Citation: Sylvia Kirchengast. Diabetes and Obesity—An Evolutionary Perspective[J]. AIMS Medical Science, 2017, 4(1): 28-51. doi: 10.3934/medsci.2017.1.28
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The theory of fixed points takes an important place in the transition from classical analysis to modern analysis. One of the most remarkable works on fixed point of functions was done by Banach [9]. Various generalizations of Banach fixed point result were made by numerous mathematicians [1,2,3,6,12,13,14,17,19,20,25,26]. One of the generalizations of the metric space is the quasi metric space that was introduced by Wilson [30]. The commutativity condition does not hold in general in quasi metric spaces. Several authors used these concepts to prove some fixed point theorems, see [7,10,15,21]. On the other hand Bakhtin [8] and Czerwik [11] generalized the triangle inequality by multiplying the right hand side of triangle inequality in metric spaces by a parameter b≥1 and defined b-metric spaces, for more results, see [4,23,24]. Fixed point results for multivalued mappings generalizes the results for single-valued mappings. Many interesting results have been proved in the setting of multivalued mappings, for example, see [5,26,29]. Kamran et al. [12] introduced a new concept of generalized b metric spaces, named as extended b-metric spaces, see also [22]. They replaced the parameter b≥1 in triangle inequality by the control function θ:X×X→[1,∞). Mlaiki et al. [16] replaced the triangle inequality in b-metric spaces by using control function in a different style and introduced controlled metric type spaces. Abdeljawad et al. [1] generalized the idea of controlled metric type spaces and introduced double controlled metric type spaces. They replaced the control function ξ(x,y) in triangle inequality by two functions ξ(x,y) and ⊤(x,y), see also [27,28]. In this paper, the concept of double controlled quasi metric type spaces has been discussed. Fixed point results and several examples are established. First of all, we discuss the previous concepts that will be useful to understand the paper.
Now, we define double controlled metric type space.
Definition 1.1.[1] Given non-comparable functions ξ,⊤:X×X→[1,∞). If △:X×X→[0,∞) satisfies:
(q1) △(ω,v)=0 if and only if ω=v,
(q2) △(ω,v)=△(v,ω),
(q3) △(ω,v)≤ξ(ω,e)△(ω,e)+⊤(e,v)△(e,v),
for all ω,v,e∈X. Then, △ is called a double controlled metric with the functions ξ, ⊤ and the pair (X,△) is called double controlled metric type space with the functions ξ,⊤.
The classical result to obtain fixed point of a mapping in double controlled metric type space is given below.
Theorem 1.2. [1] Let (X,△) be a complete double controlled metric type space with the functions ξ,⊤ :X×X →[1,∞) and let T:X→X be a given mapping. Suppose that there exists k∈(0,1) such that
△(T(x),T(y)≤k△(x,y), for all x,y∈X. |
For ω0∈X, choose ωg=Tgω0. Assume that
supm≥1limi→∞ξ(ωi+1,ωi+2)ξ(ωi,ωi+1)⊤(ωi+1,ωm)<1k. |
In the addition, assume that, for every ω∈X, we have
limg→∞ξ(ω,ωg), and limg→∞⊤(ω g,ω) exists and are finite. |
Then T has a unique fixed point ω∗∈X.
Now, we are introducing the concept of double controlled quasi metric type space and controlled quasi metric type space.
Definition 1.3. Given non-comparable functions ξ,⊤:X×X→[1,∞). If △:X×X→[0,∞) satisfies
(q1) △(ω,v)=0 if and only if ω=v,
(q2) △(ω,v)≤ ξ(ω,e)△(ω,e)+⊤(e,v)△(e,v),
for all ω,v,e∈X. Then, △ is called a double controlled quasi metric type with the functions ξ, ⊤ and (X,△) is called a double controlled quasi metric type space. If ⊤(e,v)=ξ(e,v) then (X,△) is called a controlled quasi metric type space.
Remark 1.4. Any quasi metric space or any double controlled metric type space is also a double controlled quasi metric type space but, the converse is not true in general, see examples (1.5, 2.4, 2.12 and 2.15).
Example 1.5. Let X={0,1,2}. Define △:X×X→[0, ∞) by △(0,1)=4, △(0,2)=1, △(1,0)=3=△(1,2), △(2,0)=0, △(2,1)=2, △(0,0)=△(1,1)=△(2,2)=0.
Define ξ,⊤:G×G→[1, ∞) as ξ(0,1)= ξ(1,0)= ξ(1,2)=1, ξ(0,2)=54, ξ(2,0)=109, ξ(2,1)=2019, ξ(0,0)= ξ(1,1)= ξ(2,2)=1, ⊤(0,1)=⊤(1,0)=⊤(0,2)=⊤(1,2)=1, ⊤(2,0)=32, ⊤(2,1)=118, ⊤(0,0)=⊤(1,1)=⊤(2,2)=1. It is clear that △ is double controlled quasi metric type with the functions ξ, ⊤. Let w=0, e=2, v=1, we have
△(0,1)=4>3=△(0,2)+△(2,1). |
So △ is not a quasi metric. Also, it is not a controlled quasi metric type. Indeed,
△(0,1)=4>134=ξ(0,2)△(0,2)+ξ(2,1)△(2,1). |
Moreover, it is not double controlled metric type space because, we have
△(0,1)=ξ(0,2)△(0,2)+⊤(2,1)△(2,1)=5516≠△(1,0). |
The convergence of a sequence in double controlled quasi metric type space is defined as:
Definition 1.6. Let (X,△) be a double controlled quasi metric type space with two functions. A sequence {ut} is convergent to some u in X if and only if limt→∞△(ut,u)=limt→∞△(u,ut)=0.
Now, we discuss left Cauchy, right Cauchy and dual Cauchy sequences in double controlled quasi metric type space.
Definition 1.7. Let (X,△) be a double controlled quasi metric type space with two functions.
(i) The sequence {ut} is a left Cauchy if and only if for every ε>0 such that △(um,ut)<ε, for all t>m>tε, where tε is some integer or limt,m→∞△(um,ut)=0.
(ii) The sequence {ut} is a right Cauchy if and only if for every ε>0 such that △(um,ut)<ε, for all m>t>tε, where tε is some integer.
(iii) The sequence {ut} is a dual Cauchy if and only if it is both left as well as right Cauchy.
Now, we define left complete, right complete and dual complete double controlled quasi metric type spaces.
Definition 1.8. Let (X,△) be a double controlled quasi metric type space. Then (X,△) is left complete, right complete and dual complete if and only if each left-Cauchy, right Cauchy and dual Cauchy sequence in X is convergent respectively.
Note that every dual complete double controlled quasi metric type space is left complete but the converse is not true in general, so it is better to prove results in left complete double controlled quasi metric type space instead of dual complete.
Best approximation in a set and proximinal set are defined as:
Definition 1.9. Let (ℑ,△) be a double controlled quasi metric type space. Let A be a non-empty set and l∈ℑ. An element y0∈A is called a best approximation in A if
△(l,A)=△(l,y0), where △(l,A)=infy∈A△(l,y)and △(A,l)=△(y0,l), where △(A,l)=infy∈A△(y,l). |
If each l∈ℑ has a best approximation in A, then A is know as proximinal set. P(ℑ) is equal to the set of all proximinal subsets of ℑ.
Double controlled Hausdorff quasi metric type space is defined as:
Definition 1.10. The function H△:P(E)×P(E)→[0,∞), defined by
H△(C,F)=max{supa∈C△(a,F), supb∈F△(C,b)} |
is called double controlled quasi Hausdorff metric type on P(E). Also (P(E),H△) is known as double controlled Hausdorff quasi metric type space.
The following lemma plays an important role in the proof of our main result.
Lemma 1.11. Let (X,△) be a double controlled quasi metric type space. Let (P(E),H△) be a double controlled Hausdorff quasi metric type space on P(E). Then, for all C,F∈P(E) and for each c∈C, there exists fc∈F, such that H△(C,F)≥△(c,fc) and H△(F,C)≥△(fc,c).
Let (X,△) be a double controlled quasi metric type space, u0∈X and T:X→P(X) be multifunctions on X. Let u1∈Tu0 be an element such that △(u0,Tu0)=△(u0,u1), △(Tu0,u0)=△(u1,u0). Let u2∈Tu1 be such that △(u1,Tu1)=△(u1,u2), △(Tu1,u1)=△(u2,u1). Let u3∈Tu2 be such that △(u2,Tu2)=△(u2,u3) and so on. Thus, we construct a sequence ut of points in X such that ut+1∈Tut with △(ut,Tut)=△(ut,ut+1), △(Tut,ut)=△(u t+1,ut), where t=0,1,2,⋯. We denote this iterative sequence by {XT(ut)}. We say that {XT(ut)} is a sequence in X generated by u0 under double controlled quasi metric △. If △ is quasi b-metric then, we say that {XT(ut)} is a sequence in X generated by u0 under quasi b -metric △. We can define {XT(ut)} in other metrics in a similar way.
Now, we define double controlled rational contracion which is a generalization of many other classical contractions.
Definition 2.1. Let (X,△) be a complete double controlled quasi-metric type space with the functions α,μ :X×X →[1,∞). A multivalued mapping T:X→P(X) is called a double controlled rational contracion if the following conditions are satisfied:
H△(Tx,Ty)≤k(Q(x,y)), | (2.1) |
for all x,y∈X, 0<k<1 and
Q(x,y)=max{△(x,y),△(x,Tx),△(x,Tx)△(x,Ty)+△(y,Ty)△(y,Tx)△(x,Ty)+△(y,Tx)}. |
Also, for the terms of the sequence {XT(ut)}, we have
supm≥1limi→∞α(ui+1,ui+2)α(ui,ui+1)μ(ui,um)<1k | (2.2) |
and for every u∈X
limt→∞α(u,ut) and limt→∞μ(ut,u) are finite. | (2.3) |
Now, we prove that an operator T satisfying certain rational contraction condition has a fixed point in double controlled quasi metric type space.
Theorem 2.2. Let (X,△) be a left complete double controlled quasi-metric type space with the functions α,μ:X×X→[1,∞) and T:X→P(X) be a double controlled rational contracion. Then, T has a fixed point u∗∈X.
Proof. By Lemma 1.11 and using inequality (2.1), we have
△(ut,ut+1)≤H△(Tut−1,Tut)≤k(Q(ut−1,ut)). |
Q(ut−1,ut)≤max{△(ut−1,ut),△(ut−1,ut),△(ut−1,ut)△(ut−1,Tut)+△(ut,ut+1)△(ut,Tut−1)△(ut−1,Tut)+△(ut,Tut−1)}=△(ut−1,ut). |
Therefore,
△(ut,ut+1)≤k△(ut−1,ut). | (2.4) |
Now,
△(ut−1,ut)≤H△(Tut−2,Tut−1)≤k(Q(ut−2,ut−1)). |
Q(ut−2,ut−1)=max{△(u t−2,ut−1),△(ut−2,ut−1), |
△(ut−2,ut−1)△(ut−2,Tut−1)+△(ut−1,ut)△(ut−1,Tut−2)△(ut−2,Tut−1)+△(ut−1,Tut−2)}. |
Therefore,
△(ut−1,ut)≤k△(ut−2,ut−1). | (2.5) |
Using (2.5) in (2.4), we have
△(ut,ut+1)≤k2△(ut−2,ut−1). |
Continuing in this way, we obtain
△(ut,ut+1)≤kt△(u0,u1). | (2.6) |
Now, by using (2.6) and by using the technique given in [1], it can easily be proved that {ut} is a left Cauchy sequence. So, for all natural numbers with t<m, we have
limt,m→∞△(ut,um)=0. | (2.7) |
Since (X,△) is a left complete double controlled quasi metric type space, there exists some u∗∈X such that
limt→∞△(ut,u∗)=limt→∞△(u∗,ut)=0. | (2.8) |
By using triangle inequality and then (2.1), we have
△(u∗,Tu∗)≤α(u∗,ut+1)△(u∗,ut+1)+μ(ut+1,Tu∗)△(ut+1,Tu∗) |
≤α(u∗,ut+1)△(u∗,ut+1)+μ(ut+1,Tu∗)max{△(ut,u∗),△(ut,ut+1),△(ut,ut+1)△(ut,Tu∗)+△(u∗,Tu∗)△(u∗,ut)△(ut,Tu∗)+△(u∗,ut)}. |
Using (2.3), (2.7) and (2.8), we get △(u∗,Tu∗)≤0. That is, u∗∈Tu∗. Thus u∗ is a fixed point of T.
If we take single-valued mapping instead of multivalued mapping, then we obtain the following result.
Theorem 2.3. Let (X,△) be a left complete double controlled quasi-metric type space with the functions α,μ:X×X→[1,∞) and T:X→X be a mapping such that:
△(Tx,Ty)≤k(Q(x,y)), |
for all x,y∈X, 0<k<1 and
Q(x,y)=max{△(x,y),△(x,Tx),△(x,Tx)△(x,Ty)+△(y,Ty)△(y,Tx)△(x,Ty)+△(y,Tx)}, |
Suppose that, for every u∈X and for the Picard sequence {ut}
limt→∞α(u,ut), limt→∞μ(ut,u) are finite and |
supm≥1limi→∞α(ui+1,ui+2)α(ui,ui+1)μ(ui,um)<1k. |
Then, T has a fixed point u∗∈X.
We present the following example to illustrate Theorem 2.3.
Example 2.4. Let X={0,1,2,3}. Define △:X×X→[0, ∞) by: △(0,1)=1, △(0,2)=4, △(0,3)=5, △(1,0)=0, △(1,2)=10, △(1,3)=1, △(2,0)=7, △(2,1)=3, △(2,3)=5, △(3,0)=3, △(3,1)=6, △(3,2)=2, △(0,0)=△(1,1)=△(2,2)=△(3,3)=0. Define α,μ:X×X→[1, ∞) as: α(0,1)=2, α(1,2)= α(0,2)= α(1,0)=α(2,0)=α(3,1)=α(2,3)=α(0,3)=1, α(1,3)=2, α(2,1)=73, α(3,0)=43, α(3,2)=32, α(0,0)=α(1,1)=α(2,2)=α(3,3)=1, μ(1,2)=μ(2,1)=μ(2,0)=μ(3,0)=μ(0,3)=1, μ(1,0)=32, μ(0,1)=2, μ(1,3)=3, μ(3,1)=1, μ(3,2)=4, μ(2,3)=1, μ(0,2)=52, μ(0,0)=μ(1,1)=μ(2,2)=μ(3,3)=1. Clearly (X,△) is a double controlled quasi metric type space, but it is not a controlled quasi metric type space. Indeed,
△(1,2)=10>4=α(1,0)△(1,0)+α(0,2)△(0,2). |
Also, it is not a double controlled metric type space. Take T0=T1={0}, T2=T3={1} and k=13. We observe that
△(Tx,Ty)≤k(Q(x,y)), for all x,y∈X. |
Let u0=2, we have u1=Tu0=T2=1, u2=Tu1=0, u3=Tu2=0,⋯⋅
supm≥1limi→∞α(ui+1,ui+2)α(ui,ui+1)μ(ui,um)=2<3=1k. |
Also, for every u∈X, we have
limt→∞α(u,ut)<∞ and limt→∞μ(ut,u)<∞. |
All hypotheses of Theorem 2.3 are satisfied and u∗=0 is a fixed point.
As every quasi b-metric space is double controlled quasi metric type space but the converse is not true in general, so we obtain a new result in quasi b-metric space as a corollary of Theorem 2.3.
Theorem 2.5. Let (X,△) be a left complete quasi b -metric space and T:X→X be a mapping. Suppose that there exists k∈(0,1) such that
△(Tx,Ty)≤k(Q(x,y)) |
whenever,
Q(x,y)=max{△(x,y),△(x,Tx),△(x,Tx)△(x,Ty)+△(y,Ty)△(y,Tx)△(x,Ty)+△(y,Tx)}, |
for all x,y∈X. Assume that 0<bk<1. Then, T has a fixed point u∗∈X.
Remark 2.6. In the Example 2.3, note that △ is quasi b -metric with b=103, but we can not apply Theorem 2.5 for any b=103 and k=13, because bk≮
Quasi metric version of Theorem 2.2 is given below:
Theorem 2.7. Let (X, \triangle) be a left complete quasi metric space and T:X\rightarrow P(X) be a multivalued mapping. Suppose that there exists 0 < k < 1 such that
\begin{eqnarray*} H_{\triangle }\left( Tx, Ty\right) &\leq &k\left( \max \left \{ \triangle \left( x, y\right) , \triangle \left( x, Tx\right) , \right. \right. \\ &&\left. \left. \frac{\triangle \left( x, Tx\right) \triangle \left( x, Ty\right) +\triangle \left( y, Ty\right) \triangle \left( y, Tx\right) }{ \triangle (x, Ty)+\triangle (y, Tx)}\right \} \right) \end{eqnarray*} |
for all x, y\in X. Then T has a fixed point u^{\ast }\in X.
Now, we extend the sequence \{XT(u_{\text{t}})\} for two mappings. Let ({ X}, \triangle) be a double controlled quasi metric type space, u_{0}\in X and S, T:X\rightarrow P(X) be the multivalued mappings on X . Let u_{1}\in Su_{0} such that \triangle (u_{0}, Su_{0}) = \triangle (u_{0}, u_{1}) and \triangle (Su_{0}, u_{0}) = \triangle (u_{1}, u_{0}). Now, for u_{1}\in X , there exist u_{2}\in Tu_{1} such that \triangle (u_{1}, Tu_{1}) = \triangle (u_{1}, u_{2}) and \triangle (Tu_{1}, u_{1}) = \triangle (u_{2}, u_{1}). Continuing this process, we construct a sequence u_{\text{t}}\ of points in X such that u_{2\text{t} +1}\in Su_{2\text{t}} , and u_{2\text{t}+2}\in Tu_{2\text{t}+1} with \triangle (u_{2\text{t}}, Su_{2\text{t}}) = \triangle (u_{2\text{t}}, u_{2\text{t }+1}) , \triangle (Su_{2\text{t}}, u_{2\text{t}}) = \triangle (u_{2\text{t} +1}, u_{2\text{t}}) and \triangle (u_{2\text{t}+1}, Tu_{2\text{t} +1}) = \triangle (u_{2\text{t}+1}, u_{2\text{t}+2}) , \triangle (Tu_{2\text{t} +1}, u_{2\text{t}+1}) = \triangle (u_{2\text{t}+2}, u_{2\text{t}+1}). We denote this iterative sequence by \{TS(u_{\text{t}})\} and say that \{TS(u_{ \text{t}})\} is a sequence in X generated by u_{0}.
Now, we introduce double controlled Reich type contraction.
Definition 2.8. Let X be a non empty set, (X, \triangle) be a left complete double controlled quasi-metric type space with the functions \alpha, \mu :X\times X\rightarrow \lbrack 1, \infty) and S, T:X\rightarrow P(X) be a multivalued mappings. Suppose that the following conditions are satisfied:
\begin{equation} H_{\triangle }({S}x, Ty)\leq c(\triangle (x, {y}))+k(\triangle (x, {S} x)+\triangle (y, Ty)), \end{equation} | (2.9) |
\begin{equation} H_{\triangle }({T}x, Sy)\leq c(\triangle (x, {y}))+k(\triangle (x, {T} x)+\triangle (y, Sy)), \end{equation} | (2.10) |
for each x, {y}\in X, 0 < c+2k < 1. For u_{0}\in X, choose u_{\text{t} }\in \{TS(u_{\text{t}})\} , we have
\begin{equation} \sup\limits_{m\geq 1}\lim\limits_{i\rightarrow \infty }\frac{\alpha \left( u_{i+1}, u_{i+2}\right) }{\alpha \left( u_{i}, u_{i+1}\right) }\mu \left( u_{i+1}, u_{m}\right) \lt \frac{1-k}{c+k}. \end{equation} | (2.11) |
Then the pair \left(S, T\right) is called a double controlled Reich type contraction.
The following results extend the results of Reich [18].
Theorem 2.9. Let S, T:X\rightarrow P(X) be the multivalued mappings, (X, \triangle) be a left complete double controlled quasi metric type space and \left(S, T\right) be a pair of double controlled Reich type contraction. Suppose that, for all u\in X
\begin{equation} \lim\limits_{\text{$t$}\rightarrow \infty }\alpha \left( u, u_{\text{$t$}}\right) \text{ is finite and }\lim\limits_{\text{$t$}\rightarrow \infty }\mu \left( u_{ \text{$t$}}, u\right) \lt \frac{1}{k}. \end{equation} | (2.12) |
Then, S and T have a common fixed point \dot{z} in X .
Proof. Consider the sequence \{TS(u_{\text{t}})\} . Now, by Lemma 1.11, we have
\begin{equation} \triangle (u_{2\text{$t$}}, u_{2\text{$t$}+1})\leq H_{\triangle }(Tu_{2\text{$ t$}-1}, Su_{2\text{$t$}}) \end{equation} | (2.13) |
By using the condition (2.10), we get
\begin{eqnarray*} \triangle (u_{2\text{$t$}}, u_{2\text{$t$}+1}) &\leq &c(\triangle (u_{2\text{$ t$}-1}, u_{2\text{$t$}})+k(\triangle (u_{2\text{$t$}-1}, Tu_{2\text{$t$} -1})+\triangle (u_{2\text{$t$}}, Su_{2\text{$t$}}) \\ &\leq &c(\triangle (u_{2\text{$t$}-1}, u_{2\text{$t$}})+k\left( \triangle (u_{2\text{$t$}-1}, u_{2\text{$t$}})+\triangle (u_{2\text{$t$}}, u_{2\text{$t$} +1}\right) ) \end{eqnarray*} |
\begin{equation} \triangle (u_{2\text{$t$}}, u_{2\text{$t$}+1})\leq \eta \left( \triangle (u_{2 \text{$t$}-1}, u_{2\text{$t$}})\right) , \end{equation} | (2.14) |
where \eta = \frac{c+k}{1-k}. Now, by Lemma 1.11, we have
\begin{equation*} \triangle (u_{2\text{$t$}-1}, u_{2\text{$t$}})\leq H_{\triangle }(Su_{2\text{$ t$}-2}, Tu_{2\text{$t$}-1}). \end{equation*} |
So, by using the condition (2.9), we have
\begin{eqnarray*} \triangle (u_{2\text{$t$}-1}, u_{2\text{$t$}}) &\leq &c\triangle (u_{2\text{$t $}-2}, u_{2\text{$t$}-1})+k\left( \triangle (u_{2\text{$t$}-2}, Su_{2\text{$t$} -2})+\triangle (u_{2\text{$t$}-1}, Tu_{2\text{$t$}-1})\right) \\ &\leq &c\triangle (u_{2\text{$t$}-2}, u_{2\text{$t$}-1})+k\left( \triangle (u_{2\text{$t$}-2}, u_{2\text{$t$}-1})+\triangle (u_{2\text{$t$}-1}, u_{2\text{ $t$}})\right) \end{eqnarray*} |
\begin{equation} \triangle (u_{2\text{$t$}-1}, u_{2\text{$t$}})\leq \frac{c+k}{1-k}\left( \triangle (u_{2\text{$t$}-2}, u_{2\text{$t$}-1})\right) = \eta \left( \triangle (u_{2\text{$t$}-2}, u_{2\text{$t$}-1})\right) . \end{equation} | (2.15) |
Using (2.14) in (2.15), we have
\begin{equation} \triangle (u_{2\text{$t$}}, u_{2\text{$t$}+1})\leq \eta ^{2}\triangle (u_{2 \text{$t$}-2}, u_{2\text{$t$}-1}). \end{equation} | (2.16) |
Now, by Lemma 1.11 we have
\begin{equation*} \triangle (u_{2\text{$t$}-2}, u_{2\text{$t$}-1})\leq H_{\triangle }(Tu_{2 \text{$t$}-3}, Su_{2\text{$t$}-2}). \end{equation*} |
Using the condition (2.10), we have
\begin{equation*} \triangle (u_{2\text{$t$}-2}, u_{2\text{$t$}-1})\leq c\triangle (u_{2\text{$t$ }-3}, u_{2\text{$t$}-2})+k\left( \triangle (u_{2\text{$t$}-3}, u_{2\text{$t$} -2})+\triangle (u_{2\text{$t$}-2}, u_{2\text{$t$}-1})\right) \end{equation*} |
implies
\begin{equation} \triangle (u_{2\text{$t$}-2}, u_{2\text{$t$}-1})\leq \eta \left( \triangle (u_{2\text{$t$}-3}, u_{2\text{$t$}-2})\right) . \end{equation} | (2.17) |
From (2.16) and (2.17), we have
\begin{equation} \eta ^{2}(\triangle (u_{2\text{$t$}-2}, u_{2\text{$t$}-1}))\leq \eta ^{3}(\triangle (u_{2\text{$t$}-3}, u_{2\text{$t$}-2})). \end{equation} | (2.18) |
Using (2.18) in (2.14), we have
\begin{equation*} \triangle (u_{2\text{$t$}}, u_{2\text{$t$}+1})\leq \eta ^{3}(\triangle (u_{2 \text{$t$}-3}, u_{2\text{$t$}-2})). \end{equation*} |
Continuing in this way, we get
\begin{equation} \triangle (u_{2\text{$t$}}, u_{2\text{$t$}+1})\leq \eta ^{2\text{$t$} }(\triangle (u_{0}, u_{1})). \end{equation} | (2.19) |
Similarly, by Lemma 1.11, we have
\begin{equation*} \triangle (u_{2\text{$t$}-1}, u_{2\text{$t$}})\leq \eta ^{2\text{$t$} -1}(\triangle (u_{0}, u_{1})). \end{equation*} |
Now, we can write inequality (2.19) as
\begin{equation} \triangle (u_{\text{$t$}}, u_{\text{$t$}+1})\leq \eta ^{\text{$t$}}(\triangle (u_{0}, u_{1})). \end{equation} | (2.20) |
Now, by using (2.20) and by using the technique given in [1], it can easily be proved that \left \{ u_{\text{t}}\right \} is a left Cauchy sequence. So, for all natural numbers with t < m, we have
\begin{equation} \underset{\text{$t$}, m\rightarrow \infty }{\lim }\triangle (u_{\text{$t$} }, u_{m}) = 0. \end{equation} | (2.21) |
Since (X, \triangle) is a left complete double controlled quasi metric type space. So \{u_{\text{t}}\} \rightarrow \dot{z}\in X, that is
\begin{equation} \lim\limits_{\text{$t$}\rightarrow \infty }\triangle (u_{\text{$t$}}, \dot{z}) = \lim\limits_{ \text{$t$}\rightarrow \infty }\triangle (\dot{z}, u_{\text{$t$}}) = 0. \end{equation} | (2.22) |
Now, we show that \dot{z} is a common fixed point. We claim that \triangle \left(\dot{z}, T\dot{z}\right) = 0 . On contrary suppose \triangle (\dot{z}, {T}\dot{z}) > 0. Now by Lemma 1.11, we have
\begin{equation*} \triangle (u_{2\text{$t$}+1}, T\dot{z})\leq H_{\triangle }(Su_{2\text{$t$}}, T \dot{z}). \end{equation*} |
\begin{equation} \triangle (u_{2\text{$t$}+1}, T\dot{z})\leq c(\triangle (u_{2\text{$t$}}, \dot{ z}))+k\left[ \triangle (u_{2\text{$t$}}, u_{2\text{$t$}+1})+\triangle (\dot{z} , T\dot{z}))\right] \end{equation} | (2.23) |
Taking \underset{t\rightarrow \infty }{\lim } on both sides of inequality (2.23), we get
\begin{equation*} \underset{\text{$t$}\rightarrow \infty }{\lim }\triangle (u_{2\text{$t$}+1}, T \dot{z})\leq \underset{\text{$t$}\rightarrow \infty }{c\lim }\triangle (u_{2 \text{$t$}}, \dot{z})+k\underset{\text{$t$}\rightarrow \infty }{\lim }\left[ \triangle (u_{2\text{$t$}}, u_{2\text{$t$}+1})+\triangle (\dot{z}, T\dot{z})) \right] \end{equation*} |
By using inequalities (2.21) and (2.22), we get
\begin{equation} \underset{\text{$t$}\rightarrow \infty }{\lim }\triangle (u_{2\text{$t$}+1}, T \dot{z})\leq k(\triangle (\dot{z}, T\dot{z})) \end{equation} | (2.24) |
Now,
\begin{equation*} \triangle (\dot{z}, T\dot{z})\leq \alpha (\dot{z}, u_{2\text{$t$}+1})\triangle (\dot{z}, u_{2\text{$t$}+1})+\mu (u_{2\text{$t$}+1}, T\dot{z})\triangle (u_{2 \text{$t$}+1}, T\dot{z}) \end{equation*} |
Taking \underset{t\rightarrow \infty }{\lim } and by using inequalities (2.12), (2.22) and (2.24), we get
\begin{equation*} \triangle (\dot{z}, T\dot{z}) \lt \triangle (\dot{z}, T\dot{z}). \end{equation*} |
It is a contradiction, therefore
\begin{equation*} \triangle (\dot{z}, T\dot{z}) = 0. \end{equation*} |
Thus, \dot{z}\in T\dot{z}. Now, suppose \triangle (\dot{z}, S\dot{z}) > 0. By Lemma 1.11, we have
\begin{equation*} \triangle (u_{2\text{$t$}}, S\dot{z})\leq H_{\triangle }(Tu_{2\text{$t$}-1}, S \dot{z}). \end{equation*} |
By inequality (2.10), we get
\begin{equation*} \triangle (u_{2\text{$t$}}, S\dot{z})\leq c(\triangle (u_{2\text{$t$}-1}, \dot{ z}))+k\left[ \triangle (u_{2\text{$t$}-1}, u_{2\text{$t$}})+\triangle (\dot{z} , S\dot{z})\right] . \end{equation*} |
Taking \underset{t\rightarrow \infty }{\lim } on both sides of above inequality, we get
\begin{equation} \underset{\text{$t$}\rightarrow \infty }{\lim }\triangle (u_{2\text{$t$}}, S \dot{z})\leq k(\triangle (\dot{z}, S\dot{z})). \end{equation} | (2.25) |
Now,
\begin{equation*} \triangle (\dot{z}, S\dot{z})\leq \alpha (\dot{z}, u_{2\text{$t$}})\triangle ( \dot{z}, u_{2\text{$t$}})+\mu (u_{2\text{$t$}}, S\dot{z})\triangle (u_{2\text{$ t$}}, S\dot{z}). \end{equation*} |
Taking \underset{t\rightarrow \infty }{\lim } and by using inequality (2.12), (2.22) and (2.25), we get
\begin{equation*} \triangle (\dot{z}, S\dot{z}) \lt \triangle (\dot{z}, S\dot{z}). \end{equation*} |
It is a contradiction. Hence, \dot{z}\in S\dot{z}. Thus, \dot{z} is a common fixed point for S and T.
Theorem 2.10 presents a result for single-valued mapping which is a consequence of the previous result.
Theorem 2.10. Let (X, \triangle) be a left complete double controlled quasi-metric type space with the functions \alpha, \mu :X\times X\rightarrow \lbrack 1, \infty) and S, T:X\rightarrow X be the mappings such that:
\begin{equation*} \triangle ({S}x, Ty)\leq c(\triangle (x, {y}))+k(\triangle (x, {S}x)+\triangle (y, Ty)) \end{equation*} |
and
\begin{equation*} \triangle ({T}x, Sy)\leq c(\triangle (x, {y}))+k(\triangle (x, {T}x)+\triangle (y, Sy)), \end{equation*} |
for each x, {y}\in X, 0 < c+2k < 1. Suppose that, for every u\in X and for the Picard sequence \left \{ u_{\text{t}}\right \}
\begin{equation*} \lim\limits_{\text{$t$}\rightarrow \infty }\alpha \left( u, u_{\text{$t$}}\right) \text{ is finite, }\lim\limits_{\text{$t$}\rightarrow \infty }\mu \left( u_{\text{$ t $}}, u\right) \lt \frac{1}{k}\text{ and} \end{equation*} |
\begin{equation*} \sup\limits_{m\geq 1}\lim\limits_{i\rightarrow \infty }\frac{\alpha \left( u_{i+1}, u_{i+2}\right) }{\alpha \left( u_{i}, u_{i+1}\right) }\mu \left( u_{i}, u_{m}\right) \lt \frac{1-k}{c+k}. \end{equation*} |
Then S and T have a common fixed point u^{\ast }\in X .
Quasi b -metric version of Theorem 2.10 is given below:
Theorem 2.11. Let (X, \triangle) be a left complete quasi b -metric type space with the functions \alpha, \mu :X\times X\rightarrow \lbrack 1, \infty) and S, T:X\rightarrow X be the mappings such that:
\begin{equation*} \triangle ({S}x, Ty)\leq c(\triangle (x, {y}))+k(\triangle (x, {S}x)+\triangle (y, Ty)) \end{equation*} |
and
\begin{equation*} \triangle ({T}x, Sy)\leq c(\triangle (x, {y}))+k(\triangle (x, {T}x)+\triangle (y, Sy)), \end{equation*} |
for each x, {y}\in X, 0 < c+2k < 1 and b < \frac{1-k}{c+k} . Then S and T have a common fixed point u^{\ast }\in X .
The following example shows that how double controlled quasi metric type spaces can be used where the quasi b -metric spaces cannot.
Example 2.12. Let X = \left \{ 0, \frac{1}{2}, \frac{1}{4}, 1\right \}. Define \triangle :X\times X\rightarrow \lbrack 0 , \infty) by \triangle (0, \frac{1}{2}) = 1, \triangle (0, \frac{1}{4}) = \frac{1}{3}, \triangle (\frac{1}{4}, 0) = \frac{1}{5}, \ \triangle (\frac{1}{2}, 0) = 1, \triangle \left(\frac{1}{4}, \frac{1}{2}\right) = 3, \triangle \left(\frac{ 1}{4}, 1\right) = \frac{1}{2}, \triangle \left(1, \frac{1}{4}\right) = \frac{ 1}{3}\ and \triangle (x, y) = \left \vert x-y\right \vert, if otherwise. Define \alpha, \mu :X\times X\rightarrow \lbrack 1 , \infty) as follows \alpha \left(\frac{1}{2}, \frac{1}{4}\right) = \frac{16}{5}, \alpha (0, \frac{1}{4}) = \frac{3}{2} , \ \alpha (\frac{1}{4}, 1) = 3 , \ \alpha (1, \frac{1}{ 4}) = \frac{12}{5} and \alpha (x, y) = 1, if otherwise. \mu (0, \frac{1}{2}) = \frac{14}{5} , \ \mu (1, \frac{1}{2}) = 3 and \ \mu (x, y) = 1, if otherwise. Clearly \triangle is double controlled quasi metric type for all x, y, z\in X. Let, T0 = \left \{ 0\right \}, T\frac{1}{2} = \left \{ \frac{1 }{4}\right \}, \ T\frac{1}{4} = \left \{ 0\right \}, T1 = \left \{ \frac{1}{4} \right \} , \ S0 = S\frac{1}{4} = \left \{ 0\right \} , S\frac{1}{2} = \left \{ \frac{1}{4}\right \}, S1 = \left \{ 0\right \} and c = \frac{2}{5} , k = \frac{1}{4}. Now, if we take the case x = \frac{1}{2} , y = \frac{1}{4}, we have
H_{\triangle }({S}\frac{1}{2}, T\frac{1}{4}) = H_{\triangle }\left(\left \{ \frac{1}{4}\right \}, \left \{ 0\right \} \right) = \triangle \left(\frac{1}{ 4}, 0\right) = \frac{1}{5}\leq \frac{17}{80} = c(\triangle (x, {y}))+k(\triangle (x, {S}x)+\triangle (y, Ty)). Also, satisfied for all cases x, y\in X. That are inequalities (2.9) and (2.10) hold. Take u_{0} = 1, then u_{1} = Su_{0} = 0, u_{2} = Tu_{1} = 0 , u_{3} = Su_{2} = 0\cdots.
\begin{equation*} \sup\limits_{m\geq 1}\lim\limits_{i\rightarrow \infty }\frac{\alpha \left( u_{i+1}, u_{i+2}\right) }{\alpha \left( u_{i}, u_{i+1}\right) }\mu \left( u_{i+1}, u_{m}\right) = 1 \lt \frac{15}{13} = \frac{1-k}{c+k}. \end{equation*} |
which shows that inequality (2.11) holds. Thus the pair \left(S, T\right) is double controlled Reich type contraction. Finally, for every u\in X, we obtain
\begin{equation*} \lim\limits_{\text{$t$}\rightarrow \infty }\alpha \left( u, u_{\text{$t$}}\right) \text{ is finite, }\lim\limits_{\text{$t$}\rightarrow \infty }\mu \left( u_{\text{$ t $}}, u\right) \leq \frac{1}{k}. \end{equation*} |
All hypotheses of Theorem 2.9 are satisfied and \dot{z} = 0 is a common fixed point.
Note that \triangle is quasi b -metric with b = 3, but Theorem 2.11 can not be applied because b\nless \frac{1-k}{c+k}, for all b = 3. Therefore, this example shows that generalization from a quasi b -metric spaces to a double controlled quasi metric type spaces is real.
Taking c = 0 in Theorem 2.9, we get the following result of Kannan-type.
Theorem 2.13. Let (X, \triangle) be a left complete double controlled quasi-metric type space with the functions \alpha, \mu :X\times X\rightarrow \lbrack 1, \infty) and S, T:X\rightarrow P(X) be the multivalued mappings such that:
\begin{equation*} H_{\triangle }({S}x, Ty)\leq k(\triangle (x, {S}x)+\triangle (y, Ty)) \end{equation*} |
and
\begin{equation*} H_{\triangle }({T}x, Sy)\leq k(\triangle (x, {T}x)+\triangle (y, Sy)), \end{equation*} |
for each x, {y}\in X, 0 < c+2k < 1. Suppose that, for every u\in X and for the sequence \{TS(u_{\text{t}})\} , we have
\begin{equation*} \lim\limits_{\text{$t$}\rightarrow \infty }\alpha \left( u, u_{\text{$t$}}\right) \text{ is finite, }\lim\limits_{\text{$t$}\rightarrow \infty }\mu \left( u_{\text{$ t $}}, u\right) \lt \frac{1}{k}\text{ and} \end{equation*} |
\begin{equation*} \sup\limits_{m\geq 1}\lim\limits_{i\rightarrow \infty }\frac{\alpha \left( u_{i+1}, u_{i+2}\right) }{\alpha \left( u_{i}, u_{i+1}\right) }\mu \left( u_{i}, u_{m}\right) \lt \frac{1-k}{k}. \end{equation*} |
Then S and T have a common fixed point u^{\ast }\in X . Then, S and T have a common fixed point \dot{z} in X .
Taking c = 0 and S = T in Theorem 2.9, we get the following result.
Theorem 2.14. Let (X, \triangle) be a left complete double controlled quasi-metric type space with the functions \alpha, \mu :X\times X\rightarrow \lbrack 1, \infty) and T:X\rightarrow P(X) be a multivalued mapping such that:
\begin{equation*} H_{\triangle }({T}x, Ty)\leq k(\triangle (x, {T}x)+\triangle (y, Ty)) \end{equation*} |
for each x, {y}\in X, 0 < c+2k < 1. Suppose that, for every u\in X and for the sequence \{T(u_{\text{t}})\} , we have
\begin{equation*} \lim\limits_{\text{$t$}\rightarrow \infty }\alpha \left( u, u_{\text{$t$}}\right) \text{ is finite, }\lim\limits_{\text{$t$}\rightarrow \infty }\mu \left( u_{\text{$ t $}}, u\right) \lt \frac{1}{k}\text{ and} \end{equation*} |
\begin{equation*} \sup\limits_{m\geq 1}\lim\limits_{i\rightarrow \infty }\frac{\alpha \left( u_{i+1}, u_{i+2}\right) }{\alpha \left( u_{i}, u_{i+1}\right) }\mu \left( u_{i}, u_{m}\right) \lt \frac{1-k}{k}. \end{equation*} |
Then T has a fixed point.
Now, the following example illustrates Theorem 2.14.
Example 2.15. Let X = [0, 3). Define \triangle :X\times X\rightarrow \lbrack 0 , \infty) as
\begin{equation*} \triangle (x, y) = \left \{ \begin{array}{c} 0\text{, if }x = y, \\ \left( x-y\right) ^{2}+x, \text{ if }x\neq y. \end{array} \right. \end{equation*} |
with
\begin{equation*} \alpha (x, y) = \left \{ \begin{array}{c} 2, \text{ if }x, y\geq 1, \\ \frac{x+2}{2}, \text{ otherwise}. \end{array} \right. , \text{ }\mu (x, y) = \left \{ \begin{array}{c} 1, \text{ if }x, y\geq 1\text{, } \\ \frac{y+2}{2}, \text{ otherwise}. \end{array} \right. \end{equation*} |
Clearly \left(X, \triangle \right) is double controlled quasi metric type space. Choose Tx = \left \{ \frac{x}{4}\right \} and k = \frac{2}{5} . It is clear that T is Kannan type double controlled contraction. Also, for each u \in X , we have
\begin{equation*} \lim\limits_{\text{$t$}\rightarrow \infty }\alpha \left( u, u_{\text{$t$}}\right) \lt \infty , \text{ }\lim\limits_{\text{$t$}\rightarrow \infty }\mu \left( u_{\text{$t$} }, u\right) \lt \frac{1}{k}. \end{equation*} |
Thus, all hypotheses of Theorem 2.14 are satisfied and \dot{z} = 0 is the fixed point.
Quasi b -metric version of Theorem 2.14 is given below:
Theorem 2.16. Let (X, \triangle) be a left complete quasi b -metric space and T:X\rightarrow P\left(X\right) be a mapping such that:
\begin{equation*} H_{\triangle }\left( Tx, Ty\right) \leq k\left[ \triangle \left( x, Tx\right) +\triangle \left( y, Ty\right) \right] , \end{equation*} |
for all x, y\in X, k\in \lbrack 0, \frac{1}{2}) and
\begin{equation*} b \lt \frac{1-k}{k}. \end{equation*} |
Then T has a fixed point u^{\ast }\in X .
The following remark compare, distinguish and relate the quasi b -metric with the double controlled quasi metric type spaces and illustrate the importance of double controlled quasi metric type spaces.
Remark 2.17. In the Example 2.15, \triangle (x, y) = \left(x-y\right) ^{2}+x is a quasi b -metric with b\geq 2, but we can not apply Theorem 2.16 because T is not Kannan type b -contraction. Indeed b\nless \frac{ 1-k}{k}, for all b\geq 2.
It has been shown that double controlled quasi metric is general and better than other metrics, like controlled quasi metric, controlled metric, extended quasi-b-metric, extended b-metric, quasi-b-metric and quasi metric. Also, left, right and dual completeness has been discussed. Results in dual complete spaces can be obtained as corollaries. These results may be extended to obtain results for other contractions. Double controlled quasi metric like ordered spaces can be introduced to establish new results. These results may be applied to find applications to random impulsive differential equations, dynamical systems, graph theory and integral equations.
The authors would also like to thank the editor and the reviewers for their helpful comments and suggestions. This research was funded by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT.
The authors declare that they do not have any competing interests.
[1] | Deitel M (2003) Overweight and obesity worldwide now estimated to involve 1.8 billion people. Obes Surg 13: 329-330. |
[2] | Ford ES, Mokdad AH (2008) Epidemiology of obesity in the Western hemisphere. J Clin Endocrinol Metab 93: S1-S8. |
[3] |
Hossain P, Kawar B, El Nahas M (2007) Obesity and diabetes in the developing world—a growing challenge. New England J Med 356: 213-215. doi: 10.1056/NEJMp068177
![]() |
[4] | Rheeder P (2006) Type 2 diabetes: the emerging epidemic. SA Fam Pract 48: 20. |
[5] | FAO (2008) The state of food insecurity in the world 2008. Food and Agriculture Organization of the United Nations. Rome. |
[6] | WHO (2016) Global Report on Diabetes, Geneva 2016. |
[7] | Lasserre AM, Chiolero A, Paccaud F, et al. (2007) World-wide trends in childhood obesity. Swiss Med Weekly 137: 157-158. |
[8] |
Kulie T, Slattengren A, Redner J, et al. (2011) Obesity and women's health: an evidence-based review. J Am Board Fam Med 24: 75-85. doi: 10.3122/jabfm.2011.01.100076
![]() |
[9] | Alberti KG, Zimmet P, Shaw J (2005) The metabolic syndrome—a new world wide definition. Lancet 3666: 1059-1062. |
[10] | Kaufmann F (2005) Diabesity. Bantam Books NewYork. |
[11] |
Hu FB (2011) Globalization of diabetes. Diabetes Care 34: 1249-1257. doi: 10.2337/dc11-0442
![]() |
[12] |
Spiegelman BM, Flier JS (2001) Obesity and the regulation of energy balance. Cell 104: 531-543. doi: 10.1016/S0092-8674(01)00240-9
![]() |
[13] |
Danaei G, Ding EL, Mozaffarian D, et al. (2009) The preventable causes of death in the United States: comparative risk assessment of dietary, lifestyle and metabolic risk factors. PLoS Med 6: e1000058. doi: 10.1371/journal.pmed.1000058
![]() |
[14] | WHO (1998) Obesity preventing and managing the global epidemic. Geneva WHO. |
[15] |
Misra A, Wasir JS, Vikram NK (2005) Waist circumference criteria for the diagnosis of abdominal obesity are not applicable uniformly to all populations and ethnic groups. Nutrition 21: 969-976. doi: 10.1016/j.nut.2005.01.007
![]() |
[16] | WHO (1995) Physical status the use and interpretation of anthropometry. WHO technical Reports Series 854, Geneva. |
[17] | Garg C, Khan SA, Ansari SH, et al. (2009) Prevalence of obesity in Indian women. Obes Rev 11: 105-108. |
[18] | Joshi SR (2003) Metabolic syndrome—emerging clusters of the Indian phenotype. J Assoc Physicans India 51: 445-446. |
[19] |
WHO Expert Consultation (2004) Appropriate Body mass index (BMI) for Asian populations and its implications for policy and intervention strategies. Lancet 363: 157-163. doi: 10.1016/S0140-6736(03)15268-3
![]() |
[20] | Cole TJ, Freeman JV, Preece MA (1995) Body mass reference curves for the UK, 1990. Arch Dis Child 73: 25-29. |
[21] |
Cole TJ, Bellizzi MC, Flegal KM, et al. (2000) Establishing a standard definition for child overweight and obesity worldwide: international survey. Brit Med J 320: 1-6. doi: 10.1136/bmj.320.7226.1
![]() |
[22] |
Chinn S, Rona RJ (2002) International definitions of overweight and obesity for children, a lasting solution? Ann Hum Biol 29: 306-313. doi: 10.1080/03014460110085340
![]() |
[23] |
Wang Y, Wang JQ (2000) Standard definitions of child overweight and obesity worldwide. Brit Med J 321: 1158. doi: 10.1136/bmj.321.7269.1158
![]() |
[24] |
Kromeyer-Hausschild K, Wabitsch M, Kunze D, et al. (2001) Perzentile für den Body—mass Index für das Kindes- und Jugendalter unter Heranziehung verschiedener deutscher Stichproben. Monatsschrift Kinderheilkunde 149: 807-818. doi: 10.1007/s001120170107
![]() |
[25] |
Lieberman LS (2003) Dietary, evolutionary and modernizing influences on the prevalence of type 2 diabetes. Annu Rev Nutr 23: 345-377. doi: 10.1146/annurev.nutr.23.011702.073212
![]() |
[26] | Björntorp P (1988) The associations between obesity, adipose tissue and disease. Acta Med Scand 723: 121-134. |
[27] |
Björntorp P (1997) Obesity. The Lancet 350: 423-426. doi: 10.1016/S0140-6736(97)04503-0
![]() |
[28] |
McGee DL (2005) Body mass index and mortality: a meta-analysis based on person level data from twenty-six observational studies. Ann Epidemiol 15: 87-97. doi: 10.1016/j.annepidem.2004.05.012
![]() |
[29] |
Wang Y, McPherson K, Marsh T, et al. (2011) health and economic burden of the projected obesity trends in the USA and the UK. The Lancet 378: 815-825. doi: 10.1016/S0140-6736(11)60814-3
![]() |
[30] |
Maheshwari A, Stofberg L, Bhattacharya S (2007) Effect of overweight and obesity on assisted reproductive technology—a systematic review. Hum Reprod Update 13: 433-444. doi: 10.1093/humupd/dmm017
![]() |
[31] |
Kay VJ, Barratt CLR (2009) Male obesity: impact on fertility. Brit J Diabet Vascular Dis 9: 237-241. doi: 10.1177/1474651409343132
![]() |
[32] |
Must A, Jaques PF, Dallal GE, et al. (1992) Long term morbidity and mortality of overweight adolescents: a follow up of the Harvard Growth Study of 1922 to 1935. New England J Med 327: 1350-1355. doi: 10.1056/NEJM199211053271904
![]() |
[33] | Kimm SYS (2003) Nature versus nurture. Childhood obesity: a familiar old conundrum. Am J Clin Nutr 78: 1051-1052. |
[34] |
Kimm SYS, Obarzanek E (2002) Childhood obesity: A new pandemic of the new millennium. Pediatrics 110: 1003-1007. doi: 10.1542/peds.110.5.1003
![]() |
[35] |
Ebbeling CB, Pawlak DB, Ludwig DS (2002) Childhood obesity: public-health crisis, common sense. Lancet 360: 473. doi: 10.1016/S0140-6736(02)09678-2
![]() |
[36] |
Latner JD, Stunkard AJ (2003) Getting worse: the stigmatisation of obese children. Obes Res 11: 452-456. doi: 10.1038/oby.2003.61
![]() |
[37] |
Kraig, KA, Keel PK (2001) Weight based stigmatisation in children. Int J Obes 25: 1661-1666. doi: 10.1038/sj.ijo.0801813
![]() |
[38] |
Withrow D, Alter DA (2011) The economic burden of obesity worldwide: a systematic review of the direct costs of obesity. Obes Rev 12: 131-141. doi: 10.1111/j.1467-789X.2009.00712.x
![]() |
[39] | Voland E (1993) Grundriß der Soziobiologie. Fischer Verlag Stuttgart, 1993. |
[40] |
Dobzhansky T (1973) Nothing in Biology makes sense except in the light of evolution. Am Teacher 35: 125-129. doi: 10.2307/4444260
![]() |
[41] | Belisari A (2007) Evolutionary origins of obesity Obes Rev 9: 165-180. |
[42] | Campbell BC, Cajigal A (2001) Diabetes. Energetics, development and human evolution. Med Hypotheses 57: 64-67. |
[43] |
Lev-Ran A (2001) Human obesity: an evolutionary approach to understanding our bulging waistline. Diabetes Metab Res Rev 17: 347-362. doi: 10.1002/dmrr.230
![]() |
[44] | Wells JCK (2006) The evolution of human fatness and susceptibility of obesity: an ethological approach. Biol Rrev 81: 183-205. |
[45] | Wells JCK (2009) The evolutionary biology of Human Body fatness: Thrift and control. Cambridge University Press. |
[46] |
46 Wells JCK (2012) The evolution of human adiposity and obesity: where did it all go wrong? Dis Model Mech 5: 595-60747. doi: 10.1242/dmm.009613
![]() |
[47] |
Haslam D (2007) Obesity: a medical history. Obes Rev 8: 31-36. doi: 10.1111/j.1467-789X.2007.00314.x
![]() |
[48] | Kirchengast S (2008) Adipositas und Reproduktion aus evolutionsbiologischer Sicht. Die Venus von Willendorf ein Fruchtbarkeitssymbol? Mitt Anthrop Ges Wien 138: 101-112. |
[49] | Gimbutas M (1984) The goddesses and gods of old Europe.6500-3500 B.C. Myths and cult images. Thames and Hudson, London. |
[50] |
Ulijaszek SJ, Lofink H (2006) Obesity in Biocultural perspective. Ann Rev Anthrop 35: 337-360. doi: 10.1146/annurev.anthro.35.081705.123301
![]() |
[51] |
Brown P (1991) Culture and the evolution of obesity. Hum Nature 2: 31-57. doi: 10.1007/BF02692180
![]() |
[52] | Ahmed AM (2002) History of diabetes mellitus. Saudi Med J 23: 373-378. |
[53] | Broode A (1575) The breviary of health wherin doth follow remedies for all manner of sickness and diseases the which may be in man or woman. London. |
[54] | White JR (2014) A brief history of the development of diabetes medications. Diabetes Spectrum 27: 82-86. |
[55] | Wang Y, Lobstein T (2006) Worldwide trends in childhood overweight and obesity. Int J Ped Obes 1: 11-25. |
[56] | Tremblay MS, Williams JD (2000) Secular trends in the body mass index of Canadian children. Can Med Ass J 163: 1429-1433. |
[57] | Marques-Vidal P, Madeleine G, Romain S, et al. (2008) Secular trends in height and weight among children and adolescents of the Seychelles, 1956-2006. Pub Health 8: 1-9. |
[58] | Kolle E, Steene-Johannessen J, Holme I, et al. (2009) Secular trends in adiposity in Norwegian 9-year olds from 1999-2000 to 2005. Pub Health 9: 389-399. |
[59] |
Han JC, Lawlor DA, Kimm SY (2010) Childhood obesity. The Lancet 375: 1737-1748. doi: 10.1016/S0140-6736(10)60171-7
![]() |
[60] | Pijl H (2011) Obesity: evolution of a symptom of affluence. Netherland J Med 69: 159-166. |
[61] | Pond CM (1998) The fats of life. Cambridge University Press. |
[62] |
Altmann J, Schoeller A, Altmann SA, et al. (1993) Body size and fatness of free-living baboons reflect food availability and activity levels. Am J Primatol 30: 149-161. doi: 10.1002/ajp.1350300207
![]() |
[63] | Neel V (1962) Diabetes mellitus: a thrifty genotype rendered detrimental by progress? Am J Hum Genet 14: 353-362. |
[64] | Shepard RJ, Rode A (1996) Health consequences of modernization: Evidence from the circumpolar populations. Cambridge University Press. |
[65] |
Gibbs T, Cargill K, Lieberman LS, et al. (1980) Nutrition in a slave population: an anthropological examination. Med Anthropol 4: 175-262. doi: 10.1080/01459740.1980.9965868
![]() |
[66] |
Ayub Q, Moutsianas L, Chen Y, et al. (2014) Revisting the thrifty gene hypothesis via 65 Loci associated with susceptibility to type 2 Diabetes. Am J Hum Genetics 94: 176-185. doi: 10.1016/j.ajhg.2013.12.010
![]() |
[67] |
Segurel L, Austerlitz F, Toupance B, et al. (2013) Positive selection of protective varints for type 2 diabetes from the Neolithic onward: a case study in Central Asia. Eur J Hum Genetics 21: 1146-1151. doi: 10.1038/ejhg.2012.295
![]() |
[68] | Lindsay RS, Bennett PH (2001) Type 2 diabetes, the thrifty phenotype—an overview. Brit Med Bull 60: 21-32. |
[69] |
Barker DJP, Clark PM (1997) Fetal undernutrition and disease in later life. Rev Reprod 2: 105-112. doi: 10.1530/ror.0.0020105
![]() |
[70] | Barker DJP (1999) Fetal origins of type 2 diabetes mellitus Ann Int Med 130: 322-324. |
[71] |
Yu ZB, Han SB, Zhu GZ, et al. (2011) Birth weight and subsequent risk of obesity: a systematic review and meta-analysis. Obes Rev 12: 525-542. doi: 10.1111/j.1467-789X.2011.00867.x
![]() |
[72] | Williams GC, Nesse RM (1991) The dawn of Darwinian medicine. Quart Rev Biol 66: 1-22. |
[73] | Nesse RM, Williams GC (1994) Why we get sick: the new science of Darwinian medicine. Vintage books, New York. |
[74] | Trevathan WR, McKenna JJ, Smith EO (1999) Evolutionary medicine. Oxford University Press. |
[75] | Stearns S (1999) Evolution in Health and Disease. Oxford University Press. |
[76] | Darwin C (1859) On the Origin of Species by means of natural selection. John Murray, London. |
[77] | Darwin C (1891) The Descent of Man and Selection in Relation to Sex. John Murray, London. |
[78] |
Trevathan WR (2007) Evolutionary medicine. Annu Rev Anthropol 36: 139-154. doi: 10.1146/annurev.anthro.36.081406.094321
![]() |
[79] |
Nesse RM, Stearns SC (2008) The great opportunity: Evolutionary applications to medicine and public health. Evol Applic 1: 28-48. doi: 10.1111/j.1752-4571.2007.00006.x
![]() |
[80] | Leonard WR, Snodgrass JJ, Robertson ML (2007) Effects of brain evolution on human nutrition and metabolism Ann Rev Nutr 27: 311-327. |
[81] | Navarrete A, van Schaik CP, Isler K (2011) Energetics and the evolution of human brain size Nature 480: 91-93. |
[82] | Aiello LC, Wells JCK (2002) Energetics and the evolution of the genus Homo. Ann Rev Anthropol 31: 323-338. |
[83] |
Aiello LC, Wheeler P (1995) The expensive tissue hypothesis: the brain and the digestive system in human and primate evolution. Curr Anthropol 36: 199-221. doi: 10.1086/204350
![]() |
[84] | Henke W, Hardt T (2011) The genus Homo: origin, speciation and dispersal, in: S.Condemi, G.-C.Weniger (Eds.), Continuity and Discontinuity in the peopling of Europe: One hundred fifty years of Neanderthal study, Vertebrate Paleobiology and Paleoanthropology. Springer Press, New York, pp17-43. |
[85] | Bowlby J (1969) Attachment and loss. Basic Books, New York. |
[86] |
Lieberman LS (2006) Evolutionary and anthropological perspectives on optimal foraging in obesogenic environments. Appetite 47: 3-9. doi: 10.1016/j.appet.2006.02.011
![]() |
[87] |
Eaton SB, Konner M (1985) Paleolithic nutrition: a consideration of its nature and current implications. New England J Med 312: 283-289. doi: 10.1056/NEJM198501313120505
![]() |
[88] | Konner M, Eaton SB (2010) Paleolithic nutrition Nutr Clin Pract 25: 594-602. |
[89] | O´Keefe Jr JH, Cordain L (2004) Cardio-vascular disease resulting from Diet and Lifestyle at odds with our Paleolithic genome: how to become a 21rst century hunter-gatherer. Mayo Clinic Proceed 79: 101-108. |
[90] | Howell N (2010) Life histories of the Dobe !Kung. Food, fatness and well-being over the life-span, University of California Press, Los Angeles. |
[91] | Marlowe FW (2010) The Hadza hunter gatherers of Tanzania. University of California Press, Los Angeles. |
[92] | Pontzer H, Raichlen DA, Wood BM, et al. (2012) Hunter-gatherer energetics and human obesity. Plos one 7: 340503. |
[93] |
Hockett B, Haws J (2003) Nutritional ecology and diachronic trends in Paleolithic diet and health. Evol Anthrop 12: 211-216. doi: 10.1002/evan.10116
![]() |
[94] | Bogin B (1998) The evolution of Human nutrition. In: The Anthropology of Medicine, R.Romanucci-Ross, D.Moerman, L.R.Tancredi(Eds).;Bergen and Garvey, South Hedly, pp.96-142. |
[95] | Cordain L, Eaton S, Miller J, et al. (2002) The paradoxical nature of hunter gatherer diets: meat based, yet non atherogenic. Eur J Clin Nutr 86: S42-52. |
[96] | Kious BM (2002) Hunter-gatherer nutrition and its implications for modern societies. Nutr Noteworthy 5: 1-5. |
[97] | Mann NJ (2004). Paleolithic nutrition. What can we learn from the past? Asia Pacific J Clin Nutr 13: S17. |
[98] | Cordain L, Eaton SB, Sebastian A, et al. (2005) Origins and evolution of the Western diet: health implications for the 21rst century. Am J Clin Nutr 81: 341-354. |
[99] | Ellison PT (1990) Human ovarian function and reproductive ecology. Am Anthrop 2:933-952. |
[100] |
Berbesque JC, Marlowe FW, Shaw P, et al. (2014) Hunter-gatherers have less famine than agriculturalists. Biol Lett 10: 20130853. doi: 10.1098/rsbl.2013.0853
![]() |
[101] |
Maher LA, Richter T, Macdonald D, et al. (2012) Twenty thousand year old huts at a hunter gatherer settlement in eastern Jordan. Plos ONE 7: e31447. doi: 10.1371/journal.pone.0031447
![]() |
[102] | Rosen AM, Rivera-Collazo I (2012) Climate change, adaptive cycles, and the persistence of foraging economies during the late Pleistocene/Holocene transition in the Levant. PNAS. |
[103] |
Larsen C (1995) Biological changes in human populations with agriculture. Annu Rev Anthropol 24: 185-213. doi: 10.1146/annurev.an.24.100195.001153
![]() |
[104] |
Armelagos GJ, Goodman AH, Jacobs KH (1991) The origins of agriculture: Population growth during a period of declining health. Popul Environ 13: 9-22. doi: 10.1007/BF01256568
![]() |
[105] |
Barrett R, Kuzawa CW, McDade T, et al. (1998) Emerging and re-emerging infectious diseases: The third epidemiologic transition. Ann Rev Anthropol 27: 247-271. doi: 10.1146/annurev.anthro.27.1.247
![]() |
[106] |
Omran A (1971) The epidemiologic transition. A theory of the epidemiology of population change. Milbank Memorial Fund Quart 49: 509-538. doi: 10.2307/3349375
![]() |
[107] |
Omran A (1983) The epidemiologic transition theory: a preliminary update. J Trop Ped 29: 305-316. doi: 10.1093/tropej/29.6.305
![]() |
[108] | Popkin BM (2001) The nutrition transition and obesity in the developing world. J Nutr 131: 871-873. |
[109] | 109 Popkin BM, Gordon-Larsen P (2004) The nutrition transition: worldwide obesity dynamics and their determinants. Int J Obes 28: 52-59. |
[110] |
Florentino RF (2002) The burden of obesity in Asia: Challenges in assessment, prevention and management. Asia Pacif J Clin Nutr 11: 676-680. doi: 10.1046/j.1440-6047.11.s8.4.x
![]() |
[111] |
Poston II WSC, Foreyt JP (1999) Obesity is an environmental issue. Atherosclerosis 146: 201-209. doi: 10.1016/S0021-9150(99)00258-0
![]() |
[112] | Ulijaszek SJ (2007) Obesity: a disorder of convenience. Obes Reviews 8: 183-187. |
[113] |
Oldroyd J, Banerjee M, Heald A, et al. (2005) Diabetes and ethnic minority groups. Postgrad Med J 81: 486-490. doi: 10.1136/pgmj.2004.029124
![]() |
[114] |
Popkin BM (2003) The nutrition transition in the developing world. Develop Policy Rev 21: 581-597. doi: 10.1111/j.1467-8659.2003.00225.x
![]() |
[115] |
Chatterjee P (2002) India sees parallel rise in malnutrition and obesity. The Lancet 360: 1948. doi: 10.1016/S0140-6736(02)11943-X
![]() |
[116] | Furubayashi JK, Look MA (2005) Type 2 diabetes in native Hawaiians and Pacific islanders in Hawaii. Pacific Health Sur Response 12: 103-110. |
[117] | Jowitt LM (2014) Ethnicity and type 2 diabetes in Pacific Island adults in New Zealand. Int J Diab Clin Res 1:1-5. |
[118] | Diamond JM (1992) Diabetes running wild. Nature 357: 362-363. |
[119] | Bindon JR, Crews DE, Dressler WW (1991) Lifestyle, modernization and adaptation among Samoans. Coll Antropol 15: 101-110. |
[120] | McGravey ST (1994) The thrifty gene concept and adiposity studies in biological anthropology. J Polynesian Soc 103: 29-42. |
[121] |
Hawley NL, McGarvey ST (2015) Obesity and diabetes in Pacific islanders: the current burden and the need for urgent action. Curr Diab Rep 15: 19-29. doi: 10.1007/s11892-015-0585-6
![]() |
[122] | O'Dea K, Hopper J, Patel M, et al. (1993) Obesity, Diabetes, and hyperlipidemia in a Central Australian Aboriginal community with a long history of acculturation. Diabetes Care 16: 1004-1010. |
[123] | Popkin BM, Udry JR (1998) Adolescent obesity increases significantly in second and third generation U.S.immigrants: The National Longitudinal study of adolescent health. J Nutr 128: 701-706. |
[124] |
Candib LM (2007) Obesity and diabetes in vulnerable populations: reflection on proximal and distal causes. Ann Fam Med 5: 547-556. doi: 10.1370/afm.754
![]() |
[125] | Faskunger J, Eriksson U, Johansson SE, et al. (2009) Risk of obesity in immigrants compared with Swedes in two deprived neighborhoods. Public Health 9: 304-312. |
[126] |
Kaplan MS, Huguet N, Newsom JT, et al. (2004) The association between length of residence and obesity among Hispanic immigrants. Am J Prev Med 27: 323-326. doi: 10.1016/j.amepre.2004.07.005
![]() |
[127] | Brussard JH, Erp-Baart MA van, Brants HAM, et al. (2001) Nutrition and health among migrants in the Netherlands. Pub Health Nutr 4: 659-664. |
[128] | Kirchengast S, Schober E (2005) To be an immigrant: a risk factor for developing overweight and obesity during childhood and adolescence. J biosoc Sci 38: 695-705. |
[129] |
Misra A, Ganda OP (2007) Migration and its impact on adiposity and type 2 diabetes. Nutrition 23: 696-708. doi: 10.1016/j.nut.2007.06.008
![]() |
[130] | Wolin KY, Colangelo LA, Chiu BCH, et al. (2008) Obesity and immigration among Latina women. J Immigrant Minority Health 11: 428-431. |
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