Fixed point theorems in locally convex algebras and applications to nonlinear integral equations
- Le Khanh Hung^{1}Email author
https://doi.org/10.1186/s13663-015-0310-9
© Khanh Hung; licensee Springer. 2015
Received: 9 January 2015
Accepted: 16 April 2015
Published: 29 April 2015
Abstract
In this paper, we give a fixed point theorem in locally convex algebras and present applications to the existence problem for a class of nonlinear integral equations with unbounded deviations.
Keywords
1 Introduction
The research of functional integral equations and differential equations is a main object of investigations of nonlinear functional analysis. These equations occur in physical, biological, and economic problems. The existence of solutions of them are commonly proved by suitable fixed point theorems. In this paper, we establish a new fixed point theorem for continuous mappings on locally convex algebras. These results arose out of our examination of a particular functional integral with unbounded deviations. We noticed in our study that the fixed point theorems developed in the Banach algebras (see [1–5] and the references given therein) were not always useful in establishing existence principles for the problems that we are interested in Section 4. More precisely, the well-known results in Banach algebras are not applicable to nonlinear integral equations with unbounded deviations.
2 Preliminaries
Before stating the main results, we give some useful definitions, preliminaries which will be used in the sequel.
Let X be a uniform space. Then uniform topology on X is generated by the family of uniform continuous pseudometrics on \(X\times X\) (see [6]). In this paper, by \((X, \mathcal{P})\) we mean a Hausdorff uniform space whose uniformity is generated by a saturated family of pseudometrics \(\mathcal{P}=\{ d_{\alpha}(x, y): \alpha\in I\}\), where I is an index set. Note that \((X, \mathcal{P})\) is Hausdorff if and only if \(d_{\alpha}(x, y)=0\) for all \(\alpha\in I\) implies \(x=y\).
Definition 2.1
([7])
- (1)
The sequence \(\{x_{n}\}\subset X\) is Cauchy if \(d_{\alpha}(x_{n},x_{m})\rightarrow0\) as \(m,n \rightarrow+\infty\) for every \(\alpha\in I\).
- (2)
X is said to be sequentially complete if every Cauchy sequence \(\{x_{n}\}\) in X converges to \(x\in X\).
Definition 2.2
([7])
Now, we introduce the two classes of functions which play crucial roles in the fixed point theory. Sometimes, they are called control functions.
- (i)
\(\phi:[0,+\infty)\rightarrow[0,+\infty)\) is monotone non-decreasing and continuous;
- (ii)
\(0<\phi(t)<t\) for all \(t>0\) and \(\phi(0)=0\).
- (i)
\(\psi:[0,+\infty)\rightarrow[0,+\infty)\) is monotone non-decreasing and continuous;
- (ii)
\(\psi(0)=0\).
We need the following fact for the class of functions Φ. It may be not original.
Lemma 2.3
Proof
Definition 2.4
([7])
The following is due to Angelov [8].
Theorem 2.5
- (1)
T is Φ-contractive;
- (2)for every \(\alpha\in I\) there exists a function \(\overline{\phi}_{\alpha}\in\Phi\) such thatand \(\frac{\overline{\phi}_{\alpha}(t)}{t}\) is monotone non-decreasing;$$\sup\bigl\{ \phi_{j^{n}(\alpha)}(t): n=0,1,2,\ldots\bigr\} \leq\overline{\phi}_{\alpha}(t) $$
- (3)
there is \(x_{0}\in X\) such that for every \(\alpha\in I\) there exists \(q(\alpha)>0\) such that the inequality \(d_{j^{n}(\alpha)}(x_{0}, Tx_{0})\leq q(\alpha)\) is valid for all \(n=0,1,\ldots\) .
Then T has at least one fixed point in X.
Angelov added the following properties of X for the uniqueness of fixed point.
Definition 2.6
([7])
Theorem 2.7
Suppose that the conditions of Theorem 2.5 are fulfilled. If X is j-bounded then F has a unique fixed point.
Remark 2.8
If E is a locally convex space with a saturated family of seminorms \(\{p_{\alpha}\}_{\alpha\in I}\), then the associated family of pseudometrics \(\{d_{\alpha}\}_{\alpha\in I}\) defined by \(d_{\alpha}(x, y)=p_{\alpha}(x-y)\) for every \(x, y\in E\) and \(\alpha\in I\). The uniform topology, which is generated by this family of pseudometrics \(\{d_{\alpha}\}_{\alpha\in I}\), coincides with the original topology of the space E. Therefore, as a corollary of Theorem 2.5, we obtain the fixed point theorems in locally convex spaces.
Let X be a locally convex space and \(T: X\to X\). Then T is called a compact operator if \(\overline{T(X)}\) is a compact subset of X. Again T is called totally bounded if for any bounded set S of X, \(T(S)\) is a totally bounded set of X. Further, T is called completely continuous if it is continuous and totally bounded. Note that every compact operator is totally bounded. The two notions are equivalent on a bounded set of X.
The following theorem is called the Tikhonov-Schauder fixed point theorem.
Theorem 2.9
([9])
Throughout this paper, we consider associative and commutative algebras over the field \(\Bbb{K}\) of complex numbers or real numbers.
Definition 2.10
- (1)
E is a topological vector space;
- (2)
the multiplication in E is continuous.
- (i)
\(p(x) \geq 0\), for every \(x\in E\);
- (ii)
\(p (x + y) \leq p(x) + p(y)\), for any \(x , y \in E\);
- (iii)
\(p(\lambda x) = | \lambda| p(x)\), for any \(\lambda\in K\) and \(x\in E\).
Definition 2.11
- (1)
A seminorm \(p: E\to\Bbb{R}\) is called submultiplicative if \(p(xy) \leq p(x)p(y)\) for any elements x, y in E.
- (2)
A set \(U\subset E\) is called multiplicative if \(U\cdot U\subset U\).
Definition 2.12
The topological algebra E is called a locally multiplicatively convex algebra if E has a local basis consisting of multiplicative and convex sets.
In this paper, a locally multiplicatively convex algebra is briefly called a locally convex algebra. The following remark is due to [10] and [11].
Remark 2.13
Let E be a locally convex algebra. Then one can show that its topology is defined by a saturated family \(\mathcal{P}=\{p_{\alpha}\}_{\alpha\in I}\) of submultiplicative seminorms.
Example 2.14
For more basis material as regards the theory of locally convex algebra, we refer the reader to [10, 12], and [11].
3 Fixed point theorems in locally convex algebras
Let X be a locally convex algebra with a saturated family of seminorms \(\{p_{\alpha}\}_{\alpha\in I}\).
Definition 3.1
If \(\psi_{\alpha}(t)=k_{\alpha}t\) for all \(t\geq0\), where \(k_{\alpha}\) is a real number for all \({\alpha\in I}\), then T is called Lipschitzian with the family of Lipschitz constants \(\{k_{\alpha}\}_{\alpha\in I}\).
Theorem 3.2
- (A1)
A is \(\mathcal{D}\)-Lipschitzian with the family of functions \(\{\psi_{\alpha}\}\);
- (A2)
B is completely continuous and \(x=Ax By\) implies \(x\in S\) for every \(y\in S\);
- (A3)
\(p_{j(\alpha)}(x-y)\leq p_{\alpha}(x-y)\) for every \(x,y\in S\) and \(\alpha\in I\);
- (A4)for every \(x\in X\) and for every \(\alpha\in I\), there exists \(q(\alpha, x)\) such thatfor all \(k=0,1,2,\ldots\) . In particular, \(p_{j^{k}(\alpha)}(x)\leq q(\alpha)< +\infty\) for every \(x\in S\) and for all \(k=0,1,2,\ldots\) ;$$p_{j^{k}(\alpha)}(x)\leq q(\alpha, x)< +\infty $$
- (A5)for each \(\alpha\in I\),for all \(t>0\) and there exists \(\overline{\phi}_{\alpha}\in\Phi\) such that \(\frac{\overline{\phi}_{\alpha}(t)}{t}\) is non-decreasing and$$M_{\alpha}\psi_{\alpha}(t)< t $$for every \(t>0\), where \(M_{\alpha}=\sup\{p_{\alpha}(B(x)): x\in S\}\), \(\alpha\in I\).$$\sup\bigl\{ M_{j^{k}(\alpha)}\psi_{j^{k}(\alpha)}(t): k=0,1,2,\ldots\bigr\} \leq \overline{\phi}_{\alpha}(t) $$
Then the operator equation \(x=AxBx\) has a solution.
Proof
Remark 3.3
We can immediately obtain the following corollary.
Corollary 3.4
- (B1)
A is Lipschitzian with the family of Lipschitz constants \(\{k_{\alpha}\}\);
- (B2)
B is completely continuous and \(x=Ax By\) implies \(x\in S\) for every \(y\in S\);
- (B3)
\(p_{j(\alpha)}(x-y)\leq p_{\alpha}(x-y)\) for every \(x,y\in S\) and \(\alpha\in I\);
- (B4)for every \(x\in X\) and for every \(\alpha\in I\), there exists \(q(\alpha, x)\) such thatfor all \(k=0,1,2,\ldots\) . In particular, \(p_{j^{k}(\alpha)}(x)\leq q(\alpha)< +\infty\) for every \(x\in S\) and for all \(k=0,1,2,\ldots\) ;$$p_{j^{k}(\alpha)}(x)\leq q(\alpha, x)< +\infty $$
- (B5)for each \(\alpha\in I\),and$$M_{\alpha}k_{\alpha}< 1 $$where \(M_{\alpha}=\sup \{p_{\alpha} (B(x) ): x\in S \}\), \(\alpha\in I\).$$\sup \{ M_{j^{k}(\alpha)}k_{j^{k}(\alpha)}: k=0,1,2,\ldots \}\leq r_{\alpha}< 1, $$
Then the operator equation \(x=AxBx\) has a solution.
4 Applications to nonlinear integral equations
As in [7] and [2], we shall adopt the following assumptions.
Assumption 4.1
- (C1)
The functions \(\Delta_{i}(t):\Bbb{R}_{+}\to\Bbb{R}_{+}\), \(i=1,2,\ldots,m\); \(\tau_{l}(t):\Bbb{R}_{+}\to\Bbb{R}_{+}\), \(l=1,2,\ldots,n\), are continuous and \(\Delta_{i}(t)\leq t\), \(\tau_{l}(t)\leq t\) for every \(t>0\).
- (C2)The function \(F: (t, u_{1},u_{2},\ldots,u_{m}, v_{1},\ldots ,v_{n}):\Bbb{R}_{+}\times\Bbb{R}^{m+n}\to [0, 1]\) is continuous and satisfies the conditionswhere the function \(\Omega(t, x_{1},\ldots,x_{m}, y_{1},\ldots,y_{n}):\Bbb{R}_{+}^{m+n+1}\to\Bbb{R}_{+}\) is continuous in t, non-decreasing and continuous in each \(x_{i}\), \(y_{l}\), \(\Omega(t, ay,\ldots,ay,y,\ldots,y)< y\) for every constant \(a>0\) and \(\frac{\Omega(t, ay,\ldots,ay, y,\ldots,y)}{y}\) is non-decreasing in y.$$\begin{aligned} & \bigl| F(t, u_{1},\ldots,u_{m}, v_{1}, \ldots,v_{n})-F(t, \overline{u}_{1},\ldots,\overline{u}_{m}, \overline{v}_{1},\ldots,\overline{v}_{n}) \bigr| \\ &\quad \leq\Omega \bigl(t, | u_{1}-\overline{u}_{1}|,\ldots,| u_{m}-\overline{u}_{m}|, | v_{1}-\overline{v}_{1}|,\ldots, | v_{n}-\overline{v}_{n}| \bigr), \end{aligned}$$
- (C3)
q is uniformly continuous on \(\Bbb{R}_{+}\), \(\| q\|_{\infty}=\sup_{t\in\Bbb{R}_{+}}| q(t)|< 1\) and \(\int_{0}^{+\infty} | f (s, x(s) ) |\,ds< 1-\| q\|_{\infty}\) for every \(x\in C(\Bbb{R}_{+},\Bbb{R})\) with \(| x(t)| \leq1\) for all t.
Theorem 4.2
Under assumptions (C1), (C2), and (C3), (4) has at least one solution \(x=x(t)\) which belongs to the space \(C(\Bbb{R}_{+},\Bbb{R})\).
Proof
Now, we shall check the condition (A2). Firstly, we show that B is completely continuous. Suppose \((x_{k})\subset S\) and \(x_{k}\to x\). Since S is closed, we have \(x\in S\). By the definition of seminorms \(p_{[0, n]}\), we can deduce that \((x_{k})\) uniformly convergent to x on \([0, n]\) for each \(n=1,2,\ldots\) . It follows that \(| f (s, x_{k}(s) )-f (s, x(s) ) | \to 0\) as \(k\to+\infty\) for every \(s\in[0,+\infty)\). Moreover, by the condition (C3), we infer the \(g(s):=f (s, x(s) )\) is a bounded function on \([0, n]\) for each \(n=1,2,\ldots\) .
Finally, applying Theorem 3.2, we can conclude that (4) has a solution. □
The following example is an illustration of Theorem 4.2.
Example 4.3
We will show that the equation has a solution on \(C(\Bbb{R}^{+}, \Bbb{R})\).
Declarations
Acknowledgements
The author would like to thank the referee for very deep and useful comments, and his/her suggestions that contributed to the improvement of the manuscript. My thanks go to Professor Tran Van An and Kieu Phuong Chi for their valuable suggestions.
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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