Coupled fixed point problems and applications

Fixed point theorems in b-metric spacesCoupled �xed point theorems

Applications and research directions

Coupled �xed point problems and applications

Adrian Petru³el (joint work with G. Petru³el, J.-C. Yao)

Babe³-Bolyai University Cluj-Napoca, Romania

WFPTA - University of Valencia, December 15-16, 2016

Dedicated to Enrique on the occasion of his 70th anniversary

Adrian Petru³el (joint work with G. Petru³el, J.-C. Yao) Coupled �xed point problems and applications

Outline of the talk

The purpose of this talk is to present some coupled �xed pointresults with applications.

1) �xed point results in b-metric spaces

2) coupled �xed point problems

3) applications

Outline of the talk

3) applications

Outline of the talk

3) applications

Outline of the talk

3) applications

Outline of the talk

3) applications

The concept of b-metric space

De�nition-Bakhtin, Czerwik, ...

Let X be a nonempty set and let s ≥ 1 be a given real number. Afunctional d : X × X → R+ is said to be a b-metric (quasi-metric,almost metric) on X if the following axioms are satis�ed:1) if x , y ∈ X , then d(x , y) = 0 if and only if x = y ;2) d(x , y) = d(y , x), for all x , y ∈ X ;3) d(x , z) ≤ s[d(x , y) + d(y , z)], for all x , y , z ∈ X .

A pair (X , d) with the �rst two properties is called a semi-metricspace, while a pair (X , d) with above three properties is called ab-metric space.

For the concept of b-metric space see also N. Bourbaki, D. Kurepa,L.M. Blumenthal, J. Heinonen.

Examples.

Example

The set lp(R) with 0 < p < 1, where

lp(R) := {(xn) ⊂ R|∞∑n=1

|xn|p <∞}, together with the function

d : lp(R)× lp(R)→ R,

d(x , y) := (∞∑n=1

|xn − yn|p)1/p,

is a b-metric space with constant s = 21/p > 1.

Example

For 0 < p < 1, the space Lp[a, b] of all real functions x(t),

t ∈ [a, b] such that∫ ba |x(t)|

pdt <∞, together with the function

d(x , y) := (

a|x(t)− y(t)|pdt)1/p, for each x , y ∈ Lp[a, b],

is a b-metric space. Notice that in this case s = 21/p > 1.

Example

Let E be a Banach space and P a normal cone in E withint(P) 6= ∅. Denote by "≤" the partially order generated by P .

If X is a nonempty set, then a mapping d : X × X → E is called acone metric on X if the usual axioms of the metric take place withrespect to "≤".The cone P is called normal if there is a number K ≥ 1 such that,for all x , y ∈ E , the following implication holds:

0 ≤ x ≤ y =⇒ ‖x‖ ≤ K‖y‖.

If the cone P is normal with the coe�cient of normality K ≥ 1,then the functional

d̂ : X × X → R+, d̂(x , y) := ‖d(x , y)‖

is a b-metric on X with constant s := K .

Example

Let E be a Banach space and P a normal cone in E withint(P) 6= ∅. Denote by "≤" the partially order generated by P .If X is a nonempty set, then a mapping d : X × X → E is called acone metric on X if the usual axioms of the metric take place withrespect to "≤".

The cone P is called normal if there is a number K ≥ 1 such that,for all x , y ∈ E , the following implication holds:

0 ≤ x ≤ y =⇒ ‖x‖ ≤ K‖y‖.

d̂ : X × X → R+, d̂(x , y) := ‖d(x , y)‖

Example

Let E be a Banach space and P a normal cone in E withint(P) 6= ∅. Denote by "≤" the partially order generated by P .If X is a nonempty set, then a mapping d : X × X → E is called acone metric on X if the usual axioms of the metric take place withrespect to "≤".The cone P is called normal if there is a number K ≥ 1 such that,for all x , y ∈ E , the following implication holds:

0 ≤ x ≤ y =⇒ ‖x‖ ≤ K‖y‖.

d̂ : X × X → R+, d̂(x , y) := ‖d(x , y)‖

Example

Let E be a Banach space and P a normal cone in E withint(P) 6= ∅. Denote by "≤" the partially order generated by P .If X is a nonempty set, then a mapping d : X × X → E is called acone metric on X if the usual axioms of the metric take place withrespect to "≤".The cone P is called normal if there is a number K ≥ 1 such that,for all x , y ∈ E , the following implication holds:

0 ≤ x ≤ y =⇒ ‖x‖ ≤ K‖y‖.

d̂ : X × X → R+, d̂(x , y) := ‖d(x , y)‖

Czerwik's �xed point theorem for single-valued

nonlinear contractions

Theorem. (Czerwik (1993), Kirk-Shahzad)

Let (X , d) be a complete b-metric space with constant s ≥ 1 andf : X → X be an operator, for which there exists a comparisonfunction ϕ : R+ → R+ (i.e., ϕ is increasing and lim

n→∞ϕn(t) = 0, for

every t > 0) such that

d(f (x), f (y)) ≤ ϕ(d(x , y)), ∀x , y ∈ X .

Then, f has a unique �xed point x∗ ∈ X and, for all, x ∈ Xlimn→∞

d(f n(x), x∗) = 0, i.e., f is a Picard operator.

Particular case: ϕ(t) := kt, t ∈ R+ (where k ∈ (0, 1)).

d(f (x), f (y)) ≤ ϕ(d(x , y)), ∀x , y ∈ X .

Czerwik's �xed point theorem for multi-valued nonlinearcontractions

If (X , d) is a metric space, then we denote

Hd(A,B) = max{supa∈A

infb∈B

d(a, b), supb∈B

infa∈A

d(a, b)}.

Theorem 1. (Czerwik-1998)

Let (X , d) be a complete b-metric space with constant s ≥ 1 andF : X → Pcp(X ) be a multivalued operator. Suppose that d iscontinuous and there exists a comparison function ϕ : R+ → R+

such that

Hd(F (x),F (y)) ≤ ϕ(d(x , y)), ∀x , y ∈ X .

Then, there exists x∗ ∈ X such that x∗ ∈ F (x∗).

If (X , d) is a metric space, then we denote

Hd(A,B) = max{supa∈A

infb∈B

d(a, b), supb∈B

infa∈A

d(a, b)}.

Let (X , d) be a complete b-metric space with constant s ≥ 1 andF : X → Pcp(X ) be a multivalued operator. Suppose that d iscontinuous and there exists a comparison function ϕ : R+ → R+

such that

Hd(F (x),F (y)) ≤ ϕ(d(x , y)), ∀x , y ∈ X .

Then, there exists x∗ ∈ X such that x∗ ∈ F (x∗).

Let (X , d) be a complete b-metric space with constant s ≥ 1 andF : X → Pcl(X ) be such that there exists k ∈ (0, 1s ) such that

Hd(F (x),F (y)) ≤ kd(x , y), ∀x , y ∈ X .

Then, F is a multivalued weakly Picard operator, i.e., there existsx∗ ∈ X such that x∗ ∈ F (x∗) and, for every (x , y) ∈ Graph(F ),there is a sequence of successive approximations for F starting from(x , y) which converges to x∗(x , y) ∈ Fix(F ).

Ran-Reurings type theorem

Ran-Reurings (2003)

Let X be a nonempty set endowed with a partial order "�" and dbe a complete metric on X .

Let f : X → X be a mapping for which there exists k ∈ [0, 1) suchthat

d(f (x), f (y)) ≤ kd(x , y), for all x , y ∈ X with x ≤ y .

If additionally, f is continuous and increasing (or decreasing) andthere exists an element x0 ∈ X such that x0 ≤ f (x0), then f has atleast one �xed point.

If additionally, for every x , y ∈ X there exists z ∈ X which iscomparable to x and y or every pair of elements of X has a lowerbound or an upper bound, then f is a Picard operator.

Ran-Reurings (2003)

Let X be a nonempty set endowed with a partial order "�" and dbe a complete metric on X .Let f : X → X be a mapping for which there exists k ∈ [0, 1) suchthat

Ran-Reurings (2003)

Ran-Reurings theorem in b-metric space (I)

Theorem.

Let X be a nonempty set endowed with a partial order "�" andd : X × X → X be a complete b-metric with constant s ≥ 1. Letf : X → X be an operator which has closed graph with respect tod and increasing with respect to "�".

Suppose that there exist a constant k ∈ (0, 1s ) and an elementx0 ∈ X such that:(i) d(f (x), f (y)) ≤ kd(x , y), for all x , y ∈ X with x ≤ y .(ii) x0 ≤ f (x0).

Then Fix(f ) 6= ∅ and the sequence (f n(x))n∈N converges to a �xedpoint x∗(x) of f , for each x ∈ X which is comparable to x0.Moreover, if d is continuous, then we also have

d(f n(x), x∗) ≤ skn

1− skd(x , f (x)), ∀n ∈ N∗.

Theorem.

Let X be a nonempty set endowed with a partial order "�" andd : X × X → X be a complete b-metric with constant s ≥ 1. Letf : X → X be an operator which has closed graph with respect tod and increasing with respect to "�".Suppose that there exist a constant k ∈ (0, 1s ) and an elementx0 ∈ X such that:

(i) d(f (x), f (y)) ≤ kd(x , y), for all x , y ∈ X with x ≤ y .(ii) x0 ≤ f (x0).

1− skd(x , f (x)), ∀n ∈ N∗.

Theorem.

Let X be a nonempty set endowed with a partial order "�" andd : X × X → X be a complete b-metric with constant s ≥ 1. Letf : X → X be an operator which has closed graph with respect tod and increasing with respect to "�".Suppose that there exist a constant k ∈ (0, 1s ) and an elementx0 ∈ X such that:(i) d(f (x), f (y)) ≤ kd(x , y), for all x , y ∈ X with x ≤ y .

(ii) x0 ≤ f (x0).Then Fix(f ) 6= ∅ and the sequence (f n(x))n∈N converges to a �xedpoint x∗(x) of f , for each x ∈ X which is comparable to x0.Moreover, if d is continuous, then we also have

1− skd(x , f (x)), ∀n ∈ N∗.

Theorem.

Let X be a nonempty set endowed with a partial order "�" andd : X × X → X be a complete b-metric with constant s ≥ 1. Letf : X → X be an operator which has closed graph with respect tod and increasing with respect to "�".Suppose that there exist a constant k ∈ (0, 1s ) and an elementx0 ∈ X such that:(i) d(f (x), f (y)) ≤ kd(x , y), for all x , y ∈ X with x ≤ y .(ii) x0 ≤ f (x0).

1− skd(x , f (x)), ∀n ∈ N∗.

Theorem.

Let X be a nonempty set endowed with a partial order "�" andd : X × X → X be a complete b-metric with constant s ≥ 1. Letf : X → X be an operator which has closed graph with respect tod and increasing with respect to "�".Suppose that there exist a constant k ∈ (0, 1s ) and an elementx0 ∈ X such that:(i) d(f (x), f (y)) ≤ kd(x , y), for all x , y ∈ X with x ≤ y .(ii) x0 ≤ f (x0).

1− skd(x , f (x)), ∀n ∈ N∗.

Ran-Reurings theorem in b-metric space (II)

Theorem.

Let X be a nonempty set endowed with a partial order "�" andd : X × X → X be a complete b-metric with constant s ≥ 1.

Letf : X → X be an operator which has closed graph with respect tod and is increasing with respect to "�".Suppose that there exist a comparison function ϕ : R+ → R+ andan element x0 ∈ X such that:(i) d(f (x), f (y)) ≤ ϕ(d(x , y)), for all x , y ∈ X with x � y ;(ii) x0 � f (x0);(iii) for every x , y ∈ X there exists z ∈ X which is comparable to xand y .Then f is a Picard operator.

Theorem.

Let X be a nonempty set endowed with a partial order "�" andd : X × X → X be a complete b-metric with constant s ≥ 1.Letf : X → X be an operator which has closed graph with respect tod and is increasing with respect to "�".

Suppose that there exist a comparison function ϕ : R+ → R+ andan element x0 ∈ X such that:(i) d(f (x), f (y)) ≤ ϕ(d(x , y)), for all x , y ∈ X with x � y ;(ii) x0 � f (x0);(iii) for every x , y ∈ X there exists z ∈ X which is comparable to xand y .Then f is a Picard operator.

Theorem.

Let X be a nonempty set endowed with a partial order "�" andd : X × X → X be a complete b-metric with constant s ≥ 1.Letf : X → X be an operator which has closed graph with respect tod and is increasing with respect to "�".Suppose that there exist a comparison function ϕ : R+ → R+ andan element x0 ∈ X such that:

(i) d(f (x), f (y)) ≤ ϕ(d(x , y)), for all x , y ∈ X with x � y ;(ii) x0 � f (x0);(iii) for every x , y ∈ X there exists z ∈ X which is comparable to xand y .Then f is a Picard operator.

Theorem.

Let X be a nonempty set endowed with a partial order "�" andd : X × X → X be a complete b-metric with constant s ≥ 1.Letf : X → X be an operator which has closed graph with respect tod and is increasing with respect to "�".Suppose that there exist a comparison function ϕ : R+ → R+ andan element x0 ∈ X such that:(i) d(f (x), f (y)) ≤ ϕ(d(x , y)), for all x , y ∈ X with x � y ;(ii) x0 � f (x0);(iii) for every x , y ∈ X there exists z ∈ X which is comparable to xand y .

Then f is a Picard operator.

Theorem.

Let X be a nonempty set endowed with a partial order "�" andd : X × X → X be a complete b-metric with constant s ≥ 1.Letf : X → X be an operator which has closed graph with respect tod and is increasing with respect to "�".Suppose that there exist a comparison function ϕ : R+ → R+ andan element x0 ∈ X such that:(i) d(f (x), f (y)) ≤ ϕ(d(x , y)), for all x , y ∈ X with x � y ;(ii) x0 � f (x0);(iii) for every x , y ∈ X there exists z ∈ X which is comparable to xand y .Then f is a Picard operator.

Coupled �xed point problems

The coupled �xed point problem

If (X , d) is a metric space and T : X × X → X is an operator,then, by de�nition, a coupled �xed point for T is a pair(x∗, y∗) ∈ X × X satisfying{

x∗ = T (x∗, y∗)y∗ = T (y∗, x∗) .

We will denote by CFix(T ) the coupled �xed point set for T .

Coupled �xed point theorems

Theorem.

Let (X ,≤) be a partially ordered set and let d : X × X → R+ be acomplete b-metric on X with constant s ≥ 1. Let T : X × X → Xbe an operator with closed graph which has the mixed monotoneproperty on X × X . Assume that the following conditions aresatis�ed:

(i) there exists k ∈ (0, 1s ) such that, ∀x ≤ u, y ≥ v , we have:

d(T (x , y),T (u, v))+d(T (y , x),T (v , u)) ≤ k[d(x , u)+d(y , v)];

(ii) there exist x0, y0 ∈ X such that x0 ≤ T (x0, y0) andy0 ≥ T (y0, x0).

Then, the following conclusions hold:(a) there exists (x∗, y∗) ∈ X × X a solution of the coupled �xed

point problem (7) and the sequences (xn)n∈N, (yn)n∈N in X de�nedby {

xn+1 = T (xn, yn) := T n(x0, y0)yn+1 = T (yn, xn) := T n(y0, x0),

have the property that (xn)n∈N → x∗, (yn)n∈N → y∗ as n→∞.Moreover, for every pair (x , y) ∈ X × X with x ≤ x0 and y ≥ y0(or reversely), we have that (T n(x , y))n∈N converges to x∗ and(T n(y , x))n∈N converges to y∗.

(b) In particular, if the b-metric d is continuous, then:

d(T n(x0, y0), x∗) + d(T n(y0, x0), y

∗) ≤

1− sk· [d(x0,T (x0, y0)) + d(y0,T (y0, x0))], for all n ∈ N∗.

Then, the following conclusions hold:(a) there exists (x∗, y∗) ∈ X × X a solution of the coupled �xed

point problem (7) and the sequences (xn)n∈N, (yn)n∈N in X de�nedby {

xn+1 = T (xn, yn) := T n(x0, y0)yn+1 = T (yn, xn) := T n(y0, x0),

have the property that (xn)n∈N → x∗, (yn)n∈N → y∗ as n→∞.Moreover, for every pair (x , y) ∈ X × X with x ≤ x0 and y ≥ y0(or reversely), we have that (T n(x , y))n∈N converges to x∗ and(T n(y , x))n∈N converges to y∗.(b) In particular, if the b-metric d is continuous, then:

d(T n(x0, y0), x∗) + d(T n(y0, x0), y

∗) ≤

1− sk· [d(x0,T (x0, y0)) + d(y0,T (y0, x0))], for all n ∈ N∗.

Proof.

We denote Z := X × X and the partially ordering ≤P given by

(x , y) ≤P (u, v)⇔ x ≤ u, y ≥ v .

We also introduce the functional d̃ : Z × Z → R+ de�ned by

d̃((x , y), (u, v)) := d(x , u) + d(y , v).

It is easy to see that d̃ is a b-metric on Z with the same constants ≥ 1 and if the space (X , d) is complete, then (Z , d̃) is completetoo.

Proof.

(x , y) ≤P (u, v)⇔ x ≤ u, y ≥ v .

d̃((x , y), (u, v)) := d(x , u) + d(y , v).

Proof.

(x , y) ≤P (u, v)⇔ x ≤ u, y ≥ v .

d̃((x , y), (u, v)) := d(x , u) + d(y , v).

We consider now the operator F : Z → Z given by