We prove Ekeland's variational principle in  S^{JS} - metric spaces. A generalization of Caristi fixed point theorem on S^{JS} - metric spaces is obtained as a consequenc

Beg, Ismat

Roy, Kuhal

Saha, Mantu

English

University of Niš: Facta Universitatis (E-Journals) / Универзитет у Нишу

FACTA UNIVERSITATIS (NIŠ)Ser. Math. Inform. Vol. 36, No 5 (2021), 1117- 1127https://doi.org/10.22190/FUMI210525081BOriginal Scientific PaperEKELAND’S VARIATIONAL PRINCIPLE IN SJS-METRIC SPACESIsmat Beg1, Kushal Roy2 and Mantu Saha21Centre for Mathematics and Statistical Sciences, Lahore School of Economics,Lahore-53200, Pakistan2Department of Mathematics, The University of Burdwan,Purba Bardhaman-713104, West Bengal, IndiaAbstract. We prove Ekeland’s variational principle in SJS - metric spaces. A gener-alization of Caristi fixed point theorem on SJS - metric spaces is obtained as a conse-quence.Keywords:Ekeland’s variational principle; SJS- metric space; fixed point1. IntroductionIn his classic paper Ekeland [7] proved a theorem (Ekeland’s variational principle)that asserts that there exists nearly optimal solutions to some optimization prob-lems. Ekeland’s variational principle can be applied when the lower level set of aminimization problems is not compact, so that the Bolzano–Weierstrass theoremcannot be used. Ekeland’s principle relies on Cantor intersection theorem and axiomof choice. Ekeland’s principle also leads to an elegant proof of the famous Caristifixed point theorem [5]. For further generalizations and applications of Ekeland’svariational principle we refere to [2, 8, 9, 11] and their references. Recently Beg etal. [1, 12, 13] introduced a very general notion of SJS - metric spaces (see prelimi-naries) which does not satisfy the triangle inequality and symmetry, and obtainedseveral interesting results with examples. In fact b - metric spaces [6], Sb- metricspaces [14], JS- metric spaces [10], and partial metric spaces [4] are special casesReceived May 25, 2021. accepted November 26, 2021.Communicated by Qingxiang XuCorresponding Author: Ismat Beg, Centre for Mathematics and Statistical Sciences, Lahore Schoolof Economics, Lahore-53200, Pakistan | E-mail: ibeg@lahoreschool.edu.pk2010 Mathematics Subject Classification. Primary 47H10; Secondary 54H25© 2021 by University of Nǐs, Serbia | Creative Commons License: CC BY-NC-ND1118 I. Beg, K. Roy and M. Sahaof SJS - metric spaces. The aim of this paper is to prove a variant of Ekeland’svariational principle in SJS - metric spaces and then derive Caristi fixed point the-orem as an application. The results above generalize/extend several results fromthe existing literature.2. PreliminariesIn this section, we first give the notion of SJS- metric space (X, J), due to [1], somenotations and terminology and a lemma to use in next section.Let X be a nonempty set and J : X3 → [0,∞] be a function. We define the setS(J,X, x) = {{xn} ⊂ X : limn→∞J(x, x, xn) = 0},for all x ∈ X. If J satisfies(i) J(x, y, z) = 0 implies x = y = z for any x, y, z ∈ X;(ii) there exists some s > 0 such that for any (x, y, z) ∈ X3 and {zn} ∈S(J,X, z), we haveJ(x, y, z) ≤ s lim supn→∞(J(x, x, zn) + J(y, y, zn)),then the pair (X,J) is called an SJS- metric space (with coefficient s). Severalknown examples of SJS- metric spaces are given in [1] and [13], we give anotherexamples of SJS- metric spaces in the below.Example 2.1. Let X = R and J : X3 → [0,∞] be defined by J(x, y, z) = exp(|x|) +exp(|y|) + exp(|z|) − 3 for all x, y, z ∈ X, then clearly (J1) is satisfied. For any z 6= 0,S(J,X, z) = Ø. For any {zn} ∈ S(J,X, 0), we see thatJ(x, y, 0) ≤ h lim supn→∞(J(x, x, zn) + J(y, y, zn)),where h > 12, for all x, y ∈ X. Then condition (J2) is also satisfied. So J is an SJS-metric.It is a non-symmetric SJS-metric space.Example 2.2. LetX = R and J : X3 → [0,∞] be defined by J(x, y, z) = |x−y|+|y|+2|z|for all x, y, z ∈ X, then clearly (J1) is satisfied. For any z 6= 0, S(J,X, z) = Ø. If z = 0then for any sequence {zn} ∈ S(J,X, 0), we getJ(x, y, 0) = |x− y|+ |y| 6 |x|+ 2|y| 6 2(|x|+ |y|) = 2 lim supn→∞(J(x, x, zn) + J(y, y, zn)),for all x, y ∈ X. Therefore, the condition (J2) is satisfied and J is an SJS-metric on X. Itis a non-symmetric SJS-metric space.In an SJS- metric space (X, J), a sequence {xn} ⊂ X is said to be convergent toan element x ∈ X if {xn} ∈ S(J,X, x). A sequence {xn} ⊂ X is said to be Cauchyif limn,m→∞ J(xn, xn, xm) = 0.Ekeland’s variational principle... 1119Space (X, J) is said to be complete if every Cauchy sequence in X is convergent.Open ball of center x ∈ X and radius r > 0 in X is defined as follows:BJ(x, r) = {y ∈ X : J(x, x, y) < r}.A nonempty subset U of X, with the property that for any x ∈ U there existsr > 0 such that BJ(x, r) ⊂ U is called an open set. A subset B of X is calledclosed if Bc is open.Lemma 2.1. [1][Cantor’s Intersection Theorem] Every complete SJS- metric spacehas Cantor’s intersection property.3. Ekeland’s variational principleDefinition 3.1. In an SJS-metric space (X, J), a mapping ψ : X → R is said tobe lower semi-continuous at t0 ∈ X if for any ε > 0 there exits some δε > 0 suchthat ψ(t0) < ψ(t) + ε for all t ∈ BJ(t0, δε).Definition 3.2. Let (X,J) be an SJS-metric space and {An} be a decreasingsequence of nonempty subsets of X. Then {An} is said to have vanishing diameterproperty (vd−property) if for each i ∈ N there exists some fixed ai ∈ Ai such thatJ(x, x, ai) ≤ J(ai, ai, ai)+ri for all x ∈ Ai, where {ri} ⊂ R+ with ri → 0 as i→∞.Definition 3.3. An SJS-metric space (X, J) is said to have vanishing diame-ter property if for any decreasing sequence of nonempty subsets {An} of X withvd−property we have diam(An)→ 0 as n→∞.We now establish Ekeland’s variational principle in an SJS-metric space. Let usdenote dJ(x, y) = J(x, x, y) for all x, y ∈ X.Theorem 3.1. Let (X, J) be a complete SJS-metric space with coefficient s > 1,such that dJ is continuous in both variables, sup{J(x, x, x) : x ∈ X} < ∞ and Xhas vanishing diameter property. Now let, f : X → R be a lower semi-continuous,proper and lower bounded mapping. Then for every x0 ∈ X and ε > 0 withf(x0) ≤ infx∈Xf(x) + ε(3.1)there exists a sequence {xn} ⊂ X and xε ∈ X such that:(i) xn → xε as n→∞,(ii) For all n ≥ 1,J(xε, xε, xn)− J(xn, xn, xn) ≤ε2n1120 I. Beg, K. Roy and M. Saha(iii) For all x 6= xε,f(x) +∞∑n=01snJ(x, x, xn) > f(xε) +∞∑n=01snJ(xε, xε, xn)(iv)f(xε) +∞∑n=01snJ(xε, xε, xn) ≤ f(x0) +∞∑n=01snJ(xn, xn, xn)≤ infx∈Xf(x) + ε+∞∑n=01snJ(xn, xn, xn).Proof. Consider the setSf (x0) = {x ∈ X : f(x) + dJ(x, x0) ≤ f(x0) + dJ(x0, x0)}.Since x0 ∈ Sf (x0) then Sf (x0) is nonempty. Let {zn} ⊂ Sf (x0) be such that {zn}converges to some z ∈ X. Then f(zn)+dJ(zn, x0) ≤ f(x0)+dJ(x0, x0) for all n ∈ N.Now f is lower semi-continuous at z ∈ X, so for any ε1 > 0, f(z) < f(t) + ε12 forall t ∈ BJ(z, δε1) for δε1 > 0. Also {zn} converges to some z, so there exists N1 ≥ 1such that zn ∈ BJ(z, δε1) for all n ≥ N1. Therefore f(z) < f(zn)+ ε12 for all n ≥ N1.Now continuity of dJ implies that dJ(zn, x0) → dJ(z, x0) as n → ∞. Thus for alln ≥ N2dJ(z, x0)−ε12< dJ(zn, x0) < dJ(z, x0) +ε12.Therefore, for all n ≥ N = max{N1, N2} we get,f(z) + dJ(z, x0) < f(zn) + dJ(zn, x0) + ε1∀n > N≤ f(x0) + dJ(x0, x0) + ε1.(3.2)Since ε1 > 0 is arbitrary, thus f(z) + dJ(z, x0) ≤ f(x0) + dJ(x0, x0). Thereforez ∈ Sf (x0). Hence Sf (x0) is closed. Also for any y ∈ Sf (x0) we getdJ(y, x0)− dJ(x0, x0) ≤ f(x0)− f(y)≤ f(x0)− infx∈Xf(x) ≤ ε.(3.3)We choose x1 ∈ Sf (x0) such that f(x1)+dJ(x1, x0) ≤ infx∈Sf (x0){f(x)+dJ(x, x0)}+ε2s and letSf (x1) = {x ∈ X : f(x) + dJ(x, x0) +1sdJ(x, x1) ≤ f(x1) + dJ(x1, x0) +1sdJ(x1, x1)}.(3.4)Thus x1 ∈ Sf (x1) and in a similar way as above we can prove that Sf (x1) is alsoclosed.Ekeland’s variational principle... 1121Inductively, we can suppose that xn−1 ∈ Sf (xn−2) (for n > 2) was alreadychosen and we considerSf (xn−1) = {x ∈ Sf (xn−2) : f(x) +n−1∑i=01sidJ(x, xi) ≤ f(xn−1) +n−1∑i=01sidJ(xn−1, xi)}.(3.5)Let us choose xn ∈ Sf (xn−1) such thatf(xn) +n−1∑i=01sidJ(xn, xi) ≤ infx∈Sf (xn−1){f(x) +n−1∑i=01sidJ(x, xi)}+ε2nsnand we define the setSf (xn) = {x ∈ Sf (xn−1) : f(x) +n∑i=01sidJ(x, xi) ≤ f(xn) +n∑i=01sidJ(xn, xi)}.(3.6)Clearly xn ∈ Sf (xn) and Sf (xn) is also closed. Now for each y ∈ Sf (xn) we get1sndJ(y, xn) ≤ {f(xn) +n∑i=01sidJ(xn, xi)} − {f(y) +n−1∑i=01sidJ(y, xi)}≤ {f(xn) +n∑i=01sidJ(xn, xi)} − infx∈Sf (xn−1){f(x) +n−1∑i=01sidJ(x, xi)}≤ 1sndJ(xn, xn) +ε2nsn.(3.7)Therefore, for any y ∈ Sf (xn) we havedJ(y, xn)− dJ(xn, xn) ≤ε2n∀n ∈ N.Thus the decreasing sequence of nonempty closed subsets {Sf (xn)}n≥0 has vd−property.Since X has vd− property therefore diam(Sf (xn))→ 0 as n→∞. Thus by Cantor’sintersection theorem (See Lemma 2.1) we have ∩∞n=0Sf (xn) = {xε}.Now dJ(xε, xn) ≤ diam(Sf (xn)) → 0 as n → ∞ and we have xn → xε asn→∞. From (3.7) we see thatJ(xε, xε, xn)− J(xn, xn, xn) ≤ε2n∀n ∈ N.Nowf(x1) + dJ(x1, x0) ≤ f(x0) + dJ(x0, x0),f(x2) + dJ(x2, x0) +1sdJ(x2, x1) ≤ f(x1) + dJ(x1, x0) +1sdJ(x1, x1)1122 I. Beg, K. Roy and M. Saha≤ f(x0) + dJ(x0, x0) +1sdJ(x1, x1)...f(xm) +m−1∑i=01sidJ(xm, xi) ≤ f(x0) +m−1∑i=01sidJ(xi, xi)∀m > 1.(3.8)Also xε ∈ Sf (xq) for all q ∈ N, thereforef(xε) +q∑i=01sidJ(xε, xi) ≤ f(xq) +q∑i=01sidJ(xq, xi)≤ f(x0) +q∑i=01sidJ(xi, xi)∀q ≥ 1,(3.9)which in turn implies thatf(xε) +∞∑i=01sidJ(xε, xi) ≤ f(x0) +∞∑i=01sidJ(xi, xi)≤ infx∈Xf(x) + ε+∞∑i=01sidJ(xi, xi).(3.10)Moreover for all x 6= xε, we have x /∈ ∩∞n=0Sf (xn) and thus there exists m ∈ N suchthat x /∈ Sf (xm). So x /∈ Sf (xq) for all q ≥ m. Therefore,f(x) +q∑i=01sidJ(x, xi) > f(xq) +q∑i=01sidJ(xq, xi)≥ f(xε) +q∑i=01sidJ(xε, xi)∀q ≥ m.(3.11)Hence we see thatf(x) +∞∑i=01sidJ(x, xi) > f(xε) +∞∑i=01sidJ(xε, xi).Example 3.1. Let us consider X = (−∞,+∞) and let J : X3 → [0,∞] be definedas J(x, y, z) = |x − y|2 + |y − z|2 for all x, y, z ∈ X. Then (X, J) is an SJS-metricspace for s = 3. Here dJ(x, y) = |x − y|2, which is continuous in both the variables andsup{J(x, x, x) : x ∈ X} = 0. Now we show that X has vanishing diameter property.Let {En} be a decreasing sequence of nonempty subsets ofX such that it has vd−property.Then for any i ∈ N there exists some fixed ei ∈ Ei such that J(x, x, ei) = |x − ei|2 ≤J(ei, ei, ei) + ri = ri for all x ∈ Ei, where {ri} ⊂ R+ with ri → 0 as i→∞.Ekeland’s variational principle... 1123Let x(i), y(i), z(i) ∈ Ei be arbitrary. ThenJ(x(i), y(i), z(i)) = |x(i) − y(i)|2 + |y(i) − z(i)|26 2[|x(i) − ei|2 + |y(i) − ei|2] + 2[|y(i) − ei|2 + |z(i) − ei|2]= 2[|x(i) − ei|2 + 2|y(i) − ei|2] + |z(i) − ei|2]6 8ri → 0as i→∞. This implies diam(Ai) ≤ 8ri. Since this is true for all i ∈ N we get diam(Ai)→0 as ri →∞. Thus (X, J) has vanishing diameter property.Let f : X → R be defined as f(x) = e|x|+x2 + 4|x| for all x ∈ X. Then f is continuousand lower bounded. Let us take ε > 0 as arbitrary and choose x0 ∈ X which satisfiesf(x0) ≤ infx∈X f(x) + ε. Now let us consider xε = 0, if x0 = 0 then we have to choosexn = 0 for all n ≥ 1 and clearly Theorem 3.1 follows immediately. Now if x0 6= 0 then wechoose xn =√εKrn, where K ≥ 1 and r > 2 are chosen in such a way thatε ≤ min{K(3r − 1)3r[f(x0)− 1],K}.Then we have(i) xn → xε as n→∞,(ii) For all n ≥ 1,J(xε, xε, xn)− J(xn, xn, xn) = |xε − xn|2 =εKrn<ε2n(iii) For all x 6= xε,f(x) +∑∞n=01snJ(x, x, xn)= e|x| + x2 + 4|x|+∑∞n=013n|x−√εKrn|2= e|x| + x2 + 4|x|+ 32x2 − 2√εK3r123r12−1x+ εK3r3r−1> e|x| + x2 + 4|x|+ 32x2 − 2 3r123r12−1x+ εK3r3r−1> 1 + εK3r3r−1 = f(xε) +∑∞n=01snJ(xε, xε, xn).(iv)f(xε) +∞∑n=01snJ(xε, xε, xn) = 1 +εK3r3r − 1 6 f(x0)= f(x0) +∞∑n=01snJ(xn, xn, xn)≤ infx∈Xf(x) + ε+∞∑n=01snJ(xn, xn, xn).Next we have the following consequence of Ekeland’s variational principle inSJS-metric spaces.1124 I. Beg, K. Roy and M. SahaCorollary 3.1. Let (X, J) be a complete SJS-metric space with coefficient s > 1,such that dJ is continuous in both variables, sup{J(x, x, x) : x ∈ X} < ∞ and Xhas vanishing diameter property. Now let, f : X → R be a lower semi-continuous,proper and lower bounded mapping. Then for every ε > 0 there exists a sequence{xn} ⊂ X and xε ∈ X such that:(i) xn → xε as n→∞,(ii) f(x) +∑∞n=01sn J(x, x, xn) ≥ f(xε) +∑∞n=01sn J(xε, xε, xn) for every x ∈ X,(iii) f(xε) +∑∞n=01sn J(xε, xε, xn) ≤ infx∈X f(x) + ε+∑∞n=01sn J(xn, xn, xn).As an application of Theorem 3.1 we now prove Caristi’s fixed point theorem inthe context of SJS-metric spaces.Theorem 3.2. Let (X, J) be a complete SJS-metric space with coefficient s > 1,such that dJ is continuous in both variables, sup{J(x, x, x) : x ∈ X} <∞ and X hasvanishing diameter property. Let T : X → X be an operator for which there existsa lower semi-continuous mapping, proper and lower bounded mapping f : X → Rsuch thatJ(u, u, v) + sJ(u, u, Tu) ≥ J(Tu, Tu, v)(3.12)ands2s− 1J(u, u, Tu) ≤ f(u)− f(Tu)∀u, v ∈ X.(3.13)Then T has at least one fixed point in X.Proof. Let us assume that for all x ∈ X, Tx 6= x. Using Corollary 3.1 for f , weobtain that for each ε > 0 there exists a sequence {xn} ⊂ X such that xn → xε asn→∞ andf(x) +∞∑n=01snJ(x, x, xn) > f(xε) +∞∑n=01snJ(xε, xε, xn)∀x 6= xε.If in the above inequality, we put x = T (xε) then, since T (xε) 6= xε, we get thatf(xε)− f(Txε) <∞∑n=01sn[dJ(Txε, xn)− dJ(xε, xn)]<∞∑n=01snsdJ(xε, Txε)(5.15)= s∞∑n=01sndJ(xε, Txε)=s2s− 1dJ(xε, Txε).(3.14)Ekeland’s variational principle... 1125Also from (3.13) we get s2s−1dJ(xε, Txε) ≤ f(xε)−f(Txε), a contradiction. Thereforethere exists at least one x∗ ∈ X such that Tx∗ = x∗.Definition 3.4. [14] Let X be a nonempty set and s ≥ 1 be a given number.Also let a function Sb : X3 → [0,∞) satisfy the following conditions, for eachx, y, z, w ∈ X :(i) Sb(x, y, z) = 0 if and only if x = y = z;(ii) Sb(x, y, z) ≤ s[Sb(x, x, w) + Sb(y, y, w) + Sb(z, z, w)].The pair (X,Sb) is called an Sb-metric space.Souayah and Mlaiki [14, Theorem 2.4] follows from our Theorem 3.1 as animmediate corollary.Corollary 3.2. Let (X,Sb) be a complete Sb-metric space with coefficient s > 1,such that the Sb-metric is continuous and f : X → R is a lower semi-continuous,proper and lower bounded mapping. Then for every x0 ∈ X and ε > 0 withf(x0) ≤ infx∈Xf(x) + ε,(3.15)there exists a sequence {xn} ⊂ X and xε ∈ X such that:(i) xn → xε as n→∞,(ii) Sb(xε, xε, xn) ≤ ε2n for all n ≥ 1,(iii) f(x) +∑∞n=01snSb(x, x, xn) > f(xε) +∑∞n=01snSb(xε, xε, xn) for everyx 6= xε,(iv) f(xε) +∑∞n=01snSb(xε, xε, xn) ≤ f(x0) ≤ infx∈X f(x) + ε.Proof. Let {An} be a decreasing sequence of nonempty subsets of X such that ithas vd−property. Then for each i ∈ N there exists some fixed ai ∈ Ai such thatSb(x, x, ai) ≤ Sb(ai, ai, ai) + ri = ri for all x ∈ Ai, where {ri} ⊂ R+ with ri → 0 asi→∞.Let x(i), y(i), z(i) ∈ Ai be arbitrary. ThenSb(x(i), y(i), z(i)) ≤ s[Sb(x(i), x(i), ai) + Sb(y(i), y(i), ai) + Sb(z(i), z(i), ai)]≤ 3sri.(3.16)It implies diam(Ai) ≤ 3sri. Since this is true for all i ∈ N we get diam(Ai)→ 0 asri →∞. Thus (X,Sb) has vanishing diameter property. Therefore all the conditionsof Theorem 3.1 are satisfied and the result follows immediately.Corollary 3.3. Let (X,Sb) be a complete Sb-metric space with coefficient s > 1,such that the Sb-metric is continuous and let T : X → X be an operator for which1126 I. Beg, K. Roy and M. Sahathere exists a lower semi-continuous, proper and lower bounded mapping f : X → R,such that:Sb(u, u, v) + sSb(u, u, Tu) ≥ Sb(Tu, Tu, v)(3.17)ands2s− 1Sb(u, u, Tu) ≤ f(u)− f(Tu)∀u, v ∈ X.(3.18)Then T has at least one fixed point in X.Proof. Using Theorem 3.2 and Corollary 3.2 we get the required proof.Remark 3.1. [3, Theorem 2.2] is a particular case of our Theorem 3.1.AcknowledgmentsAuthors would like to thank the editor and the learned reviewers for their valuablecomments and constructive suggestions that have helped us to significantly improvethe paper. Kushal Roy acknowledges financial support awarded by the Council ofScientific and Industrial Research, New Delhi, India, through research fellowship(Award No; 09/025(0223)/2017-EMR-I) for carrying out research work leading tothe preparation of this manuscript.REFERENCES1. I. Beg, K. Roy and M. Saha: SJS- metric and topological spaces,. J. Math.Extension, 15(4) (2021), 1–16.2. J. M. Borwein and Q. J. Zhu: Techniques of Variational Analysis, Springer,2005.3. M. Bota, A. Molnar and C. Varga: On Ekeland’s variation principle in b-metric spaces, Fixed Point Theory, 12(2) (2011), 21–28.4. M. Bukatin and R. Kopperman, S. Mathews and H. Pajoohesh: Partialmetric spaces, Amer. Math. Mon., 116(8) (2009), 708–718.5. J. Caristi: Fixed point theorems for mappings satisfying inwardness conditions,Trans. Amer. Math. Soc., 215 (1976), 241–251.6. S. Czerwik: Contraction mappings in b- metric spaces, Acta Math. Inform. Univ.Ostrav., 1 (1993), 5–11.7. I. Ekeland: On the variational principle, J. Math. Anal. Appl., 47 (1974), 324–3538. A. P. Farajzadeh, S. Plubtieng and A. Hoseinpour: A generalization ofEkeland’s variational principle by using the τ -distance with its applications, J.Inequalities and Appl., 1 (2017), 1–7.9. E. Hashemi and R. Saadati: Ekeland’s variational principle and minimizationTakahashi’s theorem in generalized metric spaces, Mathematics, 6(6) (2018), 93.Ekeland’s variational principle... 112710. M. Jleli and B. Samet: A generalized metric space and related fixed pointtheorems, Fixed Point Theory and Applications, (2015), doi:10.1186/s13663-015-0312-7.11. I. Meghea: Ekeland Variational Principle with Generalizations and Variants,Old City Publishing, 2009.12. K. Roy, I. Beg and M. Saha: Sequentially compact SJS- metric spaces, Com-mun. Optim. Theory 2020, (2020), Article ID 4.13. K. Roy, M. Saha and I. Beg: Fixed point of contractive mappings of integraltype over an SJS- metric space, Tamkang J. Math., 52(2) (2021), 267–280.14. N. Souayah and N. Mlaiki: A fixed point theorem in Sb-metric spaces, J. Math.Computer Sci., 16 (2016), 131–139.

EKELAND'S VARIATIONAL PRINCIPLE IN S^{JS}-METRIC SPACES

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EKELAND'S VARIATIONAL PRINCIPLE IN S^{JS}-METRIC SPACES

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