EXISTENCE AND UNIQUENESS OF THE SOLUTIONS OF SOME CLASSES OF INTEGRAL EQUATIONS C*-ALGEBRA-VALUED B-METRIC SPACES Текст научной статьи по специальности «Математика»

EXISTENCE AND UNIQUENESS OF THE SOLUTIONS OF SOME CLASSES OF INTEGRAL EQUATIONS C*-ALGEBRA-VALUED b-METRIC SPACES

Esad Jakupovica, Hashem P. Masihab, Zoran D. Mitrovicc, Seyede S. Razavid, Reza Saadatie

aAcademy of Sciences and Arts of the Republic of Srpska, Banja Luka,

Republic of Srpska, Bosnia and Herzegovina,

e-mail: [email protected],

ORCID ID: https://orcid.org/0000-0003-2354-5532

bK. N. Toosi University of Technology, Faculty of Mathematics, Tehran,

Islamic Republic of Iran,

e-mail: [email protected],

ORCID ID: https://orcid.org/0000-0001-9751-6828

cUniversity of Banja Luka, Faculty of Electrical Engineering, Banja Luka, Republic of Srpska, Bosnia and Herzegovina, e-mail: [email protected], corresponding author, ORCID ID: https://orcid.org/0000-0001-9993-9082

dK. N. Toosi University of Technology, Faculty of Mathematics, Tehran, Islamic Republic of Iran, e-mail: [email protected],

ORCID ID: https://orcid.org/0000-0002-9772-1140 eIran University of Science and Technology, School of Mathematics, Narmak, Tehran, Islamic Republic of Iran, e-mail: [email protected],

ORCID ID: https://orcid.org/0000-0002-6770-6951

DOI: 10.5937/vojtehg68-28632; https://doi.org/10.5937/vojtehg68-28632 FIELD: Mathematics

ARTICLE TYPE: Original scientific paper Abstract:

Introduction/purpose: The aim of the paper is to establish some coupled fixed point results in C*-algebra-valued b-metric spaces. Moreover, the obtained results are used to define the sufficient conditions for the existence of the solutions of some classes of integral equations.

Methods: The method of coupled fixed points gives the sufficient conditions for the existence of the solution of some classes of integral equations.

Results: New results were obtained on coupled fixed points in C *-algebra-valued b-metric space.

Cj26)

CD CM h-S± CL

O

Conclusion: The obtained results represent a contribution in the fixed point theory and open new possibilities of application in the theory of differential and integral ir equations.

Key words: Coupled fixed point, C*-algebra, integral equation.

Basic definitions

In this section, we review some facts of the C*-algebras which are needed in this paper. The references (Ali Abou Bakr, 2019), (Bai, 2016), (Bonsal, 1962), (Hussain etal, 2018), (Huang etal, 2018), (Hussain&Mitro-vic, 2017), (Kadelburg et al, 2016), (Kadelburg&Radenovic, 2016), (Kong-ban&Kumam, 2018), (Ma et al, 2014), (Ma&Jiang, 2015), (Mitrovic et al, 2019), (Radenovic et al, 2017), (Radenovic et al, 2019), (Vujakovic et al, 2019), (Zoto et al, 2019), (Wu&Zhao, 2018), (Cao&Xin, 2016) and (Todor-cevic, 2019) are useful.

We denote A as a unital C*-algebra with the unit 1A.

Let

Ah = {t e A : t = t*}.

We say t e A a positive element, showed it by t y 0A if t e Ah and a(t) c [0, where 0A is the zero element in A and a(t) is the spectrum of t.

On the set Ah we have a partial ordering given by v y u if and only if v - u y 0a. Also, we will denote

A+ = {t e A : t y 0a}

and

A' = {t e A : tk = kt for all k e A}.

Definition 1. Assume that X = 0. As usual suppose that 5 : XxX^A is a function fulfilling:

(1) 5(u, v) y 0A for each u and v in X;

(2) 5(v, u) = 0a if v = u;

(3) 5(u, v) = 5(v, u) for each u and v in X; a

(4) 5(u, v) y 5(u, t) + 5(t, v) for each u, v and t e X. Then 5 is a C*-algebra-valued metric (shortly, C*-AV-M). o

ro

up

727

Definition 2. Assume that X = 0. Suppose that b e A such that ||b|| > 1. A function 5b: X2 ^ A is said to be a C*-algebra-valued b-metric (in short C*-AV-BM) on X if for every u, v, t e A:

(1) 5b(u, v) ^ 0A for each u and v in X and 5b(u, v) =0 if u = v;

(2) Sb(u,v) = Sb(v,u);

(3) Sb(u,v) < b[5b(u,t) + 5b(t,v)].

Then (X, A, 5b) is a C*-AV-BM space with the coefficient b.

Example 1 (Ma&Jiang (2015)). Assume that X = R and A = Mn(R). Now, we define

5(u, v) = diag(c1\u — v|p, c2|u — v|p,..., cn|u — v|p),

where diag denotes a diagonal matrix, and where u,v e R, c1,...,cn positive constants and p e (1, +rc>). It can be shown that (X, A, 5) is a complete C*-AV-BM. We only prove the third statement of Definition 2. For this we have:

|u — v|p < 2p(|u — t|p + |t — v|p),

then 5(u, v) ^ A[5(u, t) + 5(t, v)] for every u,v,t e X, where A = 2pI e A and A> I by 2p > 1. Since |u — v|p < |u — t|p + |t — v|p is impossible for all u y t y v, (X, Mn(R),5) is not a C*-AV-M space.

Definition 3. Assume that (X, A, 5b) is a C*-AV-BM space. (u,v) e X x X is called a coupled fixed point (shortly FP) of the function ^ : XxX^X if

^(u, v) = u and ^(v, u) = v.

For some useful applications, see (Kongban&Kumam, 2018) and (Ali Abou Bakr, 2019).

Main results

The following theorem is one of our main results.

Theorem 1. Assume that (X, A, 5b) is a C*-AV-BM space and suppose that ^ : X2 ^ X is a function satisfying

5b(^(u, v),^(t, s)) ^ a*5b(u, t)a + a*5b(v, s)a, (1)

l

-2.

for every u,v,t,s e X, in which a e A with ||a|| < —. Then ^ has a unique

coupled FP. Moreover, ^ has a unique FP in X.

Proof.Let u0, v0 e X. Set ui = ^(u0,v0) and vi = ^(v0,u0). We obtain two sequences {un}, {vn} in X such that un+i = ^(un, vn) and vn+i = ^(vn, un) if we continue the above process. From (1) we have

Similarly,

5b(un,un+l) = 5b(^(un-1,vn-l),^(un,vn))

y a*5b(u,n-i,u,n)a + a*5b(v.n-i,vn)a y a*(5b(un-i,un) + 5b(vn-i,vn))a.

5b(vn,vn+l) = 5b(^(vn-1,un-l),^(vn,un))

y a*5b(v,n-i,v,n)a + a*5b(u,n-i,u,n)a y a*(5b(vn-i,vn) + 5b(un-i,un))a.

(2)

(3)

Let

5n = 5b (un,un+i) + 5b(vn,vn+i). From (2) and (3), we have

5n = 5b(un, un+i) + 5b(vn, vn+i)

y a*(5b(un-i,un) + 5b(vn-i,vn))a + a*(5b(vn-i,vn) + 5b(un-i,un))a = 2(a*(5b(u,n-i,u,n) + 5b (v,n-i,v,n ))a) y (V2a)*(5b(un-i,un) + 5b(vn-i,vn))(V2a)

y (V2a)*5n-i(V2a).

Due to the following property: (if t,k e Ah, then t y k implies u*tu y u*ku), we can obtain for any n e N,

0a y 5n y (V2a)*5n-i(V2a) y ... y ((V2a)*)n5a(V2a)n.

If 50 = 0a, then from (2) of Definition 2 we know (u0,v0) is a coupled FP of F.

Now, by letting 50 y 0A, we can obtain for m,n e N, n> m

5b(un,um) y b(5b(un,un-i) + 5b(un-i,um))

y b5b(u,n,u,n-i) + b25b(u,n-i,u,n-2) + b25b(un-2,um) y b5b(un,un-i) + b25b(u,n-i,u,n-2)

<M I

CD <M h-s± cp

O

CO

<u LLi

•O >

o cp

(0 w

oo CD

t

o" CM o CM

of

UJ

cc

ZD

o o

-J

<

O 21 X

o

LU

I—

>-

a: <

< -j

cd >o

X LU I—

o

o

+ b35b(Un-2,Un-3) + ... + bn-m5b(um+i, um)

^ b5b(Un,Un-l) + b25b(Un-l,Un-2) + ... + bn-m-l5b (Um+2, Um+i) + bn-m-l5b (Um+i,Um)

Similarly,

$b(vn, vm) ^ b5b(vn,vn-i) + b2Sb(v n—l, vn-2) + ... +

bn-m-l5b (vm+2,vm+l) + bn-m-l5b (vm+l,vm).

Therefore,

Sb(Un,Um) + 5b{vn,vm) ^ b(5b(U n, Un- l) + fib(v vn-l))

+ b2(5b(Un-l,Un-2) + Sb(vn-l,vn-2)) + ... + bn-m-l (S b (Um+2,Um+l) + S b (vm+2,vm+l)) + bn-m (Sb (Um+l,Um) + Sb (vm+l,vm)) ± bSn-l + b25n-2 + ... + bn-m5m ± b((21 a)*)n-l5o(22a)"- + b2((21 a)*)n-2So(21 a)n-2 + ... + bn-m((2 2 a)*)m5o (2 2 a)m

-<

bn-k((22a)*)k6o(22a)

k=n— l

Therefore,

ll^b(Un,Um) + 5b(v,n,vm)ll<

k=n-l <x

< E i

|n-k ll^2all2k So ln-k llV2all2k So

<

k=n-l

millV2all2(n-l)

1 - mi-lllV2all2

1 -||b||-l||^2a||2

So

llV2all2(n-l) So.

730

Since ||a|| < ^, we have §

ib

Il5b(un,um)+ 5b(vn,vm)H < '|b'' ll^2all2(n-i)5o 0, §

1 - l|b||-i||v2a||2 ^

5b(un,um) y 5b(u,n,um) + 5b(v.n,vm),

and

5b(vn,vm) y 5b(vn,vm) + 5b(u n, um),

yields that {un} and {vn} are Cauchy sequences in X, so we can find u and v in X such that un = u and vn = v. Now, we prove that

^(u, v) = u and ^(v, u) = v.

From the triangle inequality and (1), we obtain

5b(^(u,v),u) y b[5b(^(u,v),un+i) + 5b(un+i,u)}

y b[5b(^(u,v),^(un,vn)) + 5b(un+i,u) y b[a*5b(u, un)a + a*5b(v, vn)a + 5b(un+i,u)},

||5b(u,v)||<||a||2||5b(u,v)|| + M^v^^

731

o

taking n ^ to, we have that 5b(^(u, v),u) = 0A, and consequently ^(u, v) = u. In the same way ^(v, u) = v. Therefore, a coupled FP of ^ is (u, v). Assume that another coupled FP of ^ is (u', v'), so

5b(u, u') = 5b(i>(u, v),^(u', v')) y a*5b(u, u')a + a*5b(v, v')a,

5b(v,v') = 5b(^(v,u),^(v' ,u')) y a*5b(v,v')a + a*5b(u,u')a, and hence

5b(u,u')+ 5b(v,v') y (V2a)*(5b(u,u') + 5b(v,v'))(V2a). The above inequality with ||V2a|| < 1 yields that

H5b(u, u') + 5b(v, v') || < || V2aH2H5b(u, u') + 5b(v, v') ||.

The above inequality holds only when H5b(u, u')+5b(v, v') || = 0, which gives (u', v') = (u, v). So the coupled FP is unique.

Now, we prove that u = v to show that ^ has a unique FP. Note that

5b(u, v) = 5b(^(u, v) + ^(v, u)) y a*5b(u, v)a + a*5b(v, u)a,

<u

and hence w

2||£ /„. „All I IUI|2||£ /„, „.Ml >

< 2M2IMu,v)H In fact, from ||o|| < — we get H5b(U,v)ll = 0, thus u = v. □

Lemma 1. (Ma etal, 2014)

1) If u e A+ with ||u|| < 2, then 1A - u is invertible.

2) If u,v e A+ with uv = vu, then uv y 0A.

3) If u,v e Ah and t e A+, then u < v deduces tu < tv, where A'+ = A+ n A'.

Theorem 2. Assume that (X, A, 5b) is a complete C*-AV-BM space and suppose that the function ^ : X2 ^ X satisfies

fib(^(u,v),^(t,s)) ^ aSb(^(u,v),u) + b5b(^(t,s),t), Vu,v,t,s eX (4)

in which a, b, c, e A'+ with ||o|| + ||b|| < 1, ||ac|| + ||bc|| < 1, ||c|| > 1. Then ^ has a unique coupled FP. Also, ^ has a unique FP in X.

Proof. As in Theorem 1, choose {un} and {vn} in X and set un+l = ^(un,vn) and vn+l = i>(vn,Un). Then from (4),

fib(Un,Un+l) = 5b(^(Un-l,vn-l),^(Un,vn))

^ a5b(^(Un-l,vn-l),Un-l) + bfib(^(Un,vn),Un) = a5b(un,un-l) + b5b(un+l,un),

Thus,

(1a - b)5b(U",Un+l) ^ a5b(un, Un-l),

Similarly,

(1a — b)5b(vn,vn+l) ^ a5b(v n, vn l ) ,

Since a,b e A++ with ||a|| + ||b|| < 1, then 1A - b is invertible and (1A -b)-a e A+. Therefore

fib(un,un+l) ^ (1a - b)-la5b(un,un-l),

fib(vn,vn+l) ^ (1a - b)-la5b(v n, vn l ) ,

Then

Hfib(un,Un+l)H < ||(1a - b^am^u n, Un— l)||,

(wT)

H5b(vn,vn+i)H < ||(1a - b)-iaHH5b(vn,vn-i)H, £

It follows from the fact g

h-

11(1a - b)-iaH < ||(1a - b)-i||||a|| ^J|b||k||a|| = ^ < 1. &

Hence {un} and {vn} are Cauchy sequences in X. By the completeness of X, there are u,v e X such that limn^^ un = u and limn^^ vn = v. As

5b(i>(u, v),u) y c[5b(u,n+i,^(u,v)) + 5b(u,n+i,u)}

= c5b(^(un,vn),^(u,v)) + c5b(un+i,u) y ca5b(^(u,n,v,n),un) + cb5b(^(u,v),u) + c5b(u,n+i,u) y ca5b(u,n+i,u,n) + cb5b(^(u,v),u) + c5b(u,n+i,u), 0

which deduces that

5b(i>(u, v), u) y (1a - cb)-ica5b(u,n+i,u,n) + (1a - cb)-ic5b(u,n+i,u). like above

<x

||(1A - cb)-icaH < ||(1a - ^mcaU < £ ||cb||k||ca|| = < 1-

<x

i

||(1a - cb)-icH < ||(1a - cb)-i||||c|| < ^ ||cb||k||c|| = ^^ < 1.

k=0

Then 5b(^(u,v),u) = 0A or equivalently ^(u,v) = u. In the same way, ^(v, u) = v. Now if (u', v') is another coupled FP of then from (4), we get

5b(u,u') = 5b(^(u,v),^(u' ,v'))

y a5b(u', u', v')) + b5b(u, ^(u, v)) = 0a,

Hence 5b(u',u) = 0A, and thenu' = u. In the same way, we can obtain that v' = v. That is, (u, v) is the unique coupled FP of Now, we prove the uniqueness of FPs of Using (4), we obtain

733

5b(u,v) = 5b(^(u,v),^(v,u))

y a5b(^(u, v),u) + b5b(^(v, u),v) = a5b(u, u) + b5b(v, v) = 0a. lu

This yields that u = v. □

•o >

o cp

< -J

cd >o

X LLI

Theorem 3. Assume that (X, A, 5b) is a complete C*-AV-BM space and let the function F : X2 ^ X hold

fib(^(u, v),^(t, s)) ^ a5b(^(u, v),t) + b5b(^(t, s),u), Vu, v,t,s e X (5)

<1J V)

w

00 CD

1

° where a,b,c e A*+ with ||o|| + ||b|| < 1, ||ac|| + ||bc|| < 1, ||c|| > 1. then ^

™ has a unique coupled FP. Also, ^ has a unique FP in X.

ca

UJ

§ Proof. Similar to Theorem 2, choose two sequences {un} and {vn} in X § and set un+l = ^(un, vn) and vn+l = ^(vn, un). Then from (5) we obtain

fib(Un,Un+l) = 5b(^(Un-lvn-l),^(Un,vn))

<

o z

O ^ a5b(^(Un-l,vn-l),Un) + b5b(^(Un,vn),Un-l)

^ = a5b(un,un)+ bSb(U"+l,Un-l)

<C = b5b(Un+l,Un-l)

^ cb(5b(Un+l,Un) + fib(Un,Un-l)) = cb5b(Un+l,Un) + cb5b(Un,Un-l),

Thus,

(1a - cb)5b(U",Un+l) ^ cb5b(u n, Un—l )■ (6)

By the symmetry in (5),

n l , vn

fib(Un+l, Un) = fibO>(Unvn ),i>(Un-l,vn-l))

< afib(i!(Un, vn), Un-l) + bSb(^(Un-l ,vn-l),Un) ^ = a5b(U"+l,Un-l) + b5b(un,un)

= a5b(un+l,un-l) ^ ca(fib(un+l, Un) + 5b(un,un-l)) = ca5b(un+l,un) + ca5b(un,un-l),

that is,

(1a - ca)5b(U",Un+l) ^ ca5b(un, Un-l). (7)

Now, from (6) and (7) we obtain

ca + cb^ ca + cb

(1A--2-)fib(Un,Un+l) ^ -2-fib(Un,Un-l).

Sincea,b,c e A+ with ||ca+cb|| < ||ca|| +1|cb|| < 1, then (1a-ca++Cb)-1 e A+, which together with [3, Lemma 1] yields that

ca + cb -i ca + cb Ob(Un,Un+1) y (1a--2—) —2— "b(Un,u„-i).

Let t = (1A - )-1 ca+cb, then ||t|| = ||(1A - ca+cb)-1 ca+cb|| < 1. By using the same argument of Theorem 2, we obtain {un} which is a Cauchy sequence in X. Also, one can show that {vn} is a Cauchy sequence in X. Therefore by the completeness of X, there are u,v e X such that un = u and vn = v. Now, we obtain that ^>(u, v) = u and

■0(v, u) = v. To do this, we have

Sb(^(u,v),u) y c[^6(un+1,^(u,v)) + 5b(un+1,u)}

= c5b(^(v,n,v,n)A(u,v)) + c5b(un+1,u) y ca5b(^(un,vn),u) + cb5b(^(u,v),un) + c5b(u.n+1,u) y ca5b(u,n+1,u) + cb5b(^(u,v),u,n) + c5b(u,n+ 1,u),

and then

HSb(^(u,v),u)H < ||ca||||5b(un+1,u)||+||c6|pb(^(u,v),un)IMMPb(un+1,u)|| by the continuity of the metric and the norm, we get

||^(#u,v),u)||<||cb||||W(u,v),u)||.

Since ||cb|| < 1, it implies that ||^b(^(u, v),u)|| = 0, and then i>(u,v) = u. In the same way, ^(v, u) = v, which implies that (u, v) is a coupled FP of By the same reasoning in Theorem 2, we obtain u = v, which means that ^ has a unique FP in X. □

Existence and uniqueness

Consider the next equation:

x(m) = J(T1(m,n) + T2(m,n))(a(n,x(n)) + 0(n,x(n))dn + J(m), m eE

E (8) for the Lebesgue measurable set of E in which m(E) < to.

<M I

CD <M h-s±

CL

O

CO

-I—'

<u Llj

•O >

o cp

In what follows, we always let X = L^(E) denote a class of essentially bounded measurable functions on E, where E is a Lebesgue measurable set such that m(E) < to.

Now, we consider the functions Ti,T2,a,fi fulfill the following assumptions:

(i) Ti from E x E to R-0, T2 from E x E to R^°. Also, two integrable functions a and fi are from Ex R to R, and J e L™(E);

(ii) there exists £ e (0, 2) such that

0 < a(m, x) — a(m, y) < £(x — y)

and

—£(x — y) < fi(m, x) — fi(m, y) < 0 for m eE and x,y e R;

(iii) supme£ f£(Ti(m,n) — T2(m,n))dn < 1.

Theorem 4. Suppose that assumptions (i)-(iii) hold. Then the integral equation (8) has a unique solution in L^(E).

Proof. Let X = L™(E) and B(L2(E)) be the set of bounded linear operators on a Hilbert space L2(E). We endow X with the b-metric 5b : X x X ^ B(L2(E)) defined by

5b(a,fi) = M\a-p\r

where M\a-p\p is the multiplication operator on L2(E). Hence (X,B(L2(E)),5b) is a complete C*-AV-BM space. Define the self-mapping

V : X xX ^X by

V(x,y)(m) = J Ti(m,n)(a(n,x(n)) + fi(n,y(n)))dn

+ T2(m, n)(a(n, y(n)) + fi(n, x(n)))dn + J(m), for all m eE. Now, we have

5b(V(x,y), V(u,v))= Mm

We obtain,

l(V(x,y) — V(u,v))(m)lp = I J Ti(m,n)(a(n,x(n)) + fi(n,y(n)))dn

< (sup[i|x(n) — u(n)l + lly(n) - v(n)l} x / (T1(m,n) - T2(m,n))dn)p n€£ Je

<

|x - u||^ + illy - v||^} sup (T1(m,n) - T2(m,n))dn)p

me£ J£

< (i||x - ulln + illy - v||^)p

< i(||x - u|U + ||y - v|U)p.

Therefore,

H5b(^(x,y), ty(u,v))H = ||M|^(x,y)-^(„,v)|p||

= sup (M|^(x,y)-^(«,v)|p

|M|=1

= sup / K^(x,y) - ty(u,v))(m)|p<fi(m)<fi(m)dm IMM

< sup [ ^(m^dmx - u|U + i||y - v|U)p IM=1J £

< (i||x - u|U + i||y - v|U)p

< i(||x - u|U + ||y - v|U)p.

Set a = Vi1B(L2(£)), then a e B(L2(E)) and ||a|| = |Vi| < . Hence by applying Theorem 1, we get the desired result. □

737

d.

+ I T2(m,n)(a(n,y(n)) + ft(n,x(n)))dn - I T1(m,n)(a(n,u(n)) ¡5

l£ J£

eg

+ ft(n,v(n)))dn -J T2(m,n)(a(n,v(n)) + ft (n,u(n)))dn!p ^

= d I T1(m,n)(a(n,x(n)) - a(n,u(n)) + ft(n,y(n)) - ft(n,v(n)))d^ + ^ T2(m,n)(a(n,y(n)) - a(n,v(n)) + ft(n,x(n)) - ft(n,u(n)))dnDP < (J T1(m,n)|a(n,x(n)) - a(n, u(n))+ ft(n,y(n)) - ft(n, v(n))^n T2(m, ^^^n, y(n)) - a(n, v(n)) + ft(n, x(n)) - ft(n, v^n^^nY

l£

o

Funding

No funding was received. o

<1J

oo CD

O

CM

of

UJ

CC ZD

o o

-J

<

o z

X

o

LU

I—

>-

a:

21

OT <

-j

cd

■O 21 X LU I—

o

o >

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors contributed to the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

References

Ali Abou Bakr, S. M. 2019. Common fixed point of generalized cyclic Banach algebra contractions and Banach algebra Kannan types of mappings on cone quasi metric spaces. Journal of Nonlinear Sciences and Applications (JNSA), 12(10), pp.644-655. Available at: http://dx.doi.org/10.22436/jnsa.012.10.03.

Bai, C. 2016. Coupled fixed point theorems in C*-algebra-valued b-metric spaces with application. Fixed Point Theory and Applications, art.number:70. Available at: https://doi.org/10.1186/s13663-016-0560-1.

Bonsal, F.F. 1962. Lectures On Some Fixed Point Theorems Of Functional Analysis. Bombay, India: Tata Institute Of Fundamental Research [online]. Available at: http://www.math.tifr.res.in/~publ/ln/tifr26.pdf. [Accessed: 25 September 2020].

Cao, T. and Xin, Q. 2016. Common coupled fixed point theorems in C*-algebra-valued metric spaces. arXiv:1601.07168v1[math.OA], 26 January [online]. Available at: https://arxiv.org/abs/1601.07168 [Accessed: 25 September 2020].

Huang, H., Dosenovic, T. and Radenovic, S. 2018. Some fixed point results in b-metric spaces approach to the existence of a solution to nonlinear integral equations. Journal of Fixed Point Theory and Applications, 20(art.number:105). Available at: https://doi.org/10.1007/s11784-018-0577-7.

Hussain, A., Kanwal, T., Mitrovic, Z. D. and Radenovic, S., 2018. Optimal Solutions and Applications to Nonlinear Matrix and Integral Equations via Simulation Function. Filomat, 32(17), pp.6087-6106. Available at: https://doi.org/10.2298/FI L1817087H.

Hussain, N. and Mitrovic, Z. D.2017. On multi-valued weakquasi-contractions in b-metric spaces. Journal of Nonlinear Sciences and Applications (JNSA), 10(7), pp.3815-3823. Available at: http://dx.doi.org/10.22436/jnsa.010.07.35.

Kadelburg, Z., Nastasi, A., Radenovic, S. and Vetro, P. 2016. Fixed points of contractions and cyclic contractions on C*-algebra-valued b-metric spaces. Advances in Operator Theory, 1(1), pp.92-103. Available at: https://doi.org/10.22034 /aot.1610.1030.

o

Kadelburg, Z. and Radenovic, S., 2016. Fixed point results in C*-algebra-valued metric spaces are direct consequences of their standard metric counter- w parts. Fixed Point Theory and Applications, art.number:53(2016). Available at: ™ https://doi.org/10.1186/s13663-016-0544-1. g.

Kongban, C. and Kumam, P. 2018. Quadruple random common fixed point results of generalized Lipschitz mappings in cone b-metric spaces over Banach algebras. Journal of Nonlinear Sciences and Applications (JNSA), 11(1), pp.131149. Available at: http://dx.doi.org/10.22436/jnsa.011.01.10.

Ma, Z. and Jiang, L. 2015. C*-Algebra-valued b-metric spaces and related fixed point theorems. Fixed Point Theory and Applications , art.number:222(2015). Available at: https://doi.org/10.1186/s13663-015-0471-6.

Ma, Z.H., Jiang, N. and Sun, H.K. 2014. C*-Algebra-valued metric spaces and related fixed point theorems. Fixed Point Theory and Applications, art.num-ber:206(2014). Available at: https://doi.org/10.1186/1687-1812-2014-206.

Mitrovic, Z.D., Aydi, H., Md Noorani, M.S. and Qawaqneh, H. 2019. The weight inequalities on Reich type theorem in b-metric spaces. Journal of Mathematics and Computer Science (JMCS), 19(1), pp.51-57. Available at: http://dx.doi.org/10.22 436/jmcs.019.01.07.

Radenovic, S., Vetro, P., Nastasi, A. and Quan, L.T. 2017. Coupled fixed point theorems in C *-algebra-valued b-metric spaces. Scientific Publications of the State University of Novi Pazar Series A: Applied Mathematics, Informatics and mechanics, 9(1), pp.81-90. Available at: https://doi.org/10.5937/SPSUNP1701081R.

Radenovic, S., Zoto, K., Dedovic, N., Sesum Cavic, V. and Ansari, A.H. 2019. Bhaskar-Guo-Lakshmikantam-Cirictype results via new functions with applications to integral equations. Applied Mathematics and Computation, 357(C), pp.75-87. Available at: https://doi.org/10.1016/j.amc.2019.03.057.

Todorcevic, V. 2019. Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics. Springer Nature Switzerland AG. ISBN: 978-3-030-22591-9.

Vujakovic, J., Kishore, G.N.V., Rao, K.P.R., Radenovic, S. and Sadik, S.K. 2019. Existence and Unique Coupled Solution in Sb-Metric Spaces by Rational Contraction with Application. Mathematics, 7(4), p.313. Available at: https://doi. org/10.3390/math7040313.

Zoto, K., Rhoades, B.E. and Radenovic, S. 2019. Common fixed point theorems for a class of (s, q)-contractive mappings in b-metric-like spaces and applications to integral equations. Mathematica Slovaca, 69(1), pp.233-247. Available at: https://doi.org/10.1515/ms-2017-0217.

Wu, X. and Zhao, L. 2019. Fixed point theorems for generalized a - ^ type contractive mappings in b-metric spaces and applications,. Journal of Mathematics ^ and Computer Science (JMCS), 18(1), pp.49-62. Available at: http://dx.doi.org/10. 22436/jmcs.018.01.06.

•o >

o

CL

739

СУЩЕСТВОВАНИЕ И ЕДИНСТВЕННОСТЬ РЕШЕНИИ НЕКОТОРЫХ КЛАССОВ ИНТЕГРАЛЬНЫХ УРАВНЕНИИ С*-АЛГЕБРЫ В Ь-МЕТРИЧЕСКИХ ПРОСТРАНСТВАХ

Есад Якуповича, Хашем П. Масиха6, Зоран Д. Митровичв, корессподент, Сеиде С. Разави6, Реза Саадатиг

аАкадемия наук и художеств Республики Сербской, г. Баня-Лука, Республика Сербская, Босния и Герцеговина 6Технологический университет «К. Н. Тооси», математический факультет, г Тегеран, Исламская Республика Иран вУниверситет в г Баня-Лука, электротехнический факультет, г Баня-Лука, Республика Сербская, Босния и Герцеговина

гИранский научно-технологический университет, математический колледж, Нармак, г Тегеран, Исламская Республика Иран

РУБРИКА ГРНТИ: 27.00.00 МАТЕМАТИКА:

27.25.17 Метрическая теория функций, 27.33.00 Интегральные уравнения, 27.39.29 Приближенные методы функционального анализа ВИД СТАТЬИ: оригинальная научная статья

Резюме:

Введение/цель: Цель данной статьи заключается в выявлении сопряженных неподвижных точек C *-алгебры в Ь-метрических пространствах. На основании полученных результатов обсуждается существование решений интегральных уравнений.

Методы: С помощью методов сопряжения неподвижных точек представлены необходимые условия для существования решений некоторых классов интегральных уравнений.

Результаты: Получены новые результаты о сопряженных неподвижных точках C *-алгебры в Ь-метрическом пространстве.

Выводы: Полученные результаты вносят вклад в теорию неподвижных точек и открывают новые возможности применения в теории дифференциальных и интегральных уравнений.

Ключевые слова: сопряженная неподвижная точка, C *-алгебра, интегральное уравнение.

CM

КЛАСА ИНТЕГРАЛНИХ JЕДНАЧИНА У b-МЕТРИЧКИМ '

h-

ЕГЗИСТЕНЦША И JЕДИНСТВЕНОСТ РJЕШЕtoА НЕКИХ

ПРОСТОРИМА НАД С*-АЛГЕБРАМА

Есад JaKynoBM^3, Hashem P. Masiha6, Зоран Д. Митровипв, аутор за преписку, Seyede S. Razavi6, Reza Saadatir

аАкадеми]а наука и ум]етности Републике Српске, Ба^а Лука, Република Српска, Босна и Херцеговина

6Технолошки универзитет „К. N. Toosi", Математички факултет, Техеран, Исламска Република Иран

вУниверзитет у Ба^о] Луци, Електротехнички факултет, Ба^а Лука, Република Српска, Босна и Херцеговина гИрански научно-технолошки универзитет, Колец математике, Нармак, Техеран, Исламска Република Иран ОБЛАСТ: математика

ВРСТА ЧЛАНКА: оригинални научни рад

Сажетак:

Увод/цил: Циъ овог рада jecme да се do6ujy одре^ени резул-тати о придруженим непокретним тачкама у C *-алгебра b-метричким просторима. Користепи ове резултате да-ти су довол>ни услови за егзистенц^у рjеше^а неких класа интегралних 1'едначина.

Методе: Коришпеъем методе придружених фиксних тача-ка дати су довол>ни услови за егзистенц^у рjеше^а неких класа интегралних 1'едначина.

Резултати: Доб^ени су нови резултати о придруженим фиксним тачкама у b-метричком простору над C*-алгебром.

Закъучак: Доб^ени резултати предcтавn>аjу допринос теории фиксних тачака и отвараjу нове могупности за примене у теории диференцц'алних и интегралних jедна-чина.

Къучне речи: придружена фиксна тачка, C *-алгебра, инте-грална 1'едначина.

CL CL

О

Paper received on / Дата получения работы / Датум приема чланка: 30.09.2020. Manuscript corrections submitted on / Дата получения исправленной версии работы / ф

Датум достав^а^а исправки рукописа: 15.10.2020. ш

Paper accepted for publishing on / Дата окончательного согласования работы / Датум ю" коначног прихвата^а чланка за об]ав^ива^е: 16.10.2020. о

u a

ф

(Я (Я

© 2019 The Authors. Published by Vojnotehnicki glasnik / Military Technical Courier (http://vtg.mod.gov.rs, http://BTr.M0.ynp.cp6}. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license oo (http://creativecommons.org/licenses/by/3.0/rs/).

q © 2019 Авторы. Опубликовано в "Военно-технический вестник / Vojnotehnicki glasnik / Military

Technical Courier" (http://vtg.mod.gov.rs, httpV/втгмо.упр.срб). Данная статья в открытом доступе о и распространяется в соответствии с лицензией "Creative Commons" о (http://creativecommons.org/licenses/by/3.0/rs/).

Ct © 2019 Аутори. Обjавио Воjнотехнички гласник / Vojnotehnicki glasnik / Military Technical Courier (http://vtg.mod.gov.rs, http://втг.мо.yпp.cp6}. Ово jе чланак отвореног приступа и дистрибуира се у складу са Creative Commons лиценцом (http://creativecommons.org/licenses/by/3.0/rs/).

С О

X

о ш

н

^

а: <

<л С

о >о

X ш н

о —>

о >

й»