COUPLED FIXED POINT THEOREMS VIA MIXED MONOTONE PROPERTY IN AB-METRIC SPACES & APPLICATIONS TO INTEGRAL EQUATIONS Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2022-15-3-343-355 УДК 517

Coupled Fixed Point Theorems Via Mixed Monotone Property in Afr-metric Spaces & Applications to Integral Equations

K. Ravibabu*

Department of Mathematics, G.M.R.I.T Rajam, Srikakulam, India

G. N. V. Kishoret

Department of Engineering Mathematics & Humanities Sagi Rama Krishnam Raju Engineering College Chinamiram, Bhimavaram, Andhra Pradesh, India

Ch. Srinivasa Rao*

Department of Mathematics, Andhra University Visakhapatnam, India

Ch. Raghavendra Naidu§

Department of Mathematics, Govt. Degree College Palakonda, Srikakulam, India

Received 01.10.2021, received in revised form 29.11.2021, accepted 20.02.2022 Abstract. In this paper, we establish some results on the existence and uniqueness of coupled common fixed point theorems in partially ordered Ab-metric spaces. Examples have been provided to justify the relevance of the results obtained through the analysis of extant theorem. Further, we also find application to integral equations via fixed point theorems in Ab-metric spaces.

Keywords: Coupled fixed point, Mixed weakly monotone property, Ab-metric space, Integral equation.

Citation: K. Ravibabu, G.N.V. Kishore, Ch. Srinivasa Rao, Ch. Raghavendra Naidu, Coupled Fixed Point Theorems Via Mixed Monotone Property in Ab-metric Spaces & Applications to Integral Equations, J. Sib. Fed. Univ. Math. Phys., 2022, 15(3), 343-355. DOI: 10.17516/1997-1397-2022-15-3-343-355.

1. Introduction and preliminaries

The study of fixed point theory comes from wider area of non-linear function analysis. However, its study began almost a century ago in the field of algebraic topology. Fixed point theorems find applications in proving the existence and uniqueness of the solutions of certain differential and integral equations that arise in physical, engineering and other optimization problems. In the study of fixed point theory, some of the generalizations of metric space are 2-metric space, D-metric space, ^*-metric space, G-metric space, S-metric space, Rectangular metric or metric-like space, Partial metric space, Cone metric space. In 1989, I. A. Bakhtin [2] introduced the concept

*ravibabu.k@gmrit.edu.in

tgnvkishore@srkrec.ac.in, kishore.apr2@gmail.com https://orcid.org/0000-0001-7260-4071 idrcsr41@yahoo.com §ch.rvnaidu@gmail.com © Siberian Federal University. All rights reserved

of b-metric space. Consequent upon the introduction of b-metric space, many generalizations of metric spaces came into existence. In 2015, M. Abbas et al. [1] introduced the concept of n-tuple metric space and studied its topological properties. M. Ughade et al. [15] introduced the notion of Ab-metric spaces as a generalized form of n-tuple metric space. Subsequently N. Mlaiki et al. [11] obtained unique coupled common fixed point theorems in partially ordered Ab-metric spaces.

In this paper, we use the notion of a mixed weakly monotone pair of maps to state a coupled common fixed point theorem on partially ordered Ab-metric spaces. We prove some unique coupled common fixed point theorems in partially ordered Ab-metric space and also provide example to support our results.

First we recall some notions, lemmas and examples which will be useful to prove our results.

Definition 1.1 (M. Abbas et al. [1]). Let S be a non empty set and n(^ 2) be a positive integer. A function A : Sn ^ [0, œ) is called an A-metric on S, if for any Zi, a G S. i = 1, 2,... ,n, the following conditions hold.

(i) A(Cl,C2,--.,Cn-l ,Zn) > 0,

(ii) A(Ci, Z2,..., Zn-i,Zn) =0 if and only if Ci = C2 = • • • = Cn-i = Zn,

(iii) A(Zi, Z2,... Zn-i,Zn) < [A(Zi, Zi,..., Ci(„_i), a) + A(Z2,Z2,..., Z2(n-1), a) +

+----+ A(Zn-i Xn-u . . . ,Zn-in-i) ,a)+ A(Zn,Zn,■■■, Zn (n — 1) , a)].

The pair (S, A) is called an A-metric space.

Definition 1.2 (T.G.Bhaskar et al. [6]). Let X be a non empty set. A b-metric on X is a function d : X2 ^ [0, œ) such that the following conditions hold for all x,y, z G X.

(i) d(x, y) = 0 ^^ x = y,

(ii) d(x, y) = d(y, x) ,

(iii) there exists s ^ 1, such that d(x,z) ^ s[d(x,y) + d(y,z)]. The pair (X, d) is called a b-metric space.

Definition 1.3 (M. Ughade et al. [14]). Let S be a non empty set and n > 2. Suppose b > 1 is a real number. A function Ab : Sn ^ [0, œ) is called an Ab -metric on S, if for any Zi,a G S, i = 1, 2, . . . , n, the following conditions hold.

(i) Ab(Zi,Z2,...,Zn-i,Zn) > 0,

(ii) Ab(Zi,Z2, ..., Zn-i ,Zn) = 0 if and only if Zi = Z2 = • • • = Zn-i = Zn,

(iii) Ab(Zi, Z2,..., Zn-1, Zn ) < b[Ab(Zi,Zi,... ,Zin— 1), a) + Ab(x2,x2,.. .,x2(n—i) ,a) + ...

+ Ab(Zn—1, Zn-1, . . . , Zn-i(n—i),a) + Ab(Zn, Zn — 1) , a)].

The pair (S,Ab) is called an Ab-metric space.

Note: In practice we write A for Ab when there is no confusion.

Example 1.4 (M. Ughade et al. [14]). Let S = [1, œ) and n > 2. Define Ab : Sn ^ [1, œ) by

n2

Ab(Zi,Z2,...,Zn-i,Zn) = J2 J2 IZi - Zj I , for all Zi G S, i = 1, 2,... ,n. Then (S,Ab) is an

i=ii<j

Ab-metric space with b=2.

Lemma 1.5 (M. Ughade et al. [14]). Let (S, A) be Ab metric space, so that A : Sn ^ [0, œ) for some n ^ 2. Then A(Z, Z,... ,Z,y) ^ bA(y, y,... ,y, Z ), for all Z,y G S.

V V

(n-i) times (n-i) times

Lemma 1.6 (M.Ughade et al. [14]). Let (S,A) be Ab metric space, so that A : Sn ^ [0, m)

for some n ^ 2. Then A(Z, ... ,(,z) ^ (n — 1)b A(Z, (,..., (,y) + b2 A(y, y,... ,y, z), for all

\_„_✓ \_„_✓ \_„_✓

V ss ss

(n-l)times (n-l)times (n-l)times

Z,y,z e S.

Lemma 1.7 (M.Ughade et al. [14]). Let (X,A) be Ab metric space. Then (X2,DA) is Ab-metric space on X x X with Da defined by

DA((xi,yi), (x2 ,y2),..., (xn,yn)) = A(xi,x2,..., Xn) + A(yi,y2,..., yn) for all xuyi e X, i,j = 1, 2,... ,n.

Definition 1.8. Let (X,A) be Ab-metric space. A sequence {xn} in X is said to converge to a

point x e X, if A(xn, xn,..., xn, x) ^ 0 as n ^ m. That is, to each e ^ 0 there exist N e N "-v-'

(n-l)times

such that for all n ^ N, we have A(xn, xn,..., xn, x) ^ e and we write lim xn = x.

\_^_✓ n^^

(n-l)times

Note: x is called the limit of the sequence {xn}.

Lemma 1.9 (N. Mlaiki et al. [11]). Let (X,A) be Ab-metric space. If the sequence {xn} in X converges to a point x, then the limit x is unique.

Definition 1.10. Let (X,A) be Ab-metric space. A sequence {xn} in X is called a Cauchy sequence, if A(xn, xn,... ,xn, xm) ^ 0 as n,m ^ m. That is, to each e ^ 0, there exists N e N

(n-l)times

such that for all n,m ^ N, we have A(xn, xn,..., xn, xm) ^ e.

"-V-'

(n-l)times

Lemma 1.11 (N. Mlaiki et al. [11]). Every convergent sequence in a Ab-metric space is a Cauchy sequence.

Definition 1.12. A Ab-metric space (X,A) is said to be complete, if every Cauchy sequence in X is convergent.

Definition 1.13 (M.E. Gordji et al. [7]). Let (X, <) be a partially ordered set and f,g : X x X ^ X be mappings. We say that (f, g) has the mixed weakly monotone property on X, if for any x, y e X,

x < f (x,y), y > f(y,x) f (x,y) < g((f (x,y),f(y,x)), f (y,x) > g((f (y,x),f (x,y))

and

x < g(x,y), y > g(y,x) g(x,y) < f ((g(x,y),g(y,x)), g(y,x) > f (g(y,x),g(x,y)).

Definition 1.14. Let X be a non-empty set and f,g : X x X ^ X be maps on X x X.

(i) A point (x,y) e X x X is called a coupled fixed pint of f, if x = f (x, y) and y = f (y, x)

(ii) A point (x,y) e X x X is said to be a common coupled fixed pint of f and g, if x = f (x,y) = g(x, y) and y = f (y,x) = g(y,x).

Note: (x,y) is said to be a Coupled coincidence point of f and g, if f (x,y) = g(x,y) and f(y, x) = g(y, x).

We observe that a common coupled fixed pint of f and g is necessarily a Coupled coincidence point of f and g.

2. Main results

Now we prove our first main result.

Theorem 2.1. Let (X, ^,A) be a partially ordered, complete Ab-metric space and let f,g : X x X ^ X be the mappings such that

(i) the pair (f,g) has mixed weakly monotone property on X and there exists xo,yo € X such that xo < f (xo, yo), f (yo, xo) < yo or xo < g(x0, y0), g(y0, xo) < yo,

(ii) there is an a such that ab2((n — l)b + 1) < 1 and

A(f (x, y), f (x, y),■■■, f (x, y), g(u,v)) + A(f (y,x), f (y, x),..., f (y,x), g(v,u)) < aM,

where M = max

(D((u,v), (u,v), ..., (u,v), (g(u,v),g(v,u))))

(1 + D((x, y), (x, y),..., (x, y), (f (x, y), f (y, x)))) x

,D( (x,y), (x,y),..., (x,y), (u,v)),

(2.1)

(1 + D((x,y), (x,y),..., (x,y),(u,v))) {D((x,y),(x,y),... (x,y),(f (x,y),f (y,x)))+ D((u,v),(u,v),... (u,v),(g(u,v),g(v,u)))),

(D((u,v),(u,v),...(u,v),(f (x,y),f (y,x)))+ D((x,y),(x,y),... (x,y), (g(u,v),g(v,u))))

for all x,y,u,v € X with x ^ u and y ^ v, (iii) if f or g is continuous.

Then f and g have a coupled common fixed point in X.

Proof. Let (xo,yo) be a given point in X x X, satisfying (i). Write xi = f (xo, yo), yi = = f(yo,xo), x2 = g(xi,yi), y2 = g(y\,xi). Define the sequences {xn} and {yn} inductively

x2n+1 = f(x2n,y2n), y2n+1 = f (y2n,x2n) x2n+2 = g(x2n+1,y2n+l), y2n+2 = g(y2n+1, x2n+l) (2.2)

for all n € N

Since xo < f(xo,yo) and yo > f(yo,xo) and since (f,g) has mixed weakly monotone property, we have

xi = f (xo, yo) < g(f (xo, yo),f (yo, xo)) = g(xi,yi) = x2 xi < x2 and x2 = g(xi ,yi) < f (g(xi, yi), g(yi ,xi)) = f (x2, y2) = xs x2 < xs also yi = f (yo, xo) > g(f (yo, xo), f (xo, yo)) = g(yi ,xi) = y2 yi > y2 and y2 = f (yi, xi) > f (g(yi,xi),g(xi,yi)) = f (y2,x2) = ys y2 > ys. By induction,

i.e, xo ^ xi ^ x2 ^ ... ^ xn ^ xn+i ^ ...

yo > yi > y2 > ... > yn > yn+i > ... (2.3)

for all n € N

Now we show that these sequences are Cauchy. Define Dn : X x X ^ X by

Dn = D((xn, yn), (xn,yn), . . ., (xn,yn), (xn+i, yn+i))

= A(xn,xn,.. .,xn,xn+i) + A(yn,yn,.. .,yn,yn+i) for all xi,yi € X, i, j = 1, 2,...,n.

Now

D2n+1 =A(x2n+1, X2n+1, ■■■, X2n+1,X2n+2) + A(y2n+1, V2n+1, ■■■, V2n+1 ,V2n+2 ) =A(f (X2n, V2n), f (X2n, V2n), ■■ ■, f (X2n, V2n), g(x2n+1 ,V2n+1)) + + A(f (y2n, X2n), f (V2n, X2n), ■■ ■, f (V2n, X2n), g(y2n+1,X2n+1)) <

< a max

(1 + D((X2n, y2n), (X2n, y2n), ■■■, (X2n, y2n), (f (X2n, y2n ), f (y2n, X2n)))) X

(D((x2n+1, y2n+l), (x2n+l,y2n+l), ..., (x2n+l, y2n+l), (g(x2n+l, y2n+l), g(y2n+l, x2n+l)))) (1 + D((x2n, y2n), (x2n, y2n), ..., (x2n, y2n), (x2n+l,y2n+l)))

D( (x2n,y2n), (x2n,y2n), . .., (x2n,y2n), (x2n+l,y2n+l)),

D((x2n, y2n ), (x2n, y2n), ..., (x2n, y2n), (f (x2n, y2n), f (y2n, x2n))) +

+ (D((x2n+l, y2n+l), (x2n+l,y2n+l), .. ., (x2n+l, y2n+l), (g(x2n+l, y2n+l), g(y2n+l, x2n+l))))) , ((D((x2n+l,y2n+l), (x2n+l, y2n+l), .. ., (x2n+l,y2n+l), (f (x2n, y2n), f (y2n, x2n )))) + + (D((x2n, y2n), (x2n, y2n), . . . , (x2n, y2n), (g(x2n+l, y2n+l), g(y2n+l, x2n+l)))))} < < a m&x{D( (x2n+l,y2n+l), (x2n+l,y2n+l), ..., (x2n+l,y2n+l), (x2n+2,y2n+2)) , D( (x2n,y2n), (x2n,y2n), . .., (x2n,y2n), (x2n+l,y2n+l)), (D((x2n, y2n ), (x2n, y2n), .. ., (x2n, y2n), (x2n+l,y2n+l)) + + (D((x2n+l,y2n+l), (x2n+l,y2n+l), ..., (x2n+l,y2n+l), (x2n+2, y2n+2)))) ,

(D((x2n ,y2n ), (x2n, y2n), .. ., (x2n, y2n), (x2n+2,y2n+2))) }. By using Lemma 1.6, we have

D2n+l < a{ (n — 1)b[A(x2n,x2n,...,x2n,x2n+l) + A(y2n,y2n, . . . ,y2n,y2n+l)] +

2 I (2.4)

+ b [A(x2n+l,x2n+l, . . ., x2n+l ,x2n+2) + A(y2n+l, y2n+l, .. ., y2n+l, y2n+2)U.

Similarly, we get

A(y2n+l ,y2n+l, .. ., y2n+l, y2n+2) + A(x2n+l, x2n+l, ..., x2n+l,x2n+2) < < a{(n — 1)b[A(y2n, y2n, ..., y2n, y2n+l) + A(x2n, x2n, ..., x2n, x2n+l)} + (2.5)

+ b2[A(y2n+ l,y2n+l, ..., y2n+l, y2n+2) + A(x2n+l, x2n+l, . .., x2n+l, x2n+2)]}. From (2.4) and (2.5) we have,

2D2n+l = 2[A(x2n+l, x2n+l, . .., x2n+l,x2n+2) + A(y2n+l,y2n+l, ..., y2n+l ,y2n+2 )] < < 2a{(n — 1)b[A(x2n, x2n, ..., x2n, x2n+l) + A(y2n, y2n, ..., y2n, y2n+l)] + + b2[A(x2n+l , x2n+l, . . . , x2n+l, x2n+2 ) + A(y2n+l, y2n+l, . . ., y2n+l,y2n+2)]}.

Therefore

D2n+l < a{(n — 1)b[A(x2n, x2n, ..., x2n, x2n+l) + A(y2n, y2n, ..., y2n, y2n+l)] +

2 (2.6) + b [A(x2n+l,x2n+l, ..., x2n+l, x2n+2 ) + A(y2n+l, y2n+l, ..., y2n+l, y2n+2)]}.

It gives that

D . a(n — 1)b D (27) D2n+l ^ ~1-abl2~D2n. (2.7)

a(n — 1)b Put /3 = —--f-, then 0 < /3 < 1.

1 — ab2

From (2.7),

D2n+i ^ 3D2n.

Similarly we can show that

D2n+2 < 3D2n+i for n = 0, 1, 2,... .

Hence

Therefore

Define

Dn+1 < ßDn.

Dn+i < ßn+1 Do. (2.8)

Dn,m D((Xn, yn) , (xn, yn ) i • • • i (Xn, yn) ? (xmj ym )) v-v-'

(n—1) —times

A(xn ^ xn ^ • • • ^ xn^ xm ) + A(yn ^ yn, • • • j yn ^ ym ) •

\_„_✓ \_„_✓

V ss

(n—1) —times (n—1) —times

Now we have to show that Dn,m is a Cauchy sequence. By Lemma 1.6, for all n,m G N, n ^ m, we have

Dn+l,m+l = A(xri+I,xri+1, .. ., xri+i,xm+i) + A(yn+l,yn+l, ■■■, yn+l,ym+l) <

< b(n — l)[A(xn+i,xri+i,..., xn+i,xn+2) + A(yn+i,yn+i, ■■■, yn+i,yn+2)} +

+ b2 [A(xn+2,xn+2, ■■ ■, xri+2,xm+i) + A(yn+2,yn+2, ■■■, yn+2,ym+l)] = = b(n — 1)Dn+1 + b2b( n — 1)[A(xn+2, xn+2, . . . , xn+2,xn+3) + + A(yn+2,yn+2, . . . ,yn+2,yn+3)} +

+ b2 b2 [A(xn+3, xn+3,xn+3, xm+i) + A(yn+3, yn+3,yn+3, ym+i)} = = b(n — 1)Dn+i + b3(n — 1)Dn+2 + b5 (n — 1)Dn+3 ••• + + b2(m—n^—3(n — 1)[A(xm—1, xm—1, xm—1, xm) + A(ym—1, ym—1, ym—1, ym )} + + b2(m—n)—1(n — 1)[A(xm,xm,...,xm,xm+1)+ A(ym,ym,.. .,ym,ym+1)].

From (2.8), we have that

Dn+1,m+1 < b(n — 1)[ßn+1 + b2ßn+2 + b4ßn+3 + ••• + b2(m—n)—2ßm}Do < < b(n — 1)ßn+1[1 + b2ß + (b2ß)2 + ••• + (b2ß)(m—n—1)}Do =

= b(n — 1)ßn+1[1 + y + y2 + • • • + Y(m—n—1)}Do < 1

.1 — Y,

^ b(n - 1)ßn+^Y-^jDo

Thus

0 as n oo.

lim A(xn ,xn,...,xn,xm)= lim A(yn,yn,... ,yn,ym) = 0.

n.m^^ n.m^oo

Therefore {xn} and {yn} are both Cauchy sequences in X.

By the completeness of X, there exists x,y G X such that xn ^ x and yn ^ y as n ^ to. Therefore Dn,m is a Cauchy sequence.

Now we show that (x, y) is a coupled fixed point of f and g.

Without loss of generality, we may suppose that f is continuous, we have

and

x = lim X2n+1 = lim f(x2n,y2n) = f ( lim x2n, lim y2n) = f(x,y)

n—œ n—œ \n—^^o n—œ J

y = lim y2n+i = lim f (y2n,X2n) = f ( lim y2n, lim X2n) = f (y,x).

n—œ n—œ \n—œ n—œ /

Thus (x, y) is a coupled fixed point of f. From (2.1), taking x = u and y = v, we have,

A(x, x,,...,x, g(x, y)) + A(y, y,...,y, g(y, x)) =

= A(f (x, y),f (x, y)f (x, y), g(x, y)) + A(f (y,x),f (y,x),--.,f (y,x), g(y,x)) <

(D((x, y),(x, y)...(x, y), (g(x, y), g(y,x))))

< a max

(1 + D((x,y), (x,y) ... (x,y), (x, y)))-

(1+ D((x,y), (x,y)... (x,y), (x,y))) D((x, y), (x,y),..., (x, y), (x, y)), (D((x, y), (x, y),...,(x, y),(x, y)) +

+D((x, y),(x, y),...,(x, y), (g(x, y), g(y,x)))), (D((x, y),(x, y),...,(x y),(x, y)) + +D((x, y), (x, y),■■■, (x, y), (g(x, y), g(y, x))))} <

< ab ((g(x, y), g(y, x)), y),g(y, (g(x, y), g(y, x)), (x, y)).

Since ab < 1, we have (g(x,y), g(y,x)) = (x,y). Therefore g(x, y) = x and g(y, x) = y. Therefore (x, y) is a coupled fixed point of g.

Thus (x, y) is a coupled common fixed point of f and g. □

Theorem 2.2. Let (X, A) be a partially ordered, complete Ab-metric space and f, g : X x X ^ X be the mappings such that

(i) the pair (f,g) has mixed weakly monotone property on X and there exists xo,yo G X such that xo < f (xo, yo), f (yo, xo) < yo or xo < g(x0, y0), g(y0, xo) < yo,

(ii) there is an a such that ab2((n — 1)b + 1) < 1 and

A(f (x,y),f (x,y),.. .,f (x,y),g(u,v)) + A(f (y,x),f (y,x),.. .,f (y,x),g(v,u)) < aM

where

M = max

(1 + D((x, y),(x, y),...,(x, y),(f (x, y), f (y, x))))

(D((u,v), (u,v),..., (u,v), (g(u,v),g(v,u))))

(1 + D((x,y), (x,y),(x,y), (u,v)))

D((x, y), (x, y^l^.^ (x, y), (u, v)), {D((x, y), (x, y^l^.^ (x, y), (f (x, y), f (y, x))) +

+D((u, v), (u, v),■■■, (u, v), (g(u, v), g(v, u))))( (D((u, v), (u, v),■■■, (u, v), (f (x, y), f (y, x))) +

for all x,y,u,v G X with x ^ u and y ^ v, (iii) X has the following properties

(2.9)

(a) if {xn} is an increasing sequence with xn ^ x, then xn ^ x for all n G N,

(b) if {yn} is a decreasing sequence with yk ^ y, then y ^ yn for all n G N.

Then f and g have coupled common fixed points in X.

Proof. Suppose X satisfies (a) and (b), by (2.3) we get xn < x and yn > y for all n G N. Applying Lemmas 1.5 and 1.6, we have

D((xj y)j (xj y),...j (xj y)j (f (xj y)j f (yj x))) <

< b(n — 1)D((x, y), (x, y),..., (x, y), (x2n+2,y2n+2)+

+ b2D((x2n+2,y2n+2), (x2n+2, y2n+2), ..., (x2n+2,y2n+2), (f (x, y), f (y, x))) = = b(n — 1)D((x, y), (x, y),..., (x, y), (x2n+2,y2n+2)) + + b2D((g(x2n+1, y2n+1), g(y2n+1,x2n+1)), (g(x2n+1, y2n+1), g(y2n+1, x2n+1)), . . . ..., (g(x2n+1, y2n+1), g(y2n+1,x2n+1)), (f (x, y), f (y, x))).

By (2.1), we get

A( (g(x2n+1, y2n+1)), (g(x2n+1, y2n+1)), ..., (g(x2n+1, y2n+1)), (f (x, y))) + + A (g(y2n+1,x2n+1)), (g(y2n+1,x2n+1)), ..., (g(y2n+1,x2n+1)), (f (y, x))) <

^ a max 4 (1 + D((x2n+1,y2n+1), (x2n+1,y2n+1), . . . , , (x2n+1,y2n+1), (x2n+2,y2n+2))) X

(D((x, y), (x,y),■■■, (x, y), (f (x, y), f (y, x))))

(1 + D((x2n+1,y2n+l), (x2n+1,y2n+l), . .., (x2n+1,y2n+l), (x,y)))_ D( (x2n+1,y2n+1), (x2n+1,y2n+1), . .., (x2n+1,y2n+1), (x,y)) , (D((x2n+1, y2n+1), (x2n+1,y2n+1), ..., (x2n+1, y2n+\), (x2n+2,y2n+2)) + +D((x, y), (x, y),..., (x, y), (f (x, y), f (y, x)))), {D((x, y), (x, y),..., (x, y), (x2n+2,y2n+2)) + +D((x2n+1,y2n+1), (x2n+1, y2n+1), ..., (x2n+1,y2n+1), (f (x, y), f (y, x)

Taking the limit as n ^ œ in (2.9), we obtain

D((x, y), (x,y)(x, y), (f (x, y), f (y, x))) < b2a D((x, y), (x, y')^.^ (x, y), (f (x, y), f (y, x))).

Since b2a < 1, we have D((x, y), (x, y),..., (x, y), (f (x, y), f (y, x))) = 0.

That is, f (x,y) = x and f (y,x) = y. Therefore (x,y) is a coupled fixed point of f.

Similarly we can show that g(x,y) = x and g(y,x) = y. Hence f (x,y) = x = g(x,y) and

f (y,x) = y = g(y,x).

Thus (x, y) is a coupled common fixed point of f and g. □

Theorem 2.3. Suppose Theorem 2.1 or Theorem 2.2 satisfied, if further {xn} is an increasing sequence with xn ^ x and xn ^ u for each n, then x ^ u. Then f and g have a unique coupled common fixed points. Further more, any fixed point of f is a fixed point of g, and conversely.

Proof. Suppose the given condition holds. Let (x,y) and (u,v) G X x X, there exist (x*,y*) G

X x X , that is, comparable to (x,y) and (u,v).

D((x, y), (x,y),■■■, (x, y), (u, v)) =

= A(x, x,..., x, u) + A(y, y,... ,y,u) =

= A(f(x,y),f(x,y^)^.^f(x,y),g(uv)) + A(f(y,x),f(y,f(y,x),g(v,u)) <

< a max

(1 + D((x, y), (x, y),..., (x, y), (f (x, y), f (y, x))))x

(D((u, v), (u (u v), (g(u v),g(v, u))))

<D((x,y), (x,y),(x,У), (u,v)),

(1 + D((x,y), (x,y),(x,y), (u,v)))

(D((x, y), (x, y^)^.^(x, y), (f (x, y), f (y, x))) + D((u, v), (u, v),■■■, (u, v), (g(u, v), g(v, u)))), (D((u v), (u,v),■■■, (u v), (f y),f (y, x))) + D((x, y), (x, y),■■■, (x, y), v), g(v, u))))} <

< a (b + 1)D((x,y), (x,y), ..., (x,y), (u,v)).

Since a(b +1) < 1, so that

D((x, У), (x, y^)^.^ (x, У), (u, v)) = 0

(x, y) = (u, v) x = u and y = v

Suppose (x,y) and (x* ,y*) are Coupled common fixed points such that x < x* and y > y*, then x = x* and y = y*. Now

D((x, y), (x, y),..., (x, y), (x*,y*)) = A(x, x,...,x,x*)+ A(y, y,...,y,y*) =

= A(f (x, y), f (x, y^)^.^ f (x, y), g(x*,y*)) + A(f (y, x),f (y, f (y, x), g(y*,x*)) <

< a(b +1)D((x,y), (x,y),..., (x,y), (x*,y*)). Since a(b +1) < 1, so that

D((x,У), (x,У),. (x,У), (x*,y*)) = 0 (x,y) = (x*,y*)

x = x* and y = y* we show that any fixed point of f is a fixed point of g, and conversely. That is, to show that (x, y) is a fixed point of f ^^ (x, y) is a fixed point of g. Suppose that (x, y) is a coupled fixed point of f

D((x, y), (x, y^)^.^(x, y), (g(x, y), g(y, x))) = = A(f (x, y), f (x, y^)^.^ f (x, y),g(x, y)) + A(f (y, x), f (y, f (y, x), g(y, x)) <

< ab D((g(x, y), g(y, x)), (g(x, y), g(y, (g(x, y),g(y, x)), (x, y)).

Since ab < 1, we have

D((g(x, y), g(y, x)), (g(x, y), g(y, x)),(g(x, y), g(y, x)), (x, y)) = 0 (g(x, y), g(y, x)) = (x, y)

x = g(x, y) and y = g(y, x) Therefore (x, y) is a coupled fixed point of g, and conversely. □

Taking M = D((x, y), (x, y),..., (x, y), (u, v)) and g = f in Theorem 2.1, we get the following

Corollary 2.4. Let (X, A) be a partially ordered, complete Ab-metric space and let f : X x X ^ X be the mapping such that

(i) f has mixed weakly monotone property on X and there exists xo,yo G X such that xo ^

f (xo,yo), f (yo,xo) < yo,

(ii) there is an a such that a < 1 and

A(f y), f (x, y),..., f (x, y), f (u, v))+A(f (y, x), f (y, x),..., f (y, x), f (v, u)) < ^ ^

< a D((x, y),(x, y),...,(x, y),(u, v)),

for all x,y,u,v £ X with x ^ u and y ^ v,

(iii) if f is continuous.

Then f has a coupled fixed point in X.

We give an example to demonstrate the validity of the result 2.1.

Example 2.5. Let (R, ^,A) be a partially ordered complete Ab-metric space with Ab-metric

defined as X = [—to, +to] by Ab : Xn ^ [—to, +to] by

n 2

Ab(xi,x2,..., xn-i ,xn) = 'Yh\xi — xj | , for all xi £ X, i = 1, 2,..., n. Then (X, Ab) is an

i=1 i<j

Ab-metric space with b=2.

, ™ ™ , in ,, C 4x — 2y + 48n — 2 . 6x— 3y+72n — 3

Let f,g: R ^ R be two maps defined by f (x, y) =-—- and g(x, y) =-—-.

48n 72n

Then the pair (f, g) has mixed weakly monotone property on R

A(f (x, y), f (x, y^i^.^ f (x, y), g(u, v)) + A(f (y, x), f (y, f (y, x), g(v, u)) = = (n — 1)(\f(x,y) — g(u,v)l) + (n — 1)(\f (y,x) — g(v,u)l) =

4x — 2y + 48n — 2 6u — 3v + 72n — 3

= (n - 1) + (n - 1)

48n 72n

4y - 2x + 48n - 2 6v - 3u + 72n - 3

+

48n 72n

= (~n—11(\2(xx — u) — (y — v)\ + \2(y — v) — (x — u)\) < {-n—1(3 \x — u\ + 3 \y — v\) <

(n 1)

^ (\x — u\ + \y — v\) = (n ~~ 1)

= —5- D((x,y), (x,y),..., (x,y), (u,v)).

8n

For n = 2 and b=2, since ab2((n — 1)b +1) < 1 =^ a < — .

Then the contractive condition (2.1) is satisfied with a = — < — and also (1,1) is the unique

coupled common fixed point of f and g.

3. Application

The following type system of integral equations:

b

16 12

u(t) = q(t)+ i X(t,s)(f1(s,u(s)) + f2(s,v(s)))ds,

J a

v(t) = q(t)+ i \(t,s)(h(s,v(s)) + h(s,u(s)))ds,

a

(3.1)

where the space X = C([a,b], R) of continuous functions defined in [a,b]. Obviously, the space with the metric is given by

A(u,v)= max \u(t) — v(t)\, u,v £ C([a,b], R)

te[a,b]

is a complete metric space.

Let X = C([a,b], R) the natural partial order relation, that is, u,v G C ([a,b], R), u < v ^^ u(t) < v(t), t G [a,b].

Theorem 3.1. Consider the corollary 2.4 and assume that the following conditions are hold:

(i) f1,f2 : [a, b] x R ^ R are continuous;

(ii) q : [a, b] ^ R is continuous;

(iii) X : [a, b] x R ^ [0, x) is continuous;

(iv) there exist c > 0 and 0 ^ a < 1, such that for all u,v G R, v ^ u, 0 < fi(s,v) - fi(s,u) < ca(v - u)

0 < f2(s,v) - f2(s,u) < ca(v - u);

b

(v) assume that c max f X(t, s)ds ^ 1;

te[a,b] a

(vi) there exist xo,yo G X such that

xo(t) > q(t)+ X(t,s)(f (s,xo(s))+ g(s,yo(s)))ds,

a

yo(t) < q(t)+ X(t,s)(f (s,yo(s)) + g(s,xo(s)))ds.

a

Then the system of Volterra type integral equation (3.1) has a unique solution in X x X with X = C([a, b], R).

Proof. Define the mapping F : X x X ^ X by

F(u,v)(t) = q(t)+ i X(t,s)(fi(s,u(s)) + f2(s,v(s)))ds (3.2)

a

for all u,v G X and t G [a, b].

Now we have to show that all the conditions of Corollary 2.4 are satisfied. From (iv) of the Theorem 3.1, clearly F has mixed monotone property. For x,y,u,v G X with x ^ u and y ^ v, we have

A(F (x, y), F (x, y),..., F (x, y),F (u, v)) + A(F (y, x), F (y, x),..., F (y, x), F (v, u)) = = (n - 1) max(\F(x,y)(t) - F(u,v)(t)\ + \F(y,x)(t) - F(v,u)(t)\) =

te [a,b]

rb r b

= (n - 1) max

te[a,b]

X(t, s)(fi(s,x(s)) + f2(s,y(s)))ds - X(t, s)(f1(s,u(s)) + f2(s,v(s)))ds

J a

«b f b X(t, s)(fi(s,y(s)) + f2(s,x(s)))ds - X(t, s)(fi(s,v(s)) + f2(s,u(s)))ds

a

< (n - 1)max[ / \fi(s,x(s)) - f1(s,u(s))\\X(t, s)\ds +

te[a,b]\ Ja

b

+

+ (n — 1) max

te[a,b]

<

+ ! \f2(s,y(s)) - f2(s,v(s))\\X(t,s)\ds +

a

+ i \fi(s,y(s)) - fi(s,v(s))\\X(t,s)\ds + i \f2(s,x(s)) - f2(s,u(s))\\X(t,s)\ds) <

aa

^ (n — 1) max cm / \x(s) — u(s)\\A(t, s)\ds + \y(s) — v(s)\\A(t, s)\ds +

t^[a,b] \Ja J a

+ f \y(s) — v(s)\\A(t,s)\ds + / \x(s) — v(s)\\A(t,s)\ds) <

aa

^ (n — 1m max \x(t) — u(t) \ + max \y(t) — v(t)\ +

\te [a,b] te[a,b]

+ max \y(t) — v(t)\ + max \x(t) — u(t)\ ) ca / \A(t, s)\ ds ^

te[a,b] te[a,b] J Ja

< 2(n — 1) ( max \x(t) — u(t)\ + max \y(t) — v(t)^ ca f \A(t, s)\ ds <

\te[a,b] te[a,b] J Ja

^ 2(n — 1) a (A(x, x,... ,x,u) + A(y, y,... ,y,v)) = = 2(n — 1) a D((x,y), (x,y), ..., (x,y), (u,v)). Therefore

A(F (x, y), F (x, y),..., F (x, y), F (u, v)) + A(F (y, x), F (y, x),..., F (y, x), F (v, u)) < < 2(n — 1) aD((x, y), (x, y),..., (x, y), (u, v)).

For n=2, a < — < 1. Which is the contractive condition in Corollary 2.4. Thus, F has a coupled fixed point in X.

That is, the system of integral equations has a solution. □

References

[1] M.Abbas, B.Ali, Y.I.Suleiman, Generalized coupled common fixed point results in partially ordered A-metric spaces, Fixed point theory Appl., 64(2015).

DOI: 10.1186/s13663-015-0309-2

[2] I.A.Bakhtin, The contraction mapping principle in almost metric space, Functional analysis. Ulyanovsk. Gos. Ped. Inst., Ulyanovsk, 1989, 26-37 (in Russian).

[3] Jyzef Banas, Donal O'Regan, On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order, J. Math. Anal. Appl., 345(2008), no. 1, 573-582. DOI: j.jmaa.2008.04.050

[4] M.Bousselsal, M.Laid Kadri, Coupled Coincidence point for generalized monotone operators in partially ordered metric spaces, Thai Journal of Mathematics, 15(2017), no. 2, 367-385.

[5] Nguyen Van Dung, On coupled common fixed points for mixed weakly monotone maps in partially ordered S-metric spaces, Fixed Point Theory and Applications, 48(2013), 1-17. DOI: 10.1186/1687-1812-2013-48

[6] T.Gnana Bhaskar, V.Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65(2006), 1379-1393.

DOI: 10.1016/j.na.2005.10.017

[7] M.E.Gordji, E. Akbartabar, Y.J.Cho, M.Ramezani, Coupled common fixed point theorems for mixed weakly monotone mappings in partially ordered metric spaces, Fixed point theory and Appl., 95(2012). DOI: 10.1186/1687-1812-2012-95

[8] E.Karapinar, P.Kumam, W.Sintunavarat, Coupled fixed point theorems in cone metric spaces with a c-distance and applications, Fixed Point Theory and Appl., 194(2012). DOI: 10.1186/1687-1812-2012-194

[9] E.Karapinar, P.Kumam, I.Erhan, Coupled fixed points on partially ordered G-metric spaces, Fixed Point Theory and Appl., 2012(2012), 174. DOI: 10.1186/1687-1812-2012-174

[10] P.Kumam, V.Pragadeeswarar, M.Marudai, K.Sitthithakerngkiet, Coupled Best Proximity Points in Ordered Metric Spaces, Fixed Point Theory and Appl., 107(2014).

DOI: 10.1155/2014/274062

[11] N.Mlaiki, Y.Rohen, Some coupled fixed point theorem in partially ordered Аь-metric spaces, J. Nonlinear. Sci. Appl., 10(2017), 1731-1743.

[12] K.Ravibabu, Ch.Srinivasarao, Ch.Raghavendra Naidu, Coupled Fixed point and coincidence point theorems for generalized contractions in metric spaces with a partial order, Italian Journal of pure and applied mathematics, 39(2018), 434-450.

[13] K.Ravibabu, Ch.Srinivasarao and Ch.Ragavendra naidu, Applications to Integral Equations with Coupled Fixed Point Theorems in Ab-Metric Space, Thai Journal of Mathematics, (2018), 148-167.

[14] M.Ughade, D.Turkoglu, S.R.Singh, R.D.Daheriya, Some fixed point theorems in Ab-metric space, British J. Math. Comput. Sci., 19(2016), 1-24. DOI: 10.9734/BJMCS/2016/29828

Связанные теоремы о неподвижной точке через свойство смешанной монотонности в Ab-метрических пространствах и приложения к интегральным уравнениям

К. Равибабу

Кафедра математики, G.M.R.I.T. Раджам, Шрикакулам, Индия

Г.Н.В. Кишор

Департамент инженерной математики и гуманитарных наук Инженерный колледж Саги Рама Кришнам Раджу Чинамирам, Бхимаварам, Андхра-Прадеш, Индия

Ч. Шриниваса Рао

Кафедра математики, Университет Андхра Вишакхапатнам, Индия

Ч. Рагхавендра Найду

Кафедра математики, Государственный колледж Палаконда, Шрикакулам, Индия

Аннотация. В этой статье мы устанавливаем некоторые результаты о существовании и единственности связанных теорем об общей неподвижной точке в частично упорядоченных Ль-метрических пространствах. Приведены примеры для обоснования актуальности результатов, полученных в результате анализа существующей теоремы. Кроме того, мы также находим приложение к интегральным уравнениям через теоремы о неподвижной точке в Ль-метрических пространствах.

Ключевые слова: связанная неподвижная точка, смешанная слабомонотонность, Ль-метрическое пространство, интегральное уравнение.