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A generalization of DiazMargolis’s fixed point theorem and its application to the stability of generalized Volterra integral equations
 WeiShih Du^{1}Email author
https://doi.org/10.1186/s136600150931x
© Du 2015
 Received: 16 June 2015
 Accepted: 6 December 2015
 Published: 22 December 2015
Abstract
In this paper, a generalization of DiazMargolis’s fixed point theorem is established. As applications of the generalized DiazMargolis’s fixed point theorem, we present some existence theorems of the HyersUlam stability for a general class of the nonlinear Volterra integral equations in Banach spaces.
Keywords
 HyersUlam stability
 nonlinear Volterra integral equation
 \(\mathcal{MT}\)function (\(\mathcal{R}\)function)
 generalized metric space
 DiazMargolis’s fixed point theorem
MSC
 26D10
 34K20
 45D05
 47H10
1 Introduction and preliminaries
The stability of functional equations was originally raised in a famous talk given by Ulam [1] at Wisconsin University in 1940. The problem posed by Ulam was the following:
A partial answer to Ulam’s question in the case of Banach spaces was given by Hyers [2] in 1941. Later, Aoki [3] studied this problem for additive mappings and Rassias [4] generalized Hyers’ theorem for the stability of unbounded Cauchy equations. Since then the rapid growth of the study of stability of functional equations has been developed at a high rate by several authors in the last decades; for more details, we refer the readers to [3–14] and references therein.
Definition 1.1
 (GM1)
\(p(x,y)=0\) if and only if \(x=y\);
 (GM2)
\(p(x,y)=p(y,x)\) for all \(x, y\in X\);
 (GM3)
\(p(x,z)\leq p(x,y)+p(y,z)\) for all \(x,y,z\in X\).
We remark that the only one difference of the generalized metric from the usual metric is that the range of the former is permitted to include the infinity. A generalized Banach contraction principle in a complete generalized metric space proved by Diaz and Margolis [15] has played an important role in the study of stability of functional equations.
Theorem 1.1
(Diaz and Margolis [15])
 (a)
\(p(T^{n}u,T^{n+1}u)=\infty\) for all \(n\in \mathbb{N} \cup\{0\}\),
 (b)
there exists a nonnegative integer ℓ such that \(p(T^{n}u,T^{n+1}u)<\infty\) for all \(n\geq \ell\).
 (b1)
the sequence \(\{T^{n}u\}_{n\in \mathbb{N} \cup\{0\}}\) is convergent to a fixed point ŷ of T;
 (b2)ŷ is the unique fixed point of T in the set \(\mathcal{S}\), where$$ \mathcal{S}=\bigl\{ x\in X:p \bigl( T^{\ell}u,x \bigr) < \infty\bigr\} ; $$
 (b3)
\(p(x,\hat{y})\leq\frac{1}{1\lambda}p(x,Tx)\) for all \(x\in\mathcal{S}\).
The main aim of this paper is the study of the existence theorem of the HyersUlam stability for a general class of the nonlinear Volterra integral equations in Banach spaces. In Section 2, we first establish some properties for generalized metric spaces and present a generalization of DiazMargolis’s fixed point theorem. As interesting applications of the generalized DiazMargolis fixed point theorem, we establish some existence theorems of the HyersUlam stability for a general class of the nonlinear Volterra integral equations in Banach spaces in Section 3. Our new results improve and extend some known results in the literature.
2 A generalization of DiazMargolis’s fixed point theorem for \(\mathcal{MT}\)functions
In the present section, we shall establish a generalization of DiazMargolis’s fixed point theorem (i.e. Theorem 1.1) for \(\mathcal{MT}\)functions in the setting of complete generalized metric spaces. We may begin with the following definitions.
Definition 2.1
 (i)
\(\{x_{n}\}\) is said to pconverge to x if for any \(\varepsilon>0\) there exists a natural number \(n_{0}\) such that \(p(x_{n},x)<\varepsilon\) for all \(n\geq n_{0}\). We denote this by \(p\mbox{}\!\lim_{n\rightarrow\infty}x_{n}=x\) or \(x_{n}\overset {p}{\longrightarrow}x\) as \(n\rightarrow\infty\) and call x the limit of \(\{x_{n}\}\).
 (ii)
\(\{x_{n}\}\) is said to be a pCauchy sequence if for any \(\varepsilon>0\) there exists a natural number \(\mathbb{N} _{0}\) such that \(p(x_{n},x_{m})<\varepsilon\) for all \(n, m\geq \mathbb{N} _{0}\).
 (iii)
\((X,p)\) is said to be complete if every pCauchy sequence in X is pconvergent.
Definition 2.2
 (i)The pclosure of A, denoted \(cl_{p}(A)\), is defined byObviously, \(A\subseteq cl_{p}(A)\).$$ cl_{p}(A)=\bigl\{ x\in X:\exists\ \{x_{n}\}\subset A\mbox{ such that }x_{n}\overset{p}{\longrightarrow}x\mbox{ as }n \rightarrow\infty\bigr\} . $$
 (ii)
A is said to be pclosed if \(A=cl_{p}(A)\).
 (iii)
A is said to be popen if the complement \(X\setminus A\) of A is pclosed.
Theorem 2.1
Proof
Therefore, from the above, we prove that \(\mathcal{T}_{p}\) is a topology on \((X,p)\). □
According to Theorem 2.1, we can give the definition of continuity of a mapping in generalized metric spaces. Actually, the definition of continuity can transfer essentially unchanged from classical metric spaces to generalized metric spaces as follows.
Definition 2.3
The following characterization of continuous functions can easily be verified.
Theorem 2.2
Let \((X,p_{X})\) and \((Y,p_{Y})\) be generalized metric spaces and \(x_{0}\in X\). Then a mapping \(f:X\rightarrow Y\) is continuous at x̂ if and only if \(x_{n}\overset{p_{X}}{\longrightarrow}\widehat {x}\) implies \(f(x_{n})\overset{p_{Y}}{\longrightarrow}f(\widehat {x})\) as \(n\rightarrow\infty\).
The following useful auxiliary result is crucial to our proofs.
Theorem 2.3
 (a)
\(p(u,v)<\infty\) for all \(u,v\in\mathcal{W}\);
 (b)
\(\mathcal{W}\) is pclosed in \((X,p)\);
 (c)
\(\vert f(x)f(y)\vert \leq p(x,y)\) for any \(x,y\in\mathcal{W}\);
 (d)
f is uniformly continuous on \(\mathcal{W}\).
Proof
Theorem 2.4
Let \((X,p)\) be a complete generalized metric space and \(\mathcal{D}\) is a pclosed subset of X. Then \((\mathcal{D},p)\) is also complete.
Proof
Let \(\{x_{n}\}\) be a pCauchy sequence in \(\mathcal{D}\). By the completeness of \((X,p)\), there exists \(v\in X\) such that \(x_{n}\overset{p}{\longrightarrow}v\) as \(n\rightarrow\infty\). By the pclosedness of \(\mathcal{D}\), \(v\in cl_{p} ( \mathcal{D} ) =\mathcal{D}\). Hence we prove that \((\mathcal{D},p)\) is complete. □
Definition 2.4
Remark 2.1
It is obvious that if \(\varphi: [0,\infty)\rightarrow [0,1)\) is a nondecreasing function or a nonincreasing function, then φ is an \(\mathcal{MT}\)function. So the set of \(\mathcal{MT}\)functions is a rich class. In 2012, Du [20] established the following characterizations of \(\mathcal{MT}\)functions.
Theorem 2.5
([20], Theorem 2.1)
 (a)
φ is an \(\mathcal{MT}\)function.
 (b)
For each \(t\in[0,\infty)\), there exist \(r_{t}^{(1)}\in[0,1)\) and \(\varepsilon_{t}^{(1)}>0\) such that \(\varphi(s)\leq r_{t}^{(1)}\) for all \(s\in (t,t+\varepsilon_{t}^{(1)})\).
 (c)
For each \(t\in[0,\infty)\), there exist \(r_{t}^{(2)}\in[0,1)\) and \(\varepsilon_{t}^{(2)}>0\) such that \(\varphi(s)\leq r_{t}^{(2)}\) for all \(s\in [ t,t+\varepsilon_{t}^{(2)}]\).
 (d)
For each \(t\in[0,\infty)\), there exist \(r_{t}^{(3)}\in[0,1)\) and \(\varepsilon_{t}^{(3)}>0\) such that \(\varphi(s)\leq r_{t}^{(3)}\) for all \(s\in (t,t+\varepsilon_{t}^{(3)}]\).
 (e)
For each \(t\in[0,\infty)\), there exist \(r_{t}^{(4)}\in[0,1)\) and \(\varepsilon_{t}^{(4)}>0\) such that \(\varphi(s)\leq r_{t}^{(4)}\) for all \(s\in [ t,t+\varepsilon_{t}^{(4)})\).
 (f)
For any nonincreasing sequence \(\{x_{n}\}_{n\in \mathbb{N} }\) in \([0,\infty)\), we have \(0\leq\sup_{n\in \mathbb{N} }\varphi(x_{n})<1\).
 (g)
φ is a function of contractive factor; that is, for any strictly decreasing sequence \(\{x_{n}\}_{n\in \mathbb{N} }\) in \([0,\infty)\), we have \(0\leq\sup_{n\in \mathbb{N} }\varphi(x_{n})<1\).
The main result of this section is formulated in the following new fixed theorem in complete generalized metric spaces, which generalize and improve DiazMargolis’s fixed point theorem.
Theorem 2.6
 (a)
\(p(T^{n}u,T^{n+1}u)=\infty\) for all \(n\in \mathbb{N} \cup\{0\}\);
 (b)
there exists a nonnegative integer ℓ such that \(p(T^{n}u,T^{n+1}u)<\infty\) for all \(n\geq \ell\).
 (b1)
the sequence \(\{T^{n}u\}_{n\in \mathbb{N} \cup\{0\}}\) is convergent to a fixed point v of T;
 (b2)v is the unique fixed point of T in the set \(\mathcal{L}\), where$$ \mathcal{L}=\bigl\{ x\in X:p\bigl(T^{\ell}u,x\bigr)< \infty\bigr\} ; $$
 (b3)
\(p(x,v)\leq\frac{1}{1\alpha(p(x,v))}p(x,Tx)\) for all \(x\in\mathcal{L}\).
Proof
 (i)
\(w_{n}\in\mathcal{L}\),
 (ii)
\(p(w_{n},w_{n+1})<\infty\),
 (iii)
\(p(w_{n+1},w_{n+2})\leq\alpha (p(w_{n},w_{n+1}))p(w_{n},w_{n+1})\).
3 Existence of the HyersUlam stability for generalized Volterra integral equations
In this section, we study the existence theorem of the HyersUlam stability for a general class of the nonlinear Volterra integral equations in Banach spaces by applying Theorem 2.6.
Theorem 3.1
Proof
The following conclusions are immediately drawn from Theorem 3.1.
Corollary 3.1
Proof
Remark 3.1
 (a)Corollary 3.1 actually implies Theorem 3.1. Indeed, under the hypotheses of Theorem 3.1, we set \(L:=\frac{\delta}{\lambda(ba)}\). Due to (3.1), (3.2), and \(0<\delta<1\), we get the following:

\(\Vert G(x,\tau,y)G(x,\tau,z)\Vert \leq L\Vert yz\Vert \) for any \(x,\tau\in I\) and \(y,z\in E\);

\(0<\lambda L(ba)<1\).

Corollary 3.2
Proof
Remark 3.2
Recently, Jung et al. obtained an interesting result on HyersUlam stability of the linear functional equation in a single variable \(f(\phi(x))=g(x)\cdot f(x)\) on a complete metric group (for more details, see [10]). The results in this paper can be generalized further in the spirit of complete metric groups as in [10].
Declarations
Acknowledgements
The author wishes to express his cordial thanks to Professor Simeon Reich and the anonymous referees for their valuable suggestions and comments. The author was supported by Grant No. MOST 1032115M017001 and Grant No. MOST 1042115M017002 of the Ministry of Science and Technology of the Republic of China.
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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