FIXED POINT THEOREM FOR MULTIVALUED NON-SELF MAPPINGS SATISFYING JS-CONTRACTION WITH AN APPLICATION Текст научной статьи по специальности «Математика»

URAL MATHEMATICAL JOURNAL, Vol. 8, No. 1, 2022, pp. 3-12

DOI: 10.15826/umj.2022.1.001

FIXED POINT THEOREM FOR MULTIVALUED NON-SELF MAPPINGS SATISFYING JS-CONTRACTION

WITH AN APPLICATION

David Aront, Santosh Kumar^

Department of Mathematics, College of Natural and Applied Sciences, University of Dar es Salaam, Tanzania

t [email protected], [email protected]

Abstract: In this paper, we present some fixed point results for multivalued non-self mappings. We generalize the fixed point theorem due to Altun and Minak [2] by using Jleli and Sameti [9] ^-contraction. To validate the results proved here, we provide an appropriate application of our main result.

Keywords: JS-contraction mapping, Multivalued mapping, Metric space, Non-self mapping, Fixed point.

1. Introduction and preliminaries

In 1922, in Banach's PhD thesis a remarkable fixed point theorem well known as the Banach contraction principal was initiated. It's simplicity, usefulness and application made it a supreme tool in finding the existence and uniqueness of solution in numerous branches of mathematical analysis and applied sciences. Following the Banach contraction principal, some authors, Nadler [13], Assad and Kirk [4], Itoh [8] and several others have extended and generalized this theorem in several ways. In fact, Nadler [13] introduced the concept of using Hausdorff metric on multi-valued contraction of self mappings in the study of fixed points. Assad and Kirk [4] proved the Banach contraction mapping theorem for multi-valued contraction of non-self mappings and Itoh [8] generalized the theorems due to Assad and Kirk, and many other researchers have made significant contributions in this area (see [3, 7, 11]). In 2013, Alghamdi et al. [1] proved fixed point results for multivalued nonself almost contractions on convex metric spaces. Recently, Altun and Minak [2] introduced a new approach to Assad and Kirk fixed point theorem and a new real generalization of it, by using $— contractiveness of a multivalued mapping. Jleli and Samet [9] introduced $— contraction and established a new fixed point theorem for such mappings in the setting of generalized metric spaces. Following the notion $, Hussain et al. [6] supposed that O is the set of all functions $ : [0, to) ^ [1, to) satisfying the following conditions:

(^1) $ is nondecreasing and $(t) = 1 if and only if t = 0;

($2) for each sequence {tn} C (0, to), lim $(tn) = 1 if and only if the limit of lim tn = 0;

n—TO n—^^o

$(t) — 1

($3) there exists r € (0,1) and I € (0, 00] such that lim -= I;

t—0+ tr

($4) $(a + b) < $(a)$(b) for all a, b > 0.

Throughout this paper we shall denote by O the set of all functions $ satisfying ($1) — ($4).

Next, we present some definitions and preliminaries that are required to prove the main result of this paper.

Since we are dealing with multivalued mapping it is important to state a brief description of the Hausdorff metric. The Hausdorff metric measures the distance between subsets of a metric space. One among many interesting properties of this metric space, which will be our focus in this paper is that the Hausdorff induced metric space is complete if our original metric space is complete. Now, we define the Hausdorff metric as follows:

Definition 1. [5] Let (M, g) be a metric space. Denote by CB(M) the collection of non-empty closed bounded subsets of M. For A,B € CB(M) and u € M, define

p(u, A) = inf g(u, a)

aeA

and

H(A, B) = max { supp(a, B), supp(b, A)}.

aeA beB

It is seen that H is a metric on CB(M). H is called the Hausdorff metric induced by g. The completion of (M,g) implies that (CB(M),H) is a complete metric space.

Before proceeding further, it is important to know that we will need an extra condition, call it ($5), a very useful part of our tool to help us to prove our main results in multivalued mapping and O satisfies ($5).

($5) $(inf A) = inf $(A) for all A c (0, to) with inf A > 0.

The following definition is important for future work in this paper.

Definition 2. [10].

(i) A sequence {un} in a metric space (M, g) is said to converge or to be convergent if there is u € M such that

lim g(un, u) = 0.

n—y^o

(ii) A sequence {un} in a metric space (M, g) is said to be Cauchy sequence if for every e > 0 there is a number N = N(e) such that

g(un, um) < e

for every m,n> N.

(iii) A metric space (M, g) is said to be complete if every Cauchy sequence in M converges to an element of M.

The following is the description of a metrically convex metric space and some of its properties are stated.

The following definition is due to Assad and Kirk [4].

Definition 3. [4] A metric space (M,g) is said to be metrically convex if for any u,v € M with u = v, there exists a point z € M,(u = z = v) such that

g(u, v) = g(u, z) + g(z, v).

The following result is taken from Assad [4] where dK denotes the boundary of K.

Lemma 1. [4] If K is a closed subset of the complete and convex metric space M and if u € K, v / K, then there exists a point z € dK, such that

g(u, v) = g(u, z) + g(z, v), where dK denotes the boundary of K.

Assad and Kirk [4] proved the following fixed point theorem.

Theorem 1. [4] Let (M,g) be a complete and metrically convex metric space, K be a nonempty closed subset of M, T : K — CB(M) be a mapping such that, for all u,v € K,

p(Tu, Tv) < kg(u,v), for some k € (0,1). If Tu C K for each u € dK, then T has a fixed point in K.

In 2014, Jleli and Sameti [9] gave a new generalization of Banach contraction mapping theorem in the setting of Banciari metric spaces as follows:

Theorem 2. [9] Let (M, g) be a complete generalized metric space and T : M — M be a mapping. Suppose that there exist $ € O and k € (0,1) such that for u,v € M,

g(Tu, Tv) = 0 $(g(Tu, Tv)) < [$(g(u,v))]k. Then T has a unique fixed point.

Recently, Altun and Minak [2] obtained a new approach to Theorem 2 and a new generalization of it, by using $-contraction as follows:

Theorem 3. [2] Let (M, g) be a complete and metrically convex metric space, K be a nonempty closed subset of M, T : K — CB(M) be a mapping such that for all u,v € K with H(Tu, Tv) > 0,

$(H(Tu,Tv)) < [$(g(u, v))]k for some k € (0,1) and $ € O. If Tu C K for each u € dK, then T has a fixed point in K.

Suppose we want to use a different $-contraction in the above theorem. Hussain et al. [6] introduced a new concept that we can apply in the proof of the above theorem and obtain a new result.

Definition 4. [6] Let (M, g) be a metric space and let T : M — M be a mapping. T is said to be a JS-contraction whenever there is a function $ € O and positive real numbers ri,r2,t3,t4 with 0 < ti + t2 + t3 + 2t4 < 1 such that

$(g(Tu, Tv)) < [$(g(u,v))]T1 [$(g(u, Tu))]T2 [$(g(v, Tv))]T3 [$(g(u,Tv) + g(v,Tu))]T4 for all u,v € M.

2. JS-contraction fixed point theorem

We give now a definition of a generalized multivalued JS-contraction mapping.

Definition 5. Let (M, q) be a metric space and K be a nonempty closed subset of M. Let T be a mapping of K into CB(M). Then T is said to be a generalized JS-contraction mapping whenever there is a function $ € O and nonnegative real numbers t1,t2,t3, t4 with

0 < T1 + T2 + T3 + 2t4 < 1

such that

$(H(Tu, Tv)) < [$(q(u, v))]T1 [$(q(u, Tu))]T2 [$(q(v, Tv))]T3 [$(q(u, Tv) + q(v, Tu))]T4. (2.1) for all u,v € K.

We now present an extended version of Theorem 3.

Theorem 4. Let (M, q) be a complete and metrically convex metric space, K be a nonempty closed subset of M. Let T : K — CB(M) be a generalized multivalued JS-contraction mapping. If for any u € dK, Tu C K and

(1 + n + t2 + t4)(ti + t2 + r4)

(1-T3-T4)2 < '

then there is z € K such that z € T(z).

Proof. We construct two sequences {un} and {vn} in K in the following way: let u0 € K and v1 € Tu0. If v1 € K, let u1 = v1. If v1 / K, then from Lemma 1, there exists u1 € dK such that

q(u0,u1) + Q(u1,v1) = Q(u0,v1).

Thus, u1 € K. Now, we claim that q(v1 ,Tu1) > 0. Suppose q(v1,Tu1) = 0. If v1 € K, then u1 is a fixed point of T, which is a contradiction. If v1 / K, then u1 € dK and so Tu1 C K. Therefore, v1 / Tu1, which is a contradiction. Thus, q(v1 ,Tu1) > 0. Now, since q(v1 ,Tu1) < H(Tu0,Tu1), then we have

$(q(v1 ,Tu1)) < $(H(Tu0,Tu1))

< [$(Q(u0,u1))]T1 [$(Q(u0,Tu0))]T2 [$(Q(u1,Tu1))]T3 (2.2)

x [$(q(u0,Tu1) + q(u1 ,Tu0))]t4.

On the other hand, from $5 we get

$(q(v1,Tu1)) = $(inf {q(v1, m) : m € Tu1}) = inf {$(q(v1 ,m)) : m € Tu1}

and so from condition (2.2) we get

inf{$(Q(v1 ,m)) : m € Tu1} < [$(q(uq,u1))]T1 [$(q(«c,Tu0))]T2[$(q(u1,Tu1))]T3

x [$(Q(u0,Tu1) + Q(u1,Tu0))]T4.

Thus, there exists v2 € Tu1 such that

$(Q(v1, v2))<[$(Q(u0, u1))]Y1 [$(Q(u0,Tu0))]Y2 [$(q(u1 , Tu1))]Y3 [$(Q(u0, Tu1)+Q(u1, Tu0))]Y4 ,

where

0 < Ti + T2 + T3 + 2t4 <Yi + 72 + Y3 + 274 < 1. If v2 € K let u2 = v2. If v2 / K, select a point u2 € dK such that

g(ui,u2) + g(u2,v2) = g(ui,v2).

Thus, u2 € K. We can show that g(v2,Tu2) > 0. As above, we can find a point v3 € Tu2 such that

$(g(v2, v3))<[$(g(ui, u2))]71 [$(g(ui, Tui))]Y2 [$(g(u2, Tu2))]73 [$(g(ui, Tu2)+g(u2, Tui ))]Y4.

Continuing the arguments, two sequences {un} and {vn} are obtained such that for n € N we have

(i) vn+i € Tun,

(ii) $(g(vn,vn+i)) < [$(g(un-i,un))]Y1 [$(g(un-i, T^-i))]72 [$(g(un,Tun))]Y3

x [$(g(un-i,Tun) + g(un,Tun-i))]Y4,

where vn+i = un+i if vn+i € K or

g(un ,«n+i) + g(un+i,vn+i) = g(un ,vn+i) (2.3)

if vn+i / K and un+i € dK. Now, we consider sets

P = {u? € {un} : u? = v?, £ € N}, Q = {u? € {un} : u? = v?, £ € N}.

Observe that if u^ € Q for some £, then u^+i € P. Here, the intention is to estimate the distance g(un,un+i) for n > 2. Note that g(un,un+i) > 0, otherwise, T has a fixed point. For this, three cases have to be considered: Case 1. If un € P and un+i € P, then, we get

$(g(un ,un+i)) = $(g(vn ,vn+i))

< [$(g(un-i, un))]71 [$(g(un-i, Tun-i))]72 [$(g(un, Tun))]73 x [$(g(un-i,Tun) + g(un, Tun-i))]74

= [$(g(un-i, un))]71 [$(g(un-i, un))]72 [$(g(un, un+i))]73

x [$(g(un-i,un+i) + g(un,Mn))]74 = [$(g(un-i,un))]Y1 +72 [$(g(un,un+i))]Y3 [$(g(un-i,un) + g(un,un+i))]74

< [$(g(un-i,un))]71 +72[$(g(un,un+i))]73[$(g(un-i,un)]74[g(un,un+i))]74 = [$(g(un-i,un))]Y1 +72 +74 [$(g(un,un+i))]Y3 +74.

It follows that

71+72+74

ti{ß{un,un+i)) < [ti{g{un-i,un))] 1-^3-74 .

Case 2. If un € P and un+i € Q, then, from condition (2.3), we get

71+72+74

ü{g{un,un+i))<'d{ß{un, un+i)+Q{un+i, vn+i))='ß{g{vn, vn+i))<[d{ß{un-i, un))] .

Case 3. If un € Q and un+i € P, then, since

$(Q(vn, vn+1)) < [$(Q(un-1, un))]Y1 [$(Q(un-1, Tun-O)]72 [$(q(«*, Tun))]73 X [$(Q(un-1 ,Tun) + Q(un,Tun-1))]Y4 ,

Q(vn, vn+1)<(Q(un-1 ,un))(Q(un-1,Tun-1))(Q(un, Tun))(Q(un-1, Tun) + Q(un,Tun-1)). In our case, if we simplify we get

$(Q(vn, vn+1)) < [$(Q(un-1, un))]Y1 [$(Q(un-1, vn))]Y2 [$(Q(un, un+1))]73 x [$(Q(un-1,un+1) + Q(un,vn))]Y4

< [$(Q(un-1,un) + Q(un,vn))]Y1 [$(Q(un-1, vn))]Y2

X [$(Q(un,un+1))P [$(Q(un-1 ,un) + Q(un,un+1) + Q(un, vn))]Y4

< [$(Q(un-1, vn))]Y1 [$(Q(un-1, vn))]72 [$(Q(un, un+1))]73 X [$(Q(un-1,vn) + Q(un,un+1))]Y4

< [$(Q(un-1, vn))]Y1 +Y2 [$(Q(un, un+1))]Y3 X [$(Q(un-1, vn)]74 [$(Q(un, un+1 ))]Y4

= [$(Q(un-1,vn ))]Y1 +Y2+Y4 [$(Q(un ,un+1))]Y3 +Y4.

It follows that

$(Q(un,un+1)) < $(Q(un,vn) + Q(vn,un+1))

< $(Q(un-1 ,un) + Q(un,vn) + Q(vn, vn+1)) = $(Q(un-1 ,vn) + Q(vn,vn+1))

< $(Q(un-1 ,vn))$(Q(vn, vn+1))

< $(Q(un-1 ,vn))[$(Q(un-1 ,vn))]Y1 +Y2 +Y4 [$(Q(un ,un+0)]73 +74 = [$(Q(un-1,vn))]1+Y1 +Y2 +Y4 [$(Q(un,un+1))]Y3 +Y4 .

Hence

[$(Q(un,un+1))]1-Y3-Y4 < [$(Q(un-1,vn))]1+Y1 +Y2 +Y4 by Case 2, since un € Q implies un-1 € P we have

71+72+74

i9{g(un-i,vn)) < [{}(g(un-2,un-i))} 1-T3-74 .

Therefore,

(1+71+72+74) (71+72+74)

-&(6(un,un+i)) < [{}(g(un-2,un-i))} a-73-74)2 .

The case that un € Q and un+1 € Q does not occur. Since

7i + 72 + 74 < (1 + 7i + 72 + 74)(71 + 72 + 74) 1 - 73 - 74 (1 - 73 - 74)2

for n > 2 we have

$(Q(un,un+1)) < ( [$(Q(un-1,un))]Y,

[[$(Q(un-2,un-1))]7.

Now we claim that

0{Q{un,un+ i))<<^V) (2.4)

for all n € N, where

5 = max{$(Q(u0, u1)), $(q(u1 ,u2))}.

Using (2.4) we obtain

lim $(Q(un, un+1)) = 1.

n—TO

From ($2), lim Q(un,un+1) = 0 and so from ($3) there exists r € (0,1) and l € (0, to] such that

n—TO

lim &{Q{Un,Un+1)) - 1 _ l

M 00 [Q{un,un+i)]r

Suppose that l < to. In this case, let ^ = l/2 > 0. Recall from the definition of the limit, there exists n0 € N such that, for all n > n0,

't)(g(un,un+1)) - 1

<

> l - =

[Q(un,un+1)]r

This implies that, for all n > n0,

- 1

[Q(un,un+1)]r Then, for all n > n0,

n[Q(un,un+1)]r < #n[$(Q(un,un+1)) — 1], where # = 1/$. Thus, in all cases, there exist # > 0 and n0 € N such that, for all n > n0,

n[Q(un,un+1)]r < #n[$(Q(un,un+1)) — 1]. Using (2.4), we obtain, for all n > n0,

n[g(un,un+i)]r < - 1].

Letting n — to in the above inequality, we obtain

lim n[Q(un,un+1)]r = 0.

n—TO

Thus, there exists n1 € N such that n[Q(un, un+1)]r < 1 for all n > n1. So, we have, for all n > n0

Q(Un,Un+i) < -TT-- (2.5)

n1/r

In order to demonstrate that {xn} is a Cauchy sequence consider m,n € N such that m > n > n1. Now, applying the metric triangle inequality and from condition (2.5), we have

Q(un,um) <Q(un,un+1) + Q(un+1,un+2) +-----+ Q(um-1,um)

m— 1 to to 1

= Y1 iK'^+i) - Y1 ^ Y1 Tifr-

§=n §=n §=n

Since the series X^li converges, then passing to limit with n,m —> 00, we get g(un,um) —>• 0.

It is an obvious implication that the sequence {un} is a Cauchy sequence in K. Since K is closed, the sequence {un} converges to some point z € K. By our choice of {un}, there exists a subsequence {unk} of {un} such that {unk} € P that is, {unk} = {vnk}, k € N. Note that

{unk} € {unk-i} for k € N and {unk} — z as k — to. Also note that from condition (2.1) and (^i) we get

H(Tu,Tv) < (g(u,v))(g(u,Tu))(g(v,Tv))(g(v,Tu) + g(u,Tv)) for all u, v K and so, we have

g(unfc,Tz) <H(Tunk-i,Tz)

< g(unk-i,z)g(unk-i,Tunk-i)g(z,Tz){g(z,Tunk-i) + g(v,nk-i,Tz)}

which on letting k — to implies that g(z, Tz) = 0, which is a contradiction. Therefore, T has a fixed point z € K. □

Remark 1. If t2 = t3 = t4 = 0 and t1 = t in Theorem 4 we obtain Theorem 3 of Altun [2].

For particular function § selections, some significant results are obtained. First, by setting = e^1 in Theorem 4, the following corollary is obtained:

Corollary 1. Let K be a nonempty closed subset of a complete and metrically convex metric space M. Let T : K — CB(M) be a mapping such that the following condition holds:

y/H(Tu, Tv) < Tiy/g(u,v) + t2 v/g(u,Tu) + r3\Jg{v,Tv) + t4\/g(u, Tv) + g(v,Tu)

for all u,v € K, § € O and t1,t2,t3,t4 > 0 with 0 < t1 + t2 + t3 + 2t4 < 1. Then T has a unique fixed point.

And, by putting = e^ in Theorem 4, the following corollary is obtained:

Corollary 2. Let K be a nonempty closed subset of a complete and metrically convex metric space M. Let T : K — CB(M) be a mapping such that the following condition holds:

VH(Tu, Tv) < n y/Q(u,v) + r2 Tv) + r3 Tv) + r4 Tv) + g(v, Tv)

for all u,v € K, § € O and t1,t2,t3,t4 > 0 with 0 < t1 + t2 + t3 + 2t4 < 1. Then T has a unique fixed point.

3. Application to nonlinear integral equations

Nonlinear integral equations can be solved using a variety of numerical approaches. The integral equation is usually transformed into a system of nonlinear algebraic equations. Solving these systems is difficult, or the solution may be impossible to find. Therefore, in this section, we describe how the fixed point approach may be used to solve Volterra-Hammerstein integral equations. This approach does not result in a system of nonlinear algebraic equations. Now, consider the nonlinear integral equation below:

u(t) = g(t)+ [ k(t,T)H(t,u(t))dT, (3.1)

J a

where t,T € [a,b], a,b € R, u € C[a,b], g : [a,b] — R, H € C[a,b] x R — R and k € C2[a,b]2 such that k(t, t) > 0 are given functions.

Maleknejad [12] established some conditions which ensure the uniqueness of the solution and how the fixed point method approximates this solution.

Referring from Maleknejad [12] we are going to establish the following fixed point theorem.

Theorem 5. Let M = C[a, b] be a metric space endowed with the metric

q(u, v) = sup |u(t) — v(t)|.

i€[a,6]

Define the mapping T : K — CB (M) by

T(u)(t) = g(t)+ / k(t,T)H(t,u(t))dr. Ja

Let u, v € K and t € [a, b]. Assume that g € C[a, b], k € C2[a, b]2, i.e. there exists a constant

M > 0 where

fc2(i,r)dr < M< oo,

and H : [a, b] x R — R is continuous and there is $ € O so that $(sup f (t)) = sup $(f (t)) for arbitrary function f with

$( j |h(t,u(t)) — h(t,v(t))|^ < j $(|h(t,u(t)) — h(t,v(t))|)^t,

there is t € (0,1) where i = 1,2,3,4 such that

0(|H(t,u(t)) -H(t,v(t))|) < j [tf(|u(t) - v(t)|)]T1 [0(|u(i) - £k(i,r)H(r,u(r))dr|)]

r b rb

- k(t,T)H(t,v(t))dT|)] T3 [0(|u(t) - k(t,T)H(t,v(t))dT| + |v(t)

./a ■/a

- £ k(t, T)H(t, u(t))dT|)] T4} IM(b - a).

Then equation (3.1) has a unique solution.

Proof. We begin our proof by deriving the following relation where Cauchy-Schwartz inequality is used:

|Tu(t)-Tv(t)| =

a cb

Now, we have

k(t,T)(H(r,u(r)) -H(t,v(t)))dr

< / |k(i,r)||H(t,u(t))-H(r,v(r)))dr|

a

/ /• b \ 1/2 / ,b \ 1/2 <i / fc2(i,r)dr) i / H(t,u(t)) -H(t,v(t)))drj

/ /• b \ 1/2

< Mi j H(t,u(t)) -H(t,v(t)))drj

< M / |H(t,u(t))-H(r,v(r)))|dr.

a

0(|Tu(t) - Tv(t)|) = tf^M j |H(t, u(t)) -H(r,v(r))|dr

/• b

< m |H(t, u(t))-H(t,v(t))|)dT

a

b

b

< {M [tf(|u(t) - v(t)|)]Tl [tf(|u(t) - a k(t,T)H(r,u(r))dT|)]T2

r b n b

[0(|v(t) - k(t,T)H(t,v(t))dTT3 [0(|u(t) - k(t,T)H(t,v(t))dT|

J a J a

+ |v(t) - I k(t, T)H(t, u(T))dT|)] M(b - a) 1 fb

[b - a a ] [ ] [ ] [ ]

= [#(g(u, v))] T1 [#(g(u, Tu))]T2 [#(g(v, Tv))] T3 [^(g(u, Tv) + g(v, Tu))]T4.

Thus, all the conditions of Theorem 4 are satisfied. Hence the integral equation (3.1) has a solution. □

4. Conclusion

The main contribution of this study is Definition 5 and Theorem 4. This theorem is proved for multivalued non-self mappings in complete and metrically convex space. This theorem generalizes the fixed point theorem due to Altun and Minak [2] by using ^-contraction due to Jleli and Sameti [9]. To validate the results proved here, we provide an appropriate application of our main result.

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