ALGEBRAIC DISTANCES IN VARIOUS ALGEBRAIC CONE METRIC SPACES. Kamal Fallahi, Ghasem Soleimani Rad, and Stojan Radenović

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1 PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série, tome 104(118) (2018), DOI: ALGEBRAIC DISTANCES IN VARIOUS ALGEBRAIC CONE METRIC SPACES Kml Fllhi, Ghsem Soleimni R, n Stojn Renović Abstrct. We efine severl vrints of lgebric istnce in n lgebric cone metric spce. Then we obtin some new results bout their properties n compre these efinitions. 1. Introuction n preliminries Orere norme spces n cones hve mny pplictions in pplie mthemtics. Thus, fixe point theory in K-metric n K-norme spces ws evelope in the mi-20th century (see [4, 19]). In 2007, Hung n Zhng [6] reintrouce such spces uner the nme of cone metric spces by substituting the set of rel numbers by n orere norme spce s coomin of metric n obtine some fixe point results (see lso [8, 12, 14, 17] n references contine therein). In 1996, K et l. [9] efine the concept of w-istnce in metric spces. Afterwrs, mny reserchers prove some fixe point theorems uner w-istnce in complete metric spces. In the sequel, Cho et l. [2] efine the concept of c- istnce in cone metric spce, which is cone version of w-istnce. Then, some fixe point results uner c-istnce in cone metric spces n TVS-cone metric spces were obtine in [3, 5, 11, 15, 18] (see lso the references therein). On the other hn, recently, Niknm et l. [13] efine the concept of lgebric cone metric spce n stuie some of its elementry properties. Consistent with the content of [13], the following efinitions n lemm will be neee in the sequel. Let Y be rel vector spce n P be convex subset of Y. A point x P is si to be n lgebric interior point of P if for ech y Y there exists ǫ > 0 such tht x + ty P, for ll t [0, ǫ]. This mens tht point x is clle n lgebric interior point of convex set P Y if x P n for ech y Y there exists ǫ > 0 such tht [x, x+ǫy] P, where [x, x+ǫy] = {λx+(1 λ)(x+ǫy) : for ll λ [0, 1]} 2010 Mthemtics Subject Clssifiction: 46A19; 47L07; 47H10; 46B99. Key wors n phrses: lgebric cone metric spce, lgebric istnce, lgebric w-cone istnce, lgebric cone b-metric spce, lgebric b-istnce, lgebric W -b-cone istnce. Communicte by Stevn Pilipović. 89

2 90 FALLAHI, SOLEIMANI RAD, AND RADENOVIĆ (see [13]). The set of ll lgebric interior points of P is clle the lgebric interior of P n is enote by int P. Also, P is clle lgebriclly open if P = int P. Definition 1.1. [13] Let Y be vector spce with the zero vector θ. A proper nonempty n convex subset P of Y is clle n lgebric cone if: (i) P + P P ; (ii) λp P, for ech λ > 0; (iii) if x P n x P, then x = θ. Given n lgebric cone P Y, we efine prtil orering with respect to P by x y if n only if y x P. We shll write x y to men tht x y n x y. Also, we write x y if n only if y x int P. The lgebric cone P is si to be Archimeen if for ech x, y P there exists n N such tht x ny. For exmple, P = {(x, y) R 2 : x, y 0} is n lgebric cone with the Archimeen property in the rel vector spce R 2. In the sequel we ssume tht (Y, P ) hs the Archimeen property. Lemm 1.1. [13] Let Y be rel vector spce n P be n lgebric cone in Y with nonempty lgebric interior. Then we hve (i) P + int P int P ; (ii) λ int P int P, for ech λ > 0. Definition 1.2. [13] Let X be nonempty set n (Y, P ) be n lgebric cone spce with int P. Suppose tht vector vlue function : X X Y stisfies the conitions: (ACM1) θ (x, y) for ll x, y X with x y n (x, y) = θ if n only if x = y; (ACM2) (x, y) = (y, x) for ll x, y X; (ACM3) (x, z) (x, y) + (y, z) for ll x, y, z X. Then is clle n lgebric cone metric n (X, ) is clle n lgebric cone metric spce. Definition 1.3. [13] Let (X, ) be n lgebric cone metric spce, {x n } be sequence in X n x X. Then the following sttements hol: (i) {x n } converges to x if for every c Y with c int P there exists n n 0 N such tht (x n, x) c for ll n > n 0. We enote this by -lim n x n = x or x n x s n ; (ii) {x n } is clle Cuchy sequence if for every c Y with c int P there exists n n 0 N such tht (x n, x m ) c for ll m, n > n 0 ; (iii) (X, ) is complete lgebric cone metric spce if every Cuchy sequence in X is convergent. In this pper, we efine vrious lgebric istnces in vrious lgebric cone metric spces, investigte their properties n compre our results. Also, s n ppliction, we prove some fixe point results. 2. Algebric istnces In this section, we suppose tht (X, ) is n lgebric cone metric spce. Then the following properties re esy to prove n these properties re often useful.

3 ALGEBRAIC DISTANCES IN VARIOUS ALGEBRAIC CONE METRIC SPACES 91 Lemm 2.1. Let X be nonempty set, (Y, P ) be n lgebric cone spce with int P n (X, ) be n lgebric cone metric spce. Then, for ll u, v, w, c Y, the following ssertions hol: x n (p 1 ) If u v n v w, then u w. (p 2 ) If u v n v w, then u w. (p 3 ) If u v n v w, then u w. Also, if u v n v w, then u w. (p 4 ) If θ u c for ech c int P, then u = θ. (p 5 ) If u λu, where u P n 0 < λ < 1, then u = θ. (p 6 ) Let {b n } be sequence in Y, lgebric convergent to θ (i.e., b n θ), θ b n n c int P. Then there exists positive integer n 0 such tht b n c for ech n > n 0. (p 7 ) If θ u v n k is nonnegtive rel number, then θ ku kv. (p 8 ) If θ u n v n for ll n N n u n u, vn v s n, then θ u v. (p 9 ) x n x n xn y (in the lgebric cone metric) imply tht x = y. (p 10 ) Let θ c. If θ (x n, x) b n n b n θ, then eventully (x n, x) c, where x n, x re sequence n given point in X. (p 11 ) If u v + c for every c int P, then u v. (p 12 ) The fmily {N (x, c) : x X, θ c}, where N (x, c) = {y X : (y, x) c}, is subbsis for topology on X (see [13]). We enote this lgebric cone topology by τ, n note tht τ is Husorff topology. Remrk 2.1. Hung n Zhng [6] prove tht if P is norml cone then x n X converges to x X if n only if (x n, x) θ, s n, n tht x n X is Cuchy sequence if n only if (x n, x m ) θ, s n, m. It follows from (p 6 ) n (p 10 ) tht the sequence {x n } converges to x X in n lgebric cone metric spce if (x n, x) θ, s n n {x n } is Cuchy sequence if (x n, x m ) θ, s n. In our cse, we hve only one hlf of the sttements of Lemms 1 n 4 from [6]. Also, in this cse, the fct tht (x n, y n ) (x, y) if x n yn y is not pplicble (for more etils, one cn see [5, 8, 10, 13]). Now, we efine the first version of n lgebric istnce n introuce some of its properties. Definition 2.1. Let (X, ) be n lgebric cone metric spce. A function q : X X Y is clle c-lgebric istnce (or briefly, lgebric istnce) on X if the following properties re stisfie: (q 1 ) θ q (x, y) for ll x, y X; (q 2 ) q (x, z) q (x, y) + q (y, z) for ll x, y, z X; (q 3 ) for x X, if q (x, y n ) u for some u = u x n ll n 1, then q (x, y) u whenever {y n } is sequence in X converging to point y X; (q 4 ) for ll c Y with θ c, there exists e Y with θ e such tht q (z, x) e n q (z, y) e imply (x, y) c.

4 92 FALLAHI, SOLEIMANI RAD, AND RADENOVIĆ Exmple 2.1. Let Y = R, P = {x Y : x 0}, be n lgebric cone n prtil orering with respect to P be efine by x y if n only if y x P. Let X = [0, ) n efine mpping : X X P by (x, y) = x y for ll x, y X. Then (X, ) is n lgebric cone metric spce. Define mpping q : X X Y by q (x, y) = y for ll x, y X. Then, q is n lgebric istnce. In fct, (q 1 ) (q 3 ) re immeite. From (x, y) = x y x+y = q (z, x)+q (z, y), it follows tht (q 4 ) hols. Hence q is n lgebric istnce. Exmple 2.2. Let (Y, P ) be n lgebric cone spce with int P n (X, ) be n lgebric cone metric spce such tht the metric (, ) is continuous function in secon vrible. Then, q (x, y) = (x, y) is n lgebric istnce. In fct, (q 1 ) n (q 2 ) re immeite. But, property (q 3 ) is nontrivil n it follows from q (x, y n ) = (x, y n ) u, pssing to the limit when n n using continuity of. Let c Y with c int P be given n put e = c 2. Suppose tht q (z, x) e n q (z, y) e. Then (x, y) = q (x, y) q (x, z) + q (z, y) e + e = c. Using (p 1 ), this shows tht (x, y) c n thus q stisfies (q 4 ). Hence, q is n lgebric istnce. In Exmples 2.1 n 2.2, we hve introuce two known lgebric istnces in n lgebric cone metric spce. There exist other exmples of istnces in [2, 15] which the reer cn consier in lgebric version. Also, similr to Exmple 3 of Djorjević [5], one cn consier lgebric istnces which re not c-istnces in cone metric spces of [2, 15]. Remrk 2.2. Bse on Exmples 2.1 n 2.2, we hve two importnt notes: (i) For n lgebric istnce q, q (x, y) = θ is not necessrily equivlent to x = y for ll x, y X. (ii) For n lgebric istnce q, q (x, y) = q (y, x) oes not necessrily hol for ll x, y X. We recll tht sequence {u n } in n lgebric cone P is c-sequence if for every c int P there exists n 0 N such tht u n c for n n 0. It is esy to prove tht if {u n } n {v n } re c-sequences in Y n α, β > 0, then {αu n + βv n } is c-sequence. Note tht in the cse tht the cone P is norml, sequence in Y is c-sequence if n only if it is θ-sequence (see property (p 6 )). However, c-sequence nee not be θ-sequence in n rbitrry lgebric cone metric spce. Lemm 2.2. Let (X, ) be n lgebric cone metric spce n q be n lgebric istnce on X. Also, let {x n } n {y n } be sequences in X, x, y, z X, n {u n } n {v n } be two c-sequences in the respective lgebric cone P. Then the following properties hol: (qp 1 ) If q (x n, y) u n n q (x n, z) v n for n N, then y = z. In prticulr, if q (x, y) = θ n q (x, z) = θ, then y = z. (qp 2 ) If q (x n, y n ) u n n q (x n, z) v n for n N, then {y n } converges to z. (qp 3 ) If q (x n, x m ) u n for m > n, then {x n } is Cuchy sequence in X. (qp 4 ) If q (y, x n ) u n for n N, then {x n } is Cuchy sequence in X.

5 ALGEBRAIC DISTANCES IN VARIOUS ALGEBRAIC CONE METRIC SPACES 93 Proof. (qp 1 ) It is enough to prove tht (y, z) c for ech c int P in orer to prove tht y = z. For the given c, choose e int P such tht property (q 4 ) is stisfie. Since {u n } n {v n } re c-sequences, there exists n 0 N such tht u n e n v n e for ech n n 0. By (p 1 ), since u n e n q (x n, y) u n, we hve q (x n, y) e. Similrly, we hve q (x n, z) e. Now, using (q 4 ), we get (y, z) c. (qp 2 ) Let gin c int P be rbitrry n choose corresponing e int P stisfying property (q 4 ). If n 0 N such tht u n e n v n e for ech n n 0, then (p 1 ) implies tht q (x n, y n ) e n q(x n, z) e for n n 0. Hence, by (q 4 ), we hve (y n, z) c for ll n n 0 ; tht is, y n z s n. (qp 3 ) Let c Y with c int P. As in the proof of (qp 1 ), choose e Y with e int P. Then there exists positive integer n 0 N such tht q (x n, x n+1 ) e n q (x n, x m ) e for ny m > n n 0 n hence (x n+1, x m ) c (by (q 4 )). This implies tht {x n } is Cuchy sequence in X. As in the proof of (qp 3 ), we cn esily prove (qp 4 ). Our min result in this section is the following theorem of Chtterje type (see [1]) uner n lgebric istnce in n lgebric cone metric spce. Theorem 2.1. Let (X, ) be complete lgebric cone metric spce, q be n lgebric istnce on X n T : X X be continuous mpping. Suppose tht there exist nonnegtive constnts α, β, γ such tht (2.1) (2.2) q (T x, T y) αq (x, y) + βq (x, T y) + γq (y, T x), q (T y, T x) αq (y, x) + βq (T y, x) + γq (T x, y) for ll x, y X, where α + 2β + 2γ < 1. Then T hs fixe point in X. If T u = u, then q (u, u) = θ. Proof. Let x 0 be n rbitrry point of X. If T x 0 = x 0, then x 0 is fixe point of T n the proof is finishe. Suppose tht T x 0 x 0. Then we construct sequence {x n } in X such tht x n = T n x 0 = T x n 1. In orer to prove tht it is Cuchy sequence, put x = x n n y = x n 1 in (2.1). We hve (2.3) q (x n+1, x n ) = q (T x n, T x n 1 ) αq (x n, x n 1 ) + βq (x n T x n 1 ) + γq (x n 1, T x n ) = αq (x n, x n 1 ) + βq (x n, x n ) + γq (x n 1, x n+1 ) αq (x n, x n 1 ) + β[q (x n, x n+1 ) + q (x n+1, x n )] + γ[q (x n 1, x n ) + q (x n, x n+1 )] = αq (x n, x n 1 ) + βq (x n+1, x n ) + (β + γ)q (x n, x n+1 ) + γq (x n 1, x n ). Similrly, putting x = x n n y = x n 1 in (2.2), we hve (2.4) q (x n, x n+1 ) αq (x n 1, x n ) + βq (x n, x n+1 ) + (β + γ)q (x n+1, x n ) + γq (x n, x n 1 ).

6 94 FALLAHI, SOLEIMANI RAD, AND RADENOVIĆ Aing up (2.3) n (2.4), we hve q (x n+1, x n ) + q (x n, x n+1 ) (α + γ)[q (x n, x n 1 ) + q (x n 1, x n )] + (2β + γ)[q (x n+1, x n ) + q (x n, x n+1 )]. Set u n = q (x n+1, x n ) + q (x n, x n+1 ). We get u n (α + γ)u n 1 + (2β + γ)u n. Thus, we hve u n λu n 1, where λ = (α+γ) 1 (2β+γ) < 1 (by α + 2β + 2γ < 1). By repeting the proceure, we get u n λ n u 0 (by (p 3 )) for ll n N. Thus, (2.5) q (x n, x n+1 ) v n λ n [q (x 1, x 0 ) + q (x 0, x 1 )]. Let m > n. In the usul wy, it follows from (2.5) n λ [0, 1) tht q (x n, x m ) q (x n, x n+1 ) + + q (x m 1, x m ) λ n 1 λ [q (x 1, x 0 ) + q (x 0, x 1 )] = v n, where {v n } is c-sequence. Lemm 2.2 (qp 3 ) implies tht {x n } is Cuchy sequence in the lgebric cone metric spce X n, since X is complete, x n x X s n. Continuity of T implies tht x n+1 = T x n T x, n since the limit of sequence in n lgebric cone metric spce is unique, we obtin T x = x ; tht is, x is fixe point of T. Now, we suppose tht T u = u. It follows from (2.1) tht q (u, u) = q (T u, T u) αq (u, u)+βq (u, T u)+γq (u, T u) = (α+β +γ)q (u, u). Since α + β + γ < α + 2β + 2γ < 1, so by Lemm (p 5 ), we hve q (u, u) = θ. Now, we efine nother version of lgebric istnce ccoring to Ćirić et l. [3]. Let (X, ) be n lgebric cone metric spce. Then G : X P is lower semicontinuous t x X if for ech c Y with c int P, there is n 0 N such tht G(x) G(x n ) + c for ll n n 0, whenever {x n } is sequence in X n x n x. Definition 2.2. Let (X, ) be n lgebric cone metric spce. A function p : X X P is clle n lgebric w-cone istnce on X if it stisfies: (w 1 ) p (x, z) p (x, y) + p (y, z) for ll x, y, z X; (w 2 ) for ny x X, p (x, ): X P is lower semicontinuous; (w 3 ) for ll c Y with θ c, there exists e Y with θ e such tht p (z, x) e n p (z, y) e imply (x, y) c. Exmple 2.3. Let (X, ) be n lgebric cone metric spce. Clerly, stisfies (w 1 ) n (w 3 ). So we hve only to prove tht stisfies (w 2 ). Suppose tht {y n } is sequence in X such tht y n y n c Y with θ c is rbitrry. Since y n y, then there exists N0 N such tht (y n, y) c for ll n N 0. Define G: X P by G(y) = (x, y) where x X. Then for ll n N 0 we hve G(y) = (x, y) (x, y n ) + (y n, y) (x, y n ) + c = G(y n ) + c. Therefore p (x, ) = (x, ) is lower semicontinuous. Hence, is n lgebric w-cone istnce on X.

7 ALGEBRAIC DISTANCES IN VARIOUS ALGEBRAIC CONE METRIC SPACES 95 Concerning the work using lgebric w-cone istnces, note tht the following properties hol: (1) Using the notion of c-sequence, conition (w 2 ) of the previous efinition cn be formulte in the following sttement: If y n, y X, y n y s n n g(y) = p (x, y), then g(y) g(y n ) is c-sequence. (2) Definition 2.1 of n lgebric istnce is ifferent from Definition 2.2 of n lgebric w-cone istnce in the wy tht the (q 3 ) is use inste of (w 2 ). Also, it is cler tht ech lgebric w-cone istnce is c-lgebric istnce, but the converse oes not hol. (3) There exists n extension of Lemm 2.2 for n lgebric istnce (or w-istnce of K et l. [9]) to n lgebric w-cone istnce. Also, similrly s in the cse of n lgebric istnce, Lemm 2.2 n Theorem 2.1 cn be prove for n lgebric w-cone istnce. 3. Algebric b-istnce In this section, we efine n lgebric b-istnce on lgebric cone metric spces s nother version of lgebric istnce in n lgebric cone metric spce. Very recently, Rhimi et l. [16] efine n lgebric cone b-metric spce. Definition 3.1. [16] Let X be nonempty set, (Y, P ) be n lgebric cone spce with int P n s 1 be given rel number. Suppose tht vectorvlue function D : X X Y stisfies the following conitions: (ACbM1) θ D (x, y) for ll x, y X n D (x, y) = θ if n only if x = y; (ACbM2) D (x, y) = D (y, x) for ll x, y X; (ACbM3) D (x, z) s[d (x, y) + D (y, z)] for ll x, y, z X. Then D is clle n lgebric cone b-metric with prmeter s 1 n (X, D ) is clle n lgebric cone b-metric spce. Definition 3.2. [16] Let (X, D ) be n lgebric cone b-metric spce, {x n } be sequence in X n x X. Then (i) {x n } converges to x if, for every c Y with c int P there exists n n 0 N such tht D (x n, x) c for ll n > n 0. We enote this by D D -lim n x n = x or x n x s n ; (ii) {x n } is clle Cuchy sequence if, for every c Y with c int P there exists n n 0 N such tht D (x n, x m ) c for ll m, n > n 0 ; (iii) (X, D ) is complete lgebric cone b-metric spce if every Cuchy sequence in X is convergent. Now, we efine nother version of n lgebric istnce in lgebric cone b- metric spce. Definition 3.3. Let (X, D ) be n lgebric cone b-metric spce with prmeter s 1. A function Q : X X Y is clle c-b-lgebric istnce (or briefly, lgebric b-istnce) on X if it stisfies the following properties: (Q 1 ) θ Q (x, y) for ll x, y X;

8 96 FALLAHI, SOLEIMANI RAD, AND RADENOVIĆ (Q 2 ) Q (x, z) s[q (x, y) + Q (y, z)] for ll x, y, z X; (Q 3 ) for x X, if Q (x, y n ) u for some u = u x n ll n 1, then Q (x, y) su whenever {y n } is sequence in X converging to point y X; (Q 4 ) for ll c Y with θ c, there exists e Y with θ e such tht Q (z, x) e n Q (z, y) e imply D (x, y) c. Exmple 3.1. Let Y = R, X = [0, 1] n P = {x Y : x 0}. Define mpping D : X X Y by D (x, y) = x y 2 for ll x, y X. Then (X, D ) is n lgebric cone b-metric spce with prmeter s = 2. Define mpping Q : X X Y by Q (x, y) = y 2 for ll x, y X. Then Q is n lgebric b-istnce on X. Lemm 3.1. Let (X, D ) be n lgebric cone b-metric spce with prmeter s 1 n Q be n lgebric b-istnce on X. Also, let {x n } n {y n } be sequences in X n x, y, z X, n {u n } n {v n } be two c-sequences in lgebric cone P. Then the following properties hol: (Qp 1 ) If Q (x n, y) u n n Q (x n, z) v n for n N, then y = z. Prticulrly, if Q (x, y) = θ n q (x, z) = θ, then y = z. (Qp 2 ) If Q (x n, y n ) u n n Q (x n, z) v n for n N, then {y n } converges to z. (Qp 3 ) If Q (x n, x m ) u n for m > n, then {x n } is Cuchy sequence in X. (Qp 4 ) If Q (y, x n ) u n for n N, then {x n } is Cuchy sequence in X. Proof. The proof is similr to Lemm 2.2. Our min result in this section is the following fixe point theorem uner n lgebric b-istnce in n lgebric cone b-metric spce. Theorem 3.1. Let (X, D ) be complete lgebric cone b-metric spce with prmeter s 1, Q be n lgebric b-istnce on X n T : X X be continuous mpping. Suppose tht there exist nonnegtive rel numbers α, β, γ such tht the following conitions hol: (t 1 ) s(α + β) + γ < 1; (t 2 ) for ll x, y X, (3.1) Q (T x, T y) αq (x, y) + βq (x, T x) + γq (y, T y). Then T hs fixe point in X. If T u = u, then Q (u, u) = θ. Proof. Let x 0 be n rbitrry point in X. If T x 0 = x 0, then x 0 is fixe point of T n the proof is finishe. Suppose tht T x 0 x 0. Then we construct sequence {x n } in X such tht x n = T n x 0 = T x n 1. In orer to prove tht it is Cuchy sequence, put x = x n 1 n y = x n in (3.1) n use (t 1 ). We hve Q (x n, x n+1 ) = Q (T x n 1, T x n ) αq (x n 1, x n ) + βq (x n 1, T x n 1 ) + γq (x n, T x n ) = (α + β)q (x n 1, x n ) + γq (x n, x n+1 ),

9 ALGEBRAIC DISTANCES IN VARIOUS ALGEBRAIC CONE METRIC SPACES 97 which implies tht Q (x n, x n+1 ) this process, we hve (3.2) Q (x n, x n+1 ) δ n Q (x 0, x 1 ) α+β 1 γ Q (x n 1, x n ) for ll n N. Repeting for ll n N, where 0 δ = α+β 1 γ < 1 s by (t 1). Let m > n. In the usul wy, it follows from (3.2) tht Q (x n, x m ) sq (x n, x n+1 ) + s 2 Q (x n+1, x n+2 ) + + s m n Q (x m 1, x m ) (sδ n + + s m n δ m 1 )Q (x 0, x 1 ) = sδ n (1 + sδ + + s m n 1 δ m n 1 )Q (x 0, x 1 ) sδ n 1 sδ Q (x 0, x 1 ) = v n, where {v n } is c-sequence. Lemm 3.1(Qp 3 ) implies tht {x n } is Cuchy sequence in the lgebric cone b-metric spce X n, since X is complete, x D n x D X s n. Continuity of T implies tht x n+1 = T x n T x, n since the limit of sequence in n lgebric cone b-metric spce is unique, we obtin T x = x ; tht is, x is fixe point of T. Now, we suppose tht T u = u. It follows from (3.1) tht Q (u, u) = Q (T u, T u) αq (u, u) + βq (u, T u) + γq (u, T u) = (α + β + γ)q (u, u), which is, by property (p 5 ) n (t 1 ), possible only if Q (u, u) = θ. Exmple 3.2. Consier Y, P, X, D, s n Q s in Exmple 3.1. Also, let mpping T : X X be efine by T x = x2 4 for ll x X. Tke α = 1 16, β = 1 7 n γ = 0. Now, we hve (i) s(α + β) + γ = 2( ) = < 1; (ii) for ll comprble x, y X, Q (T x, T y) = (T y) 2 = y4 16 αq (x, y) + βq (x, T x) + γq (y, T y). Moreover, T is continuous on X. Therefore, ll the conitions of Theorem 3.1 re stisfie n hence T hs fixe point (which is x = 0 with Q (0, 0) = 0). The following corollry is n lgebric b-istnce version of Theorems 3 n 6 of Rhimi et l. [16]. Corollry 3.1. Let (X, D ) be complete lgebric cone b-metric spce with prmeter s 1 n Q be n lgebric b-istnce on X. If continuous mpping T : X X stisfies the contrctive conition Q (T x, T y) αq (x, y) for ll x, y X, where α [0, 1 s ), then T hs fixe point in X. If T u = u, then Q (u, u) = θ. Now, we efine nother version of n lgebric b-istnce ccoring to Ćirić et l. s work [3]. Let us recll tht rel-vlue function F : X P efine on n lgebric cone b-metric spce X is si to be lower b-semicontinuous t

10 98 FALLAHI, SOLEIMANI RAD, AND RADENOVIĆ point x X if for ech c Y with c int P, there is n 0 N such tht D F(x) sf(x n )+c for ll n n 0, whenever {x n } is sequence in X n x n x. Definition 3.4. Let (X, D ) be n lgebric cone b-metric spce. A function P : X X P is clle n lgebric W -b-cone istnce on X if the following conitions re stisfie: (W 1 ) P (x, z) s[p (x, y) + P (y, z)] for ll x, y, z X; (W 2 ) for ny x X, P (x, ): X P is lower b-semicontinuous; (W 3 ) for ll c Y with θ c, there exists e Y with θ e such tht P (z, x) e n P (z, y) e imply D (x, y) c. Exmple 3.3. Let (X, D ) be n lgebric cone b-metric spce. Then D is n lgebric W -b-cone istnce on X. Remrk 3.1. Let s = 1. Then n lgebric b-istnce of Definition 3.3 is n lgebric istnce of Definition 2.1 n n lgebric W -b-cone istnce of Definition 3.4 is n lgebric w-cone istnce of Definition 2.2. Remrk 3.2. Concerning the work using lgebric W -b-cone istnces, note tht the following properties hol: (1) Definition 3.3 of n lgebric b-istnce is ifferent from Definition 3.4 of n lgebric W -b-cone istnce in the wy tht the (Q 3 ) is use inste of (W 2 ). Also, it is cler tht ech lgebric W -b-cone istnce is c-b-lgebric istnce, but the converse oes not hol. (2) There exists n extension of Lemm 3.1 for n lgebric istnce (or wtistnce of Hussin et l. [7]) to n lgebric W -b-cone istnce. Also, similrly s for lgebric istnces, Lemm 3.1 n Theorem 3.1 cn be prove for n lgebric W -b-cone istnce. In Theorem 3.1, tke s = 1. Then we obtin the following corollry for lgebric istnces. Theorem 3.2. Let (X, ) be complete lgebric cone metric spce, q be n lgebric istnce on X n T : X X be continuous mpping. Suppose tht there exist nonnegtive constnts α, β, γ such tht the following conitions hol: (t 1 ) α + β + γ < 1; (t 2 ) for ll x, y X, q (T x, T y) αq (x, y) + βq (x, T x) + γq (y, T y). Then T hs fixe point in X. If T u = u, then q (u, u) = θ. Acknowlegement. The uthors re grteful to the Eitoril Bor n the referees for their ccurte reing n helpful suggestions. Also, the first n the secon uthors re thnkful to the Deprtment of Mthemtics of Pyme Noor University.

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