In this paper we give some theorems of existence of best points of proximity for a multifunctional non-self-mapping in a partial metric space and some approximations on the sets of the best points of proximity. Other results are also given.The authors are partially supported by operator laboratory, Eloued, Algeria.Publisher's Versio

Djedidi Mostefa

Mansour, Abdelouahab

TWMS Journal of Applied and Engineering Mathematics

English

Isik University Academic Open Access

TWMS J. App. and Eng. Math. V.11, N.2, 2021, pp. 319-328EXISTENCE OF FIXED POINT AND BEST POINT OF PROXIMITYFOR MULTIFUNCTIONAL NON SELF MAPPINGS IN A PARTIALMETRIC SPACEMOSTEFA DJEDIDI1, ABDELOUHAB MANSOUR1, §Abstract. In this paper we give some theorems of existence of best points of proximityfor a multifunctional non-self-mapping in a partial metric space and some approxima-tions on the sets of the best points of proximity. Other results are also given.Keywords: Partial metric space; Fixed point; best point; proximity.AMS Subject Classification: 47H09; 54E50; 37C25; 47H10.1. IntroductionLet A,B be two nonempty subsets of a metric space (X, d). The aim of this paperis to establish the existence theorems of a best proximity point x ∈ A, which satisfiesinf{p(x, y) : y ∈ F (x)} = dist(A,B) for a non-self-mapping multifunction F : A→ 2B.In this article, we prove that the results obtained in [3] can be enhanced and managed onpartial metric spaces. We defined(x,B) = infy∈Bd(x, y), e(A,B) = supx∈Ad(x,B)andD(A,B) = max(e(A,B), e(B,A)).Here e(A,B) is the excess of A over B and D(A,B) is the Pompeiu-Hausdorff distancebetween A and B. And letA0 = {x ∈ A : d(x, y) = d(A,B), for some y ∈ B},B0 = {y ∈ B : d(x, y) = d(A,B), for somex ∈ A}.1 Department of Mathematics, El Oued University, Algeria.e-mail: djedidimostefa@yahoo.fr; ORCID: https://orcid.org/0000-0003-3242-4716.e-mail: amansour@math.univ-lyon1.fr; ORCID: https://orcid.org/0000-0002-7530-9921.§ Manuscript received: February 27, 2019; accepted: February 14, 2021.TWMS Journal of Applied and Engineering Mathematics, Vol.11, No.4 © Işık University, Departmentof Mathematics, 2021; all rights reserved.The authors are partially supported by operator laboratory, Eloued, Algeria.319320 TWMS J. APP. ENG. MATH. V.11, N.2, 2021Definition 1.1. [3] Let (X, d) be a partial metric space and (A,B) a pair of nonemptysubsets of X, and A 6= ∅.We say that the pair (A,B) satisfies the P-property if and only if{d(x1, y1) = d(A,B)d(x2, y2) = d(A,B)⇒ d(x1, y1) = d(x2, y2).Where x1, x2 ∈ A0 and y1, y2 ∈ B0.Theorem 1.1. [3] Let (X, d) be a complete metric space and A,B two closed and nonemptysubsets of X such that A0 6= ∅ and the pair (A,B) satisfies the P-property. We assumethat F : A→ 2B a multifunction with bounded and closed value. If there exists a constantθ ∈ (0, 1) such that[D(F (x), F (y)) ≤ θd(x, y), ∀x, y ∈ A]and F (x) ⊆ B0 for allx ∈ A0. Then F has a best proximity point x in A.Definition 1.2. We say that the function p : X ×X → [0,+∞[ is a partial metric on Xif the following conditions are satisfied:(a) p(x, x) = p(y, y) = p(x, y) if and only if x = y for all x, y ∈ X,(b) p(x, x) ≤ p(x, y) for all x, y ∈ X,(c) p(x, y) = p(y, x) for all x, y ∈ X,(d) p(x, z) ≤ p(x, y) + p(y, z)− p(y, y) for all x, y, z ∈ X.Then (X, p) is called a partial metric space.For a partial metric p on X, the function ps : X ×X → R+ given byps(x, y) = 2p(x, y)− p(x, x)− p(y, y) (1)andpw(x, y) = p(x, y)−min{p(x, x), p(y, y)} (2)are metrics on X. Each partial metric p on X Generates a T0-topology τp with a basis thefamilies of open p-balls {Bp(x, ε) : x ∈ X, ε > 0}, where Bp(x, ε) = {y ∈ X : p(x, y) <p(x, x) + ε} for all x ∈ X and ε > 0.Definition 1.3. [17](i) A sequence {xn} in a partial metric space (X, p) converges to x ∈ X if and only ifp(x, x) = limn→∞ p(x, xn).(ii) A sequence {xn} in a partial metric space (X, p) is called Cauchy sequence if andonly if limn,m→∞ p(xn, xm) exists and finite.(iii) A partial metric space (X, p) est is complete if any Cauchy sequence {xn} in Xconverges to a point x ∈ X such that p(x, x) = limn,m→∞ p(xn, xm).Lemma 1.1. [17](a1) A sequence {xn} is Cauchy in a partial metric space (X, p) if and only if {xn} isCauchy in the space (X, ps).(a2) A partial metric space (X, p) is complete if and only if the space (X, ps) Is complete.Morewerlimn→∞ps(x, xn) = 0⇔ p(x, x) = limn→∞p(x, xn) = limn,m→∞p(xn, xm). (3)M. DJEDIDI, A. MANSOUR: EXISTENCE OF FIXED POINT AND BEST POINT ... 321Let A,B be two nonempty subsets of a partial metric space (X, p), the partial surplusep(A,B) from A to B and Dp(A,B) are the partial distance of Pompeiu-Hausdorff betweenA and B defined as follows:p(x,B) = infy∈Bp(x, y),p(A,B) = supx∈Ap(x,B)andp(A,B) = max(ep(A,B), ep(B,A)).Remark 1.1. [6] Let (X, p) be a partial metric space and A a nonempty subset of X, thena ∈ A⇔ p(a,A) = p(a, a).Lemma 1.2. [7] Let (X, p) be a partial metric space and A,B two closed, nonempty andbounded subsets of X, and h > 1. Then for each a ∈ A, there exists b ∈ B, such thatp(a, b) ≤ hDp(A,B).We next apply the notationsA0 = {x ∈ A : p(x, y) = p(A,B), for some y ∈ B}andB0 = {y ∈ B : p(x, y) = p(A,B), for somex ∈ A}.Definition 1.4. Let (X, p) be a partial metric space and (A,B) a pair of nonempty subsetsof X, and A 6= ∅.We say that the pair (A,B) satisfies the P-property if and only if{p(x1, y1) = p(A,B)p(x2, y2) = p(A,B)⇒ p(x1, y1) = p(x2, y2).Where x1, x2 ∈ A0 and y1, y2 ∈ B0.2. Main results2.1. Existence of best proximity point for non-self mapping multifunction.Theorem 2.1. Let (X, p) be a complete partial metric space and A,B two closed andnonempty subsets of X such that A0 6= ∅ and the pair (A,B) satisfies the P-property . Weassume that F : A→ 2B a multifunction with bounded and closed values. If there exists aconstant θ ∈ (0, 1) such thatDp(F (x), F (y)) ≤ θp(x, y), ∀x, y ∈ A.And F (x) ⊆ B0 for each x ∈ A0. Then F has a best proximity point x in A.322 TWMS J. APP. ENG. MATH. V.11, N.2, 2021Proof. Let x0 ∈ A0, and y0 ∈ F (x0) ⊆ B0. Then there exists x1 ∈ A0, such thatp(x1, y0) = p(A,B). On the other hand, by lemma 1.2 we obtain y1 ∈ F (x1) ⊆ B0 such thatp(y0, y1) ≤ Dp(F (x0), F (x1)) + θ.Futher more, there exists x2 ∈ A0, such that p(x1, y1) = p(A,B), and y2 ∈ F (x2) ⊆ B0such thatp(y1, y2) ≤ Dp(F (x1), F (x2)) + θ2.One continues for getting xn ∈ A0, such that p(xn, yn−1) = p(A,B), and yn ∈ F (xn) ⊆ B0such thatp(yn−1, yn) ≤ Dp(F (xn−1), F (xn)) + θn.Since p(xn+1, yn) = p(A,B) and p(xn, yn−1) = p(A,B), by the P-property, we obtainp(xn, xn+1) = p(yn−1, yn). Which givesp(xn, xn+1) = p(yn−1, yn) ≤ Dp(F (xn−1), F (xn)) + θn≤ θp(xn−1, xn) + θn≤ θp(yn−2, yn−1) + θn)≤ θ(Dp(F (xn−2), F (xn−1)) + θn−1) + θn≤ θ2p(xn−2, xn−1) + 2θn...≤ θnp(x0, x1) + nθn → 0, whenn→ +∞.Next by definition of ps, we get ps(xn, xn+1) ≤ 2p(xn, xn+p)→ 0, when n→ +∞. We nowprove that (xn) is a Cauchy sequence in (X, ps). Suppose that there exist an ε > 0 andk ∈ N, there exist mk, nk ∈ N such that mk > nk > k and p(xmk , xnk) > ε. We havep(A,B) ≤ p(ynk−1, xmk)≤ p(ynk−1, xnk) + p(xnk , xmk)− p(xnk , xnk)≤ p(ynk−1, xnk) + p(xnk , xmk)≤ p(ynk−1, xnk) + p(xnk , xnk+1) + ...+ p(xmk−1, xmk).Taking the limits on k we getlimk→∞p(ynk−1, xnk) + limk→∞p(xnk , xmk) = p(A,B).Which gives limk→∞p(xnk , xmk) = 0. Since ps(xnk , xmk) < 2p(xnk , xmk) → 0, then (xn) is aCauchy sequence in (X, ps), and since (X, p) is complete then by lemma 1.1, (X, ps) is acomplete metric space and so the sequence (xn) converges in X. Let x = limnxn. Since Ais closed, we have x ∈ A.Now, since p(xn, xn+1) = p(yn−1, yn), the sequence (yn) is convergent in X. Let y = limnyn.M. DJEDIDI, A. MANSOUR: EXISTENCE OF FIXED POINT AND BEST POINT ... 323Sine B is closed, we have y ∈ B. On the other hand, by lemma 1.1, we havep(x, x) = limn→+∞p(xn, x) = limn,m→+∞p(xn, xm) = 0andp(y, y) = limn→+∞p(yn, y) = limn,m→+∞p(yn, ym) = 0and p(x, y) = p(A,B). Hence,0 ≤ p(yn, F (x))≤ Dp(F (xn), F (x))≤ θp(xn, x).We take the limit n → +∞, we obtain p(y, F (x)) = 0 which gives p(y, F (x)) = p(y, y),using remark 1.1, to get y ∈ F (x) = F (x). One obtains that x is a best proximity pointin A which satisfies p(x, F (x)) = p(A,B). As a direct result of Theorem 2.1, we have the following result.Corollary 2.1. [7] Let (X, p) be a complete partial metric space and F : X → 2X be amultifunction with bounded and closed values. If there exists a constant θ ∈ (0, 1) suchthatDp(F (x), F (y)) ≤ θp(x, y), ∀x ∈ X and ∀y ∈ X.Then F has a fixed point x in X.In the following, let PFi be the set of best proximity points for a multifunction Fi.Theorem 2.2. Let (X, d) be a complete metric space and A,B two closed nonemptysubsets of X such that A0 6= ∅ and the pair (A,B) satisfies the P-property. Let Fi :A→ 2B, i = 1, 2 be two multifunctions with compact non empty values. If there exist twoconstants θ1, θ2 ∈ (0, 1) such thatD(F1(x), F1(y)) ≤ θ1d(x, y) ∀x ∈ Aand ∀y ∈ A,D(F2(x), F2(y)) ≤ θ2d(x, y) ∀x ∈ Aand ∀y ∈ Aand Fi(x) ⊆ B0, i = 1, 2, for each x ∈ A0. ThenD(PF1, PF2) ≤11−max{θ1, θ2}[supx∈AD(F1(x), F2(x))].Proof. Let ε > 0, we choose β > 0 such that β∑nθn2 < 1 and α =βε(1− θ2). Letx0,1 = x0 ∈ PF1, there exists y0,1 ∈ F1(x0) ⊆ B0 such that d(x0, y0,1) = d(A,B). Onthe other hand, there exist y0 ∈ F2(x0) ⊆ B0 and x1 ∈ A0 such that d(x1, y0) = d(A,B).By the P-property, we get d(x0, x1) = d(y0,1, y0). Which givesd(x0, x1) ≤ D(F1(x0), F2(x0)) + ε.We take y1 ∈ F (x1) ⊆ B0 such thatd(y0, y1) ≤ D(F2(x0), F2(x1)) + θ2α.324 TWMS J. APP. ENG. MATH. V.11, N.2, 2021Furthermore, there exists x1 ∈ A0, such that d(x1, y1) = d(A,B), and there existsy2 ∈ F2(x2) ⊆ B0 such thatd(y1, y2) ≤ D(F2(x1), F2(x2)) + θ22α.We continue to find xn ∈ A0, such that d(xn, yn−1) = d(A,B), and there exists yn ∈F2(xn) ⊆ B0 such thatd(yn−1, yn) ≤ D(F2(xn−1), F2(xn)) + θn2α.Since d(xn+1, yn) = d(A,B) and d(xn, yn−1) = d(A,B), by the P-property, we get d(xn, xn+1) =d(yn−1, yn). Which givesd(xn, xn+1) = d(yn−1, yn) ≤ D(F2(xn−1), F2(xn)) + θn2α≤ θ2d(xn−1, xn) + θn2α≤ θ2d(yn−2, yn−1) + θn2α≤ θ2(D(F2(xn−2), F2(xn−1)) + θn−12 ) + θn2α≤ θ22d(xn−2, xn−1) + 2θn2α...≤ θn2 d(x0, x1) + nθn2α→ 0, whenn→ +∞.(4)On the other hand, we prove that (xn) is a Cauchy sequence in A. One suppose thatthere exists ε > 0 and for all k ∈ N, there exist mk, nk ∈ N such that mk > nk > k andd(xmk , xnk) > ε. We haved(A,B) ≤ d(ynk−1, xmk)≤ d(ynk−1, xnk) + d(xnk , xmk)≤ d(ynk−1, xnk) + d(xnk , xmk)≤ d(ynk−1, xnk) + d(xnk , xnk+1) + ...+ d(xmk−1, xmk).(5)Taking the limit, we getlimk→∞d(ynk−1, xnk) + limk→∞d(xnk , xmk) = d(A,B).Which gives limk→∞d(xnk , xmk) = 0. We get (xn) a Cauchy sequence in a complete metricspace and so the sequence (xn) is convergent in X. Let x2 = limnxn. Since A is closed, weget x2 ∈ A.Since d(xn, xn+1) = d(yn−1, yn), the sequence (yn) converges in X. Let y = limnyn. SinceB is closed, we get x2 ∈ B and0 ≤ d(yn, F2(x2))≤ D(F2(xn), F2(x2))≤ θ2d(xn, x2).M. DJEDIDI, A. MANSOUR: EXISTENCE OF FIXED POINT AND BEST POINT ... 325Taking the limit, n → +∞, we get d(y, F2(x2)) = 0. We obtain x2 as the best proximitypoint in A which satisfies d(x2, F2(x2)) = d(A,B). Then we getd(x0, x2) ≤∞∑n=0d(xn, xn+1)≤∞∑n=0(θn2 d(x0, x1) + nθn2α)≤ 11− θ2d(x0, x1) +∞∑n=1nθn2α)≤ 11− θ2(d(x0, x1) + ε)≤ 11− θ2(D(F1(x0), F2(x0)) + 2ε)(6)We obtain for all ε > 0,d(x0,1, x2) ≤11− θ2(D(F1(x0,1), F2(x0,1)) + ε).As previously, let x0,2 ∈ PF2, there exists x1 ∈ PF1 such thatd(x0,2, x1) ≤11− θ1(D(F1(x0,1), F2(x0,1)) + ε), ∀ε > 0.Theorem 2.3. Let (X, p) a complete partial metric space and A,B two closed nonemptysubsets of X such that A0 6= ∅ and the pair (A,B) satisfies the P-property. Let Fi :A → 2B, i = 1, 2 two multifunctions with compact values. If there exist two constantsθ1, θ2 ∈ (0, 1) such thatDp(F1(x), F1(y)) ≤ θ1p(x, y) ∀x ∈ A , ∀y ∈ A,Dp(F2(x), F2(y)) ≤ θ2p(x, y) ∀x ∈ A , ∀y ∈ A,and Fi(x) ⊆ B0, i = 1, 2, for each x ∈ A0. ThenDp(PF1, PF2) ≤11−max{θ1, θ2}[supx∈ADp(F1(x), F2(y))].Proof. Let ε > 0, we choose β > 0 such that β∑nθn2 < 1 and α =βε(1− θ2). Letx0,1 = x0 ∈ PF1, there exist y0,1 ∈ F1(x0) ⊆ B0 such that d(x0, y0,1) = d(A,B). Other-wise, there exist y0 ∈ F2(x0) ⊆ B0 and x1 ∈ A0 such that p(x1, y0) = p(A,B). By theP-property, we get p(x0, x1) = p(y0,1, y0), which givesp(x0, x1) ≤ Dp(F1(x0), F2(x0)) + ε.We take y1 ∈ F (x1) ⊆ B0 such thatp(y0, y1) ≤ Dp(F2(x0), F2(x1)) + θ2.Else, there exists x1 ∈ A0, such that p(x1, y1) = p(A,B), and there exists y2 ∈ F2(x2) ⊆ B0326 TWMS J. APP. ENG. MATH. V.11, N.2, 2021such thatp(y1, y2) ≤ Dp(F2(x1), F2(x2)) + θ22.We continue to find xn ∈ A0, such that p(xn, yn−1) = p(A,B), and yn ∈ F2(xn) ⊆ B0 suchthatp(yn−1, yn) ≤ Dp(F2(xn−1), F2(xn)) + θn2 .Since p(xn+1, yn) = p(A,B) and p(xn, yn−1) = p(A,B), by the P-property, we get p(xn, xn+1) =p(yn−1, yn). Which givesp(xn, xn+1) = p(yn−1, yn) ≤ Dp(F2(xn−1), F2(xn)) + θn2≤ θ2p(xn−1, xn) + θn2≤ θ2(Dp(F2(yn−2), F2(yn−1)) + θn−12 )≤ θ2(Dp(F2(xn−2), F2(xn−1)) + θn−12 ) + θn2≤ θ22p(xn−2, xn−1) + 2θn2...≤ θn2 p(x0, x1) + nθn2 → 0, whenn→ +∞.By definition of ps, we get ps(xn, xn+1) ≤ 2p(xn, xn+p)→ 0, when n→ +∞. On the otherhand, we prove that (xn) is a Cauchy in (X, ps). One suppose that there exist ε > 0 andfor all k ∈ N, there exists mk, nk ∈ N such that mk > nk > k and p(xmk , xnk) > ε. Wehavep(A,B) ≤ p(ynk−1, xmk)≤ p(ynk−1, xnk) + p(xnk , xmk)− p(xnk , xnk)≤ p(ynk−1, xnk) + p(xnk , xmk)≤ p(ynk−1, xnk) + p(xnk , xnk+1) + ...+ p(xmk−1, xmk).Taking the limit we getlimk→∞p(ynk−1, xnk) + limk→∞p(xnk , xmk) = p(A,B).Which gives limk→∞p(xnk , xmk) = 0. Since ps(xnk , xmk) < 2p(xnk , xmk) → 0, we get (xn) aCauchy sequence in (X, ps), and since (X, p) is complete then by lemma 1.1, (X, ps) is acomplete metric space and the sequence (xn) converges in X. Let x2 = limnxn. Since A isclosed we have x2 ∈ A. Next, since p(xn, xn+1) = p(yn−1, yn), the sequence (yn) convergesin X. Let y = limnyn. Since B is closed we have y ∈ B.Otherwise, by lemma 1.1, we havep(x2, x2) = limn→+∞p(xn, x2) = limn,m→+∞p(xn, xm) = 0,M. DJEDIDI, A. MANSOUR: EXISTENCE OF FIXED POINT AND BEST POINT ... 327p(y, y) = limn→+∞p(yn, y) = limn,m→+∞p(yn, ym) = 0,and p(x2, y) = p(A,B). As result,0 ≤ p(yn, F2(x2))≤ Dp(F2(xn), F2(x2))≤ θ2p(xn, x2).Taking the limit n → +∞, we get p(y, F2(x2)) = 0 which gives p(y, F2(x2)) = p(y, y),using remark 1.1, to get y ∈ F2(x2) = F2(x2). Then we obtain x2 as a best proximitypoint in A which satisfies p(x2, F2(x2)) = p(A,B). Then we getp(x0, x2) ≤∞∑n=0p(xn, xn+1)−∞∑n=1p(xn, xn)≤∞∑n=0p(xn, xn+1)≤∞∑n=0(θn2 p(x0, x1) + nθn2α)≤ 11− θ2p(x0, x1) +∞∑n=1nθn2α)≤ 11− θ2(p(x0, x1) + ε)≤ 11− θ2(Dp(F1(x0), F2(x0)) + 2ε).(7)We obtain for all ε > 0,p(x0,1, x2) ≤11− θ2(Dp(F1(x0,1), F2(x0,1)) + ε).As previously, let x0,2 ∈ PF2, there exists x1 ∈ PF1 such thatp(x0,2, x1) ≤11− θ1(Dp(F1(x0,1), F2(x0,1)) + ε), ∀ε > 0.3. 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Neammance, A. kaewkhao,Fixed points and Best Proximity Points for multivalued mapping Sat-isfing Cyclical Condition, Int J.Math.Sciences and Applications Vol. 1, (2011)[20] S. Oltra and O. Valero, Banach’s fixed point theorem for partial metric spaces, Rend. Istit. Mat. Univ.Trieste 36, (1-2), 17-26 (2004)[21] S. Romaguera, Fixed point theorems for generalized contractions on partial metric spaces. Topol.Appl. 159, 194-199 (2012)[22] S. Sadiq Basha, Best Proximity Points: Optimal Solutions. J. Optim. Theory Appl. 151, (1), 210-216(2011)Mostefa Djedidi is currently working as an associate professor at the University ofEl-Oued, Algeria. His research interests are nonlinear analysis, theory methods andapplications, fixed point theory and numerical methods. He has published researcharticles in different international reputed journals of mathematics.Abdelouahab Mansour for the photography and short autobiography, see TWMS J. Appl. Eng. Maths.,V.10, N.3, 2020.

Existence of fixed point and best point of proximity for multifunctional non self mappings in a partial metric space

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Existence of fixed point and best point of proximity for multifunctional non self mappings in a partial metric space

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