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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 979586, 10 pages doi:10.1155/2011/979586 Research Article Fixed-Point Results for Generalized Contractions on Ordered Gauge Spaces with Applications Cristian Chifu and Gabriela Petrus¸el Faculty of Business, Babes¸-Bolyai University, Horia Street no. 7, 400174 Cluj-Napoca, Romania Correspondence should be addressed to Cristian Chifu, cochifu@tbs.ubbcluj.ro Received 6 December 2010; Accepted 31 December 2010 Academic Editor: Jen Chih Yao Copyright q 2011 C. Chifu and G. Petrus¸el. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The p urpose of this paper is to present some fixed-point results for single-valued ϕ-contractions on ordered and complete gauge space. Our theorems generalize and extend some recent results in the literature. As an application, existence results for some integral equations on the positive real axis are given. 1. Introduction Throughout this paper will denote a nonempty set E endowed with a separating gauge structure D {d α } α∈Λ ,whereΛ is a directed set see 1 for definitions.Let : {0, 1, 2, } and ∗ : \{0}.Wealsodenoteby the set of all real numbers and by :0, ∞. Asequencex n of elements in E is said to be Cauchy if for every ε>0andα ∈ Λ, there is an N with d α x n ,x np ≤ ε for all n ≥ N and p ∈ ∗ .Thesequencex n is called convergent if there exists an x 0 ∈ X such tha t for every ε>0andα ∈ Λ,thereisanN ∈ ∗ with d α x 0 ,x n ≤ ε,foralln ≥ N. A gauge space is called complete if any Cauchy sequence is convergent. A subset of X is said to be closed if it contains the limit of any convergent sequence of its elements. S ee also Dugundji 1 for other definitions and details. If f : E → E is an operator, then x ∈ E is called fixed point for f if and only if x fx. The set F f : {x ∈ E | x fx} denotes the fixed-point set of f. On the other hand, Ran and Reurings 2 proved the following Banach-Caccioppoli type principle in ordered metric spaces. Theorem 1.1 Ran and Reurings 2. Let X be a partially ordered set such that every pair x, y ∈ X has a lower and an upper bound. Let d be a metric on X such that the metric space X, d is complete. 2 Fixed Point Theory and Applications Let f : X → X be a continuous and monotone (i.e., either decreasing or increasing) operator. Suppose that the following two assertions hold: 1 there exists a ∈ 0, 1 such that dfx,fy ≤ a · dx, y,foreachx, y ∈ X with x ≥ y; 2 there exists x 0 ∈ X such that x 0 ≤ fx 0 or x 0 ≥ fx 0 . Then f has an unique fixed point x ∗ ∈ X,thatis,fx ∗ x ∗ ,andforeachx ∈ X the sequence f n x n∈ of successive approximations of f starting from x converges to x ∗ ∈ X. Since then, several authors considered the problem of existence and uniqueness of a fixed point for contraction-type operators on partially ordered sets. In 2005, Nieto and Rodrguez-L ´ opez proved a modified variant of Theorem 1.1,by removing the continuity of f. The case of decreasing operators is treated in Nieto and Rodrguez-L ´ opez 3, where some interesting applications to ordinary differential equations with periodic boundary conditions are also given. Nieto, Pouso, and Rodrguez-L ´ opez, in a very recent paper, improve some results given by Petrus¸el and Rus in 4 in the s etting of abstract L-spaces in the sense of Fr ´ echet, see, for example, 5,Theorems3.3and3.5.Another fixed-point result of this type was given by O’Regan and Petrus¸el in 6 for the case of ϕ- contractions in ordered complete metric spaces. The aim of this paper is to present some fixed-point theorems for ϕ-contractions on ordered and complete gauge space. As an application, existence results for some integral equations on the positive real axis are given. Our theorems generalize the above-mentioned theorems as well as some other ones in the recent literature see; Ran and Reurings 2 ,Nieto and Rodrguez-L ´ opez 3, 7,Nietoetal.5,Petrus¸el and Rus 4, Agarwal et al. 8,O’Regan and Petrus¸el 6,etc.. 2. Preliminaries Let X be a nonempty set and f : X → X be an operator. Then, f 0 : 1 X , f 1 : f, ,f n1 f ◦ f n , n ∈ denote the iterate operators of f.LetX be a nonempty set and let sX : {x n n∈N | x n ∈ X, n ∈ N}.LetcX ⊂ sX asubsetofsX and Lim : cX → X an operator. By definition the triple X, cX, Lim is called an L-space Fr ´ echet 9 if the following conditions are satisfied. i If x n x,foralln ∈ N,thenx n n∈N ∈ cX and Limx n n∈N x. ii If x n n∈N ∈ cX and Limx n n∈N x, then for all subsequences, x n i i∈N ,of x n n∈N we have that x n i i∈N ∈ cX and Limx n i i∈N x. By definition, an element of cX is a convergent sequence, x : Limx n n∈N is the limit of this sequence and we also write x n → x as n → ∞. InwhatfollowwedenoteanL-space by X, → . In this setting, if U ⊂ X × X,thenanoperatorf : X → X is called orbitally U- continuous see 5 if x ∈ X and f ni x → a ∈ X,asi → ∞ and f ni x,a ∈ U for any i ∈ imply f ni1 x → fa,asi → ∞. In particular, if U X × X,thenf is called orbitally continuous. Let X, ≤ be a partially ordered set, that is, X is a nonempty set and ≤ is a reflexive, transitive, and antisymmetric relation on X.Denote X ≤ : x, y ∈ X × X | x ≤ y or y ≤ x . 2.1 Fixed Point Theory and Applications 3 Also, if x, y ∈ X,withx ≤ y then by x, y ≤ we will denote the ordered segment joining x and y,thatis,x, y ≤ : {z ∈ X | x ≤ z ≤ y}. In the same setting, consider f : X → X. Then, LF f : {x ∈ X | x ≤ fx} is the lower fixed-point set of f, while UF f : {x ∈ X | x ≥ fx} is the upper fixed-point set of f. Also, if f : X → X and g : Y → Y, then the cartesian product of f and g is denoted by f × g, and it is defined in the following way: f × g : X × Y → X × Y, f × gx, y :fx,gy. Definition 2.1. Let X be a nonempty set. By definition X, → , ≤ is an ordered L-space if and only if iX, → is an L-space; iiX, ≤ is a partially ordered set; iii x n n∈ → x, y n n∈ → y and x n ≤ y n ,foreachn ∈ ⇒ x ≤ y. If :E, D is a gauge space, then the convergence structure is given by the family of gauges D {d α } α∈Λ .Hence,E, D, ≤ is an ordered L-space, and it will be called an ordered gauge space, see also 10, 11. Recall that ϕ : → is said to be a comparison function if it is increasing and ϕ k t → 0, as k → ∞. As a consequence, we also have ϕt <t,foreacht>0, ϕ00 and ϕ is right continuous at 0. For example, ϕtat where a ∈ 0, 1, ϕtt/1 t and ϕtln1 t, t ∈ are comparison functions. Recall now the following important abstract concept. Definition 2.2 Rus 12.LetX, → be an L-space. An operator f : X → X is, by definition, aPicardoperatorif i F f {x ∗ }; iif n x n∈ → x ∗ as n →∞,forallx ∈ X. Several classical results in fixed-point theory can be easily transcribed in terms of the Picard operators, see 4, 13, 14.InRus12 the basic theory of Picard operators is presented. 3. Fixed-Point Results Our fi rst main result is the following existence, uniqueness, and approximation fixed-point theorem. Theorem 3.1. Let E, D, ≤ be an ordered complete gauge space and f : E → E be an operator. Suppose that i for each x, y ∈ E with x, y /∈ X ≤ there exists cx, y ∈ E such that x, cx, y ∈ X ≤ and y, cx, y ∈ X ≤ ; ii X ≤ ∈ If × f; iii if x, y ∈ X ≤ and y, z ∈ X ≤ ,thenx, z ∈ X ≤ ; iv there exists x 0 ∈ X ≤ such that x 0 ,fx 0 ∈ X ≤ ; 4 Fixed Point Theory and Applications v f is orbitally continuous; vi there exists a comparison function ϕ : → such that, for each α ∈ Λ one has d α f x ,f y ≤ ϕ d α x, y , for each x, y ∈ X ≤ . 3.1 Then, f is a Picard operator. Proof. Let x 0 ∈ E be such that x 0 ,fx 0 ∈ X ≤ . Suppose first that x 0 / fx 0 . Then, from ii we obtain f x 0 ,f 2 x 0 , f 2 x 0 ,f 3 x 0 , , f n x 0 ,f n1 x 0 , ,∈ X ≤ . 3.2 From vi,byinduction,weget,foreachα ∈ Λ,that d α f n x 0 ,f n1 x 0 ≤ ϕ n d α x 0 ,f x 0 , for each n ∈ . 3.3 Since ϕ n d α x 0 ,fx 0 → 0asn → ∞, for an arbitrary ε>0 we can choose N ∈ ∗ such that d α f n x 0 ,f n1 x 0 <ε−ϕε,foreachn ≥ N.Sincef n x 0 ,f n1 x 0 ∈ X ≤ for all n ∈ , we have for all n ≥ N that d α f n x 0 ,f n2 x 0 ≤ d α f n x 0 ,f n1 x 0 d α f n1 x 0 ,f n2 x 0 <ε− ϕ ε ϕ d α f n x 0 ,f n1 x 0 ≤ ε. 3.4 Now since f n x 0 ,f n2 x 0 ∈ X ≤ see iii we have for any n ≥ N that d α f n x 0 ,f n3 x 0 ≤ d α f n x 0 ,f n1 x 0 d f n1 x 0 ,f n3 x 0 <ε− ϕ ε ϕ d α f n x 0 ,f n2 x 0 ≤ ε. 3.5 By induction, for each α ∈ Λ,wehave d α f n x 0 ,f nk x 0 <ε, for any k ∈ ∗ ,n≥ N. 3.6 Hence f n x 0 n∈ is a Cauchy sequence in . From the completeness of the gauge space we have f n x 0 n∈ → x ∗ ,asn → ∞. Let x ∈ E be arbitrarily chosen. T hen; 1 If x, x 0 ∈ X ≤ then f n x,f n x 0 ∈ X ≤ and thus, for each α ∈ Λ,wehave d α f n x,f n x 0 ≤ ϕ n d α x, x 0 ,foreachn ∈ . Letting n → ∞ we obtain that f n x n∈ → x ∗ . Fixed Point Theory and Applications 5 2 If x, x 0 /∈ X ≤ then, by i,thereexistscx, x 0 ∈ E such that x, cx, x 0 ∈ X ≤ and x 0 ,cx, x 0 ∈ X ≤ . From t he second relation, a s before, we get, for each α ∈ Λ,thatd α f n x 0 ,f n cx, x 0 ≤ ϕ n d α x 0 ,cx, x 0 ,foreachn ∈ and hence f n cx, x 0 n∈ → x ∗ ,asn → ∞. Then, using the first relation we infer that, for each α ∈ Λ,wehaved α f n x,f n cx, x 0 ≤ ϕ n d α x, cx, x 0 ,foreachn ∈ . Letting again n → ∞,weconcludef n x n∈ → x ∗ . By the orbital continuity of f we get that x ∗ ∈ F f .Thusx ∗ fx ∗ .Ifwehavefyy for some y ∈ E, then from above, we must have f n y → x ∗ ,soy x ∗ . If fx 0 x 0 ,thenx 0 plays the role of x ∗ . Remark 3.2. Equivalent representation of condition iv are as follows. iv’Thereexistsx 0 ∈ E such that x 0 ≤ fx 0 or x 0 ≥ fx 0 iv” LF f ∪ UF f / ∅. Remark 3.3. Condition ii can be replaced by each of the following assertions: ii’ f : E, ≤ → E, ≤ is increasing, ii” f : E, ≤ → E, ≤ is decreasing. However, it is easy to see that assertion ii in Theorem 3.1. is more general. As a consequence of Theorem 3.1, we have the following result very useful for applications. Theorem 3.4. Let E, D, ≤ be an ordered complete gauge space and f : E → E be an operator. One supposes that i for each x, y ∈ E with x, y /∈ X ≤ there exists cx, y ∈ E such that x, cx, y ∈ X ≤ and y, cx, y ∈ X ≤ ; ii f : E, ≤ → E, ≤ is increasing; iii there exists x 0 ∈ E such that x 0 ≤ fx 0 ; iv a f is orbitally continuous or b if an increasing sequence x n n∈ converges to x in E,thenx n ≤ x for all n ∈ ; v there exists a comparison function ϕ : → such that d α f x ,f y ≤ ϕ d α x, y , for each x, y ∈ X ≤ ,α∈ Λ; 3.7 vi if x, y ∈ X ≤ and y, z ∈ X ≤ ,thenx, z ∈ X ≤ . Then f is a Picard operator. 6 Fixed Point Theory and Applications Proof. Since f : E, ≤ → E, ≤ is increasing and x 0 ≤ fx 0 we immediately have x 0 ≤ fx 0 ≤ f 2 x 0 ≤ ···f n x 0 ≤ ···.Hencefromv we obtain d α f n x 0 ,f n1 x 0 ≤ ϕ n d α x 0 ,fx 0 ,foreachn ∈ . By a similar approach as in the pr oof of Theorem 3.1 we obtain d α f n x 0 ,f nk x 0 <ε, for any k ∈ ∗ ,n≥ N, 3.8 Hence f n x 0 n∈ is a Cauchy sequence in . From the completeness of the gauge space we have that f n x 0 n∈ → x ∗ as n → ∞. By the orbital continuity of the operator f we get that x ∗ ∈ F f .Ifivb takes place, then, since f n x 0 n∈ → x ∗ , given any >0thereexistsN ∈ ∗ such that for each n ≥ N we have d α f n x 0 ,x ∗ <. On the other hand, for each n ≥ N ,sincef n x 0 ≤ x ∗ ,wehave, for each α ∈ Λ that d α x ∗ ,f x ∗ ≤ d α x ∗ ,f n1 x 0 d α f f n x 0 ,f x ∗ ≤ d α x ∗ ,f n1 x 0 ϕ d α f n x 0 ,x ∗ < 2. 3.9 Thus x ∗ ∈ F f . The uniqueness of the fixed point follows by contradiction. Suppose there exists y ∗ ∈ F f ,withx ∗ / y ∗ . There are two possible cases. a If x ∗ ,y ∗ ∈ X ≤ ,thenwehave0 <d α y ∗ ,x ∗ d α f n y ∗ ,f n x ∗ ≤ ϕ n d α y ∗ ,x ∗ → 0asn → ∞, which is a contradiction. Hence x ∗ y ∗ . b If x ∗ ,y ∗ /∈ X ≤ then there exists c ∗ ∈ E such that x ∗ ,c ∗ ∈ X ≤ and y ∗ ,c ∗ ∈ X ≤ . The monotonicity condition implies that f n x ∗ and f n c ∗ are comparable as well as f n c ∗ and f n y ∗ .Hence0 <d α y ∗ ,x ∗ d α f n y ∗ ,f n x ∗ ≤ d α f n y ∗ ,f n c ∗ d α f n c ∗ ,f n x ∗ ≤ ϕ n d α y ∗ ,c ∗ ϕ n d α c ∗ ,x ∗ → 0as n → ∞, which is again a contradiction. Thus x ∗ y ∗ . 4. Applications We will apply the above result to nonlinear integral equations on the real axis x t t 0 K t, s, x s ds g t ,t∈ . 4.1 Theorem 4.1. Consider 4.1. Suppose that i K : × × n → n and g : → n are continuous; ii Kt, s, · : n → n is increasing for each t, s ∈ ; iii there exists a comparison function ϕ : → ,withϕλt ≤ λϕt for each t ∈ and any λ ≥ 1,suchthat | K t, s, u − K t, s, v | ≤ ϕ | u − v | , for each t, s ∈ ,u,v∈ n ,u≤ v; 4.2 Fixed Point Theory and Applications 7 iv there exists x 0 ∈ C , n such that x 0 t ≤ t 0 K t, s, x 0 s ds g t , for any t ∈ . 4.3 Then the integral equation 4.1 has a unique solution x ∗ in C0, ∞, n . Proof. Let E : C0, ∞, n and the family of pseudonorms x n : max t∈0,n | x t | e −τt , where τ>0. 4.4 Define now d n x, y : x − y n for x, y ∈ E. Then D :d n n∈ ∗ is family of gauges on E.ConsideronE the partial order defined by x ≤ y if and only if x t ≤ y t for any t ∈ . 4.5 Then E, D, ≤ is an ordered and complete gauge space. Moreover, for any increasing sequence x n n∈ in E converging to some x ∗ ∈ E we have x n t ≤ x ∗ t,foranyt ∈ 0, ∞. Also, for every x, y ∈ E there exists cx, y ∈ E which is comparable to x and y. Define A : C0, ∞, n → C0, ∞, n ,bytheformula Ax t : t 0 K t, s, x s ds g t ,t∈ . 4.6 First observe that from ii A is increasing. Also, for each x, y ∈ E with x ≤ y and for t ∈ 0,n,wehave Ax t − Ay t ≤ t 0 K t, s, x s − K t, s, y s ds ≤ t 0 ϕ x s − y s ds t 0 ϕ x s − y s e −τs e τs ds ≤ t 0 e τs ϕ x s − y s e −τs ds ≤ ϕ d n x, y t 0 e τs ds ≤ 1 τ ϕ d n x, y e τt . 4.7 Hence, for τ ≥ 1weobtain d n Ax, Ay ≤ ϕ d n x, y , for each x,y ∈ X, x ≤ y. 4.8 From iv we have that x 0 ≤ Ax 0 . The conclusion follows now from Theorem 3.4. 8 Fixed Point Theory and Applications Consider now the following equation: x t t −t K t, s, x s ds g t ,t∈ . 4.9 Theorem 4.2. Consider 4.9. Suppose that i K : × × n → n and g : → n are continuous; ii Kt, s, · : n → n is increasing for each t, s ∈ ; iii there exists a comparison function ϕ : → ,withϕλt ≤ λϕt for each t ∈ and any λ ≥ 1,suchthat | K t, s, u − K t, s, v | ≤ ϕ | u − v | , for each t, s ∈ ,u,v∈ n ,u≤ v; 4.10 iv there exists x 0 ∈ C , n such that x 0 t ≤ t −t K t, s, x 0 s ds g t , for any t ∈ . 4.11 Then the integral equation 4.9 has a unique solution x ∗ in C , n . Proof. We consider the gauge space E :C , n , D :d n n∈ where d n x, y max −n≤t≤n x t − y t · e −τ|t| ,τ>0, 4.12 and the operator B : E → E defined by Bx t t −t K t, s, x s ds g t . 4.13 As before, consider on E the partial order defined by x ≤ y iff x t ≤ y t for any t ∈ . 4.14 Then E, D, ≤ is an ordered and complete gauge space. Moreover, for any increasing sequence x n n∈ in E converging to a certain x ∗ ∈ E we have x n t ≤ x ∗ t,foranyt ∈ . Also, for every x, y ∈ E there exists cx, y ∈ E which is comparable to x and y.Noticethat ii implies that B is increasing. Fixed Point Theory and Applications 9 From condition iii,forx, y ∈ E with x ≤ y,wehave Bx t − By t ≤ t −t ϕ x s − y s e −τ|s| e τ|s| ds ≤ t −t e τ|s| ϕ x s − y s e −τ|s| ds ≤ ϕ d n x, y t −t e τ|s| ds ≤ ϕ d n x, y |t| − | t | e τ|s| ds ≤ 2 τ ϕ d n x, y e τ|t| ,t∈ −n; n . 4.15 Thus, for any τ ≥ 2, we obtain d n B x ,B y ≤ ϕ d n x, y , ∀ x, y ∈ E, x ≤ y, for n ∈ . 4.16 As before, from iv we have that x 0 ≤ Bx 0 . The conclusion follows again by Theorem 3.4. Remark 4.3. It is worth mentioning that it could be of interest to extend the above technique for other metrical fixed-point theorems, see 15, 16, and so forth. References 1 J. Dugundji, Topology, Allyn and Bacon, Boston, Mass, USA, 1966. 2 A. C. M. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435–1443, 2004. 3 J. J. Nieto and R. Rodr ´ ıguez-L ´ opez, “Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations,” Acta Mathematica Sinica, vol. 23, no. 12, pp. 2205–2212, 2007. 4 A. Petrus¸el and I. A. Rus, “Fixed point theorems in ordered L-spaces,” Proceedings of the American Mathematical Society, vol. 134, no. 2, pp. 411–418, 2006. 5 J. J. Nieto, R. L. Pouso, and R. Rodr ´ ıguez-L ´ opez, “Fixed point theorems in ordered abstract spaces,” Proceedings of the American Mathematical Society, vol. 135, no. 8, pp. 2505–2517, 2007. 6 D. O’Regan and A. Petrus¸el, “Fixed point theorems for generalized contractions in ordered metric spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 1241–1252, 2008. 7 J. J. Nieto and R. R odr ´ ıguez-L ´ opez, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223–239, 2005. 8 R. P. Agarwal, M. A. El-Gebeily, and D. O’Regan, “Generalized contractions in partially ordered metric spaces,” Applicable Analysis, vol. 87, no. 1, pp. 109–116, 2008. 9 M. Fr ´ echet, Les Espaces Abstraits, Gauthier-Villars, Paris, France, 1928. 10 G. Petrus¸el, “Fixed point results for multivalued contractions on ordered gauge spaces,” Central European Journal of Mathematics, vol. 7, no. 3, pp. 520–528, 2009. 11 G. Petrus ¸el and I. Luca, “Strict fixed point results for multivalued contractions on gauge spaces,” Fixed Point Theory, vol. 11, no. 1, pp. 119–124, 2010. 12 I. A. Rus, “Picard operators and applications,” Scientiae Mathematicae Japonicae, vol. 58, no. 1, pp. 191– 219, 2003. 13 I. A. Rus, “The theory of a metrical fixed point theoremml: theoretical and applicative relevances,” Fixed Point Theory, vol. 9, no. 2, pp. 541–559, 2008. 14 I. A. Rus, A. Petrus¸el, and G. Petrus¸el, Fixed Point Theory, Cluj University Press, Cluj-Napoca, Romania, 2008. 10 Fixed Point Theory and Applications 15 J. Caballero, J. Harjani, and K. Sadarangani, “Contractive-like mapping principles in ordered metric spaces and application to ordinary differential equations,” Fixed Point Theory and Applications,vol. 2010, Article ID 916064, 14 pages, 2010. 16 J. Harjani and K. Sadarangani, “Fixed point theorems for weakly contractive mappings in partially ordered sets,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3403–3410, 2009. . Corporation Fixed Point Theory and Applications Volume 2011, Article ID 979586, 10 pages doi:10.1155/2011/979586 Research Article Fixed-Point Results for Generalized Contractions on Ordered Gauge Spaces. fixed-point results for single-valued ϕ -contractions on ordered and complete gauge space. Our theorems generalize and extend some recent results in the literature. As an application, existence results for. in ordered complete metric spaces. The aim of this paper is to present some fixed-point theorems for ϕ -contractions on ordered and complete gauge space. As an application, existence results for
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