Coincidence Points of a Sequence of Multivalued Mappings in Metric Space with a Graph

Transcription

1 mahemaics Aricle Coincidence Poins of a Sequence of Mulivalued Mappings in Meric Space wih a Graph Muhammad Nouman Aslam Khan,2,, Akbar Azam 2, and Nayyar Mehmood 3, *, School of Chemical and Maerials Engineering, Naional Universiy of Sciences and Technology, H-2, Islamabad 44, Pakisan; mnouman@scme.nus.edu.pk 2 Deparmen of Mahemaics, COMSATS Insiue of Informaion Technology, Chak Shahzad, Islamabad 44, Pakisan; akbarazam@yahoo.com 3 Deparmen of Mahemaics and Saisics, Inernaional Islamic Universiy, H-, Islamabad 44, Pakisan * Correspondence: nayyar.mehmood@iiu.edu.pk; Tel.: These auhors conribued equally o his work. Academic Edior: Lokenah Debnah Received: 24 March 27; Acceped: 22 May 27; Published: 26 May 27 Absrac: In his aricle he coincidence poins of a self map and a sequence of mulivalued maps are found in he seings of complee meric space endowed wih a graph. A novel resul of Asrifa and Verivel is generalized and as an applicaion we obain an exisence heorem for a special ype of fracional inegral equaion. Moreover, we esablish a resul on he convergence of successive approximaion of a sysem of Bernsein operaors on a Banach space. Keywords: graphic conracion; coincidence poins; sequence of mulivalued maps; Bernsein operaors. Inroducion and Preliminaries For he meric space (X, d), using he noions of Nadler [] and Hu [2], denoe CB(X), C (X) and 2 X by he collecion of nonempy closed and bounded, compac and all nonempy subses of X respecively. Consider A, B CB(X) he disance beween ses A and B is defined by d(a, B) = d(x, y), which does no allow o enjoy he properies of meric on CB (X) herefore a inf x A, y B well known idea of Hausdorff Pompeiu disance H on CB(X) induced by d is used o define a meric on CB (X) as follows: H(A, B) = inf{ɛ > : A N(ɛ, B), B N(ɛ, A)}, where: N(ɛ, A) = {x X : d(x, a) < ɛ, for some a A}. In 969, Nadler [] proved fixed poin resuls for mulivalued mappings in complee meric spaces, using he Hausdorff disance H, which was he generalizaion of Banach conracion principle in he seings of se-valued mappings. Coviz and Nadler [3] exended he idea of mulivalued mappings in he generalized meric spaces. Reich [4] in 972 published a fixed poin resul for he mulivalued maps on he compac subses of a complee meric space and posed he quesion, can C (X) be replaced by CB (X)?. In 989, Mizoguchi and Takahashi answered his quesion in Theorem 5 of [5] and hey also provide some Carisi ype heorems for mulivalued operaors. Whereas Hu [2] in 98 exended he mulivalued fixed poin resuls from complee meric space o complee ε-chainable meric space. Azam and Arshad [6] have exended he Theorem 6 of [] by finding he fixed poins of a sequence of locally conracive mulivalued maps in ε-chainable meric space. Furher Feng and Liu [7] used Mahemaics 27, 5, 3; doi:.339/mah523

2 Mahemaics 27, 5, 3 2 of he concep of lower semi-coninuiy and a generalized conracive condiion o exend he resul of Nadler [] and Carisi ype heorems as defined in [5]. For more references he readers are referred o he work of Ciric [8], Klim and Wardowski [9,], Nicolae []. Jachymski [2] in 27 unified and exended he work of Nieo [3] and Ran and Reuring [4] by defining a new class of conracions (G-conracion ) on meric space (X, d) endowed wih a graph. The conneciviy of he graph brings more aracions regarding a necessary and sufficien condiion for any G-conracive operaor o be a Picard operaor. In he presen aricle, fascinaed by [6] he exisence of coincidence poins of a sequence of mulivalued maps wih a self map are aken ino accoun wih a generalized form of G-conracion. This provides a new way o generalize many exising resuls in he lieraure (see [,6] and he references herein). Le us recall some definiions from graph heory wih he perspecive of using hem in our ideas and resuls. For a meric space (X, d) le be he diagonal of he Caresian produc X X. Consider a direced graph G such ha X = V(G), where V(G) is he se of verices of G. The se E(G) of edges of G conains all he loops. If G has no parallel edge hen we can idenify G wih he pair (V(G), E(G)). Furher, he graph G can be deal wih as a weighed graph if each edge is assigned by he disance beween is edges. Consider a direced graph G, hen G denoe he graph obained from G by reversing he direcion of edges and if we ignore he direcion of edges in graph G we ge an undireced graph G. The pair (V, E ) is said o be a subgraph of G if V V (G) and E E (G) and for any edge (a, b) E for all a, b V. Recall some fundamenal definiions regarding he conneciviy of graphs, which can be found in [5]. Definiion. A pah in G from he verex p o q of lengh K, is a sequence {p i } of K + verices such ha p = p,...,p K = q and (p j, p j ) E(G) for j =, 2,..., K. Definiion 2. A graph G is called conneced if here is a pah beween any wo verices. Graph G is weakly conneced if G is conneced. Definiion 3. For a, b and c in V (G), [a] G denoe he equivalence class of he relaion defined on V (G) by he rule: b c if here is a pah in G from b o c. For v V(G) and K N {} by [v] K G we denoe he se [v] K G := {u V(G) : here is a pah of lengh K from v o u }. Following is he definiion of G-conracion by Jachymski [2]. Definiion 4. [2] Le (X, d) be a meric space endowed wih a graph G. We say ha a mapping T : X X is a G-conracion if T preserves edges of G i.e., and here exiss some α [, ) such ha: (x, y) E(G) (Tx, Ty) E(G), x,y X (x, y) E(G) d(tx, Ty) αd(x, y). x,y X Mizoguchi and Takahashi [5] had defined a MT funcion as follows:

3 Mahemaics 27, 5, 3 3 of Definiion 5. [6] A funcion ϕ: [, + ) [, ) is said o be a MT funcion if i saisfies Mizoguchi and Takahashi s condiion (i.e., lim sup ϕ(r) < for all [, + )). Clearly, if ϕ: [, + ) [, ) is a r + nondecreasing funcion or a nonincreasing funcion, hen i is a MT funcion. Now we sae some resuls from he basic heory of mulivalued mappings. Lemma. [7] Le (X, d) be a meric space and A, B CB(X), wih H(A, B) < ɛ, hen for each a A, here exiss an elemen b B such ha: d(a, b) < ɛ. Lemma 2. [8] Le (X, d) be a meric space and A, B CB(X), hen for each a A: d(a, B) H(A, B). Lemma 3. [9] Le {A n } be a sequence in CB(X) and here exiss A CB(X) such ha lim n H(A n, A). If x n A n (n =, 2, 3,...) and here exiss x X such ha lim n d(x n, x) hen x A. 2. Main Resuls Definiion 6. [2] A mulivalued mapping F : X CB (X) is said o be Mizoguchi-Takahashi G-conracion if for all x, y in X, x = y wih (x, y) E (G) : (i) H (F (x), F (y)) ϕ (d (x, y)) d (x, y) ; (ii) If u F (x) and v F (y) are such ha d (u, v) d (x, y), hen (u, v) E (G). Moivaed by he Definiion 2. of [2], in a more general seings, we define he sequence of mulivalued G f -conracion as follows: Definiion 7. Le f : X X be a edge preserving surjecion. A sequence of mulivalued mappings {T q } q= from X ino CB(X) is said o be sequence of mulivalued G f -conracion if ( f u, f v) E(G), implies: H(T q (u), T r (v)) µ(d( f u, f v))d( f u, f v), for all q, r N. () For x T q (u) and y T r (v) saisfying d( f x, f y) d( f u, f v) implies ( f x, f y) E(G), where µ: [, ) [, ) is a MT-funcion. The nex heorem provides he way o find he coincidence of a self map and a sequence of mulivalued maps. Theorem. Le (X, d) a complee meric space,{t q } q= a sequence of mulivalued G f -conracion from X ino CB(X) and f : X X a surjecion. If here exis m N and v X, such ha: (i) T (v ) [ f v ] m G = φ; (ii) For any sequence {v n } in X, if v n v and v n T n (v n ) [v n ] m G for all n N, hen here exiss a subsequence { } ( v nk such ha vnk, v ) E(G) for all k N. Then f and sequence of mappings {T q } q= have a coincidence poin, i.e., here exiss v X such ha f v T q (v ). q N Proof. Choose any v X such ha f v T (v ) [ f v ] m G hen here exiss a pah from f v o f v, i.e., f v = f u () (,... f u() m = f v T (v ), and f u () i, f u () ) i+ E(G) for all i =,, 2,..., m.

4 Mahemaics 27, 5, 3 4 of Wihou any loss of generaliy, assume ha f u () k = f u () j for each k, j {,, 2,..., m} wih k = j. Since ( f u (), f u() ) E(G), so: H(T (u () ), T 2(u () )) µ(d( f u(), f u() < µ(d( f u (), f u() Rename f v as f u (2). As f u(2) T (u () such ha: d( f u (2), f u(2) Since ( f u (), f u() 2 ) E(G), so: < d( f u (), f u() ) ))d( f u(), f u() ) ))d( f u(), f u() ) ), and using Lemma one can find some f u(2) T 2 (u () ) ) < d( f u(), f u() ). H(T 2 (u () ), T 2(u () 2 )) µ(d( f u(), f u() 2 ))d( f u(), f u() 2 ) < d( f u (), f u() 2 ). Similarly since f u (2) T 2 (u () ), again using Lemma one can find some f u(2) 2 T 2 (u () 2 ) such ha: d( f u (2), f u(2) 2 ) < d( f u(), f u() 2 ). Thus we obain { f u (2), f u(2), f u(2) 2,, f u(2) m } of m + verices of X such ha f u (2) T (u () ) and f u (2) s T 2 (u () s ) for s =, 2,...., m, wih: d( f u (2) s, f u (2) s+ ) < d( f u() s, f u () s+ ), for s =,, 2,...., m. As ( f u () s, f u () s+ ) E(G) for all s =,, 2,...., m, hus ( f u(2) s, f u (2) s+ ) E(G) for all s =,, 2,...., m. Le f u m (2) = f v 2. Thus he se of poins f v = f u (2), f u(2), f u(2) 2,, f u(2) m = f v 2 T 2 (v ) is a pah from f v o f v 2. Rename f v 2 as f u (3). Then by he same procedure we obain a pah: from f v 2 o f v 3. Inducively, obained: wih: f v 2 = f u (3), f u(3), f u(3) 2,, f u(3) m = f v 3 T 3 (v 2 ) f v h = f u (h+), f u (h+), f u (h+) 2,, f u (h+) m = f v h+ T h+ (v h ) d( f u (h+), f u (h+) + ) < d( f u (h) + ), (2) hence ( f u (h+), f u (h+) + ) E(G) for =,, 2,...., m. Consequenly, consruc a sequence { f v h } h= of poins of X wih: f v = f u () m = f u (2) T (v ), f v 2 = f u (2) m = f u (3) T 2 (v ), f v 3 = f u (3) m = f u (4) T 3 (v 2 ), f v h+ = f u (h+) m. = f u (h+2) T h+ (v h ),

5 Mahemaics 27, 5, 3 5 of for all h N. For each {,, 2,..., m }, and from (2), clearly {d( f u (h) of non-negaive real numbers and so here exiss a such ha: lim h max r=,...,k d( f u(h) + ) = a. + )} h= is a decreasing sequence By assumpion, lim sup a + µ() <, so here exiss k N such ha µ(d( f u (h) + )) < ω(a ) for all h k where lim sup a + µ() < ω(a ) <. Now pu: { } Θ = max µ(d( f u (r), f u (r) + ω(a )), ). Then, for every h > k, consider: d( f u (h+), f u (h+) + ) < Puing q = max{k : =,, 2,..., m }, gives: Now for p > h > q, consider: < µ(d( f u (h) + ))d( f u(h) ω(a )d( f u (h) Θ d( f u (h) + ) + ) (Θ ) 2 d( f u (h ), f u (h ) + )... (Θ ) h d( f u (), f u () + ). d( f v h, f v h+ ) = d( f u (h+), f u (h+) m ) < m = m = d( f u (h+), f u (h+) + ) (Θ ) h d( f u (), f u () + + ) ), for all h > q. d( f v h, f v p ) d( f v h, f v h+ ) + d( f v h+, f v h+2 ) + + d( f v p, f v p ) < m (Θ ) h d( f u (), f u () m + ) + + (Θ ) p d( f u (), f u () + ). (3) = = Since Θ < for all {,, 2,..., m }, i follows ha { f v h = f u (h) m } is a Cauchy sequence. Using compleeness of X, find v X such ha f v h f v. Now using he fac ha f v n T (v n ) [ f v n ] m G for all n N, find a subsequence { f v nk } of { f vh } such ha ( f v nk, f v ) E(G) for all k N. Now for any q N : d ( f v, T q (v ) ) d ( f v, f v h+ ) + d ( f v h+, T q (v ) ) d ( f v, f v h+ ) + H ( T h+ (v h ), T q (v ) ) d ( f v, f v h+ ) + µ (d ( f v h, f v )) d ( f v h, f v ). Leing h in he above inequaliy, gives d ( f v, T q (v ) ), which implies f v T q (v ) for all q N. Hence, f v T q (v ) as required. q N

6 Mahemaics 27, 5, 3 6 of { } Example. Le X = {} q n : n N {} for q N. Consider he graph G such ha V (G) = X and for all x and y in X : E (G) = {(x, y) : x = y}. For q N, le T q : X CB(X) be defined by: } {, { q +, } T q (x) = +, q{ n+ q + } i f x =, i f x = q n, n N, i f x =. If we assume f : X X as an ideniy map hen sequence of mulivalued mappings {T q } q= from X ino CB(X) is a sequence of mulivalued G f -conracion. I saisfies he condiions of Theorem and X is he fixed poin of sequence of mulivalued maps T q for q N. The nex heorem provides a way o find he coincidence poin of a hybrid pair. Theorem 2. Le (X, d) be a complee meric space, T : X CB(X) and f : X X a surjecion. If u, v X (wih u = v) such ha ( f u, f v) E(G), implies: H(T(u), T(v)) µ(d( f u, f v))d( f u, f v), (4) where µ : [, ) [, ) is a MT-funcion, if here exis m N and v X, such ha: (i) T(v ) [ f v ] m G = φ; (ii) For any sequence {v n } in X, if v n v and v n T(v n ) [v n ] m G for all n N and j =, 2,..., hen here exiss a subsequence { } ( v nk such ha vnk, v ) E(G) for all k N. Then f and T have a coincidence poin, i.e., here exiss v X such ha f v T(v ). Proof. Take T q := T for all q N in Theorem and proof is following he same procedure. Corollary. Le (X, d) be a complee meric space,{t q } q= a sequence of he self mappings on X and f : X X a surjecion. If u, v X (wih u = v) such ha ( f u, f v) E(G), implies: d(t q (u), T r (v)) µ(d( f u, f v))d( f u, f v), (5) for all q, r N, where µ : [, ) [, ) is a MT funcion, if here exis m N and v X, such ha: (i) T (v ) [ f v ] m G = φ; (ii) For any sequence {v n } in X, if v n v and v n = T n (v n ) [v n ] m G for all n N, hen here exiss a subsequence { v nk } such ha ( vnk, v ) E(G) for all k N. Then f and sequence of mappings {T q } q= have a coincidence poin, i.e., here exiss v X such ha f v = T q (v ). q N Corollary 2. Le (X, d) be a complee meric space, T : X CB(X) and if u, v X (wih u = v) such ha (u, v) E(G), implies: H(T(u), T(v)) µ(d(u, v))d(u, v), (6) where µ: [, ) [, ) is a MT-funcion, if here exis m N and v X, such ha: (i) T(v ) [v ] m G = φ;

7 Mahemaics 27, 5, 3 7 of (ii) For any sequence {v n } in X, if v n v and v n T(v n ) [v n ] m G for all n N and j =, 2,..., hen here exiss a subsequence { } ( v nk such ha vnk, v ) E(G) for all k N. Then T has a fixed poin, i.e., v = T(v ). The following are he consequence of he Theorem and Theorem 2 for he case of self mappings. Corollary 3. Le (X, d) be a complee meric space, T : X X and f : X X a surjecion. If u, v X (wih u = v) such ha ( f u, f v) E(G), implies: d(t(u), T(v)) µ(d( f u, f v))d( f u, f v), (7) where µ : [, ) [, ) is a MT funcion, if here exis m N and v X, such ha: (i) T(v ) [ f v ] m G = φ; (ii) For any sequence {v n } in X, if v n v and v n = T(v n ) [v n ] m G for all n N and j =, 2,..., hen here exiss a subsequence { } ( v nk such ha vnk, v ) E(G) for all k N. Then f and T have a coincidence poin, i.e., here exiss v X such ha f v = T(v ). Corollary 4. Le (X, d) be a complee meric space, T : X X and if u, v X (wih u = v) such ha (u, v) E(G), implies: d(t(u), T(v)) µ(d(u, v))d(u, v), (8) where µ : [, ) [, ) is a MT-funcion, if here exis m N and v X, such ha: (i) T(v ) [v ] m G = φ; (ii) For any sequence {v n } in X, if v n v and v n = T(v n ) [v n ] m G for all n N and j =, 2,..., hen here exiss a subsequence { } ( v nk such ha vnk, v ) E(G) for all k N. Then T has a fixed poin, i.e., v = T(v ). The nex remark highlighs he applicaions of all he above resuls in seings of complee meric spaces, complee meric spaces endowed wih parial order and ε-chainable complee meric spaces. Remark. Consider he following cases: R. Le (X, d) be a complee meric space, consider he graph G wih: E (G ) = X X. R2. Le (X, d) be a complee meric space wih parial order on X, consider he graphs G and G 2 wih: E (G ) = {(x, y) X X : x y}, and: E (G 2 ) = {(x, y) X X : x y or y x}. R3. Le ε > and (X, d) be a complee ε-chainable meric space, consider he graph: G 3 := {(x, y) X X : < d (x, y) < ε, for ε > }. We remark ha all above resuls are valid under he above consrucion of remarks (R), (R2) and (R3). Furher, in an applicaion of Theorem we generalize he Theorem 6 of [2]. I esablishes he convergence of successive approximaions of operaors on a Banach space, which consequenly yields

8 Mahemaics 27, 5, 3 8 of he Kelisky-Rivlin heorem on ieraes of Bernsein operaors on he space C (I), where I is he closed uni inerval. Theorem 3. Le X be a Banach space and X be a closed subspace of X. Le T, f : X X be maps such ha f is surjecion and: Tx Ty ϕ ( f x f y ) f x f y whenever f x f y X, x = y. (9) If (I f ) (X) X and ( f T) (X) X, hen for all x X, {T n x} converges o Coin {T, f }, where Coin {T, f } = {x X : Tx = f x}. Proof. Consider he graph G = (V (G), E (G)) where V (G) = X and E(G) = {(x, y) X X : x y X }. Clearly, E(G), G = G and G has no parallel edges. Consider (x, y) E (G), hen f x f y = (y f y) (x f x) + (x y) X, since (I f ) (X) X. Hence and by given conracive condiion (9), we see ha (x, y) E (G) wih x = y, (6) holds. Also Tx Ty = ( f y Ty) ( f x Tx) + ( f x f y) X, since ( f T) (X) X. The use of ( f T) (X) X, implies ha ( f x, Tx) E (G) for x in X. Therefore condiion (i) of Corollary 4 holds wih x = v = x and N =. Thus we are able o generae a sequence such ha Tx n = f x n for all n N. Assume ha Tx n v X bu since f is surjecion so here exiss some v in X such ha v = f v. Here also Tx n [Tx n ] G for all n N, which implies ha (Tx n, Tx n ) E (G) for all n N. Now using he ouline of he proof of Theorem 4. of [2], (Tx n, f v) E (G) for all n N. Now assume: f v Tv f v f x n+ + f x n+ Tv () = f v f x n+ + Tx n Tv. Since(Tx n, f v) E (G) for all n N, hus from (9) and () we have: f v Tv f v f x n+ + ϕ ( f x n f v ) f x n f v. As n, we ge f v = Tv. Thus v is he coincidence poin of f and T, by using Corollary 4. For he uniqueness of he coincidence poin we le wo coincidence poins u, v of f and T, hen: This implies ha Tu = Tv. ( ϕ ( f u f v )) Tu Tv. Tu Tv ϕ ( f u f v ) f u f v In he nex resul, we discussed he generalizaion of fracional differenial equaion described in [2]. For he closed inerval I = [, ], assume funcion g C (I, R) and f : I R R is a coninuous funcion. The fracional differenial equaion is given as follows: D α x () + f (, g (x ())) = (, α > ) () wih boundary condiions x () = x () =. I is o be noed ha associaed Green s funcion wih he problem () is: { ( ( s)) α ( s) α s, G (, s) =, s. (( s)) α Γ(α) where Γ (.) represens he Gamma funcion. Theorem 4. Consider he surjecive funcion g C (I, R) and f : I R R saisfies: (i) ( f (s, g (x (s))) f (s, g (y (s)))) g (x (s)) g (y (s)) for all s I;

9 Mahemaics 27, 5, 3 9 of (ii) sup I G (, s) ds k <. Then, problem () has a unique soluion. Proof. Assume space X = C (I, R), and we have d (x, y) = max x () y () for x and y in X. I is [,] well known ha x X is a soluion of () if and only if i is a soluion of he inegral equaion: x () = G (, s) f (s, (gx) (s)) ds for all I. Define he operaor F : X X by: Fx () = G (, s) f (s, (gx) (s)) ds for all I, and S : X X by: Sx = gx, wih (Sx) () = (gx) () for I. Thus, for finding a soluion of (), i is sufficien o show ha F has a coincidence poin wih g. Now le x, y C (I) for all s I. Here we have: Fx () Fy () = This implies ha for each x, y X, we have: G (, s) ( f (s, (gx) (s)) f (s, (gy) (s))) ds G (, s) ( f (s, (gx) (s)) f (s, (gy) (s))) ds G (, s) (gx) (s) (gy) (s) ds G (, s) (Sx) (s) (Sy) (s) ds G (, s) d (Sx, Sy) ds d (Sx, Sy) sup G (, s) ds I kd (Sx, Sy). d (Fx, Fy) kd (Sx, Sy). Now he use of Corollary 3 wih graph G = G, we have x X such ha Fx = Sx wih (Sx ) () = (gx ) () for I. Thus x is he required coincidence poin of F and g. Auhor Conribuions: All auhors conribued equally o he wriing of his paper. All auhors read and approve he final manuscrip.

10 Mahemaics 27, 5, 3 of Conflics of Ineres: The auhors declare no conflic of ineres. References. Nadler, B. Mulivalued conracion mappings. Pacific J. Mah. 969, 3, Hu, T. Fixed poin heorems for mulivalued mappings. Can. Mah. Bull. 98, 23, Coviz, H.; Nadler, SB, Jr. Muli-valued conracion mappings in generalized meric spaces. Israel J. Mah. 97, 8, Reich, S. Fixed poins of conracive funcions. Boll. Unione Ma. Ial. 972, 5, Mizoguchi, N.; Takahashi, W. Fixed poin heorems for mulivalued mappings on complee meric spaces. J. Mah. Anal. Appl. 989, 4, Azam, A.; Arshad, M. Fixed poins of a sequence of locally conracive mulivalued maps. Comp. Mah. Appl. 29, 57, Feng, Y.; Liu, S. Fixed poin heorems for muli-valued conracive mappings and muli-valued Carisi ype mappings. J. Mah. Anal. Appl. 26, 37, Ciric, L.B. Muli-valued nonlinear conracion mappings. Nonlinear Anal. 29, 7, Klim, D.; Wardowski, D. Fixed poin heorems for se-valued conracions in complee meric spaces. J. Mah. Anal. Appl. 27, 334, Klim, D.; Wardowski, D. Fixed poins of dynamic processes of se-valued F-conracions and applicaion o funcional equaions. Fixed Poin Theory Appl. 25,, 22.. Nicolae, A. Fixed poin heorems for muli-valued mappings of Feng-Liu ype. Fixed Poin Theory 2, 2, Jachymski, J. The conracion principle for mappings on a meric space wih a graph. Proc. Amer. Mah. Soc. 28, 36, Nieo, J.; Rodríguez-López, R. Conracive mapping heorems in parially ordered ses and applicaions o ordinary differenial equaions. Order 25, 3, Ran, A.C.M.; Reurings, M.C.B. A fixed poin heorem in parially ordered ses and some applicaions o marix equaions. Proc. Am. Mah. Soc. 24, Johnsonbaugh, R. Discree Mahemaics; Prenice-Hall, Inc.: Upper Saddle River, NJ, USA, Du, W.S. On coincidence poin and fixed poin heorems for nonlinear mulivalued maps. Topol. Appl. 22, 59, Hu, T.; Rosen, H. Locally conracive and expansive mappings. Boll. Unione Ma. Ial. 99, 7, Bailey, B.F. Some heorems on conracive mappings. J. Lond. Mah. Soc. 966, 4, Assad, N.A.; Kirk, W.A. Fixed poin heorems for se valued mappings of conracive ype. Pacific J. Mah. 972, 43, Asrifa, S.; Verivel, V. Fixed poins of Mizoguchi-Takahashi conracion on a meric space wih a graph and applicaions. J. Mah. Anal. Appl. 24, 47, Baleanu, D.; Shahram, R.; Hakimeh, M. Some exisence resuls on nonlinear fracional differenial equaions. Philos. Trans. A Mah. Phys. Eng. Sci. 23, 37, c 27 by he auhors. Licensee MDPI, Basel, Swizerland. This aricle is an open access aricle disribued under he erms and condiions of he Creaive Commons Aribuion (CC BY) license (hp://creaivecommons.org/licenses/by/4./).