/compile/output.dvi Fixed Point Results for α−ψ−ϕ− Contractive Type Mappings in b-Metric-Like Spaces M. A. Akturk1,∗, M. Kır2 and E. Yolacan 3 1 Department of Engineering Sciences, Istanbul University, 34320 Istanbul, Turkey 2 Department of Civil Engineering, Faculty of Engineering Şırnak University, Şırnak, Turkey 3 Department of Mathematics, Faculty of Science, Ataturk University, Erzurum, 25240, Turkey Abstract. In this paper, we introduce the concept of α−ψ−ϕ- contractive type mappings in b-metric- like spaces and state some related fixed point theorems. Our results generalize related results in the literature. Furthermore, an example and an application to integral equations are provided to illustrate the usability of obtained results. 2010 Mathematics Subject Classifications: 47H10, 54H25 Key Words and Phrases: b-Metric-Like, α-Admissible Mappings, Fixed Point, Integral Equations 1. Introduction There are a lot of generalizations of the concept of metric space in the literature. The no- tion of b-metric-like space was initiated by Alghamdi [1] in 2013 as a new generalization of metric-like space. Recently, Hussain et al. [4] examined topological structure of these spaces and presented some fixed point results in b-metric-like space. Very recently, Chen et al. [3] established some fixed point theorems in b-metric-like space and showed existence of a solu- tion for an integral equation. In this paper we introduce the concept of α−ψ−ϕ-contractive type mappings in b-metric-like spaces and state some related fixed point theorems. Our re- sults generalize related results in the literature. Furthermore, an example and an application to integral equations are provided to illustrate the usability of obtained results. 2. b-Metric-Like Spaces Definition 1 ([1]). Let X be a nonempty set and κ≥ 1 a given real number. A function ς : X × X → R+ is b-metric-like if, for all x , y, z ∈ X , the following conditions are satisfied: ∗Corresponding author. Email addresses: mehmetaliakturk@yandex.com (M. Akturk), mehmetkir04@gmail.com (M. Kır),yolacanesra@gmail.com (E. Yolacan) http://www.ejpam.com 175 c© 2016 EJPAM All rights reserved. EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 9, No. 2, 2016, 175-185 ISSN 1307-5543 – www.ejpam.com M. Akturk, M. Kır, E. Yolacan / Eur. J. Pure Appl. Math, 9 (2016), 175-185 176 (A1) if ς � x , y � = 0⇒ x = y; (A2) ς � x , y � = ς � y, x � ; (A3) ς � x , y � ≤ κ � ς (x , z) + ς � y, z �� . A b-metric-like space is a pair (X ,ς) such that X is nonempty set and ς is b-metric-like on X . The number κ is called the coefficient of (X ,ς). Each b-metric-like ς on X generates a topology τς on X whose base is the family of all open ς− balls � Dς (x ,ǫ) : x ∈ X , ǫ > 0 , where Dς (x ,ǫ) = {a ∈ X : |ς (x , a)− ς (x , x)|< ǫ} for all x ∈ X and ǫ > 0. Definition 2 ([1]). Let (X ,ς) be a b-metric-like space with coefficient κ, and let � xn be any sequence in X and x ∈ X . Then (a) a sequence � xn is convergent to x with respect to τς, if limn→∞ ς � xn, x � = ς (x , x); (b) a sequence � xn is a Cauchy sequence in (X ,ς) if limn,m→∞ ς � xn, xm � exists and is finite; (c) (X ,ς) is a complete b-metric-like space if for every Cauchy sequence � xn in X there exists x ∈ X such that limn,m→∞ ς � xn, xm � = limn→∞ ς � xn, x � = ς (x , x). It is obvious that the limit of a sequence in b-metric-like space is usually not unique (see [3, Remark 1.1]). Lemma 1 ([4]). Let (X ,ς) be a b-metric-like space with coefficient κ, and suppose that � xn and � yn are convergent to x and y, respectively. Then one has 1 κ2 ς � x , y � − 1 κ ς (x , x)− ς � y, y � ≤ lim inf n→∞ ς � xn, yn � ≤ lim sup n→∞ ς � xn, yn � ≤κς (x , x) + κ2ς � y, y � + κ2ς � x , y � . In particular, if ς � x , y � = 0, then one has limn→∞ ς � xn, yn � = 0. Moreover, for each z ∈ X one has 1 κ ς (x , z)− ς (x , x) ≤ lim inf n→∞ ς � xn, z � ≤ lim sup n→∞ ς � xn, z � ≤κς (x , z) + κς (x , x) . 3. Preliminaries Let Ψ be the family of function ψ : [0,∞)→ [0,∞) satisfying the following conditions: (i) ψ is continuous and nondecreasing; (ii) ψ (t) = 0 if and only if t = 0. M. Akturk, M. Kır, E. Yolacan / Eur. J. Pure Appl. Math, 9 (2016), 175-185 177 Samet et al. [5] introduced the class of α−admissable mappings. Definition 3 ([5]). For a nonempty set X , let T : X → X and α : X × X → [0,∞) be given mappings. We say that T is α-admissible if for all x , y ∈ X , we have α � x , y � ≥ 1⇒ α �� T x , T y �� ≥ 1. Now, we establish the α−ψ−ϕ-contractive type mapping on b-metric-like space. Definition 4. Let (X ,ς) be a b-metric-like space with coefficient κ ≥ 1. We say that T : X → X is an α−ψ−ϕ- contractive type mapping if there exists three functions α : X × X → [0,∞) and ψ,ϕ ∈ Ψ such that α � x , y � ψ � κς � T x , T y �� ≤ψ � M � x , y �� −ϕ � M � x , y �� (1) where M � x , y � =max ¨ ς � x , y � ,ς (x , T x) ,ς � y, T y � , ς � x , T y � + ς � y, T x � 2κ « (2) for all x , y ∈ X . 4. Main Results Theorem 1. Let (X ,ς) be a complete b-metric-like space with the constant κ ≥ 1 and T : X → X be an α−ψ−ϕ-contractive mapping. Suppose that (i) T is α-admissible; (ii) there exists x0 ∈ X such that α � x0, T x0 � ≥ 1; (iii) T is continuous and if ς (x , x) = 0 for some x ∈ X , then α (ω,ω) ≥ 1. Then, such ω is a fixed point of T , that is Tω =ω. Proof. From condition (ii), there exists x0 ∈ X such that α � x0, T x0 � ≥ 1. Define xn+1 = T xn = T n+1 x0 for all n ≥ 0. If xn0 = xn0+1 for some n0, then it is clear that xn0 is a fixed point of T . Suppose that xn 6= xn+1 for all n. Observe that α � x0, T x0 � = α � x0, x1 � ≥ 1=⇒ α � T x0, T x1 � = α � x1, x2 � ≥ 1, since T is α-admissible. By repeating the process above, we derive α � xn, xn+1 � ≥ 1, for all n ∈ N. (3) Using (1) and (3) for all n ∈ N, we have ψ � κς � xn+1, xn+2 �� =ψ � κς � T xn, T xn+1 �� ≤α � xn, xn+1 � ψ � κς � T xn, T xn+1 �� M. Akturk, M. Kır, E. Yolacan / Eur. J. Pure Appl. Math, 9 (2016), 175-185 178 ≤ψ � M � xn, xn+1 �� −ϕ � M � xn, xn+1 �� (4) where M � xn, xn+1 � =max ¨ ς � xn, xn+1 � ,ς � xn, T xn � ,ς � xn+1, T xn+1 � , ς � xn, T xn+1 � + ς � xn+1, T xn � 2κ « =max ¨ ς � xn, xn+1 � ,ς � xn+1, xn+2 � , ς � xn, xn+2 � + ς � xn+1, xn+1 � 2κ « ≤max ¨ ς � xn, xn+1 � ,ς � xn+1, xn+2 � , κς � xn, xn+1 � + κς � xn+1, xn+2 � + ς � xn+1, xn+1 � 2κ « . Since ς (x , x) ≤ ς � x , y � ≤ kς � x , y � for each x , y ∈ X , we arrive at M � xn, xn+1 � =max ¨ ς � xn, xn+1 � ,ς � xn+1, xn+2 � , 2ς � xn, xn+1 � + ς � xn+1, xn+2 � 2 « =max ¨ ς � xn+1, xn+2 � , 2ς � xn, xn+1 � + ς � xn+1, xn+2 � 2 « . (5) If for some n, M � xn, xn+1 � = ς � xn+1, xn+2 � ( 6= 0) then (4) and (5) turn into ψ � κς � xn+1, xn+2 �� ≤ψ � ς � xn+1, xn+2 �� −ϕ � ς � xn+1, xn+2 �� <ψ � ς � xn+1, xn+2 �� , which is a contraction. Hence, M � xn, xn+1 � = ς � xn, xn+1 � for all n ∈ N and (4) with (5) we obtain ψ � κς � xn+1, xn+2 �� ≤ψ � ς � xn, xn+1 �� −ϕ � ς � xn, xn+1 �� . (6) Consequently, the sequence � ς � xn+1, xn+2 � is non-increasing for all n ∈ N. Hence, there exists a ≥ 0 such that limn→∞ ς � xn+1, xn+2 � = a. Taking n→∞ in (6), the continuity ofψ and ϕ and limn→∞ ς � xn+1, xn+2 � = a show that ψ (κa)≤ψ (a)−ϕ (a), yielding a = 0. So, we have lim n→∞ ς � xn+1, xn+2 � = 0. (7) Next, we show that � xn is a Cauchy sequence. If it is not, then there exists ǫ > 0 for which we can find subsequences � xmk and � xnk of sequence � xn where nk is the smallest index for which nk > mk > k with ς � xmk , xnk � ≥ ǫ. (8) Then ς � xmk , xnk−1 � < ǫ. (9) Using (8) and (9), we obtain ǫ ≤ ς � xmk , xnk � ≤ κ � ς � xmk , xnk−1 � + ς � xnk−1, xnk �� < κǫ + κς � xnk−1, xnk � . (10) M. Akturk, M. Kır, E. Yolacan / Eur. J. Pure Appl. Math, 9 (2016), 175-185 179 Taking the upper and lower limits as k→∞, we conclude ǫ ≤ lim inf k→∞ ς � xmk , xnk � ≤ lim sup k→∞ ς � xmk , xnk � ≤ κǫ. (11) By using (A3) and we deduce ς � xmk+1, xnk � ≤ κς � xmk+1, xmk � + κ2ς � xmk , xnk−1 � + κ2ς � xnk−1, xnk � , (12) with taking the upper limit as k→∞ in (12), we obtain lim sup k→∞ ς � xmk+1, xnk � ≤ κ2ǫ. (13) Use (A3) and we find ς � xmk+1, xnk−1 � ≤ κς � xmk+1, xmk � + κς � xmk , xnk−1 � (14) by taking the upper limit as k→∞ in (14), we get lim sup k→∞ ς � xmk+1, xnk−1 � ≤ κǫ. (15) On the other hand, ς � xmk , xnk � ≤ κς � xmk , xmk+1 � + κ2ς � xmk+1, xnk−1 � + κ2ς � xnk−1, xnk � . (16) Using (11) and (7), we obtain ǫ κ2 ≤ lim inf k→∞ ς � xmk+1, xnk−1 � . (17) Moreover, ǫ ≤ ς � xmk , xnk � ≤ κς � xmk , xmk+1 � + κς � xmk+1, xnk � , (18) with taking the upper limit as k→∞ in (18), we have ǫ κ ≤ lim sup k→∞ ς � xmk+1, xnk � . (19) By using (1), we have ψ � κς � xmk+1, xnk �� ≤α � xmk , xnk−1 � ψ � κς � T xmk , T xnk−1 �� ≤ψ � M � xmk , xnk−1 �� −ϕ � M � xmk , xnk−1 �� (20) where M � xmk , xnk−1 � =max § ς � xmk , xnk−1 � ,ς � xmk , xmk+1 � ,ς � xnk−1, xnk � (21) , ς � xmk , xnk � + ς � xnk−1, xmk+1 � 2κ ª M. Akturk, M. Kır, E. Yolacan / Eur. J. Pure Appl. Math, 9 (2016), 175-185 180 from on taking the upper limit as k→∞, from (7), (9), (11) and (15) we obtain lim k→∞ sup M � xmk , xnk−1 � =max § ǫ, 0, 0, κǫ + κǫ 2κ ª = ǫ. (22) Thus, from (19) and (20), we have ψ � κ ǫ κ � ≤ψ (ǫ)−ϕ (ǫ) (23) which is a contradiction. Hence � xn is a Cauchy sequence in X . Since X is complete, there exists ω ∈ X such that 0= lim n,m→∞ ς � xn, xm � = lim n→∞ ς � xn,ω � = ς (ω,ω) . (24) By using (A3), we deduce ς (ω, Tω) ≤ κς � ω, T xn � + κς � T xn, Tω � . (25) Taking the upper limit as n→∞ in (25) and using the continuity of T we have ς (ω, Tω)≤ κς (Tω, Tω) . (26) Since α (ω,ω)≥ 1 and using (1) we have ψ (κς (Tω, Tω))≤ α (ω,ω)ψ (κς (Tω, Tω))≤ψ (M (ω,ω))−ϕ (M (ω,ω)) (27) where M (ω,ω) =max § ς (ω,ω) ,ς (ω, Tω) ,ς (ω, Tω) , ς (ω, Tω) + ς (ω, Tω) 2κ ª = ς (ω, Tω) . (28) Hence, ψ (κς (Tω, Tω))≤ α (ω,ω)ψ (κς (Tω, Tω))≤ψ (ς (ω, Tω))−ϕ (ς (ω, Tω)) . (29) The property of ψ, we obtain κς (Tω, Tω) ≤ ς (ω, Tω) . (30) Here we deduce ϕ (ς (ω, Tω)) = 0. Hold ς (Tω,ω) = ς (Tω, Tω) = ς (Tω,ω) = 0 and Tω =ω. Hence, ω is a fixed point of T . If we replace the continuity condition (iii), Theorem 1 remains true. This statement is given as follows. M. Akturk, M. Kır, E. Yolacan / Eur. J. Pure Appl. Math, 9 (2016), 175-185 181 Theorem 2. Let (X ,ς) be a complete b-metric-like space with the constant κ≥ 1 and let T : X → X be an α−ψ−ϕ-contractive type mapping. Suppose that (i) T is α-admissible; (ii) There exists x0 ∈ X such that α � x0, T x0 � ≥ 1; (iii) If � xn is a sequence in X such that α � xn, xn+1 � ≥ 1 for all n and xn→ x ∈ X as n→∞, then there exists a subsequence � xnk of � xn such that α � xnk , x � ≥ 1 for all k. Then, such ω is a fixed point of T , that is Tω =ω. Proof. From proof of Theorem 1, we know that the sequence � xn defined by xn+1 = T xn for all n ∈ N is Cauchy in (X ,ς) and converges to some ω ∈ X . Consider (24), lim k→∞ ς � xnk+1, Tω � = ς (ω, Tω) (31) holds. By the assumption on X , we have ψ � κς � xnk+1, Tω �� ≤α � xnk ,ω � ψ � κς � T xnk , Tω �� ≤ψ � M � xnk ,ω �� −ϕ � M � xnk ,ω �� (32) where M � xnk ,ω � =max ¨ ς � xnk ,ω � ,ς � xnk , T xnk � ,ς (ω, Tω) , ς � xnk , Tω � + ς � ω, T xnk � 2κ « =max ¨ ς � xnk ,ω � ,ς � xnk , xnk+1 � ,ς (ω, Tω) , ς � xnk , Tω � + ς � ω, xnk+1 � 2κ « . With (7) and (31), we have lim k→∞ M � xnk ,ω � = ς (ω, Tω) . (33) Since α � xn,ω � ≥ 1 we have ψ (ς (Tω,ω))≤ψ � κ � ς � Tω, T xn � + ς � T xn,ω ��� ≤ψ � κς � Tω, T xn �� +ψ � κς � T xn,ω �� ≤α � ω, xn � ψ � κς � Tω, T xn �� +ψ � κς � T xn,ω �� ≤ψ � M � ω, xn �� −ϕ � M � ω, xn �� (34) Let n→∞ in (34), we haveψ (ς (Tω,ω))≤ 0. Henceω is a fixed point of T , or equivalently, ω = Tω. M. Akturk, M. Kır, E. Yolacan / Eur. J. Pure Appl. Math, 9 (2016), 175-185 182 Corollary 1. Let (X ,ς) be a b-metric-like space with coefficient κ ≥ 1 and T : X → X be such that κς � T x , T y � ≤ M � x , y � −ϕ � M � x , y �� for all x , y ∈ X where M � x , y � defined by (2). Then, T has a fixed point. To prove Corollary 1 it suffices to take α � x , y � = 1 and ψ (t) = t in Theorem 2. Corollary 2. Let (X ,ς) be a b-metric-like space with coefficient κ ≥ 1 and T : X → X be such that κς � T x , T y � ≤ sM � x , y � for all x , y ∈ X where s ∈ (0,1) and M � x , y � defined by (2). Then, T has a fixed point. To prove Corollary 2 it suffices to take ϕ (t) = (1− s) t in Corollary 1. Remark 1. 1. Note that b-metric-like spaces are a proper extension of metric-like and b-metric spaces. There- fore, it is clear that one can easily state the Analog of Theorem 1, Theorem 2, Corollary 1, and Corollary 2 in the setting of metric-like and b-metric spaces. 2. Theorem 1, Theorem 2, and Corollary 2 improve and generalized Theorem 2.1 in [2], Theorem 2.2 in [2] and Corollary 3.2 in [2], respectively. Example 1. Let X = [0,∞) and ς on X be given by ς � x , y � = x2+ y2+ � �x − y � � 2 for all x , y ∈ X . (X ,ς) is a complete b-metric-like space with coefficient κ = 2 (see [4, Example 14]). Define the mappings ψ,ϕ : [0,∞)→ [0,∞) by ψ (t) = t, ϕ (t) = t 2 and α : X × X → [0,∞) by α � x , y � = ¨ 1 if x , y ∈ [0,1] , 0 otherwise. Let T : X → X be defined by T x = ln(x+1) 2 . It is easy to see that T is a continuous on X . Understandably T is an α−ψ−ϕ-contractive type mapping with ψ (t) = t and ϕ (t) = t 2 for all t ≥ 0, for x , y ∈ X , α � x , y � ψ � κς � T x , T y �� =ψ � 2ς � T x , T y �� =2ς � T x , T y � =2 � T2 x + T2 y + � �T x − T y � � 2 � =2 � ln (x + 1) 2 �2 + � ln � y + 1 � 2 �2 ! + 2   � � � � � ln (x + 1) 2 − ln � y + 1 � 2 � � � � � 2   M. Akturk, M. Kır, E. Yolacan / Eur. J. Pure Appl. Math, 9 (2016), 175-185 183 ≤2 � x2 4 + y2 4 + � � � x 2 − y 2 � � � 2 � = 1 2 ς � x , y � =ς � x , y � − 1 2 ς � x , y � =ψ � ς � x , y �� −ϕ � ς � x , y �� ≤ψ � M � x , y �� −ϕ � M � x , y �� . (i) Now, we claim that T is α-admissible. Let � x , y � ∈ X × X such that α � x , y � ≥ 1. From the definition of T and α we have both T x = ln(x+1) 2 and T y = ln(y+1) 2 are in [0,1]. Therefore, α � T x , T y � = 1 ≥ 1. Then T is α-admissible. (ii) Taking x0 = 0 and T x0 = T0= ln(0+1) 2 = 0, we have α � x0, T x0 � = α (0, T0) = 1≥ 1. It is also obvious that hypothesis (iii) of Theorem 1 is satisfied. Thus, we apply Theorem 1 and so T has a fixed point, which is ω = 0. 5. Existence of the Solution for Nonlinear Fredholm Integral Equations In this section we will present an existence theorem for solution of nonlinear Fredholm integral equations. Define the nonlinear Fredholm integral equations by x (s) = T ∫ 0 G (s, r, x (r)) dr where T > 0. (35) We will examine (35) under the following conditions: (a) G : [0, T]× [0, T]× R→ R is continuous; (b) for all (s, r) ∈ [0, T]2 and x , y ∈ R, there exists a continuous a : [0, T]× [0, T]→ R such that |G (s, r, x)|+ � �G � s, r, y �� �≤ � 1 κ3 � 1 p a (s, r) � |x |+ � �y � � � (36) and sup s∈[0,T] T ∫ 0 a (s, r)≤ 1. (37) M. Akturk, M. Kır, E. Yolacan / Eur. J. Pure Appl. Math, 9 (2016), 175-185 184 Let X = C [0, T] be the set of continuous real functions defined on [0,1]. We endow X with the b-metric-like ς (u, v) = max s∈[0,1] (|u (s)|+ |v (s)|)p for all u, v ∈ X where p > 1. Also, (X ,ς) is complete b-metric-like space with the constant κ= 2p−1 (see more details [3]). Theorem 3. Under conditions (a) and (b), (35) has a unique solution in C [0, T]. Proof. By (36) and (37), we have κς � T x (s) , T y (s) � =κ � |T x (s)|+ � �T y (s) � � �p =κ    � � � � � � � T ∫ 0 G (s, r, x (r)) dr � � � � � � � + � � � � � � � T ∫ 0 G � s, r, y (r) � dr � � � � � � �    p ≤κ   T ∫ 0 |G (s, r, x (r))| dr + T ∫ 0 � �G � s, r, y (r) �� � dr   p ≤κ   T ∫ 0 � 1 κ3 � 1 p a (s, r) � � �� �x (r) + y (r) � � �p � 1 p � dr   p ≤κ   T ∫ 0 � 1 κ3 � 1 p a (s, r)ς 1 p � x (r) , y (r) � dr   p ≤ 1 κ2 ς � x (r) , y (r) �   T ∫ 0 a (s, r) dr   p ≤ 1 κ2 ς � x (r) , y (r) � ≤ 1 κ2 max � ς � x (r) , y (r) � ,ς (x (r) , T x (r)) ,ς � y (r) , T y (r) � , ς � x (r) , T y (r) � + ς � y (r) , T x (r) � 2κ « =sM � x (r) , y (r) � where s = 1 κ2 ∈ (0,1). 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