FIXED POINTS OF SET-VALUED F -CONTRACTION OPERATORS IN QUASI-ORDERED METRIC SPACES WITH AN APPLICATION TO INTEGRAL EQUATIONS Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2021-14-2-150-158 УДК 517.9

Fixed Points of Set-valued F-contraction Operators in Quasi-ordered Metric Spaces with an Application to Integral Equations

Ehsan Lotfali Ghasab* Hamid Majani*

Department of Mathematics Shahid Chamran University of Ahvaz Ahvaz, Iran

Ghasem Soleimani Rad*

Young Researchers and Elite club, West Tehran Branch

Islamic Azad University Tehran, Iran

Received 01.01.2020, received in revised form 22.09.2020, accepted 20.11.2020 Abstract. In this paper, we prove some new fixed point theorems involving set-valued F-contractions in the setting of quasi-ordered metric spaces. Our results are significant since we present Banach contraction principle in a different manner from that which is known in the present literature. Some examples and an application to existence of solution of Volterra-type integral equation are given to support the obtained results.

Keywords: fixed point, sequentially complete metric spaces, F-contraction, ordered-close operator. Citation: E.L. Ghasab, H.Majani, G.S.Rad, Fixed Points of Set-valued F-contraction Operators in Quasi-ordered Metric Spaces with an Application to Integral Equations, J. Sib. Fed. Univ. Math. Phys., 2021, 14(2), 150-158. DOI: 10.17516/1997-1397-2021-14-2-150-158.

1. Introduction and preliminaries

It is well known that the Banach contraction principle is a very useful and classical tool in nonlinear analysis [3]. After that, the generalization of this principle has been a heavily investigated. For example, in 1969, Nadler [10] extended the Banach contraction principle for set-valued mapping as follows:

Theorem 1.1. Let (X,d) be a complete metric space and T : X ^ CB(X) be a set-valued operator. Also, let H : N(X)2 ^ [0, +œ] be the Hausdorff metric on N(X) which defined by

H(A, B) = max sup D(a, B), sup D(b, A) i , laeA beB J

where D(a, B) = D(B,a) = inB d(a,b). Assume that there exists a G [0,1) such that

H(Tx, Ty) ^ ad(x, y) for all x,y G X. Then T has a fixed point in X.

*e-lotfali@stu.scu.ac.ir https://orcid.org/0000-0002-8418-9351

1 Correspondent: h.majani@scu.ac.ir; majani.hamid@gmail.com https://orcid.org/0000-0001-7022-6513 ^gha.soleimani.sci@iauctb.ac.ir; gh.soleimani2008@gmail.com https://orcid.org/0000-0002-0758-2758 © Siberian Federal University. All rights reserved

Then Ciric [6] extended Nadler's result as follows:

Theorem 1.2. Let (X,d) be a complete metric space and T : X ^ CB(X) be a set-valued operator. Assume that there exists a G [0,1) such that H(Tx,Ty) ^ aM(x,y) for all x,y G X, where

M(x, y) = max Id(x, y),D(x, Tx), D(y, Ty), ±[D(x, Ty) + D(y, Tx)]

Then T has a fixed point in X.

In 2011, Amini-Harandi [2] considered some fixed point theorem for set-valued quasi-contraction mappings in metric spaces.

Theorem 1.3 ([2]). Let (X,d) be a complete metric space and T : X ^ CB(X) be a k-set-valued quasi-contraction with k G [0,1) ; that is,

H (Tx, Ty) < k max {d(x, y), D(x, Tx), D(y, Ty), D(x, Ty), D(y, Tx)}

for all x,y G X. Then T has a fixed point in X.

On the other hands, Ran and Reurings [12], and Nieto and Rodriguez-Lopez [11] studied the Banach contraction principle distinctly from another point of view. They imposed a partial order to the metric space (X, d) and discussed on the existence and uniqueness of fixed points for contractive conditions and for the comparable elements of X (also, see [1,4,6-8,13,15]). Moreover, in 2012, Wardowski [14] obtained a new fixed point theorem concerning ^-contraction for single-valued mapping.

Theorem 1.4 ( [14]). Let (X, d) be a complete metric space and T : X ^ X be an F-contraction. Then T has an unique fixed point x* G X and for every x0 G X a sequence {T™x0}„eN is convergent to x*.

In this paper, we obtain several fixed point results for set-valued F-contraction mappings in quasi-ordered metric spaces. Also, we prepare some examples and an application to the existence of a solution for Volterra-type integral equation. Throughout this paper, the family of all nonempty closed and bounded subsets of X is denoted by CB(X ), and the family of all nonempty subsets of X by N(X).

Definition 1.1 ( [9]). Let (X, d) be a metric space with a quasi-order " ^ " (pre-order or pseudoorder; that is, a reflexive and transitive relation). We say that X is sequentially complete if every Cauchy sequence whose consecutive terms are comparable in X converges.

Definition 1.2 ( [9]). Let (X, d) be a metric space with a quasi-order " ^ ". For two subsets A, B of X, we say that A Ç B if each a G A and each b G B imply that a ^ b.

Definition 1.3 ( [9]). Let (X,d) be a metric space with a quasi-order " ^ ".

(i) A subset D c X is said to be approximative, if the set-valued mapping PD (x) = {p G D : d(x, D) = d(p, x)} for all x G X has nonempty value.

(ii) The set-valued mapping G : X —> N(X) is said to be have approximative values (for short, AV), if Gx is approximative for each x G X.

(iii) The set-valued mapping G : X —> N(X) is said to be have comparable approximative values (for short, CAV), if Gx has approximative values for each x G X and for each z G X, there exists y G PGz(x) such that y is comparable to z.

(iv) The set-valued mapping G : X —> N(X) is said to be have upper comparable approximative values (for short, UCAV), if Gx has approximative values and for each z £ X, there exists y £ PGz (x) such that y ^ z.

(v) The set-valued mapping G : X —> N(X) is said to be have lower comparable approximative values (for short, LCAV), if Gx has approximative values and for each z £ X, there exists y £ PGz (x) such that y < z.

Definition 1.4 ([9]). The set-valued mapping G is said to has a fixed point if there exists x £ X such that x £ Gx.

2. Main result

From the idea of Wardowski [14], we consider a new type of F-contraction for set-valued operator in quasi-ordered metric spaces as follows.

Definition 2.1. Let H : N(X)2 ^ [0, be the Hausdorff metric on N(X) and F : R+ —> R be a mapping satisfying the following conditions:

(F1) F is increasing, i.e., for all a,b £ R+ such that a < b, then F(a) < F(b);

(F2) for each sequence [an}nFN of positive numbers lim an =0 if and only if lim F(an) = —to;

(F3) there exists k £ (0,1) such that lim akF(a) = 0.

A mapping G : X —> CB(X) is said to be an F-contraction if there exists t > 0 such that

H(Gx, Gy) > 0=^ t + F(H(Gx, Gy)) < F(d(x, y)) (1)

for all x, y £ X.

Example 2.1. If F(a)= lna + a for all a> 0 and H : N(X)2 ^ [0, is the Hausdorff metric on N(X), then F satisfies (F1)-(F3) and each mapping G : X —> CB(X) is an F-contraction such that H(Gx, Gy)eH(Gx'Gy)-d(x'y) < e-Td(x,y) for all x,y £ X.

Example 2.2. If F(a)= ln a for all a > 0 and H : N(X)2 ^ [0, is the Hausdorff metric on N(X), then F satisfies (F1)-(F3) and each mapping G : X —> CB(X) is an F-contraction such that H(Gx, Gy) ^ e-Td(x, y) for all x,y £ X.

Definition 2.2. Ordered-close operator is set-valued operator G : X ^ CB(X) if for two monotone sequences {xn} , {yn} C X and x0, y0 £ X; xn ^ x0, yn ^ y0 and yn £ G(xn) imply y0 £ G(x0).

Theorem 2.1. Let (X,d, <) be a sequentially complete metric space. Also, let the mapping G : X —>• CB(X) be an ordered-close set-valued F-contraction and has UCAV. Then G has a fixed point x* £ X.

Proof. Let x0 £ X. If x0 £ Gx0, then our proof is complete. Otherwise, since G has UCAV, there exists xi £ Gx0 with x0 = xi and x0 ^ xi such that d(x0,x{) = inf d(x0,x) = D(x0,Gx0).

xEGXq

Continue this procedure, we obtain a non-decreasing sequence {xn}, where

xn £ Gxn-

i with

xn-i ^ xn and xn-i = xn such that d(xn,xn+i) = inf d(xn,x) = D(xn, Gxn). On the other hand,

D(xn,Gxn) ^ sup D(x,Gxn) ^ H(Gxn,Gxn-i).

xEGxn-i

Therefore, d(xn,xn+1) ^ H(Gxn,Gxn-1). From (F1), we have F(d(xn,xn+1)) ^ F(H(Gxn,Gxn-1)). In addition, G is F-contraction. Thus,

F(d(xn,xn+i)) < F(H(Gxn , gxn- 1)) < F(d(xn,xn-i)) - t

< F(d(xn-2, Xn-i)) - 2t

<

(2)

^ F(d(xo, xi)) — ut. We obtain lim F(d(xn,xn+1)) = —x> that together with (F2) gives

lim d(xn,xn+i) = 0. Denote Yn = d(xn,xn+1). By (F3), there exists k € (0,1) such that

lim YknF(Yn) = 0.

(3)

(4)

By (2), we have

YknF(Yn) - lknF(70) < Ykn(F(70) - nr) - ^F(70) = ^nr < 0

< -k< 1

(5)

for all u € N. Letting u ^ <x in (5), and applying (3) and (4), we obtain lim nYn = 0. Hence,

n^w

there exists n1 € N such that uyk < 1 for each u > u1. Consequently, we have

(6)

for all n > n1. In order to show that {xn} is a Cauchy sequence, let m,n € N with m > n > n1. From the definition of the metric and (6), we obtain

d(Xn,Xm) < 7m-i + 7m-2 +-----+ 7n < V"7i ^y] T^ •

— V»

//-rt-l -Ï-T> "

(7)

w 1

From (7) and the convergence of the series , we conclude that {xn} is Cauchy sequence.

i=n V i

From the completeness of X, there exists x* € X such that lim xn ^ x*. Since G is ordered-

n^w

close operator, {xn} is monotone and xn+1 € G(xn), we deduce x* € G(x*) and x* is a fixed point of G. □

Theorem 2.2. Let (X, d, <) be a sequentially complete metric space. Also, let the mapping G : X ^ CB(X) be an ordered-close set-valued F-contraction and has LCAV. Then G has a fixed point x* € X.

Proof. The proof is similar to Theorem 2.1.

Example 2.3. Consider the sequence {Sn}neN by S1 = 1 and Sn = 1 + 2 + ■ ■ ■ + n

for all n € N. Let X = {Sn : n € N} and d(x,y) = \x — y\ for all x,y € X. Also, we define the relation " ^ " on X by x < y ^ Sp ^ Sq for all x = Sp, y = Sq € X. Then (X, d, ^) is a sequentially complete metric space. Also, let the mapping G : X ^ CB(X) be a ordered-close

□

n(n + 1) 2

set-valued mapping and has LCAV defined by G(S1) = {Si} and G(Sn) = [1, S^^ for all n > 1. Then G is an ^-contraction with F as in Example 2.1 and t = 1. To see this, let us consider the following calculations:

For each m,n G N with m > 2 and n = 1, we have

H(G(Sm),G(S\ )) = max( sup D(a,G(S\)), sup D(b,G(Sm))\ = d(Sm_i,S\)

laiG(Sm) biG(S1) J

and

H(G(Sm),G(Si)) eH(G(Sm),G(S1))_d(Sm,Si) = d(Sm_USl) ed(Sm-1,S1)_d(Sm,S1) d(Sm,S1) d(Sm ,S1)

Sm 1

_eSm- — 1 Sm

m — m — 2 _1

= -- e m <e m <e 1.

m2 + m — 2

Now, for each m,n G N with m > n > 1, we have

H(G(Sm),G(Sn))=maxi sup D(a,G(Sn)), sup D(b,G(Sm))\ = d(Sm_1,Sn_1)

I aeG(Sm) beG(Sn) J

and

H(G(Sm),G(Sn)) eH(G(Sm),G(Sn))_d(Sm,Sn) = d(Sm_1, Sn_1) ed(Sm—1,Sn—1)_d(Sm,Sn) =

d(Sm,Sn) d(Sm,Sn)

= Sm_1 — Sn_1 eSn_Sn — 1+Sm — 1_Sm =

s - s e

= m + n — 1 en_m <en_m <e_1

m + n + 1

Therefore, by Theorem 2.2, S1 is a fixed point of G.

Theorem 2.3. Let (X,d, <) be a sequentially complete metric space. Suppose that the mapping G : X ^ CB(X) is an ordered-close set-valued F-contraction and has AV. If there exists x0 G X such that {x0} Q Gx0, then G has a fixed point x* G X.

Proof. If x0 G Gx0, then the proof is finished. Otherwise, by Definition 1.2, we have x ^ x0 for any x G Gx0. Since G has approximative values, there exists x1 G Gx0 with x1 y x0 and x0 = x1 such that d(x0,x1) = D(x0,Gx0). Continue this procedure, we have a non-decreasing sequence {xn} with xn_1 < xn, where xn G Gxn_1 and xn = xn_1 such that d(xn,xn+1) = = inf d(xn, x) = D(xn, Gxn). The rest of this proof is the same as that of Theorem 2.1. □

Theorem 2.4. Let (X,d, <) be a sequentially complete metric space. Suppose that the mapping G : X ^ CB(X) be an ordered-close set-valued F-contraction and has AV. If there exists x0 G X such that Gx0 Q {x0}, then G has a fixed point x* G X.

Proof. The proof is similar to Theorem 2.2. □

Theorem 2.5. Let (X, d, <) be a sequentially complete metric space. Also, let the mapping G : X —>• CB(X) be an ordered-close set-valued and has UCAV. If we have

F(H(Gx,Gy)) < F(M(x,y)) — t (8)

for all x,y € X, where

M(x,y) = max |d(x,y), D(x, Gx), D(y,Gy), 1[D(x, Gy) + D(y, Gx)] j , then G has a fixed point x* € X.

Proof. Let xo € X. If xo € Gxo, then the proof is complete. Otherwise, Since G has UCAV, there exists x1 € Gxo with xo = x1 and xo ^ x1 such that d(xo,x1) = inf d(xo,x) = D(xo, Gxo).

Gxo

Continue this procedure, we obtain a non-decreasing sequence {xn} with xn-1 ^ xn, where xn € Gxn-1 and xn = xn-1 such that d(xn,xn+1) = inf d(xn,x) = D(xn,Gxn). On the other

hand,

D(xn,Gxn) ^ sup D(x,Gxn) ^ H(Gxn,Gxn-1).

Therefore, d(xn,xn+1) < H(Gxn,Gxn-1). Now, from (F1) and (8) we have

F(d(xn, xn+1)) < F(H(Gxn, Gxn-1)) < F(M(xn, xn-1)) — t for all n € N, where M (xn,xn-1) =

= max |d(xn, xn-1), D(xn, Gxn), D(xn-1, Gxn-1), 1[D(xn, Gxn-1) + D(xn-1,Gxn)]^j . Once more, note that xn+1 € Gxn and D(xn, Gxn) = d(xn, xn+\). Hence, we have

M(xnxn-1) ^ max ^^^ <xn-1,xn) 22[d(xnxn) + d^-n-1,xn+1)} <

< max jd(xn, xn-1), d(xn, xn+1), 2[d(xn-1, xn) + d(xn,xn+1<

< max {d(xn,xn-1), d(xn,xn+1)} .

If max {d(xn, xn-1), d(xn, xn+1)} = d(xn,xn+1), then F(d(xn,xn+1)) < F(d(xn,xn+1)) — t, which contradicts with t > 0. Thus, we have F(d(xn, xn+1)) < F(d(xn, xn-1)) — t. The rest of the proof is in the similar manner given in Theorem 2.1. □

Theorem 2.6. Let (X,d, <) be a sequentially complete metric space. Assume that the mapping G : X —>• CB(X) is an ordered-close set-valued and has LCAV, and F(H(Gx,Gy)) ^ F(M(x, y)) — t for all x,y € X, where

M(x, y) = max |d(x, y), D(x, Gx), D(y, Gy), 1[D(x, Gy) + D(y, Gx)] j .

Then G has a fixed point x* € X.

Proof. Let x0 € X. If x0 € Gx0, then the proof is complete. Otherwise, Since G has LCAV, there exists x1 € Gxo with xo = x1 and x1 ^ xo such that d(xo,x{) = inf d(xo,x) = D(xo, Gxo).

Gxo

Continue this procedure, we obtain a non-increasing sequence {xn} with xn < xn-1, where xn € Gxn-1 and xn = xn-1 such that d(xn,xn+{) = inf d(xn,x) = D(xn,Gxn). The rest of

x£Gxn

this proof is the same as that of Theorem 2.5. □

Theorem 2.7. Let (X,d, <) be a sequentially complete metric space. Assume that the mapping G : X ^ CB(X) is an ordered-close set-valued and has AV, and F(H(Gx, Gy)) ^ F(M(x, y))—T for all x, y G X, where

M(x,y) = max |d(x,y),D(x, Gx), D(y, Gy), 1[D(x,Gy) + D(y,Gx)]

If there exists x0 G X such that {x0} Q Gx0, then G has a fixed point x* G X.

Proof. If x0 G Gx0, then the proof is finished. Otherwise, by Definition 1.2, we have x y x0 for any x G Gx0. Since G has approximative values, there exists x1 G Gx0 with x1 y x0 and x0 = x1 such that d(x0,x1) = D(x0,Gx0). Continue this procedure, we have a non-decreasing sequence {xn} with xn_1 < xn, where xn G Gxn_1 and xn = xn_1 such that d(xn,xn+1) = = inf d(xn, x) = D(xn, Gxn). The rest of this proof is the same as that of Theorem 2.5. □

xiGxn

Theorem 2.8. Let (X,d, <) be a sequentially complete metric space. Assume that the mapping G : X ^ CB(X) is an ordered-close set-valued and has AV, and F(H(Gx, Gy)) ^ F(M(x, y))—T for all x, y G X, where

M(x,y) = max |d(x,y),D(x, Gx), D(y, Gy), 1[D(x,Gy) + D(y,Gx)]

If there exists x0 G X such that Gx0 Q {x0}, then G has a fixed point x* G X.

Proof. If x0 G Gx0, then the proof is finished. Otherwise, by Definition 1.2, we have x0 y x for any x G Gx0. Since G has approximative values, there exists x1 G Gx0 with x0 y x1 and x0 = x1 such that d(x0,x1) = D(x0,Gx0). Continue this procedure, we have a non-increasing sequence {xn} with xn < xn_1, where xn G Gxn_1 and xn = xn_1 such that d(xn,xn+1) = = inf d(xn, x) = D(xn, Gxn). The rest of this proof is the same as that of Theorem 2.5. □

3. Application to integral equation

As an application of our results, we will consider the following Volterra integral equation:

x(t) = [ K(t,s,x(s))ds + g(t), (9)

0

where I = [0,1], K G C(I x I x R, R) and g G C(I, R) for all t G I.

Let C(I, R) be the Banach space of all real continuous functions defined on I with the sup norm \\x\\^ = maxtei |x(t)| for all x G C(I,R) and C(I x I x C(I,R),R) be the space of all continuous functions defined on I x I x C(I, R). Alternatively, the Banach space C(I, R) can be endowed with Bielecki norm \\x\\B = supteJ{\x(t)\e_Tt} for all x G C(I, R) and t > 0, and the

induced metric dB(x,y) = \ \x—y\\B for all x,y G C(I, R) (see [5]). Also, let f : C(I,R) ^ C(I,R)

t

defined by fx(t) = J K(t,s,x(s))ds + g(t) and g G C(I, R). Moreover, we define the relation

0

" ^ " on C(I,R) by x ^ y & \ \x\\^ < \\y\\x for all x,y G C(I,R). Clearly the relation " ^ " is a quasi-order relation.

Theorem 3.1. Let (C(I, R),dB, <) be a sequentially complete metric space. Suppose that G : C(I, R) ^ CB(C(I, R)) is a set-valued operator such that G(x) = {fx(t)} and has UCAV. Let K G C(I x I x R, R) be an operator satisfying the following conditions:

(i) K is continuous; t

(ii) f K(t, s,.) for all t, s € I is increasing;

0

(iii) there exists t > 0 such that \K(t, s, x(s)) — K(t, s, y(s))\ ^ e-T\x(s) — y(s)\ for all x,y € C(I, R) and all t, s € I.

Then, the Volterra-type integral equation (9) has a solution in C(I, R).

Proof. By definition of G, we have H(Gx,Gy) = dB (f (x),f (y)) for all x,y € C(I, R). Thus,

H(Gx,Gy) = ds(f(x),f(y)) = sup< / K(t, s,x(s))ds — K(t, s,y(s))ds

tei I Jo Jo

< sup! i \K(t, s, x(s)) — K (t,s,y(s))\e-TtdsX tei Wo J

< sup < / e-T \x(s) — y(s)\e-Ttds\ tei I Jo J

< \\x — y\\s sup s / e-Tds >

tei I Jo J

= e-T ds (x,y).

Taking logarithms, we have ln(H(Gx,Gy)) < \n(e-TdB(x,y)), which implies that (t+ + ln(H(Gx, Gy))) < ln(dB (x,y)). Now, consider the function F(t) = ln(t) for all t € C(I, R) and t > 0. Then, all conditions of Theorem 2.1 are satisfied. Consequently, Theorem 2.1 ensures the existence of fixed point of G that this fixed point is the solution of the integral equation. □

We are grateful to the Research Council of Shahid Chamran University of Ahvaz for financial support (Grant number: SCU.MM99.25894).

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Неподвижные точки многозначных операторов F-сжатия в квазиупорядоченных метрических пространствах с приложением к интегральным уравнениям

Эхсан Л. Гасаб Хамид Маджани

Университет Шахида Чамрана в Ахвазе

Ахваз, Иран

Гасем С. Рад

Исламский университет Азад Тегеран, Иран

Аннотация. В этой статье мы докажем некоторые новые теоремы о неподвижных точках, включающие многозначные F-сжатия в условиях квазиупорядоченных метрических пространств. Наши результаты важны, поскольку мы представляем принцип банахового сжатия иначе, чем тот, который известен в настоящей литературе. Для подтверждения полученных результатов приведены некоторые примеры и приложение к существованию решения интегрального уравнения типа Воль-терра.

Ключевые слова: неподвижная точка, F-сжатие, секвенциально полные метрические пространства, оператор упорядоченного замыкания.