In this paper we give Caristi type fixed point theorem in complete modular metric spaces. Moreover we give a theorem which can be derived from Caristi type. Also an application for the bounded solution of funcional equations is investigated

Duran Turkoglu

Emine Kilinc

International Journal of Analysis and Applications

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Some Fixed Point Results for Caristi Type Mappings in Modular Metric Spaces with an Application

International Journal of Analysis and ApplicationsISSN 2291-8639Volume 12, Number 1 (2016), 15-21http://www.etamaths.comSOME FIXED POINT RESULTS FOR CARISTI TYPE MAPPINGS INMODULAR METRIC SPACES WITH AN APPLICATIONDURAN TURKOGLU1 AND EMINE KILINÇ2,∗Abstract. In this paper we give Caristi type fixed point theorem in complete modular metricspaces. Moreover we give a theorem which can be derived from Caristi type. Also an application forthe bounded solution of funcional equations is investigated.1. IntroductionFixed point theory is one of the very popular tools in various fields. Since Banach intoruced thistheory in 1922[6], ıt has been extended and generalized by several authors. Caristi type fixed pointtheorem is one of these genealizations. It is a modification of ε−variational principle of Ekeland[13].It iscrucial in nonlinear analysis, in particular, optimization, variational inequalites, differantial equationsand contol theory.The notion of modular space was introduced by Nakano [20] and was intensively developed by Koshi,Shimogaki, Yamamuro (see [18, 21]) and others. A lot of mathematicians are interested in fixed pointof modular space. In 2008, Chistyakov introduced the notion of modular metric space generated byF-modular and developed the theory of this space [8], on the same idea was defined the notion ofa modular on an arbitrary set and developed the theory of metric space generated by modular suchthat called the modular metric spaces in 2010 [9]. Afrah A. N. Abdou [1] studied and proved somenew fixed points theorems for pointwise and asymptotic pointwise contraction mappings in modularmetric spaces. Azadifer et. al. [3] introduced the notion of modular G-metric spaces.Azadifer et. al.[5] proved the existence and uniqueness of a common fixed point of compatible mappings of integraltype in this space. Kılınç and Alaca [14] defined (ε, k)−uniformly locally contractive mappings andη-chainable concept and proved a fixed point theorem for these concepts in a complete modular metricspaces. Kılınç and Alaca [15] proved that two main fixed point theorems for commuting mappings inmodular metric spaces. Recently, many authors [4, 7, 10, 11, 12, 19] studied on different fixed pointresults for modular metric spaces. In 2014 Khamsi and Abdou investigated Hausdorff modular metricin modular metric spaces[16], and proved fixed point theorem for multivalued mappings.In this paper we investigate Caristi type fixed point theorems for multivalued mappings in modularmetric spaces, which is more general than the results of Khojasteh, Karapinar and Khandani[17]. Andwe also give an application of results for functional equations.2. PreliminariesIn this section, we will give some basic concepts and definitions about modular metric spaces.Definition 2.1 [[9], Definition 2.1] Let X be a nonempty set, a function w : (0,∞)×X×X → [0,∞]is said to be a metric modular on X if satisfying, for all x, y, z ∈ X the following condition holds:(i) wλ (x, y) = 0 for all λ > 0 ⇔ x = y;(ii) wλ (x, y) = wλ (y, x) for all λ > 0;(iii) wλ+µ (x, y) ≤ wλ (x, z) +wµ (z, y) for all λ, µ > 0.If instead of (i), we have only the condition(i)́wλ (x, x) = 0 for all λ > 0, then w is said to be a (metric) pseudomodular on X.2010 Mathematics Subject Classification. 46A80, 47H10, 54E35.Key words and phrases. modular metric spaces; fixed point; Hausdorff metric.c©2016 Authors retain the copyrights oftheir papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.1516 TURKOGLU, AND KILINÇThe main property of a metric modular [[9]] w on a set X is the following: given x, y ∈ X, thefunction 0 < λ 7→ wλ (x, y) ∈ [0,∞] is nonincreasing on (0,∞). In fact, if 0 < µ < λ, then (iii),(i)́and (ii) implywλ (x, y) ≤ wλ−µ (x, x) + wµ (x, y) = wµ (x, y) .It follows that at each point λ > 0 the right limit wλ+0 (x, y) = limµ→λ+0wµ (x, y) and the left limitwλ−0 (x, y) = limε→+0wλ−ε (x, y) exist in [0,∞] and the following two inequalities hold:wλ+0 (x, y) ≤ wλ (x, y) ≤ wλ−0 (x, y) .Theorem 2.1 [[19]] Let Xw be a complete modular metric space and T a contraction on Xw. Then,the sequence (Tnx)n∈N converges to the unique fixed point of T in Xw for any initial x ∈ Xw.Now we give some definitions, which are useful for our main results.Definition 2.2 Let Xw be a modular metric space. Then following definitions exist:(1) The sequence (xn)n∈N in Xw is said to be convergent to x ∈ Xw if w1 (xn, x)→ 0, as n→∞(2) The sequence (xn)n∈N in Xw is said to be Cauchy if w1 (xm, xn)→ 0, as m,n→∞(3) A subset C of Xw is said to be closed if the limit of a convergent sequence of C always belongto C.(4) A subset C of Xw is said to be complete if any Cauchy sequence in C is a convergent sequenceand its limit is in C.(5) A subset C of Xw is said to be w−bounded ifδw(C) = sup {w1(x, y);x, y ∈ C} <∞.(6) A subset C of Xw is said to be w−compact if for any (xn) in C there exists a subset sequence(xnk) and x ∈ C such that w1(xnk , x)→ 0(7) w is said to satisfy the Fatou property if and only if for any sequence (xn)n∈N inXw w−convergentto x, we havew1(x, y) ≤ lim infn→∞w1(xn, y),for any y ∈ Xw.Now we will give some basic properties and notions of multivalued mappings in modular metricspaces which was given in [2] For a subset M of modular metric space Xw set(i) CB(M) = {C : C is nonempty w − closed and w − bounded subset of M} ;(ii) K(M) = {C : C is nonempty w − compact subset of M} ;(iii) the Haussdorf modular metric is defined onCB(M) byHw(A,B) = max{supx∈Aw1(x,B), supy∈Bw1(y,A)},where w1(x,B) = infy∈Bw1(x, y).Definition 2.3 Let Xw be a complete modular metric space and M be a nonempty subset of Xw.A mapping T : M → CB(M) is called a multivalued Lipschitzian mapping, if there exists a constantk > 0 such thatHw(Tx, Ty) ≤ kw1(x, y),for any x, y ∈M.A point x ∈ M is called fixed point of T whenever x ∈ Tx.The set of fixed points of T will bedenoted by Fix(T )It was shown in[2] that Definition 2.4 is more general than Theorem 2.1.SOME FIXED POINT RESULTS IN MODULAR METRIC SPACES 173. main resultsIn this section we will give a fixed point theorem for Caristi type mappings and a generalization ofthe theorem in modular metrc spaces. This works are more general than the results of [17].Theorem Let Xw be a complete modular metric space and T : Xw → CB(Xw) be a nonexpansivemapping such that for each x ∈ Xw and for all y ∈ Tx, there exists z ∈ Ty such that(3.1) wλ(x, y) ≤1λ(ϕw(x, y)− ϕw(y, z))specificially for λ = 1 we can writew1(x, y) ≤ ϕw(x, y)− ϕw(y, z)where ϕ : Xw× Xw → [0,∞] is lower semicontinuous with respect to the firs variable. Then forw1(xn, xn+1) <∞, T has a fixed point.Proof. Let x0 ∈ Xw and x1 ∈ Tx0. If x1 = x0, then x0 is a fixed point and theorem is satisfied.Otherwise, let x0 6= x1. By assumption there exists x2 ∈ Tx1 such thatw1(x0, x1) ≤ ϕw(x0, x1)− ϕw(x1, x2)Alternatively, one can choose xn ∈ Txn−1 such that xn 6= xn−1 and find xn+1 ∈ Txn such that(3.2) 0 < w1(xn−1, xn) ≤ ϕw(xn−1, xn)− ϕw(xn, xn+1),which means that (ϕw(xn−1, xn))n is a nonincreasing sequence in fact0 < w1(xn−1, xn) + ϕw(xn, xn+1) ≤ ϕw(xn−1, xn)0 < ϕw(xn, xn+1) ≤ ϕw(xn−1, xn)Thus it is bounded below, so it converges to some r ≥ 0. By taking the limit on both sides of (3.2) wegetlimn→∞w1(xn−1, xn) ≤ limn→∞(ϕw(xn−1, xn)− ϕw(xn, xn+1))limn→∞w1(xn−1, xn) ≤ limn→∞ϕw(xn−1, xn)− limn→∞ϕw(xn, xn+1)limn→∞w1(xn−1, xn) = r − rlimn→∞w1(xn−1, xn) = 0Now we show that w1(xn, xm−n) is Cauchy sequence . For all m,n ∈ N with m > n,(3.3) w1(xn, xm) ≤m∑i=n+1w 1m−n(xi−1, xi)On the other hand from the main property of metric modular one can choose thatw1(xn−1, xn) ≤ w 1m−n(xn−1, xn)w 1m−n(xn−1, xn) ≤1(m− n)ϕw(t)when we write this in (3.3) we getw1(xn, xm) ≤1m− n(ϕw(xn−1, xn)− · · · − ϕw(xm, xm+1))w1(xn, xm) ≤1m− n(ϕw(xn−1, xn)− ϕw(xm, xm+1))18 TURKOGLU, AND KILINÇWhen we take the limsup on both sides of the inequalities above, we havelimn→∞(sup {w1(xn, xm) : m > n} ≤ limn→∞(sup1(m− n)(ϕw(xn−1, xn)− ϕw(xm, xm+1)))limn→∞(sup {w1(xn, xm) : m > n} ≤ limn→∞(sup1(m− n))(r − r)limn→∞(sup {w1(xn, xm) : m > n} = 0Thus (xn) is a Cauchy sequence. Since Xw is complete it converges to u ∈ Xw. Now we show thatu is a fixed point of T.We havew1(u, Tu) ≤ w 12(u, xn+1) + w 12(Tu, xn+1)≤ w 12(u, xn+1) +Hw(Tu, Txn)≤ w 12(u, xn+1) + w 12(u, xn)When we take the limit on both sides of the inequalities above, we getlimn→∞w1(u, Tu) ≤ limn→∞w 12(u, xn+1) + limn→∞w 12(u, xn)limn→∞w1(u, Tu) ≤ w 12(u, u) + w 12(u, u)limn→∞w1(u, Tu) = 0Hence we get u ∈ Tu. Therefore u is fixed point of T. Now let us give the theorem which is a generalized version of the theorem above.Theorem Let Xw be a complete modular metric space and T : Xw → CB(Xw) be a multivaluedmapping such thatHw(Tx, Ty) ≤ η(w1(x, y))for all x, y ∈ Xw; where η : [0,∞]→ [0,∞] is a lower semicontunious map such that η(t) < t for allt ∈ [0,∞] , η(t)t is nondecreasing. Then T has a fixed point.Proof. Let x ∈ Xw and y ∈ Tx. If x = y then T has a fixed point and the proof is complete, so wesuppose that x 6= y. Let define thatθ(t) =η(t) + t2, for all t ∈ [0,∞]Then we haveHw(Tx, Ty) ≤ η(w1(x, y))Since η is nondecreasing and η(t) < t we getη(t) < t⇔ η(w1(x, y)) ≤ w1(x, y)θ(w1(x, y)) =η(w1(x, y)) + w1(x, y)2≤ w1(x, y)θ(w1(x, y)) =η(w1(x, y)) + w1(x, y)2≥ η(w1(x, y))Then we get(3.4) Hw(Tx, Ty) ≤ η(w1(x, y)) ≤ θ(w1(x, y)) ≤ w1(x, y)Thus there exists ε0 > 0 such that θ(w1(x, y) = Hw(Tx, Ty)+ ε0. So there exists z ∈ Ty such thatSOME FIXED POINT RESULTS IN MODULAR METRIC SPACES 19(3.5) w1(y, z) ≤ Hw(Tx, Ty) + ε0 = θ(w1(x, y)) ≤ w1(x, y)Again suppose that y 6= z.Thenw1(x, y)− θ(w1(x, y)) ≤ w1(x, y)− w1(y, z)or we can rewrite this inequality as(3.6) w1(x, y) ≤w1(x, y)1− θ(w1(x,y))w1(x,y)− w1(y, z)1− θ(w1(x,y))w1(x,y)Since θ(t)t is nondecreasing and w1(y, z) < w1(x, y) we rewrite the (3.5) asw1(x, y) ≤w1(x, y)1− θ(w1(x,y))w1(x,y)− w1(y, z)1− θ(w1(y,z))w1(y,z)Defineφw(x, y) ={w1(x,y)1− θ(w1(x,y))w1(x,y), x 6= y0 , x = yTherefore T satisfies (3.1) of Theorem 1, so we conclude that T has a fixed point u ∈ Xw and theproof is complete. 4. an application to functional equationsWe use fixed point theory in many fields of mathematics. One of theese fields is mathematical opti-mization.Dynamic programing is useful for mathematical optimizationand it is related to a multistageprocess reduces to solving functional equation p(x). In this section we try to give an application ofTheorem 2 to a functional equation defined as(4.1) p(x) = supy∈T{f(x, y) + =(x, y, p(η(x, y)))} , x ∈ Z,where η : Z × T → Z, f : Z × T → R, and = : Z × T × R→ R. We assume that M and N areBanach spacesZ ⊂Mand T ⊂ NNow we study the existence of the bounded solution of the functional equation (4.1). LetB(Z) denote the set of all bounded real-valued functions on Z, and for an arbitrary h ∈ B(Z), define‖h‖ = supx∈Z|h(x)| . Clearly, (B(Z), ‖.‖) endowed with the metric modular(4.2) wλ(h, k) =11 + λsupx∈Z|h(x)− k(x)|specificially for λ = 1 writen asw1(h, k) =12supx∈Z|h(x)− k(x)|for all h, k ∈ B(Z), is Banach space, so the convergence in this space according to w1(h, k) is uniformwhich means a Cauchy sequence in B(Z) is uniformly convergent to a function say h∗, that is boundedand so h∗ ∈ B(Z)Now we will give the theorem that gives the solution of the functional equation defined in (4.1). Togive the solution let us define an operator as follows:(4.3) S(h)(x) = supy∈T{f(x, y) + =(x, y, h(η(x, y)))}for all h∈ B(Z) and x ∈ ZNow we give the theorem for the bounded solution of functional equation given in (4.4)20 TURKOGLU, AND KILINÇTheorem Let S : B(Z) → B(Z)be an upper semicontinuous operator defined by (4.4) and assumethat the following conditions are satisfied:(i) f :Z × T → R, and = : Z×T × R→ R are continuous and bounded;(ii) for all h, k ∈ B(Z), if0 < w1(h, k) < 1implies12|=(x, y, h(η(x, y)))−=(x, y, k(η(x, y)))| ≤ w21(h, k)w1(h, k) ≥implies12|=(x, y, h(η(x, y)))−=(x, y, k(η(x, y)))| ≤ w1(h, k).where x ∈ Z and y ∈ T . Then the functional equation (4.1) has a bounded solution.Proof. ClearlyB(Z) is a complete modular metric space .Let µ > 0 be arbitrary, x ∈ Z, and h1, h2 ∈B(Z), then there exists y1, y2 ∈ T such thatS(h1)(x) < f(x, y) + =(x, y, h1(η(x, y))) + µS(h2)(x) < f(x, y) + =(x, y, h2(η(x, y))) + µS(h1)(x) ≥ f(x, y) + =(x, y, h1(η(x, y)))S(h2)(x) ≥ f(x, y) + =(x, y, h2(η(x, y)))Let % : [0,∞]→ [0,∞] be defined as%(t) ={t2, 0 < t < 1t, t ≥ 1Then we get12|=(x, y, h(η(x, y)))−=(x, y, k(η(x, y)))| ≤ %(w1(h, k))for all h, k ∈ B(Z).It is clear that %(t) < t for all t > 0 and %(t)t is nondecreasing function. Thereforewhen we use the inequalities above we get12(S(h1)(x)− S(h2)(x)) <12{=(x, y, h1(η(x, y)))−=(x, y, h2(η(x, y)))}+ µ≤ 12|=(x, y, h1(η(x, y)))−=(x, y, h2(η(x, y)))|+ µ≤ %(w1(h1, h2)) + µThen inequality turns into12{S(h1)(x)− S(h2)(x)} < %(w1(h1, h2)) + µAnalogously we get12{S(h2)(x)− S(h1)(x)} < %(w1(h1, h2)) + µHence theese inequalities equal to12|S(h1)(x)− S(h2)(x)| < %(w1(h1, h2)) + µwhich means thatw1(S(h1)(x), S(h2)(x)) < %(w1(h1, h2)) + µSOME FIXED POINT RESULTS IN MODULAR METRIC SPACES 21Since the above inequality turns intow1(S(h1)(x), S(h2)(x)) ≤ %(w1(h1, h2)),hence we get that S is a %−contraction. And S satisfies the conditions in Theorem 2 and S has a fixedpoint say h∗ ∈ B(Z), which means is a bounded solution of the functional equation (4.1). References[1] Afrah A.N. Abdou, On asymptotic pointwise contractions in modular metric spaces, Abstract and Applied Analysis2013(2013), Art. 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Book Ser, vol.1, Maruzen Co, Tokyo (1950).[21] S. Yamammuro, On conjugate space of Nakano space, Trans. Amer. Math. Soc., 90(1959), 291-311.1Department of Mathematics, Faculty of Science, Gazi University, Ankara, Turkey2Department of Mathematics, Institute of Natural and Applied Science, Gazi University, Ankara, Turkey∗Corresponding author: eklnc07@gmail.com

Some Fixed Point Results for Caristi Type Mappings in Modular Metric Spaces with an Application

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