NEW TYPE OF MULTIVALUED CONTRACTIONS WITH RELATED RESULTS AND APPLICATIONS

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1 U.P.B. Sci. Bull., Series A, Vol. 80, Iss., 018 ISSN NEW TYPE OF MULTIVALUED CONTRACTIONS WITH RELATED RESULTS AND APPLICATIONS Hüseyin Işik 1 nd Cristin Ionescu In this pper, we focus on chieving new results bout the existence of fixed points for new type of multivlued contrctions. We furnish n exmple which demonstrte the supremcy of our results to the existing ones in the literture. We derive new fixed point results on metric spce endowed with prtil ordering/grph by using the results obtined herein. We lso discuss sufficient conditions to ensure the existence of solutions of integrl equtions s n ppliction of our results. Keywords: θ-contrction, dmissible mpping, α-ψ-contrction, prtil order, grph, integrl eqution. MSC010: 47H10, 54H5. 1. Introduction In 1937, Von Neumnn [19] gve strt the fixed point theory for multivlued mppings in the study of gme theory. In prticulr, the fixed point theorems for multivlued mppings re rther dvntgeous in optiml control theory nd hve been frequently used to solve mny problems of economics nd gme theory. Consecutively, Ndler [9] initited the development of the geometric fixed point theory for multivlued mppings by using the notion of the Husdorff metric nd extended Bnch contrction principle to multivlued mppings, which is known s Ndler s multivlued contrction principle. Recently, new types of single vlued week contrctive mppings with control functions, clled s θ-contrction nd α-ψ-contrction respectively, re introduced in [7] nd [11] long mny others in the literture: for instnce, plese see [1, 13]. This pproch llowed to estblish existence nd uniqueness results for fixed points, which improve Bnch contrction principle [5, 6], nd the development of numericl lgorithms for suitble clsses of problems with rel world pplictions [14, 15, 16, 17]. Lter on, by using these concepts, severl reserchers extended the results in [7] nd [11] to multivlued mppings, see, for exmple, Ali et l. [1], Ali nd Kirn [], Asl et l. [3], Mohmmdi et l. [8] nd Vetro [18]. In this study, we introduce new type of multivlued contrctions to estblish existence results for fixed points of this new type of contrctions on complete metric spces. Our results improve nd extend the results in Asl et l. [3], Mohmmdi et l. [8], Vetro [18] nd mny others in the literture. An exmple is constructed in order to illustrte the generlity of our results. As pplictions of the obtined results, some new fixed point theorems re presented on metric spce endowed with prtil ordering/grph nd sufficient conditions re discussed to ensure the existence of solutions of integrl equtions. 1 Deprtment of Mthemtics, Fculty of Science nd Arts, Muş Alprsln University, Muş 49100, Turkey, e-mil: isikhuseyin76@gmil.com Deprtment of Mthemtics nd Computer Science, University Politehnic of Buchrest, Buchrest, Romni, e-mil: cristinionescu58@yhoo.com 13

2 14 Hüseyin Işik, Cristin Ionescu. Preliminries nd Bckground Here, we recollect some bsic definitions, lemms, nottions nd some known theorems which re helpful for understnding of this pper. Let (X, d) be metric spce nd denote the fmily of nonempty, closed nd bounded subsets of X by CB(X). For A, B CB(X), define H : CB(X) CB(X) [0, + ) by H(A, B) = mx { sup A d(, B), sup b B } d(b, A) where d(, B) = inf {d(, x): x B}. Such function H is clled the Pompeiu-Husdorff metric induced by the metric d, for more detils, see [4]. Also, denote the fmily of nonempty nd closed subsets of X by CL(X) nd the fmily of nonempty nd compct subsets of X by K(X). Note tht H : CL(X) CL(X) [0, + ] is generlized Pompeiu-Husdorff metric, tht is, H(A, B) = + if mx {sup A d(, B), sup b B d(b, A)} does not exist. Lemm.1 ([18]). Let (X, d) be metric spce nd A, B CL(X) with H(A, B) > 0. Then, for ech h > 1 nd for ech A, there exists b = b() B such tht d(, b) < hh(a, B). By the properties of closed sets, one deduces the following lemm. Lemm.. Let (X, d) be metric spce. For A CL(X) nd x X, d(x, A) = 0 if nd only if x A. Firstly, Asl et l. [3] dpted the notion of α-dmissible to multivlued mppings s α -dmissible. Afterwrds, Mohmmdi et l. [8] introduced the concept of α-dmissible for multivlued mppings. Let (X, d) be metric spce nd α: X X [0, + ) be given mpping. A mpping T : X CL(X) is n (1) α -dmissible, if α(x, y) 1 implies α (T x, T y) 1, where α (T x, T y) = inf {α(, b) : T x, b T y}; () α-dmissible, if for ech x X nd y T x with α(x, y) 1, we hve α(y, z) 1 for ll z T y. One cn esily see tht ech α -dmissible mpping is lso α-dmissible, but the converse is not true in generl. Let Ψ be the fmily of nondecresing functions ψ : [0, + ) [0, + ) such tht + n=1 ψn (t) < + for ll t > 0. If ψ Ψ, then it is esy to see tht ψ(t) < t for ll t > 0. Let (X, d) be metric spce. A mp T : X CL(X) is (1) multivlued α -ψ-contrction, if () multivlued α-ψ-contrction, if α (T x, T y)h(t x, T y) ψ(d(x, y)); α(x, y)h(t x, T y) ψ(d(x, y)), for ll x, y X where ψ Ψ nd α : X X [0, + ). Theorem.1 ([3, 8]). Let (X, d) be complete metric spce, ψ Ψ be strictly incresing function nd T : X CL(X) be given mpping. Assume tht the following conditions re stisfied: (i) T is α-dmissible nd multivlued α-ψ-contrction (or α -dmissible nd multivlued α -ψ-contrction); (ii) There exist x 0 X nd x 1 T x 0 such tht α (x 0, x 1 ) 1; (iii) T is continuous or

3 New type of multivlued contrctions with relted results nd pplictions 15 (iv) X is α-regulr, tht is, for every sequence {x n } in X such tht x n x X nd α (x n, x n+1 ) 1 for ll n N, then α (x n, x) 1 for ll n N. Then T hs fixed point, tht is, there exists u X such tht u T u. In 014, Jleli nd Smet [7] introduced new type of contrctive mppings, known s θ-contrction. Following the results in [7], Vetro [18] presented fixed point results for multivlued mppings. Definition.1 ([7, 18]). Let (X, d) be metric spce. A mp T : X CL(X) is clled wek θ-contrction, if there exist k (0, 1) nd θ Θ such tht θ(h(t x, T y)) [θ(d(x, y))] k, (1) for ll x, y X with H(T x, T y) > 0, where Θ is the set of functions θ : (0, + ) (1, + ) stisfying the following conditions: (θ1) θ is non-decresing; (θ) for ech sequence {t n } (0, + ), lim n + θ(t n ) = 1 if nd only if lim n + t n = 0; θ(t) 1 (θ3) there exist r (0, 1) nd λ (0, + ] such tht lim t 0 + t r = λ. The following functions θ i : (0, + ) (1, + ) for i {1, }, re the elements of Θ. Furthermore, substituting in (1) these functions, we obtin some contrctions known in the literture: for ll x, y X with H(T x, T y) > 0, θ 1 (t) = e t, θ (t) = e te t, H(T x, T y) k d(x, y), H(T x, T y) e H(T x,t y) d(x,y) k. d(x, y) Theorem. ([18]). Let (X, d) be complete metric spce nd T : X K(X) be wek θ-contrction. Then T hs fixed point. Note tht Theorem. is invlid, if we tke CB(X) insted of K(X). In the reference [18], Vetro showed tht Theorem. is still true for T : X CB(X), whenever θ Θ is right continuous. 3. The Results We begin this section with the following definition. Definition 3.1. Let (X, d) be metric spce nd α: X X [0, + ) be given function. A mpping T : X CL(X) is clled multivlued (α-θ-ψ)-contrction, if there exist θ Θ, ψ Ψ nd k (0, 1) such tht for ll x, y X with α(x, y) 1 nd H(T x, T y) > 0. θ(h(t x, T y)) [θ(ψ(d(x, y)))] k, () Remrk 3.1. Let (X, d) be metric spce. If T : X CL(X) is multivlued (α-θ-ψ)- contrction, then by (θ1) nd (), we deduce tht H(T x, T y) < ψ(d(x, y)), for ll x, y X with α(x, y) 1 nd H(T x, T y) > 0. Hence, we hve for ll x, y X with α(x, y) 1. H(T x, T y) ψ(d(x, y)), Now, we cn stte the first result of this pper.

4 16 Hüseyin Işik, Cristin Ionescu Theorem 3.1. Let (X, d) be complete metric spce nd T : X K(X) be multivlued (α-θ-ψ)-contrction. Assume tht the following conditions re stisfied: (i) T is α-dmissible; (ii) There exist x 0 X nd x 1 T x 0 such tht α (x 0, x 1 ) 1; (iii) T is continuous or X is α-regulr. Then T hs fixed point. Proof. By the ssumption (ii), there exist x 0 X nd x 1 T x 0 such tht α (x 0, x 1 ) 1. If x 0 = x 1 or x 1 T x 1, then x 1 is fixed point of T nd so the proof is completed. Becuse of this, ssume tht x 0 x 1 nd x 1 / T x 1, then d(x 1, T x 1 ) > 0 nd hence H(T x 0, T x 1 ) > 0. Since T x 1 is compct, there exists x T x 1 such tht d(x 1, x ) = d(x 1, T x 1 ). Now, considering () nd (θ1), we infer 1 < θ(d(x 1, x )) = θ(d(x 1, T x 1 )) θ(h(t x 0, T x 1 )) [θ(ψ(d(x 0, x 1 )))] k < [θ(d(x 0, x 1 ))] k. Following the previous procedures, we cn ssume tht x 1 x nd x / T x. Then d(x, T x ) > 0, nd so H(T x 1, T x ) > 0. Since, α(x 0, x 1 ) 1 nd T is n α-dmissible multivlued mpping, we derive tht α(x 1, x ) 1 for x T x 1. Also, since T x is compct, there exists x 3 T x such tht d(x, x 3 ) = d(x, T x ). Regrding (θ1) nd (), we deduce 1 < θ(d(x, x 3 )) = θ(d(x, T x )) θ(h(t x 1, T x )) [θ(ψ(d(x 1, x )))] k < [θ(d(x 1, x ))] k. Repeting this process, we cn constitute sequence {x n } X such tht x n x n+1 T x n, α(x n, x n+1 ) 1 nd 1 < θ(d(x n, x n+1 )) < [θ(d(x n 1, x n ))] k, (3) for ll n N. Letting ρ n := d(x n, x n+1 ) for ll n N {0}, from (3), we get 1 < θ(ρ n ) < [θ(ρ 0 )] kn, for ll n N. (4) This implies tht lim θ(ρ n) = 1, n + nd by (θ), we hve lim ρ n = 0. (5) n + To prove tht {x n } is Cuchy sequence, let us consider condition (θ3). Then there exist r (0, 1) nd λ (0, + ] such tht θ(ρ n ) 1 lim n + (ρ n ) r = λ. (6) Tke δ (0, λ). By the definition of limit, there exists n 0 N such tht [ρ n ] r δ 1 [θ(ρ n ) 1], for ll n > n 0. Using (4) nd the bove inequlity, we deduce This implies tht n[ρ n ] r δ 1 n([θ(ρ 0 )] kn 1), for ll n > n 0. lim n[ρ n] r = lim n[d(x n, x n+1 )] r = 0. n + n +

5 New type of multivlued contrctions with relted results nd pplictions 17 Thence, there exists n 1 N such tht d(x n, x n+1 ) 1 n 1/r, for ll n > n 1. (7) Let m > n > n 1. Then, using the tringulr inequlity nd (7), we hve d(x n, x m ) m 1 k=n d(x k, x k+1 ) m 1 k=n 1 k 1/r k=n 1 k 1/r nd hence {x n } is Cuchy sequence in X. From the completeness of (X, d), there exists u X such tht x n u s n +. If T is continuous, then which gives tht d(u, T u) = lim H(T x n, T u) = 0, n + lim d(x n+1, T u) lim H(T x n, T u) = 0, n + n + nd so d(u, T u) = 0. Since T u is closed, we obtin tht u T u, tht is, u is fixed point of T. If X is α-regulr, then α(x n, u) 1 for ll n N. If there exists k N such tht d(x k+1, T u) = 0, then from the uniqueness of limit, d(u, T u) = 0. So the proof is finished. Hence, there exists n N such tht d(x n+1, T u) > 0 nd so H(T x n, T u) > 0 for ll n > n. Considering Remrk 3.1, we hve nd so d(x n+1, T u) H(T x n, T u) < ψ(d(x n, u)) < d(x n, u), 0 < d(x n+1, T u) < d(x n, u), for ll n > n. Pssing to limit s n + in the bove inequlity, we obtin d(u, T u) = 0 nd so u T u. In the next theorem, we replce K(X) with CB(X) by considering n dditionl condition for the function θ. Theorem 3.. Let (X, d) be complete metric spce nd T : X CB(X) be multivlued (α-θ-ψ)-contrction with right continuous function θ Θ. Assume tht the following conditions re stisfied: (i) T is α-dmissible; (ii) There exist x 0 X nd x 1 T x 0 such tht α (x 0, x 1 ) 1; (iii) T is continuous or X is α-regulr. Then T hs fixed point. Proof. Strting with (ii), there exist x 0 X nd x 1 T x 0 such tht α (x 0, x 1 ) 1. Arguing similr lines in Theorem 3.1, we cn ssume tht x 0 x 1 nd x 1 / T x 1. By (θ1) nd (), we get θ(h(t x 0, T x 1 )) [θ(ψ(d(x 0, x 1 )))] k < [θ(d(x 0, x 1 ))] k. By the property of right continuity of θ Θ, there exists rel number h 1 > 1 such tht θ(h 1 H(T x 0, T x 1 )) [θ(d(x 0, x 1 ))] k. (8) From d(x 1, T x 1 ) < h 1 H(T x 0, T x 1 ), by Lemm.1, there exists x T x 1 such tht d(x 1, x ) h 1 H(T x 0, T x 1 ). Then, using (θ1), (8) nd lst inequlity, we infer tht θ(d(x 1, x )) θ(h 1 H(T x 0, T x 1 )) [θ(d(x 0, x 1 ))] k.

6 18 Hüseyin Işik, Cristin Ionescu In view of the fct tht T is α-dmissible nd α(x 0, x 1 ) 1, we hve α(x 1, x ) 1 for x T x 1. Assume tht x / T x. Since θ is right continuous, there exists h > 1 such tht θ(h H(T x 1, T x )) [θ(d(x 1, x ))] k. (9) From d(x, T x ) < h H(T x 1, T x ), by Lemm.1, there exists x 3 T x such tht d(x, x 3 ) h H(T x 1, T x ). Then, using (θ1), (9) nd lst inequlity, we deduce tht θ(d(x, x 3 )) θ(h H(T x 1, T x )) [θ(d(x 1, x ))] k [θ(d(x 0, x 1 ))] k. Continuing in this mnner, we build two sequences {x n } X nd {h n } (1, + ) such tht x n x n+1 T x n, α(x n, x n+1 ) 1 nd Hence, 1 < θ(d(x n, x n+1 )) θ(h n H(T x n 1, T x n )) [θ(d(x n 1, x n ))] k, for ll n N. 1 < θ(d(x n, x n+1 )) [θ(d(x 0, x 1 ))] kn, for ll n N. which gives tht lim θ(d(x n, x n+1 )) = 1. n + From (θ), we obtin lim d(x n, x n+1 ) = 0. n + The rest of the proof is like in the proof of Theorem 3.1. The following exmple illustrtes Theorem 3. (resp. Theorem 3.1) where Theorems.1 nd. re not pplicble. Exmple 3.1. Let X = [0, + ) with the usul metric d (x, y) = x y for ll x, y X. Define T : X CB(X) nd α: X X [0, + ) by [ 0, x ] 9, if x [0, 3],, if x, y [0, 3], 8 T x = nd α (x, y) = [0, x], if x > 3, 0, otherwise. Obviously, α(x, y) 1 nd H(T x, T y) > 0 for ech x, y [0, 3] with x y. Firstly, we clim tht T is multivlued (α-θ-ψ)-contrction with k = 1, ψ(t) = t nd θ(t) = e te t. For ll x, y [0, 3] with x y, ( ) x y θ (H (T x, T y)) = θ 8 = e e 1 = e 1 x y 8 e x y 8 x y e x y ψ(d(x,y))e ψ(d(x,y)) = [θ(ψ(d(x, y)))] k, tht is, the condition () is stisfied. Moreover, it is esy to see tht T is n α-dmissible multivlued mpping nd there exist x 0 = 3 nd x 1 = 3/8 T x 0 such tht α (x 0, x 1 ) 1. For ech sequence {x n } in X with x n x X s n + nd α (x n, x n+1 ) 1 for ll n, we hve x, x n [0, 3] for ll n. Hence, α (x n, x) 1 for ll n, tht is, X is α-regulr. Consequently, ll conditions of Theorem 3. (resp. Theorem 3.1) re stisfied. Then T hs fixed point in X. Note tht the set of fixed points of T is not finite.

7 New type of multivlued contrctions with relted results nd pplictions 19 On the other side, for x = 0 nd y = 4, we hve θ (H (T x, T y)) = θ (H (T 0, T 4)) = θ (4) > [θ (4)] k = [θ (d(x, y))] k, for ll θ Θ nd k (0, 1). Therefore, T is not wek θ-contrction nd hence Theorem. cn not pplied to this exmple. Also, for x = 0 nd y = 3, we get α(x, y)h (T x, T y) = α(0, 3)H (T 0, T 3) = = 7 16 > 3 = ψ(d(x, y)), for ψ(t) = t. Thus, T is not multivlued α-ψ-contrction nd so Theorem.1 cn not pplied to this exmple. Since ech α -dmissible mpping is lso α-dmissible, we obtin following result. Corollry 3.1. Let (X, d) be complete metric spce, α: X X [0, + ) be function nd T : X CB(X) (resp. K(X)) be multivlued mpping. Assume tht the following ssertions hold: (i) T is n α -dmissible; (ii) There exist x 0 X nd x 1 T x 0 such tht α (x 0, x 1 ) 1; (iii) T is continuous or X is α-regulr; (iv) There exist k (0, 1), ψ Ψ nd θ Θ such tht θ(h(t x, T y)) [θ(ψ(d(x, y)))] k, for ll x, y X with α (T x, T y) 1 nd H(T x, T y) > 0. Then T hs fixed point. Corollry 3.. Let (X, d) be complete metric spce, α: X X [0, + ) be function nd T : X CB(X) (resp. K(X)) be n α-dmissible multivlued mpping nd the following ssertions hold: (i) There exist x 0 X nd x 1 T x 0 such tht α (x 0, x 1 ) 1; (ii) T is continuous or X is α-regulr; (iii) there exist k (0, 1), ψ Ψ nd θ Θ such tht x, y X, H(T x, T y) > 0 θ(α(x, y)h(t x, T y)) [θ(ψ(d(x, y)))] k. (10) Then T hs fixed point. Proof. Let x, y X such tht α (x, y) 1 nd H(T x, T y) > 0. Using (θ1) nd (10), we hve nd hence θ(h(t x, T y)) θ(α(x, y)d(t x, T y)) [θ(ψ(d(x, y)))] k, θ(h(t x, T y)) [θ(ψ(d(x, y)))] k, for ll x, y X with α (x, y) 1 nd H(T x, T y) > 0. This implies tht the inequlity () holds. Thus, the rest of proof follows from Theorem 3. (resp. Theorem 3.1). 4. Some Consequences In this section we give new fixed point results on metric spce endowed with prtil ordering/grph, by using the results provided in previous section. Define { 1, if x y, α: X X [0, + ), α (x, y) = 0, otherwise. Then the following result is direct consequence of our results.

8 0 Hüseyin Işik, Cristin Ionescu Theorem 4.1. Let (X,, d) be complete ordered metric spce nd T : X CB(X) (resp. K(X)) be multivlued mpping. Assume tht the following ssertions hold: (i) For ech x X nd y T x with x y, we hve y z for ll z T y; (ii) There exist x 0 X nd x 1 T x 0 such tht x 0 x 1 ; (iii) T is continuous or, for every sequence {x n } in X such tht x n x X nd x n x n+1 for ll n N, we hve x n x for ll n N; (iv) There exist k (0, 1), ψ Ψ nd θ Θ such tht θ(h(t x, T y)) [θ(ψ(d(x, y)))] k, for ll x, y X with x y nd H(T x, T y) > 0. Then T hs fixed point. Now, we present the existence of fixed point for multivlued mppings from metric spce X, endowed with grph, into the spce of nonempty closed nd bounded subsets of the metric spce. Consider grph G such tht the set V (G) of its vertices coincides with X nd the set E (G) of its edges contins ll loops; tht is, E (G), where = {(x, x) : x X}. We ssume G hs no prllel edges, so we cn identify G with the pir (V (G), E (G)). If we define the function { 1, if (x, y) E (G), α: X X [0, + ), α (x, y) = 0, otherwise, then the following result is direct consequence of our results. Theorem 4.. Let (X, d) be complete metric spce endowed with grph G nd T : X CB(X) (resp. K(X)) be multivlued mpping. Assume tht the following conditions re stisfied: (i) For ech x X nd y T x with (x, y) E(G), we hve (y, z) E(G) for ll z T y; (ii) There exist x 0 X nd x 1 T x 0 such tht (x 0, x 1 ) E(G); (iii) T is continuous or, for every sequence {x n } in X such tht x n x X nd (x n, x n+1 ) E(G) for ll n N, we hve (x n, x) E(G) for ll n N; (iv) There exist k (0, 1), ψ Ψ nd θ Θ such tht θ(h(t x, T y)) [θ(ψ(d(x, y)))] k, for ll x, y X with (x, y) E(G) nd H(T x, T y) > 0. Then T hs fixed point. 5. An Appliction Consider the following integrl eqution: p (r) = q(r) + λ H (r, z) f(z, p(z))dz, r I = [, b], (11) where q : I R, H : I I R, f : I R R re given continuous functions. In this section, we estblish the existence of solutions for the integrl eqution (11) tht belongs to the spce X := C(I, R) of the continuous functions defined on I nd with rel vlues. Let X be endowed with the metric d defined by Then (X, d) is complete metric spce. d(x, y) = x y for ll x, y X.

9 New type of multivlued contrctions with relted results nd pplictions 1 We define n opertor T : X X, T p(r) := q(r) + λ H (r, z) f(z, p(z))dz, r I, then the existence of solutions of (11) is equivlent to the existence of fixed points of T. We will nlyze (11) under the following ssumptions: (A) λ 1; (B) For ech z I nd for ll x, y X with (x, y) E(G) nd x y, there exists β (0, + ) such tht nd f(z, x(z)) f(z, y(z)) ξ(x, y)( x(z) y(z) ) H (r, z) ξ(x, y)dz e β ; (C) x, y X, (x, y) E(G) implies (T x, T y) E(G); (D) There exists x 0 X such tht ξ (x 0, T x 0 ) E(G); (E) if {x n } is sequence in X such tht x n x X nd (x n, x n+1 ) E(G) for ll n N, then (x n, x) E(G) for ll n N. Theorem 5.1. Under the ssumptions (A)-(E), the integrl eqution (11) hs t lest one solution in X. Proof. Let (x, y) E(G) nd T x T y. On ccount of (A), for ll r I ( T x(r) T y(r) = λ H (r, z) f(z, x(z))dz H (r, z) f(z, y(z))dz) = λ H (r, z) [f(z, x(z)) f(z, y(z))] dz Thus, we hve nd hence λ λ H (r, z) f(z, x(z)) f(z, y(z)) dz H (r, z) ξ(x, y) x(z) y(z) dz λ x y H (r, z) ξ(x, y)dz x y H (r, z) ξ(x, y)dz. T x T y x y H (r, z) ξ(x, y)dz, d(t x, T y) e β d(x, y). Since θ(t) = e t Θ for ll t > 0, by the lst inequlity, we infer [ e d(t (x,y)) e β d(x,y) d(x,y) e e ] k

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