n})_{n\in\mathbb{N}}\) is uniformly convergent to x on the interval \([0,T]\);
 (d)
a bounded subset M of \(C(\mathbb{R}_{+},\mathbb{R}^{d})\) is relatively compact if and only if, for each \(T>0\), the restrictions to \([0,T]\) of all functions from M form an equicontinuous set.
 (\(\mathcal{H}\)1):

\(u:\mathbb{R}_{+}^{2}\times\mathbb{R}^{d}\rightarrow\mathbb {R}^{d}\) is continuous, and there exists a continuous function \(\kappa _{1}:\mathbb{R}_{+}^{2}\rightarrow\mathbb{R}_{+}\) such that, for all \((t,s)\in \mathbb{R}_{+}^{2}\) and \(x\in\mathbb{R}^{d}\),$$ \bigl\vert u(t,s,x)\bigr\vert \leq\kappa_{1}(t,s)\vert x\vert ; $$
 (\(\mathcal{H}\)2):

there exists a continuous function \(h:\mathbb {R}_{+}\rightarrow\mathbb{R}_{+}\) such that the improper integral \(\int _{0}^{\infty}\kappa_{1}(t,s)h(s)\,ds\) is uniformly convergent on all parameters \(t\in[0,T]\) for each \(T>0\), i.e. for arbitrarily given \(\varepsilon>0\) there exists \(S=S(\varepsilon, T)>0\) such that, for all \(t\in[0,T]\),$$ \int_{S}^{\infty}\kappa_{1}(t,s)h(s)\,ds< \varepsilon; $$
 (\(\mathcal{H}\)3):

\(v:\Delta\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}\) is continuous, and there exists a continuous function \(\kappa_{2}:\Delta \rightarrow\mathbb{R}_{+}\) such thatfor all \((t,s)\in\Delta\) and \(x\in\mathbb{R}^{d}\), where \(\Delta:=\{ (t,s)\in\mathbb{R}_{+}^{2}:0\leq t\leq s\}\);$$ \bigl\vert v(t,s,x)\bigr\vert \leq\kappa_{2}(t,s)\vert x\vert , $$
 (\(\mathcal{H}\)4):

\(f:\mathbb{R}_{+}\times(\mathbb{R}^{d})^{3}\rightarrow\mathbb {R}^{d}\) is continuous, and there exist three continuous functions \(\alpha :\mathbb{R}_{+}\rightarrow[0,1)\), \(\beta:\mathbb{R}_{+}\rightarrow\mathbb {R}_{+}\) and \(\gamma:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}\) such thatfor all \(x_{i},\overline{u}_{i},\overline{v}_{i}\in\mathbb{R}^{d}\) (\(i=1,2\)) and \(t\in\mathbb{R}_{+}\);$$ \bigl\vert f(t,x_{1},\overline{u}_{1}, \overline{v}_{1})f(t,x_{2},\overline{u}_{2}, \overline {v}_{2})\bigr\vert \leq\alpha(t)\vert x_{1}x_{2} \vert +\beta(t)\vert \overline{u}_{1}\overline {u}_{2} \vert +\gamma(t)\vert \overline{v}_{1}\overline{v}_{2} \vert , $$
 (\(\mathcal{H}\)5):

the following inequality holds for all \(t\in\mathbb{R}^{+}\):$$ \bigl\vert f(t,0,0,0)\bigr\vert +\alpha(t)h(t)+\beta(t) \int_{0}^{\infty}\kappa_{1}(t,s)h(s)\,ds + \gamma(t) \int_{0}^{t}\kappa_{2}(t,s)h(s)\,ds\leq h(t); $$
 (\(\mathcal{H}\)6):

the following conditions are satisfied:$$ \lim_{t\rightarrow\infty}\frac{\beta(t)}{1\alpha(t)} \int_{0}^{\infty}\kappa _{1}(t,s)h(s)\,ds=0 \quad \mbox{and}\quad \lim_{t\rightarrow\infty}\frac{\gamma(t)}{1\alpha(t)} \int _{0}^{t}\kappa_{2}(t,s)h(s)\,ds=0. $$
Lemma 3.2
Proof
Lemma 3.3
Proof
Let \(\Delta_{T}=\{(t,s):0\leq s\leq t\leq T\}\) for an arbitrarily given \(T>0\). It is easily known that the function Vx is continuous on \(\mathbb{R}^{+}\) for a given \(x\in M\) since v is continuous and the function \(v(t,\cdot,x(\cdot))\) is integrable on \([0,t]\) for each \(t\in [0,T]\). Thus the operator \(V:C(\mathbb{R}_{+},\mathbb{R}^{d})\rightarrow C(\mathbb{R}_{+},\mathbb{R}^{d})\) is well defined.
We are now in a position by applying our Theorem 2.1 to prove the existence and global attractivity of solutions of equation (1.2).
Theorem 3.4
Under the assumptions (\(\mathcal{H}\)1)(\(\mathcal{H}\)5) equation (1.2) has at least one solution \(x=x(t)\) which belongs to the spaces \(C(\mathbb{R}_{+};\mathbb{R}^{d})\); furthermore if (\(\mathcal{H}\)6) is also satisfied then the solutions of equation (1.2) are globally attractive.
Proof
(1) According to Lemma 3.2 and Lemma 3.3, for an arbitrarily given \(x\in X\) we easily infer \(Ax=(Ux,Vx)\in Y\) since \((Ux)(t)\) and \((Vx)(t)\) are all continuous functions defined on \(\mathbb {R}_{+}\); and the continuity of operator A is also easily obtained from the continuity of operators U and V.
Further, from Lemma 3.2 and Lemma 3.3 we infer that \(A(M)\subseteq U(M)\times V(M)\) is relatively compact by the Tychonoff product theorem, and therefore the condition (i) of Theorem 2.1 is satisfied.
(2) Obviously, for arbitrarily given \(x\in X\) and \(y\in Y\) the continuity of \(F(x,y)(t)\) on \(\mathbb{R}_{+}\) may easily be inferred from continuity of the function \(f(t,x(t),\overline{u}(t),\overline{v}(t))\).
By now we showed that the solution of equation (1.2) is globally attractive and finish the proof. □
Declarations
Acknowledgements
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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