Set-valued mappings: fixed point results with $\beta$-function and some applications Текст научной статьи по специальности «Математика»

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta

2024. Volume 63. Pp. 61-79

MSC2020: 47H09, 47H10 © Y. Touail

SET-VALUED MAPPINGS: FIXED POINT RESULTS WITH ^-FUNCTION AND SOME APPLICATIONS

Via the so-called ^-function, some fixed-point results for nonexpansive set-valued mappings are obtained. In this study, the results are considered in the context of complete metric spaces which are neither uniformly convex nor compact. Our theorems extend, unify, and improve several recent results in the existing literature. In the end, we apply our new results to ensure the existence of a solution for a nonlinear integral inclusion. Moreover, we approximate the fixed point by a faster iterative process.

Keywords: fixed point, set-valued mappings, Tp-multivalued contractive mapping, Tp-weakly contractive multivalued mapping, integral inclusion, iterative process.

DOI: 10.35634/2226-3594-2024-63-05 Introduction

In fixed point theory for single-valued mappings, Banach contraction principle (for short, BCP) [4] (1922) is perhaps the most celebrated and useful tool in all of analysis, in particular in nonlinear analysis. It has been extended in different directions and many fixed point theorems.

In 1962, Edelstein mentioned in [9], to obtain a fixed point of strict contractive mappings on a metric space (X, d) (d(Tx,Ty) < d(x,y) for all x = y G X), it is necessary to add the compactness assumption of the space.

In 1965, Browder [6] and Gohde [12] independently showed one of the most interesting extensions of BCP by proving that every nonexpansive mapping whose Lipschitz's constant equal to 1 (that is ||Tx — Ty|| < ||x — y|| for all x, y G X) of a closed convex and bounded subset of the Banach space X has a fixed point, if the subset is supposed to be uniformly convex (for each 0 < e < 2, there exists 5 > 0 such that for all ||x|| < 1, ||y|| < 1 the condition ||x — y|| > e implies that < 1 - 5 see [8]).

For multivalued mappings (T: X ^ 2X), fixed point theory plays a major role in various fields of pure and applied mathematics because of its many applications, for instance, game theory, real and complex analysis, optimal control problems as well as integral inclusions etc.

Nadler (1969) [20] was the first mathematician who combined the concept of contraction (see condition below) with the notion of multivalued mappings. In other words, the author in [20] proved the existence of a fixed point u G X (i.e., u G Tu) on a metric space (X,d) for a multivalued mapping T satisfying the following condition

H(Tx,Ty) < kd(x,y),

for all x, y G X where H is the Hausdorff metric induced by the metric d.

In 2012, Samet et al. [26] introduced the concept of a-admissible functions and a-^-contrac-tive mappings and established various fixed point theorems for such mappings in the setting of complete metric spaces. Asl et al. [3] extended the notion of a-admissibility to a*-admissibility for multivalued mappings and showed that a*-admissible function is also a-admissible, but the converse may not be true as shown in [3]. For more detail see [14,16,25].

On the other hand, very recently, in 2020 the authors in [29] introduced a new class of strict contraction for multivalued mappings as follows

inf {d(x,y) — H(Tx,Ty)} > 0 (0.1)

x=y€X

and proved some fixed point results without the compactness assumption of the space. In this direction, recent works can be found in [28,30-35].

Note that the fixed point theory of nonexpansive multivalued mappings is more difficult but more important than the corresponding theory of single-valued mappings and the strict class of multivalued mappings. So, it is a very natural question to ask: if we can extend (0.1) to

inf {d(x,y) - H(Tx,Ty)}> 0

x=y€X

and prove, in this area, a fixed point theorem for a new class of nonexpansive multivalued mappings. For further information on this topic, interested readers are directed to the latest papers [2,13,36].

In this paper, via the concept of «»-admissible functions [3] and the notion of symmetric spaces discussed in [11] we give an affirmative answer to the above-asked question. In other words, we introduce the concept of Tp-contractive multivalued mappings and prove a fixed point theorem for this type of contractions which is a class of nonexpansive multivalued mappings without using neither the compactness nor the uniform convexity of the space X. Also, some examples are presented to show the importance of the proven results.

Furthermore, motivated by the notion of T-weakly contractive multivalued mappings in [29] (see also, weakly contractive maps defined in [1,7]), we employ our first result to prove a fixed point theorem for the newly called Tp-weakly contractive multivalued mappings.

Moreover, as an application of our studies, we prove the existence of a solution for Volterra-type integral inclusion

x(t) e f(t)+ / K(t,s,x(s)) ds, t e [0, t], (0.2)

J 0

where K: [0,t] x [0,t] x R ^ Pcv(R), where Pcv(R) denotes the class of nonempty compact and convex subsets of R. We point out that the study of (0.2) is under new and weak conditions.

After existence theorems, it is natural to find an iteration scheme to approximate the fixed point. In this paper, we propose the following scheme:

x1 = x e X,

(1 - Pn) + PnU n yn (1 an)xn + an vni

,Xn+1 = Wn, n e N,

where sequences {an}, {^n} are real sequences in (0,1) such that un e PT(xn), vn e PT(zn), wn e PT(yn) and we denote by PT(x) = {y e Tx: ||x — y|| = d(x, Tx)} for a multivalued mapping T: X ^ C(X), where C(X) denotes the family of all closed subsets of the space (X, d).

Our scheme converges faster compared to the existing ones in the literature, which is always preferred in the practice.

§ 1. Preliminaries

The main purpose of this section is to introduce some concepts and results required in this article. We begin with the following definition.

Definition 1.1 (see [37]). Let X be a nonempty set. A symmetric on X is a nonnegative real valued function D on X x X such that

(i) D(x, y) = 0 if and only if x = y,

(ii) D(x,y) = D(y,x).

A symmetric space is a pair (X, D) where D is a symmetric on X.

<

Definition 1.2 (see [11, Definition 2.1.1]). Let (X, D) be a symmetric space and A be a nonempty subset of X.

(i) A is called D-closed if and only if AD = A, where

AD = {x G X: D(x, A) = 0} and D(x,A) = inf{D(x,y): y G A}.

(ii) A is called D-bounded if and only if SD (A) < to, where

SD(A) = sup{D(x,y): x,y G A}.

Definition 1.3 (see [11, Definition 2.1.2]). Let (X, D) be a D-bounded symmetric space and let CD (X) be the class of all nonempty D-closed sets of (X, D). Consider the function

HD: 2X x 2X ^ R+ defined by

Hd (A,B) = max {sup D(a, B), sup D(b, A)},

aeA b&B

for all A, B G CD(X). HD is called the symmetric Hausdorff distance induced by D.

Remark 1.1. (CD (X ),HD) is a symmetric space [11, Remark 2.1.1]. Recall that:

• A sequence in a symmetric space (X, D) is called a D-Cauchy sequence if it satisfies the usual metric condition lim D(xn,xm) = 0.

• A symmetric space (X, D) is called S-complete if for every D-Cauchy sequence {xn}, there exists x G X such that lim D(x, xn) = 0.

• A symmetric space (X, D) satisfies the axiom (W.4) given by Wilson [37]: if {xn} C X, {yn} C X and x G X are such that lim D(xn,x) = 0 and lim D(xn,yn) = 0, then

lim D(yn, x) = 0. We state the following result.

Theorem 1.1 (see [11, Theorem 2.2.1]). Let (X, D) be D-bounded and S-complete symmetric space satisfying (W.4) and T: X ^ CD(X) be a multivalued mapping such that

HD(Tx,Ty) < kD(x,y), k G [0,1), Vx,y G X.

Then there exists u G X such that u G Tu.

Definition 1.4 (see [19]). If 0: X ^ 2Y, then a selection for 0 is a continuous mapping f: X ^ Y such that f (x) G 0(x), where X and Y are two topological spaces.

Definition 1.5 (see [19]). A mapping T: X ^ 2Y is called lower semicontinuous provided that, whenever x G X and V is an open set such that Tx n V = 0, there exists an open neighborhood U of x such that for all y G U, we have Ty n V = 0.

Theorem 1.2 (Michael's selection theorem [19]). Let X be a paracompact space and Y be a Banach space. Let F: X ^ 2Y be a lower semicontinuous multivalued map with nonempty convex closed values. Then there exists a continuous selection f: X ^ Y of F.

Lemma 1.1 (see [27, Lemma 2.2]). Let (X, d) be a metric space and A, B G CB(X), where CB(X) denotes the family of all bounded and closed subsets of (X, d). If there exists Y > 0 such that

(i) for each a G A, there is b G B such that d(a, b) < y,

(ii) for each b G B, there is a G A such that d(b, a) < 7, then H(A, B) < 7.

In 2012, Asl et al. [3] introduced the notion of a*-—-contractive multifunction type and established some fixed point theorems for these mappings. Let us denote by ^ the family of nondecreasing functions —: [0, to) ^ [0, to) such that —n(t) < to for all t > 0, where —n is the nth iterate of —.

Definition 1.6 (see [3]). Let (X, d) be a metric space, T: X ^ 2X be a closed-valued multifunction, — G ^ and a: X xX ^ [0, to) be a function. We say that T is an a*-—-contractive multifunction whenever a*(x, y)H(Tx, Ty) < —(d(x, y)) for x, y G X, where H is the Hausdorff metric and a*(A,B) = inf{a(a,b): a G A, b G B}. Also, we say that T is a*-admissible whenever a(x,y) > 1 implies a*(Tx,Ty) > 1.

Theorem 1.3 (see [3]). Let (X, d) be a complete metric space, a: X x X ^ R+ be a function, — G ^ be a strictly increasing map and T be a closed-valued, a*-admissible and a*-—-contractive multivalued mapping on X. Suppose that there exist x0 G X and xi G Txo such that a(x0,xi) > 1. Assume that if {xn} is a sequence in X such that a(xn, xn+i) > 1 for all n and {xn} converges to x, then a(xn, x) > 1 for all n G N. Then T has a fixed point.

At the end of this section, we recall the proven results in [29].

Lemma 1.2 (see [29, Lemma 2]). Let (X, d) be a bounded metric space.

(i) Let D(x, y) = ed(x'y) — 1 for all x, y G X, then (X, D) is a D-bounded symmetric space.

(ii) Let HD (A, B) = eH (A'B) — 1 for all A, B G CD (X), then the function HD is a symmetric Hausdorff distance.

Theorem 1.4 (see [29, Theorem 3]). Let (X, d) be a bounded complete metric space and let T be a multivalued mapping from X into C (X). Suppose that

inf {d(x,y) — H(Tx,Ty)} > 0.

x=yex

Then T has a fixed point.

Definition 1.7 (see [29, Definition 5]). Let (X, d) be a metric space and T: X ^ C (X) be a multivalued mapping. T will be called a T-weakly contractive multivalued mapping if for

all x, y G X

H(Tx,Ty) < d(x,y) — 0[1 + d(x,y)],

where 0: [1, to) ^ [0, to) is a function satisfying:

(i) 0(1) = 0;

(ii) inf 0(t) > 0. t>i

Theorem 1.5 (see [29, Theorem 4]). Let T: X ^ C(X) be a T-weakly contractive multivalued mapping of a bounded complete metric space (X, d). Then there exists u G X such that u G Tu.

§ 2. Main results

In this section, we prove two auxiliary lemmas that we will need for the proof of our main results.

L e m m a 2.1. Let (X, d) be a bounded complete metric space. Then (X, D) is a D-bounded S-complete symmetric space, where D = ed — 1.

Proof. Let (X, d) be a complete metric space and {xn} C X a D-Cauchy sequence. Then lim D(xn, xm) = 0, and hence lim d(xn, xm) = 0. Since X is complete, there exists u G X

such that lim d(u,xn) = 0. Finally, we deduce that lim D(u,xn) = 0. □

n^-x

Lemma 2.2. Zet (X, D) be a D-bounded and S-complete symmetric space such that (W.4) is satisfied and let T: X ^ C(X) be an a^-admissible mapping satisfying

a(x,y)HD(Tx,Ty) < kD(x,y),

for all x,y G X and for a constant k G [0,1). Assume that there exist x0 and xi G Tx0 such that a(x0,xi) > 1. Suppose that if {xn} C X is a sequence convergent to x G X such that a(xn, xn+i) > 1 for all n G N, then a(xn, x) > 1 for all n G N. Then T has a fixed point; that is, u G Tu.

Proof. We suppose that D(x0, xi) > 0 and xi / Txi, otherwise, the proof is finished. Let

a G (k, 1), so

0 < D(xi,Txi) < a(x0,xi)HD(Tx0,Txi) < kD(x0,xi) < aD(x0,xi).

It follows that there exists x2 G Txi such that

D(xi,x2) < aD(x0,xi).

Since a*(Tx0,Txi) > 1, xi G Tx0 and x2 G Txi, we achieve a(xi,x2) > 1. If x2 G Tx2, the proof is completed. Assume that x2 / Tx2, hence,

0 < D(x2,Tx2) < a(xi;x2)HD(Txi,Tx2) < kD(xi;x2) < aD(xi,x2),

which implies that there exists x3 G Tx2 such that

D(x2,x3) < aD(xi;x2).

Repeating this process, we get a sequence {xn} C X satisfying: xn G Txn-i, a(xn-i,xn) > 1 and

D(x„,xn+i) < anD(xo,xi), for all n G N. Let n, m G N, we obtain

D(x„ ,x„+m) < anD(xo ,xm) < an^D (X),

where (X) = sup{D(x,y): x,y G X}. Then {xn} is a D-Cauchy sequence which implies that there exists u G X such that lim D(u, xn) = 0.

Since a(xn, u) > 1, so there exists {yn} C Tu such that

D(xn+i,Tu) < a(xn,u)HD(Txn, Tu) < kD(xn,u),

for all n G N. Hence lim D(xn+1,Tu) = 0. Then there exists {yn} C Tu such that lim D(xn+1,yn) = 0. Using (W.4), we obtain lim D(u,yn) = 0, which implies that

n^-x n^-x

u G Tm = Tu. □

Now, we introduce the notion of Tp-contractive multivalued mapping.

Definition 2.1. Let T be a multivalued mapping of a bounded metric space (X, d). T is said to be Tp-contractive multivalued mapping if

We are ready to state and prove our main results.

Theorem 2.1. Let (X, d) be a bounded complete metric space and T: X ^ C (X) be a Tp -contractive multivalued mapping such that:

(i) there exist x0 £ X and x1 £ Tx0 such that ß(x0, £i) < 0;

(ii) if {xn} C X is a sequence convergent to x £ X such that ß(xn, xn+1) < 0 for all n £ N, then ß (xn, x) <0 for all n £ N;

(iii) ß(a, b) < inf I d(x, y) - H(Tx, Ty) + ß(x, y) i for all a, b £ X.

[ J

Then T has a fixed point u £ X.

Proof. T is a Tp-contractive mapping, then there exists a function ß: X x X ^ R such that

where ß: X x X ^ R is a function satisfying

ß(x,y) < 0 ß*(Tx, Ty) < 0,

with

ß*(Tx, Ty) = sup{ß(a, b): a £ Tx, b £ Ty}.

We put

a(x,y)eH(Tx'Ty) - e-Y < k(ed(x'y) - 1). On the other hand, it follows from (iii) that —a(x, y) = — e-ß(x'y) < — e-Y. Then

a(x,y)HD(Tx,Ty) < kD(x,y),

for all x, y £ X, with D(x, y) = ed(x'y) — 1 and HD defined in Lemma 1.2.

Finally, we deduce from Lemma 2.1 and Lemma 2.2 that T has a fixed point u £ X. □

Example 2.1. Let X = {0,1}2 be endowed with the metric d((xi,yi), (x2,y2)) = = ||(x1 ,y1) — (x2,y2)|^ = |x1 — x2| + |y1 — y2|. We note that (X, d) is not a uniform convex space, indeed:

For e = 1, x = (1,0) and y = (0,1):

INK = bill = llx — y\\i = 2 > 1 = £ and \\\x + y||i = 1 > 1 — 5 for each 5 > 0. Define the following multivalued mapping T by

T(0, 0) = T(1, 0) = T(0,1) = {(0, 0)}, T(1,1) = {(1, 0), (0,1), (0, 0)},

and a function 0: X x X ^ R by

0((0, 0), (0, 0)) = 0((0, 0), (1, 0)) = 0((1, 0), (0, 0)) = 0, 0((0, 0), (0,1)) = 0((0,1), (0, 0)) = 0, 0((1, 0), (1, 0)) = 0((0,1), (0,1)) = 0, 0((1,1), (0, 0)) = 0((0, 0), (1,1)) = 0((0,1), (1,1)) = 1/3, 0((1,1), (1, 0)) = 0((1, 0), (1,1)) = 0((1,1), (1, 0)) = 1/3,

and

0((1,1), (1,1)) = 1/4.

So, we have, for all x, y e X,

inf d(x, y) — H (Tx, Ty) + 0 (x, y) > 1/3.

[ J

Then T satisfies all conditions of Theorem 2.1 and has the fixed point (0,0).

Remark 2.1. Theorem 2.1 is a real and an important extension of Theorem 1.4. Indeed, the multivalued mapping defined in the above example is nonexpansive (i. e., H(Tx, Ty) < d(x, y)), therefore our result ensures the existence of a fixed point for a class of nonexpansive multivalued contractions without adding the uniform convexity of the space.

Example 2.2. Let X = B(0,1) x [0,1], where B(0,1) is the unit closed ball of a real Banach space of infinite dimension. Define the following metric on X by

W ' M J1 + |y — y'L ifx = <

d( (x, y), (x , y ) ) = < '

[|y — y L if x = x •

Define a multivalued mapping T by T(x,y) = B(0,1) x {1 — y} for all (x,y) e X. Define a function 0: X x X ^ R by

Off w ' i1/2, if (x,y) = (x',y'),

0((x,y), (x ,y )) = <

if (x,y) = (x ,y ).

So, we have

0((x,y), (x',y')) < 0 0*(T(x,y),T(x',y')) < 0,

for all (x,y), (x',y') e X. Then T satisfies all assumptions of Theorem 2.1 and T has the fixed point which is equal to (0,1/2). Moreover, we observe that X is not compact and

inf d((x,y), (x',y')) — H(T(x,y),T(x',y')) + 0((x,y), (x',y')) > 1/2 > 0.

(x,y)=(x',y')€X [ J

Then T is a multivalued Tp-contractive mapping such that

0 (a,b) < inf \ 4 (x,y^ (x',y S — H(T (x,y),T (x',y S + 0 (foy^ (x',y))

for all a, b e X.

Remark 2.2. Since for all x,y g X we have d(x,y) — H(Tx,Ty) = 0, Theorem 1.4 does not ensure the existence of the fixed point. Also, we note that X is not compact.

In the following, p: X x X ^ R is the function defined (on a metric space X) in the Definition 2.1 and Theorem 2.1. In order to state the second result, we first give the following definition.

Definition 2.2. $ is the class of all functions 0: [1, +to) ^ R satisfying:

(i) inf 0(t) > 0; t>i

(ii) 0(1) < P(x,x) for all x G X.

D e f i n i t i o n 2.3. Let T: X ^ C(X) be a multivalued mapping of a metric space (X, d). T will be called a Tp-weakly contractive multivalued mapping if there exists 0 G $ such that

H(Tx,Ty) — P(x,y) < d(x,y) — 0(1 + d(x,y)),

for all x, y G X.

Theorem 2.2. Let T: X ^ C(X) be a Tp-weakly contractive multivalued mapping of a bounded complete metric space (X, d). Then T has a fixed point.

Proof. Let x = y G X, which implies by Definition 2.3 that

0 < inf 0(t) < 0(1 + d(x,y)) < d(x,y) — H(Tx,Ty) + p(x,y).

Then

inf {d(x, y) — H(Tx, Ty) + p(x, y)} > 0. By Theorem 2.1, T has a fixed point in X. □

Example 2.3. Let X = {0,1, 2} with the usual metric d(x, y) = |x — y| for all x, y G X. Define a multivalued mapping T by

T0 = {0} and T1 = {0,1} = T2,

a function P : X x X ^ R by

p (x,y)

and a function 0: [1, to) ^ R by

0(t) =

We have 0(1) = —1 < P(x, x) for all x G X. Further

1/2, if x = y, 3/2, if x = y,

— 1, if t = 1, 1, if t> 1.

P(a,b) < 3/2 < inf < d(x, y) — H(Tx,Ty) + p(x,y)

for all a, b G X.

Hence, we have the following cases:

Casel: H(T0,T 1) — p(0,1) = —1/2 < 0 = d(0,1) — 0(1 + d(0,1)). Case2: H(t0,T2) — p(0,2) = —1/2 < 1 = d(0,2) — 0(1 + d(0,2)). Case3: H(t 1,T2) — p(1,2) = —3/2 < 0 = d(1,2) — 0(1 + d(1,2)). Then, T satisfies all assumptions in Theorem 2.2 and 0 G T0.

§3. Applications

§ 3.1. Integral inclusion

Different fixed point theorems have been used in several contexts to ensure the existence of a solution for integral inclusions (see [10,19,23,24,27,29]). Throughout this section we assume that X = C([0, t], R) is the space of all continuous functions from [0, t] (t > 0) into R and Pcv (R) is the class of nonempty compact and convex subsets of R. If X is equipped with d(x, y) = sup |x(t) — y(t)|, then (X, d) is a complete metric space. In this section, we first

te[0,T ]

consider the following Volterra-type inclusion

x(t) e /(t)+ / K(t,s,x(s)) ds, t e [0, t], (3.1)

0

where / e X and K: [0, t] x [0, t] x R ^ Pcv(R). We suppose that the multi-valued mapping Kx(t, s) := K(t, s, x(s)), (t, s) e [0, t]2 is lower semicontinuous for each x e X.

Now, our main purpose is to weaken some conditions of Theorem 5 [29], so we define a multivalued operator T by

Tx(t) = jv e X: v(t) e /(t) + j0 K(t, s, x(s)) ds, t e [0, t]

for all x e X.

According to Michael's selection Theorem [19], for all x e X there exists a continuous operator : [0,t] x [0,t] ^ R (selection) such that kx(t,s) e Kx(t,s) for any t, s e [0,t] and hence /(t) + J0i kx(t, s) ds e Tx(t) which leads to Tx = 0. Moreover, Tx is a closed set (see [27,29]).

We now have all the tools needed to prove the next result.

Theorem 3.1. Suppose that there exist M > 0 and a function n: X x X ^ R such that for all s, t e [0, t ], n e N, and x = y e X the following hypotheses hold: (i)1

rj(x,y)> 0 ==> H(K(t,s,x(s)),K(t,s,y(s))) < ~[\x(s) - y(s)\- M],

T ! (3.2)

V(x,y)< 0 H(K(t,s,x(s)),K(t,s,y(s))) < -\x{s)-y{s)\-,

t

(ii) n(x,y) > 0 n*(Tx,Ty) > 0, with n*(Tx,Ty) = inf{^(a, b): a e Tx, b e Ty};

(iii) there exist x0 e X and xi e Tx0 such that n(x0, xi) > 0;

(iv) if {xn} C X is a sequence convergent to x e X with n(xn, xn+i) > 0, then n(xn, x) > 0. Then the integral inclusion (3.1) has a solution.

Proof. Let x,y e X be such that a e Tx. Hence, there exists kx(t,s) e Kx(t, s) for t, s e [0,t] with a(t) = /(t) + Jq kx(t,s) ds. So, the condition (3.2) implies that there exists

b(t, s) e Ky (t, s) such that

r?(x,y)>0 Ifc^Ct, s) — 6(t, s)| < -[|x(s)-y(s)|-M],

t

v(xi ?/) < 0 |fcx(i,s)-6(i,s)| < -y(s)|,

t

for all t,s e [0, t].

In this step, let us consider the set-valued map S defined as follows:

if n(x, y) > 0

S(t, s) = Ky(t, s) Pi {w G E: \kx(t, s) - w\ < -[|x(t, s) - y(t, s)| - M]},

T

if y) < 0

S(t, s) = s) Pi {w G E: |fcx(t, s) - < -\x(t, s) - y(t, s)|},

T

for all t,s e [0, t].

Since S is a lower semicontinuous multivalued mapping, then, in view of Michael's selection theorem, there exists a continuous mapping : [0,t] x [0,t] ^ R such that (t, s) e S(t, s), for all t, s e [0, t]. Therefore, we get

c(t) = f (t)+ [ ky(t,s) ds e f (t)+ [ K(t, s, y(s)) ds, t e [0,t],

and for any t G [0, t], we have

|a(t) - c(t)|

kx(t, s) ds — / ky(t, s) ds

•'x I ""y\

10 Jo

Then, we have:

if n(x, y) > 0

|a(t) — c(t)| < / |kx(t, s) — ky(t,s)| ds < d(x,y) — M,

if n(x, y) < 0 which implies that:

|a(t) — c(t)| < |kx(t, s) — ky(t,s)| ds < d(x,y),

0

if n(x, y) > 0 if n(x, y) < 0

d(a, c) < d(x, y) — M, d(a, c) < d(x, y),

for all x = y G X.

Now, define 0 : X x X ^ R by

0(x,y)

0, if n(x, y) > 0,

1, otherwise.

It is straightforward to check the implication

0(x,y) < 0 0*(x,y) < 0. Also, by interchanging the role of x and y and by using Lemma 1.1, we conclude that

inf {d(x, y) — H(Tx,Ty) + 0(x,y)} > 0,

and hence T is a Tp-contractive multivalued mapping. Moreover, it is easy to see that the conditions (i), (ii) and (iii) of Theorem 2.1 are satisfied. This theorem ensures the existence of a solution for the Volterra-type integral inclusion (3.1). □

Remark 3.1. If we take n(x,y) = 0 in Theorem 3.1 we obtain the statement of Theorem 5 proved in [29] (that is: if there exists M > 0 such that for all s, t g [0, t] and x = y g X we have H(K(t, s, x(s)),K(t, s, y(s))) < — y(s)| — M], then the integral inclusion (3.1) has a solution).

Therefore, our result is a real enhancement of this theorem.

0

0

t

t

t

0

t

§ 3.2. Approximation of the fixed point via a new and faster iterative process

In 2000, Noor [21] introduced the following iterative process for a mapping T: X — X by

'ti = t e X,

(1 — )tn + a„Ty„

m+i =

yn = (1 — Pn)t n + PnTzn,

zn (1 — Yn)tn + Yn Ttn,

(3.3)

n e N,

where {an}, {Pn} and {Yn} are real sequences in (0,1).

In 2019, Okeke [22] gave the so-called Picard-Krasnoselskii iterative process as follows

'xi = x e X,

xn+i Tvn,

vn (1 an)xn + an P^n

„ Un = (1 — Pn)xn + PnTxn, n e N,

(3.4)

where {an}, {Pn} are real sequences in (0,1). The author in [22] showed that this iterative scheme converges faster than the scheme introduced by Noor [21] and other known schemes in the literature [15,17,18].

In the following, we give some needed definitions for our approach.

Definition 3.1 (see [5]). Let {an} and {bn} be two sequences of real numbers converging to a and b respectively. If

v K ~ a\ n lim li.-iJ = °>

n^x |bn — b|

then {an} converges faster than {bn}.

Definition 3.2 (see [5]). Suppose that for two fixed point iterative processes {un} and {vn}, both converging to the same fixed point p, the error estimates

||un — p|| < an, ||vn — p|| < bn for all n e N,

are satisfied where {an} and {bn} are two sequences of positive numbers converging to zero. If {an} converges faster than {bn}, then {un} converges faster than {vn} to p.

Throughout this section, let (X, || ■ ||) be a bounded Banach space. For x e X, we denote by PT(x) = {y e Tx: ||x — y|| = d(x,Tx)} a multivalued mapping T: X — C(X). Now, we introduce the multivalued version of the scheme (3.4) as follows

xi = x e X,

zn (1 — Pn) + Pn U n

yn (1 an )xn + an vn

(3.5)

xn+i = Wn, n e N,

where {an}, {Pn} are real sequences in (0,1) such that

Un e Pt(xn), vn e Pt(zn) and Wn e Pt(yn)-Also, we can give the multivalued version of the Noor iterative process (3.3) as follows:

ti = t e X,

tn+i (1 an)tn + anan> yn (1 Pn )tn + Pnbn, zn (1 Yn)tn + Yncn, n e N,

(3.6)

where {an}, {0«} and {y«} are real sequences in (0,1) such that

a« G Pt(y«), b« G Pt(z«) and c« G Pt(¿«). Now, we are able to state the following result

Theorem 3.2. Let X be a real Banach space such that T: X ^ C(X) with F(T) := := {x G X: x G Tx} = 0. We suppose that PT is a T-weakly contractive multivalued mapping (see Theorem 1.5). Let {xn} be a sequence generated by the iteration scheme (3.5) such that a«0« = ro. Then {xn} converges to u G F(T).

Proof. Since F(T) = 0, let u e F(T). From (3.5) it follows that

||xn+1 - u|| = ||w« - u|| < H(Pt(yn), Pt(u)) < ||yn - u|| - 0(1 + ||yn - u||) < ||yn - u||. (3.7)

From the assumptions that PT is a T-weakly contractive multivalued mapping and (3.5), we obtain

||yn - u|| = ||(1 - «n)xn + anVn - u||

< (1 - an)||x« - u || + a«||v« - u||

< (1 - an)||x« - u|| + anH(Pt(z«),Pt(u)) (3.8)

< (1 - an)||x« - u|| + a«||z« - u|| - a«0(1 + ||z« - u||)

< (1 - an)||x« - u || + an||zn - u ||.

We have also

||Zn - u| = ||(1 - 0«)x« + 0«u« - u|

< (1 - 0«)||x« - u| + 0« ||u« - u||

< (1 - 0«)||x« - u|| + 0«H(Pt(x«),Pt(u)) (3.9)

< (1 - 0n)||x« - u|| + 0« ||x« - u|| - 0«0(1 + ||x« - u||)

< | xn - u| .

By combining with (3.8) and (3.9), we get

||y« - u|| < (1 - a«)||x« - u|| + a«||x« - u|| < ||x« - u||. (3.10)

From (3.7) and (3.10), we have

||x«+i - u|| < ||x« - u||. (3.11)

Therefore, {dn = ||xn - u|| + 1} converges to d > 1. Now, ^om (3.7), (3.10) and (3.11), we get

d«+i < d« - 0(d«), (3.12)

where d« = 1 + ||y« - u||.

In the same manner, we can show that {d«} converges to d' > 1. If d' = 1, so, by (3.7), {xn} converges to u.

If d' > 1, the inequality (3.12) implies that 0(d«) < dn - d«+1. Since 0 < a«0n < 1, we have

a.

a„inf < Y^ ara0„0(dn) < Y^(dn - dn+1) < di - d.

lnyn 1J"1"L YK^n/ neN

n=1 n=1 n=1

<x

This leads to a contradiction with the fact that an0n = œ. Therefore d' =1, and hence d =1

n=1

which implies that {xn} converges to u. □

In the following, we take 0(1 +1) = 1 +1 and we prove that our new scheme converges faster than the multivalued version of the Noor iterative process in the sense of Definition 3.2.

Theorem 3.3. The new introduced scheme (3.5) converges faster than the multivalued version of the Noor iterative process (3.6).

Proof. From (3.5) and the proof of Theorem 3.2, we obtain

||xn+1 -u|| < ||xn-u||-a„P„0(1 + ||xn-u||) < (1-«nPn)||x„-u|| < ... < (1-a„P„)n||xi-u||, for all n e N. Let

A„ = (1 - a„p„)n|xi - u||. (3.13)

On the other hand, from (3.6) it follows that

(3.14)

+ вп||&п - u||

+ в„Я (Pr (z„),PT (u)) (3.15)

+ вП 11 U 11

||tn+i - u|| < (1 - a

Also by (3.6), we have

— u|| + an||an — u|

< (1 - an)||tn - u|| + a„H(Pr(yra),Pr(u))

< (1 - a„)||i„ - u|| + a„||y„ - u||.

- u|| < (1 - вп) - u|

< (1 - вп)||*п - u|

< (1 - вп)

- u|

п

п

and

||zn - u|| < ||(1 - Yn)|tn - u|| + Yn|cn - u||

< (1 - Yn)|tn - u|| + YnH(Pt(tn),PT(u)) (3.16)

< ||tn - u|| - Yn|tn - u||.

By combining with (3.14), (3.15) and (3.16), we get

||tn+i - u|| < (1 - anenYn)||tn - u|| < ... < (1 - anenYn)n||ti - u||, for all n e N. Let

Bn =(1 - anenYn)n||ti - u||. (3.17)

Since схп/Зп^п < OLn, we conclude from (3.13) and (3.17) that lim = 0. □

Bn

Remark 3.2. In this application, the iterative process is more general than Noor's iterative process (see [21]) because of the following two reasons:

(1) the convergence is obtained only in a Banach space; compared to Noor iterative process, the convergence is proved over a Hilbert space; in other words, we didn't employ the inner product in any step;

(2) we get the convergence in the setting of multivalued mappings, while Noor iterative process is achieved only for single-valued mappings.

REFERENCES

1. Alber Ya. I., Guerre-Delabriere S. Principle of weakly contractive maps in Hilbert spaces, New results in operator theory and its applications, Basel: Birkhauser, 1997, pp. 7-22. https://doi.org/10.1007/978-3-0348-8910-0_2

2. Ali F., Ali J., Nieto J. J. Some observations on generalized non-expansive mappings with an application, Computational and Applied Mathematics, 2020, vol. 39, no. 2, article number: 74. https://doi.org/10.1007/s40314-020-1101-4

3. Asl J. H., Rezapour S., Shahzad N. On fixed points of a-^-contractive multifonctions, Fixed Point Theory and Applications, 2012, vol. 2012, no. 1, article number: 212. https://doi.org/10.1186/1687-1812-2012-212

4. Banach S. Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fundamenta Mathematicae, 1922, vol. 3, issue 1, pp. 133-181. https://eudml.org/doc/213289

5. Berinde V. Iterative approximation of fixed points, Berlin-Heidelberg: Springer, 2007. https://doi.org/10.1007/978-3-540-72234-2

6. Browder F. E. Nonexpansive nonlinear operators in a Banach space, Proceedings of the National Academy of Sciences of the United States of America, 1965, vol. 54, no. 4, pp. 1041-1044.

https ://www .jstor. org/stable/73047

7. Cho Seonghoon. Fixed point theorems for generalized weakly contractive mappings in metric spaces with applications, Fixed Point Theory and Applications, 2018, vol. 2018, no. 1, article number: 3. https://doi.org/10.1186/s 13663-018-0628-1

8. Clarkson J. A. Uniformly convex spaces, Transactions of the American Mathematical Society, 1936, vol. 40, no. 3, pp. 396-414. https://doi.org/10.2307/1989630

9. Edelstein M. On fixed and periodic points under contractive mappings, Journal of the London Mathematical Society, 1962, vol. s1-37, issue 1, pp. 74-79. https://doi.org/10.1112/jlms/s1-37.1.74

10. El-Sayed A.M. A., Al-Issa Sh.M. On a set-valued functional integral equation of Volterra-Stieltjes type, Journal ofMathematics and Computer Science, 2020, vol. 21, issue 4, pp. 273-285. https://doi.org/10.22436/jmcs.021.04.01

11. El Moutawakil D. A fixed point theorem for multivalued maps in symmetric spaces, Applied Mathematics E-Notes, 2004, vol. 4, pp. 26-32. https://www.emis.de/journals/AMEN/2004/030314-1.pdf

12. Gohde D. Zum Prinzip der kontraktiven Abbildung, Mathematische Nachrichten, 1965, vol. 30, issues 3-4, pp. 251-258. https://doi.org/10.1002/mana.19650300312

13. Hajisharifi H. R. On some generalization of multivalued nonexpansive mappings, Bulletin of the Iranian Mathematical Society, 2020, vol. 46, no. 2, pp. 557-571. https://doi.org/10.1007/s41980-019-00275-7

14. Hussain N., Parvaneh V., Hoseini Ghoncheh S. J. Generalized contractive mappings and weakly a-admissible pairs in G-metric spaces, The Scientific World Journal, 2014, vol. 2014, article ID: 941086. https://doi.org/10.1155/2014/941086

15. Ishikawa S. Fixed points by a new iteration method, Proceedings of the American Mathematical Society, 1974, vol. 44, no. 1, pp. 147-150. https://doi.org/10.1090/S0002-9939-1974-0336469-5

16. Karapinar E., Kumam P., Salimi P. On a-^-Meir-Keeler contractive mappings, Fixed Point Theory and Applications, 2013, vol. 2013, no. 1, article number: 94. https://doi.org/10.1186/1687-1812-2013-94

17. Krasnosel'skii M. A. Two remarks on the method of successive approximations, Uspekhi Matematich-eskikh Nauk, 1955, vol. 10, issue 1 (63), pp. 123-127. https://www.mathnet.ru/eng/rm7954

18. Mann W. R. Mean value methods in iteration, Proceedings of the American Mathematical Society, 1953, vol. 4, no. 3, pp. 506-510. https://doi.org/10.1090/S0002-9939-1953-0054846-3

19. Michael E. Continuous selection. I, Annals of Mathematics, 1956, vol. 63, no. 2, pp. 361-382. https://doi.org/10.2307/1969615

20. Nadler S. B. Multi-valued contraction mappings, Pacific Journal ofMathematics, 1969, vol. 30, no. 2, pp. 475-488. https://doi.org/10.2140/pjm.1969.30.475

21. Noor M. A. New approximation schemes for general variational inequalities, Journal of Mathematical Analysis and Applications, 2000, vol. 251, issue 1, pp. 217-229. https://doi.org/10.1006/jmaa.2000.7042

22. Okeke G. A. Convergence analysis of the Picard-Ishikawa hybrid iterative process with applications, Afrika Matematika, 2019, vol. 30, no. 5-6, pp. 817-835. https://doi.org/10.1007/s13370-019-00686-z

23. Pietkun R. Structure of the solution set to Volterra integral inclusions and applications, Journal of Mathematical Analysis and Applications, 2013, vol. 403, issue 2, pp. 643-666. https://doi.org/10.1016/jjmaa.2013.02.059

24. Sadygov M. A. An extreme problem for a Volterra type integral inclusion, Open Access Library Journal, 2019, vol. 6, no. 8, pp. 1-8. https://doi.org/10.4236/oalib.1105605

25. Salimi P., Latif A., Hussain N. Modified a-^-contractive mappings with applications, Fixed Point Theory and Applications, 2013, vol. 2013, no. 1, article number: 151. https://doi.org/10.1186/1687-1812-2013-151

26. Samet B., Vetro C., Vetro P. Fixed point theorems for a-^-contractive type mappings, Nonlinear Analysis: Theory, Methods and Applications, 2012, vol. 75, issue 4, pp. 2154-2165. https://doi.org/10.1016/j.na.2011.10.014

27. Sîntamarian A. Integral inclusions of Fredlhom type relative to multivalued ^-contractions, Seminar on Fixed Point Theory Cluj-Napoca, 2002, vol. 3, pp. 361-368. https://zbmath.org/1028.47043

28. Touail Y., El Moutawakil D., Bennani S. Fixed point theorems for contractive selfmappings of a bounded metric space, Journal of Function Spaces, 2019, vol. 2019, article ID: 4175807. https://doi.org/10.1155/2019/4175807

29. Touail Y., El Moutawakil D. Fixed point results for new type of multivalued mappings in bounded metric spaces with an application, Ricerche di Matematica, 2020, vol. 71, no. 2, pp. 315-323. https://doi.org/10.1007/s11587-020-00498-5

30. Touail Y., El Moutawakil D. New common fixed point theorems for contractive self mappings and an application to nonlinear differential equations, International Journal of Nonlinear Analysis and Applications, 2021, vol. 12, issue 1, pp. 903-911. https://doi.org/10.22075/IJNAA.2021.21318.2245

31. Touail Y., El Moutawakil D. Fixed point theorems for new contractions with application in dynamic programming, Vestnik of Saint Petersburg University. Mathematics, 2021, vol. 54, no. 2, pp. 206-212. https://doi.org/10.1134/S1063454121020126

32. Touail Y., El Moutawakil D. Some new common fixed point theorems for contractive selfmappings with applications, Asian-European Journal of Mathematics, 2021, vol. 15, no. 4, article number: 2250080. https://doi.org/10.1142/S1793557122500802

33. Touail Y., El Moutawakil D. Fixed point theorems on orthogonal complete metric spaces with an application, International Journal of Nonlinear Analysis and Applications, 2021, vol. 12, issue 2, pp. 1801-1809. https://doi.org/10.22075/IJNAA.2021.23033.2464

34. Touail Y., Jaid A., El Moutawakil D. New contribution in fixed point theory via an auxiliary function with an application, Ricerche di Matematica, 2021, vol. 72, no. 1, pp. 181-191. https://doi.org/10.1007/s11587-021-00645-6

35. Touail Y. On multivalued -contractions on generalized orthogonal sets with an application to integral inclusions, Problemy Analiza —Issues of Analysis, 2022, vol. 11 (29), issue 3, pp. 109-124. https://www.mathnet.ru/eng/pa363

36. Ullah K., Khan M. S.U., Muhammad N., Ahmad J. Approximation of endpoints for multivalued nonexpansive mappings in geodesic spaces, Asian-European Journal of Mathematics, 2020, vol. 13, no. 8, article number: 2050141. https://doi.org/10.1142/S1793557120501417

37. Wilson W. A. On semi-metric spaces, American Journal of Mathematics, 1931, vol. 53, no. 2, pp. 361-373. http://doi.org/10.2307/2370790

Received 21.04.2023

Accepted 23.02.2024

Youssef Touail, Doctor of Mathematics, Professor, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah, Fez, Morocco. ORCID: https://orcid.org/0000-0003-3593-8253 E-mail: [email protected]

Citation: Y. Touail. Set-valued mappings: fixed point results with ß-function and some applications,

Izvestiya Instituta Matematiki i Informatik Udmurtskogo Gosudarstvennogo Universiteta, 2024, vol. 63, pp. 61-79.

Ю. Туаиль

Многозначные отображения: результаты о неподвижной точке с использованием в-функции и некоторые приложения

Ключевые слова: неподвижная точка, многозначные отображения, Тр-многозначное сжимающее отображение, Тр-слабо сжимающее многозначное отображение, интегральное включение, итерационный процесс.

УДК: 517.98

DOI: 10.35634/2226-3594-2024-63-05

С помощью так называемой 0-функции получены некоторые результаты о неподвижной точке для нерасширяющих многозначных отображений. В данной работе результаты рассматриваются в контексте полных метрических пространств, которые не являются ни равномерно выпуклыми, ни компактными. Полученные результаты расширяют, объединяют и улучшают несколько недавних результатов в существующей литературе. В заключение мы применяем наши новые результаты, чтобы обеспечить существование решения для нелинейного интегрального включения. Кроме того, мы аппроксимируем неподвижную точку более быстрым итерационным процессом.

СПИСОК ЛИТЕРАТУРЫ

1. Alber Ya. I., Guerre-Delabriere S. Principle of weakly contractive maps in Hilbert spaces // New results in operator theory and its applications. Basel: Birkhauser, 1997. P. 7-22. https://doi.org/10.1007/978-3-0348-8910-0_2

2. Ali F., Ali J., Nieto J. J. Some observations on generalized non-expansive mappings with an application // Computational and Applied Mathematics. 2020. Vol. 39. No. 2. Article number: 74. https://doi.org/10.1007/s40314-020-1101-4

3. Asl J. H., Rezapour S., Shahzad N. On fixed points of a-^-contractive multifunctions // Fixed Point Theory and Applications. 2012. Vol. 2012. No. 1. Article number: 212. https://doi.org/10.1186/1687-1812-2012-212

4. Banach S. Sur les operations dans les ensembles abstraits et leur application aux equations integrales // Fundamenta Mathematicae. 1922. Vol. 3. Issue 1. P. 133-181. https://eudml.org/doc/213289

5. Berinde V. Iterative approximation of fixed points. Berlin-Heidelberg: Springer, 2007. https://doi.org/10.1007/978-3-540-72234-2

6. Browder F. E. Nonexpansive nonlinear operators in a Banach space // Proceedings of the National Academy of Sciences of the United States of America. 1965. Vol. 54. No. 4. P. 1041-1044.

https ://www .jstor. org/stable/73047

7. Cho Seonghoon. Fixed point theorems for generalized weakly contractive mappings in metric spaces with applications // Fixed Point Theory and Applications. 2018. Vol. 2018. No. 1. Article number: 3. https://doi.org/10.1186/s13663-018-0628-1

8. Clarkson J. A. Uniformly convex spaces // Transactions of the American Mathematical Society. 1936. Vol. 40. No. 3. P. 396-414. https://doi.org/10.2307/1989630

9. Edelstein M. On fixed and periodic points under contractive mappings // Journal of the London Mathematical Society. 1962. Vol. s1-37. Issue 1. P. 74-79. https://doi.org/10.1112/jlms/s1-37.1.74

10. El-Sayed A.M. A., Al-Issa Sh.M. On a set-valued functional integral equation of Volterra-Stieltjes type // Journal of Mathematics and Computer Science. 2020. Vol. 21. Issue 4. P. 273-285. https://doi.org/10.22436/jmcs.021.04.01

11. El Moutawakil D. A fixed point theorem for multivalued maps in symmetric spaces // Applied Mathematics E-Notes. 2004. Vol. 4. P. 26-32. https://www.emis.de/journals/AMEN/2004/030314-1.pdf

12. Gohde D. Zum Prinzip der kontraktiven Abbildung // Mathematische Nachrichten. 1965. Vol. 30. Issues 3-4. P. 251-258. https://doi.org/10.1002/mana.19650300312

13

14.

15.

16.

17.

18.

19.

20.

21

22

23

24

25

26

27

28

29

30

31

32

Hajisharifi H. R. On some generalization of multivalued nonexpansive mappings // Bulletin of the Iranian Mathematical Society. 2020. Vol. 46. No. 2. P. 557-571. https://doi.org/10.1007/s41980-019-00275-7

Hussain N., Parvaneh V., Hoseini Ghoncheh S.J. Generalized contractive mappings and weakly a-admissible pairs in G-metric spaces // The Scientific World Journal. 2014. Vol. 2014. Article ID: 941086. https://doi.org/10.1155/2014/941086

Ishikawa S. Fixed points by a new iteration method // Proceedings of the American Mathematical Society. 1974. Vol. 44. No. 1. P. 147-150. https://doi.org/10.1090/S0002-9939-1974-0336469-5 Karapinar E., Kumam P., Salimi P. On a-^-Meir-Keeler contractive mappings // Fixed Point Theory and Applications. 2013. Vol. 2013. No. 1. Article number: 94. https://doi.org/10.1186/1687-1812-2013-94

Красносельский М. А. Два замечания о методе последовательных приближений // Успехи математических наук. 1955. Т. 10. Вып. 1 (63). С. 123-127. https://www.mathnet.ru/rus/rm7954 Mann W. R. Mean value methods in iteration // Proceedings of the American Mathematical Society. 1953. Vol. 4. No. 3. P. 506-510. https://doi.org/10.1090/S0002-9939-1953-0054846-3 Michael E. Continuous selection. I //Annals of Mathematics. 1956. Vol. 63. No. 2. P. 361-382. https://doi.org/10.2307/1969615

Nadler S. B. Multi-valued contraction mappings // Pacific Journal of Mathematics. 1969. Vol. 30. No. 2. P. 475-488. https://doi.org/10.2140/pjm.1969.30.475

Noor M. A. New approximation schemes for general variational inequalities // Journal of Mathematical Analysis and Applications. 2000. Vol. 251. Issue 1. P. 217-229. https://doi.org/10.1006/jmaa.2000.7042 Okeke G. A. Convergence analysis of the Picard-Ishikawa hybrid iterative process with applications // Afrika Matematika. 2019. Vol. 30. No. 5-6. P. 817-835. https://doi.org/10.1007/s13370-019-00686-z Pietkun R. Structure of the solution set to Volterra integral inclusions and applications // Journal of Mathematical Analysis and Applications. 2013. Vol. 403. Issue 2. P. 643-666. https://doi.org/10.1016/j.jmaa.2013.02.059

Sadygov M.A. An extreme problem for a Volterra type integral inclusion // Open Access Library Journal. 2019. Vol. 6. No. 8. P. 1-8. https://doi.org/10.4236/oalib.1105605

Salimi P., Latif A., Hussain N. Modified a-^-contractive mappings with applications // Fixed Point Theory and Applications. 2013. Vol. 2013. No. 1. Article number: 151. https://doi.org/10.1186/1687-1812-2013-151

Samet B., Vetro C., Vetro P. Fixed point theorems for a-^-contractive type mappings // Nonlinear Analysis: Theory, Methods and Applications. 2012. Vol. 75. Issue 4. P. 2154-2165. https://doi. org/10.1016/j.na.2011.10.014

Sintamarian A. Integral inclusions of Fredlhom type relative to multivalued ^-contractions // Seminar on Fixed Point Theory Cluj-Napoca. 2002. Vol. 3. P. 361-368. https://zbmath.org/1028.47043 Touail Y., El Moutawakil D., Bennani S. Fixed point theorems for contractive selfmappings of a bounded metric space // Journal of Function Spaces. 2019. Vol. 2019. Article ID: 4175807. https://doi.org/10.1155/2019/4175807

Touail Y., El Moutawakil D. Fixed point results for new type of multivalued mappings in bounded metric spaces with an application // Ricerche di Matematica. 2020. Vol. 71. No. 2. P. 315-323. https://doi.org/10.1007/s11587-020-00498-5

Touail Y., El Moutawakil D. New common fixed point theorems for contractive self mappings and an application to nonlinear differential equations // International Journal of Nonlinear Analysis and Applications. 2021. Vol. 12. Issue 1. P. 903-911. https://doi.org/10.22075/IJNAA.2021.21318.2245 Туаль Ю., Аль-Мутавакиль Д. Теоремы о неподвижной точке для новых сжимающих отображений с приложением в динамическом программировании // Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия. 2021. Т. 8. № 2. С. 338-348. https://doi.org/10.21638/spbu01.2021.213

Touail Y., El Moutawakil D. Some new common fixed point theorems for contractive selfmappings with applications // Asian-European Journal of Mathematics. 2021. Vol. 15. No. 4. Article number: 2250080. https://doi.org/10.1142/S1793557122500802

33. Touail Y., El Moutawakil D. Fixed point theorems on orthogonal complete metric spaces with an application // International Journal of Nonlinear Analysis and Applications. 2021. Vol. 12. Issue 2. P. 1801-1809. https://doi.org/10.22075/IJNAA.2021.23033.2464

34. Touail Y., Jaid A., El Moutawakil D. New contribution in fixed point theory via an auxiliary function with an application // Ricerche di Matematica. 2021. Vol. 72. No. 1. P. 181-191. https://doi.org/10.1007/s11587-021-00645-6

35. Touail Y. On multivalued -contractions on generalized orthogonal sets with an application to integral inclusions // Проблемы анализа — Issues of Analysis. 2022. Т. 11 (29). Вып. 3. С. 109-124. https ://www. mathnet. ru/rus/pa3 63

36. Ullah K., Khan M. S. U., Muhammad N., Ahmad J. Approximation of endpoints for multivalued nonexpansive mappings in geodesic spaces // Asian-European Journal of Mathematics. 2020. Vol. 13. No. 8. Article number: 2050141. https://doi.org/10.1142/S1793557120501417

37. Wilson W. A. On semi-metric spaces // American Journal of Mathematics. 1931. Vol. 53. No. 2. P. 361-373. https://doi.org/10.2307/2370790

Поступила в редакцию 21.04.2023

Принята к публикации 23.02.2024

Туаиль Юсеф, д. ф.-м. н., профессор, отделение математики, факультет наук Дхар Эль Махраз, университет Сиди Мохамед Бен Абделла, Марокко, г. Фез. ORCID: https://orcid.org/0000-0003-3593-8253 E-mail: [email protected]

Цитирование: Ю. Туаиль. Многозначные отображения: результаты о неподвижной точке с использованием 0-функции и некоторые приложения // Известия Института математики и информатики Удмуртского государственного университета. 2024. Т. 63. С. 61-79.