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 Open Access
Positive fixed points for convex and decreasing operators in probabilistic Banach spaces with an application to a twopoint boundary value problem
 Mohamed Jleli^{1} and
 Bessem Samet^{1}Email author
https://doi.org/10.1186/s136630150338x
© Jleli and Samet 2015
 Received: 28 February 2015
 Accepted: 22 May 2015
 Published: 12 June 2015
Abstract
We establish a fixed point theorem for decreasing and convex operators in a probabilistic Banach space partially ordered by a normal cone. We give an application to the study of the existence and uniqueness of positive solutions to a twopoint boundary value problem.
Keywords
 positive fixed point
 probabilistic Banach space
 cone
 convex and decreasing operator
 boundary value problem
MSC
 47H07
 33E30
1 Introduction and preliminaries
The concept of probabilistic Banach spaces was introduced by Serstnev [1] by adapting the idea of Menger [2] to linear spaces. Fixed point theory in such spaces was studied and developed by many authors (see [3–8] and the references mentioned therein).
This paper deals with the existence and uniqueness of fixed points for a certain class of convex and decreasing operators defined in a probabilistic Banach space partially ordered by a cone.
For the sake of convenience, we first give some definitions and known results from the existing literature. For more details, we refer to [4, 9].
Definition 1.1
 (i)
f is nondecreasing;
 (ii)
f is leftcontinuous;
 (iii)
\(\inf_{t\in\mathbb{R}}f(t)=0\) and \(\sup_{t\in\mathbb{R}}f(t)=1\).
Definition 1.2
 (i)
\(T(a,1)=a\);
 (ii)
\(T(a,b)=T(b,a)\);
 (iii)
\(c\geq a, d\geq b\Longrightarrow T(c,d)\geq T(a,b)\);
 (iv)
\(T(T(a,b),c)=T(a,T(b,c))\).
As standard examples, \(T_{m}(a,b)=\min\{a,b\}\) and \(T_{p}(a,b)=ab\) on \([0,1]\) are Tnorms.
Definition 1.3
 (i)
\(N_{x}(0)=0\) for every \(x\in X\);
 (ii)
\(N_{x}(t)=1\) for all \(t>0\) iff \(x=0\);
 (iii)
\(N_{\alpha x}(t)=N_{x} (\frac{t}{\alpha} )\) for all \(x\in X\) and \(\alpha\in\mathbb{R}\backslash\{0\}\);
 (iv)
\(N_{x+y}(s+t) \geq T(N_{x}(s),N_{y}(t))\) for all \(x,y\in X\) and \(s,t\geq0\).
In the above definition, for \(x \in X\), the distribution function \(N(x)\) is denoted by \(N_{x}\) and \(N_{x}(t)\) is the value \(N_{x}\) at \(t\in \mathbb{R}\).
Example 1.4
Example 1.5
Now, let us recall some topological properties of probabilistic normed spaces.
Definition 1.6
Definition 1.7
Definition 1.8
Let \((X,N,T)\) be a probabilistic normed space. It is said to be a Banach probabilistic normed space (or complete) if every Cauchy sequence in X is convergent to a point in X.
Definition 1.9
Let \((X,N,T)\) be a probabilistic normed space. A subset A of X is said to be closed if every convergent sequence in A converges to an element of A.
Definition 1.10
 (i)
P is closed and convex;
 (ii)
if \(p\in P\), \(tp\in P\) for every \(t\geq0\);
 (iii)
if both p and −p are in P, then \(p=0\).
Definition 1.11
Definition 1.12
Definition 1.13
The paper is organized as follows. In Section 2, we study the existence and uniqueness of positive fixed points for a certain class of decreasing and convex operators \(A: P\to P\). In Section 3, we study the existence and uniqueness of positive solutions to the nonlinear functional equation \(x=x_{0}+Bx\), where \(x_{0}\in P\) and \(B: P\to P\) is a given operator satisfying certain conditions. Section 4 contains a Banach version of our man result established in Section 2. Finally, in Section 5, we present an application of our main result to the study of the existence and uniqueness of positive solutions to a nonlinear differential equation of second order with twopoint boundary value problem.
2 Main result and proof
Before stating our main result, we need some lemmas.
Lemma 2.1
Let \((X,N,T)\) be a probabilistic Banach space and \(\{x_{n}\}\) be a sequence in X that converges to some \(x\in X\). Then any subsequence of \(\{x_{n}\}\) converges to x.
Proof
Lemma 2.2
(see [9])
 (i)
if \(\{x_{n}\}\) converges to \(x\in X\) and \(\{y_{n}\}\) converges to \(y\in X\), then \(\{x_{n}+y_{n}\}\) converges to \(x+y\);
 (ii)
if \(\{\alpha_{n}\}\) converges to some \(\alpha\in\mathbb {R}\) and \(\{x_{n}\}\) converges to some \(x\in X\), then \(\{\alpha_{n}x_{n}\}\) converges to αx.
Lemma 2.3
Proof
Lemma 2.4
Proof
The following result is an immediate consequence of Lemma 2.2 and Lemma 2.4.
Lemma 2.5
Now, we are ready to state and prove our main result.
 (\(\mathcal{A}1\)):

\(0\prec A0\);
 (\(\mathcal{A}2\)):

A is a convex and decreasing operator;
 (\(\mathcal{A}3\)):

there exist \(\gamma\in(0,1)\) and \(m_{0},n_{0}\in\mathbb {N}\) with \(n_{0}>m_{0}\) such thatand$$ A^{2m_{0}+2}0A^{2m_{0}}0\succeq\gamma \bigl(A^{2m_{0}+3}0A^{2m_{0}}0\bigr) $$(2.3)$$ A^{2n_{0}}0\succeq\frac{1}{2} \bigl(A^{2m_{0}+1}0+A^{2m_{0}}0 \bigr). $$(2.4)
Theorem 2.6
 (i)
A has a unique fixed point \(x^{*} \in P\);
 (ii)for any initial value \(x_{0}\in P\), the Picard sequence \(\{ x_{n}\}\) in X defined byconverges to \(x^{*}\);$$x_{n} =Ax_{n1}, \quad n\geq1 $$
 (iii)we have the estimatesfor every \(n>n_{0}+2m_{0}\), \(t\in\mathbb{R}\), and$$\begin{aligned}& N_{x_{2(m_{0}+n)}x^{*}}(t) \\& \quad \geq T \biggl(N_{A0} \biggl( \frac {(n2n_{0}+m_{0})t}{K^{2}} \biggr),N_{A0} \biggl(\frac {(n2n_{0}+m_{0})t}{K^{2}} \biggr) \biggr) \end{aligned}$$(2.5)for every \(n>n_{0}+1m_{0}\), \(t\in\mathbb{R}\).$$\begin{aligned}& N_{x_{2(m_{0}+n)+1}x^{*}}(t) \\& \quad \geq T \biggl(N_{A0} \biggl( \frac {(n1n_{0}+m_{0})t}{K^{2}} \biggr),N_{A0} \biggl(\frac {(n1n_{0}+m_{0})t}{K^{2}} \biggr) \biggr) \end{aligned}$$(2.6)
Proof
3 Positive solutions for the nonlinear functional equation: \(x=x_{0}+Bx\)
 (\(\mathcal{B}1\)):

\(B0=0\);
 (\(\mathcal{B}2\)):

B is a convex and decreasing operator.
We have the following result.
Theorem 3.1
Let \(B\in\mathcal{B}\) and \(x_{0}\in P\) such that \(0\prec x_{0}\). Then the operator Eq. (3.1) has a unique solution \(x^{*}\in P\).
Proof
Hence \(A\in\mathcal{A}\) and the result follows from Theorem 2.6. □
4 The case of Banach spaces
 (i)
\(0\prec A0\);
 (ii)
A is a convex and decreasing operator;
 (iii)there exist \(\gamma\in(0,1)\) and \(m_{0},n_{0}\in\mathbb{N}\) with \(n_{0}>m_{0}\) such thatand$$A^{2m_{0}+2}0A^{2m_{0}}0\succeq\gamma\bigl(A^{2m_{0}+3}0A^{2m_{0}}0 \bigr) $$$$A^{2n_{0}}0\succeq\frac{1}{2} \bigl(A^{2m_{0}+1}0+A^{2m_{0}}0 \bigr). $$
Lemma 4.1
P is a normal cone in the probabilistic Banach space \((X,N,T_{m})\) with normal constant K.
Proof
Now, using Theorem 2.6 and Lemma 4.1, we obtain the following fixed point result in Banach spaces ([14], Theorem 2.1).
Corollary 4.2
 (I)
A has a unique fixed point \(x^{*} \in P\);
 (II)for any initial value \(x_{0}\in P\), the Picard sequence \(\{ x_{n}\}\) in X defined byconverges to \(x^{*}\);$$x_{n} =Ax_{n1},\quad n\geq1 $$
 (III)we have the estimatesfor every \(n>n_{0}+2m_{0}\), and$$\bigl\Vert x_{2(m_{0}+n)}x^{*}\bigr\Vert \leq\frac{K^{2}\A0\}{n2n_{0}+m_{0}} $$for every \(n>n_{0}+1m_{0}\).$$\bigl\Vert x_{2(m_{0}+n)+1}x^{*}\bigr\Vert \leq\frac{K^{2}\A0\}{n1n_{0}+m_{0}} $$
5 An application to a twopoint boundary value problem
In this section, we present an application of Theorem 2.6 to a twopoint boundary value problem.
 (a_{1}):

a is a continuous function;
 (a_{2}):

\(a(x(1x))=a(x)\) for every \(x\in[0,1]\);
 (a_{3}):

\(0< m\leq a(x)\leq M\) for every \(x\in[0,1]\).
Example 5.1
Let \(a: [0,1]\to\mathbb{R}\) be a positive constant function. Then a satisfies conditions (a_{1})(a_{4}).
Example 5.2
 (f_{1}):

f is a continuous function, \(f\geq0\);
 (f_{2}):

f is a decreasing and convex function;
 (f_{3}):

\(f(0)=1\), \(0< f (\frac{M}{8} )<1\);
 (f_{4}):

\(f(\gamma x)\geq\frac{1}{2}\) for every \(x\in [0,\frac {M}{8} ]\), where$$\gamma=1\frac{1}{6}f \biggl(\frac{M}{8} \biggr) \biggl(1f \biggl(\frac {m}{2} \biggr) \biggr). $$
Example 5.3
Theorem 5.4
The boundary value problem (5.1) has a unique positive solution \(u^{*}\in P\).
Proof
Clearly, the operator A is convex and decreasing with respect to the partial order ⪯.
Now, the desired result follows from Theorem 2.6 with \(m_{0}=0\) and \(n_{0}=2\). □
Declarations
Acknowledgements
This project was funded by the National Plan for Science, Technology and Innovation (MAARIFAH), King Abdulaziz City for Science and Technology, Kingdom of Saudi Arabia, Award Number (12MAT 291302).
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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