Bachar and Khamsi Fixed Point Theory and pplications (15) 15:11 DOI 1.1186/s13663-15-36-x RESERCH Open ccess Fixed points of monotone mappings and application to integral equations Mostafa Bachar1* and Mohamed mine Khamsi,3 * Correspondence: mbachar@ksu.edu.sa 1 Department of Mathematics, King Saud University, Riyadh, Saudi rabia Full list of author information is available at the end of the article bstract In this work, we discuss the existence of fixed points of monotone nonexpansive mappings defined on partially ordered Banach spaces. This work is a continuity of the previous works of Ran and Reurings, Nieto et al., and Jachimsky done for contraction mappings. s an application, we discuss the existence of solutions to an integral equations. MSC: Primary 46B; 45D5; secondary 47E1; 341 Keywords: fixed point; integral equation; Krasnoselskii iteration; Lebesgue measure; monotone mapping; nonexpansive mapping 1 Introduction Banach s contraction principle [ ] is remarkable in its simplicity, yet it is perhaps the most widely applied fixed point theorem in all of analysis. This is because the contractive condition on the mapping is simple and easy to test, because it requires only a complete metric space for its setting, and because it finds almost canonical applications in the theory of differential and integral equations. Over the years, many mathematicians tried successfully to extend this fundamental theorem. Recently a version of this theorem has been given in partially ordered metric spaces [, ] (see also [, ]) and in metric spaces with a graph [ ]. In this work, we discuss the case of nonexpansive mappings defined in partially ordered Banach spaces. Nonexpansive mappings are those which have Lipschitz constant equal to. The fixed point theory for such mappings is rich and varied. It finds many applications in nonlinear functional analysis [ ]. It is worth mentioning that such investigation is new and has never been carried. Monotone nonexpansive mappings Let (X, ) be a Banach vector space. ssume that we have a partial order defined on X such that order intervals are convex and τ -closed, where τ is a Hausdorff topology on X. Recall that an order interval is any of the subsets [a, b] = {x X; a x b}, [a, ) = {x X; a x}, (, a] = {x X; x a} for any a, b X. Definition. Let C be a nonempty subset of X. Let T : C C be a map. ( ) T is said to be monotone if T(x) T(y) whenever x y for any x, y C. ( ) T is said to be monotone nonexpansive if and only if T is monotone and T(x) T(y) x y, whenever x y. 15 Bachar and Khamsi. This article is distributed under the terms of the Creative Commons ttribution 4. International License (http://creativecommons.org/licenses/by/4./), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Bachar and Khamsi Fixed Point Theory and pplications (15) 15:11 Page of 7 The point x C is called a fixed point of T if T(x)=x.ThesetoffixedpointsofT will be denoted by Fix(T). Throughout the paper we assume that C is convex and bounded not reduced to one point. Let T : C C be a monotone nonexpansive mapping. Fix λ (, 1) and x C. The Krasnoselskii [8, 9]iterationsequence{x n } C is defined by x n+1 = λx n +(1 λ)t(x n ), n. (KIS) The following lemma holds. Lemma.1 Under the above assumptions, if we assume that x T(x ), then we have x n x n+1 T(x n ) T(x n+1 ) (KI) for any n. Moreover, if {x n } has two subsequences which τ -converge to z and w respectively, then we must have z = w. Proof First note that if x y holds, then we have x λx +(1 λ)y y for any x, y X since order intervals are convex. Therefore it is enough to only prove x n T(x n ) for any n. By assumption, we have x T(x ). ssume that x n T(x n )forn 1. Then we have x n λx n +(1 λ)t(x n ) T(x n ), i.e., x n x n+1 T(x n ). Since T is monotone, we get T(x n ) T(x n+1 ). By induction, we conclude that the inequalities (KI) hold for any n. Next let {x φ(n) } be a subsequence of {x n } which τ -converges to z. Clearly, {[x n, ); n N} is a decreasing family of sets. Consequently, if U z τ is a neighborhood of z, then U z [x n, ) for any n N. Therefore, z belongs to all sets [x n, ) astheyareclosed. Let w be the τ -limit of another subsequence of {x n }.IfU w τ is a neighborhood of w, then U w contains many points from the sequence {x n } since w is a τ -limit of one of its subsequences. Hence U w (, z]. Therefore, w belongs to (, z] asitisclosed,i.e., w z. By reversing the roles of z and w,wegetz w. The properties of the partial order will force z = w as claimed. Remark.1 Note that under the assumptions of Lemma.1, if we assume T(x ) x, then we will have T(x n+1 ) T(x n ) x n+1 x n for any n.theconclusionontheτ -convergence limits of {x n } will also hold. The following result is found in [1, 11]. Proposition.1 Under the above assumptions, we have ( 1+n(1 λ) ) T(x i ) x i T(x i+n ) x i + λ n( T(x i ) x i T(x i+n ) x i+n ) (GK)
Bachar and Khamsi Fixed Point Theory and pplications (15) 15:11 Page 3 of 7 for any i, n N. This inequality implies lim xn T(x n ) =. Proof The first part of this proposition is easy to prove via an induction argument on the index i. s for the second part, note that { x n T(x n ) } is decreasing. Indeed we have x n+1 x n =(1 λ)(t(x n ) x n ) for any n 1. Therefore { x n T(x n ) } is decreasing if and only if { x n+1 x n } is decreasing, which holds since x n+ x n+1 λ x n+1 x n +(1 λ) T(xn+1 ) T(x n ) xn+1 x n for any n. Set lim x n T(x n ) = R. Thenweleti + in the inequality (GK) to obtain ( 1+n(1 λ) ) R δ(c) for any n N,whereδ(C)=sup{ x y, x, y C} <+.Hence R δ(c) (1 + n(1 λ)), n =1,,..., which implies R =,i.e., lim x n T(x n ) =. Before we state the main result of this work, let us recall the definition of Opial condition [1]. Definition. X is said to satisfy the τ -Opial condition if for any sequence {y n } in X which τ -converges to y,wehave lim sup y n y < lim sup y n z for any z X such that z y. Now we are ready to state the main result of this section. Theorem.1 Let X be a Banach space. Let τ be a topology on X such that X satisfies the τ -Opial condition. Let be a partial order on X such that order intervals are convex and τ -closed. Let C be a bounded convex τ -compact nonempty subset of X. Let T : C Cbea monotone nonexpansive mapping. ssume that there exists x Csuchthatx and T(x ) are comparable. Then T has a fixed point. Proof Without loss of any generality, we assume that x T(x ). Consider the (KIS) sequence {x n } which starts at x.sincec is τ -compact, then {x n } will have a subsequence {x kn } which τ -converges to some point w C. Lemma.1 implies that {x n } τ -converges to w and x n w for any n N. Consider the type function r(x)=lim sup x n x, x C.
Bachar and Khamsi Fixed Point Theory and pplications (15) 15:11 Page 4 of 7 Then Proposition.1 implies r(x) =lim sup T(x n ) x for any x C. SinceT is monotone nonexpansive, we get r ( T(w) ) = lim sup T(xn ) T(w) lim sup x n w = r(w). In fact we have r(t(x)) r(x) for any x C such that x n and x are comparable for any n N. Finally, if X satisfies the τ -Opial condition, then we must have T(w)=w, i.e., w is a fixed point of T. The following results are direct consequences of Theorem.1. Corollary.1 Let C be a bounded closed convex nonempty subset of l p,1<p <+. Let τ be the weak topology. Consider the pointwise partial ordering in l p, i.e., (α n ) (β n ) iff α n β n for all n 1. Then any monotone nonexpansive mapping T : C Chasafixed point provided there exists a point x Csuchthatx and T(x ) are comparable. Remark. The case of p = 1 is not interesting for the weak topology since l 1 is a Schur Banach space. But if we consider the weak* topology σ (l 1, c )onl 1 or the pointwise convergence topology, then l 1 satisfies the Opial condition for these topologies. Note that these two topologies are Hausdorff. In this case we have a similar conclusion of Corollary.1 for l 1. Recall the definitions of l p and c spaces: (i) l p = {(α n ) R N, n α n p <+ } for 1 p <+ ; (ii) c = {(α n ) R N, lim α n =}. 3 pplication to integral equations Let us consider the following integral equation of the form x(t)=g(t)+ where F ( t, s, x(s) ) ds, t [, 1], (IE) (i) g is in L ([, 1], R), (ii) F :[,1] [, 1] L ([, 1], R) R is measurable and satisfies the condition F(t, s, x) F(t, s, y) x y, (3.1) where t, s [, 1],andx, y L ([, 1], R) such that y x. Recall that for any u, v L ([, 1], R), we have u v u(t) v(t) almosteverywheret [, 1]. Condition (3.1) represents the monotonicity of the flow of the integral equation. comprehensive study of the monotonicity of the flow can be found in the book of Smith [13].
Bachar and Khamsi Fixed Point Theory and pplications (15) 15:11 Page 5 of 7 ssume that there exists a non-negative function h(, ) L ([, 1] [, 1]) and M < 1 such that F(t, s, x) h(t, s)+m x, (3.) where t, s [, 1] and x L ([, 1], R). Let B = { y L ( [, 1], R ),suchthat x L ([,1],R) ρ }, where ρ is sufficiently large, i.e., B is the closed ball of L ([, 1], R) centered at with radius ρ. Consider the operator defined by F(t)(y)(s)=F ( t, s, y(s) ), (3.3) and define the operator J : L ([, 1], R) L ([, 1], R)by (Jy)(t)=g(t)+ F(t)(y)(s) ds. (3.4) We have J(B) B. Indeed let x B, then by using the Cauchy-Schwarz inequality, condition (3.) and the quadratic inequality (a + b) a +b for any a, b R,wehave Jx L ([,1],R) = Jx(t) dt = g(t)+ F(t)(x)(s) ds dt 1 g(t) dt + F(t)(x)(s) ds dt = = g(t) dt + 1 g(t) dt +4 1 g(t) dt +4 g(t) dt +4 1 g(t) dt +4 Since M < 1/, choose ρ such that (1 4M ) g(t) 4 dt + (1 4M ) h(t, s)+m x(s) ds dt h(t, s) ds dt +4M h(t, s) ds dt +4M x(s) ds dt x(s) ds h(t, s) ds dt +4M x L ([,1],R) h (t, s) ds dt +4M ρ. h (t, s) ds dt ρ, we will get J(x) B as claimed. Next we prove that J is monotone nonexpansive. First from condition (3.1), J is obviously monotone. Let x, y L ([, 1], R) suchthaty x. Usingthe
Bachar and Khamsi Fixed Point Theory and pplications (15) 15:11 Page 6 of 7 Cauchy-Schwarz inequality, we have Jx Jy L ([,1],R) = ( ) Jx(t) Jy(t) dt ( = ( F(t)(x)(s) F(t)(y)(s) ) ) ds dt ( ) ( ) x(s) y(s) ds dt ( x(s) y(s) ) ds = x y L ([,1],R), which implies that J is a monotone nonexpansive operator as claimed. In order to use Theorem.1, we need to check its assumptions. First note that X = L ([, 1], R)isaHilbert space. If we choose τ to be the weak topology, then X satisfies the weak Opial condition. It is easy to check that order intervals are convex. In order to show that order intervals are closed, we will show that if {u n } is a non-negative sequence of elements in X which convergesweaklytou, thenu is positive. Let a <, then the set = {t [, 1]; u(t) a} has measure. Indeed, we have u(t) dt = lim u n (t) dt because of weak convergence. So lim u n (t) dt = u(t) dt am(). Hence m()=.set D = n 1 { t [, 1]; u(t) 1 }. n Then D has measure, which implies that u(t) for almost every t [, 1]. Using Theorem.1, we get the following result. Theorem 3.1 Under the above assumptions, we conclude that (i) the integral equation (IE) has a non-negative solution provided we assume that g(t)+ F(t, s,)ds for almost every t [, 1] (which implies J() ); (ii) the integral equation (IE) has a non-positive solution provided we assume that g(t)+ F(t, s,)ds for almost every t [, 1] (which implies J() ). Competing interests The authors declare that they have no competing interests. uthors contributions ll authors contributed equally to the writing of this paper. ll authors read and approved the final manuscript. uthor details 1 Department of Mathematics, King Saud University, Riyadh, Saudi rabia. Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX 79968, US. 3 Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran 3161, Saudi rabia.
Bachar and Khamsi Fixed Point Theory and pplications (15) 15:11 Page 7 of 7 cknowledgements The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this Research group No. (RG-1435-79). Received: 7 February 15 ccepted: 3 June 15 References 1. Banach, S: Sur les opérations dans les ensembles abstraits et leur application. Fundam. Math. 3, 133-181 (19). Nieto, JJ, Rodríguez-López, R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order,3-39 (5) 3. Ran, CM, Reurings, MCB: fixed point theorem in partially ordered sets and some applications to matrix equations. Proc.m.Math.Soc.13, 1435-1443 (4) 4. Turinici, M: Fixed points for monotone iteratively local contractions. Demonstr. Math. 19,171-18 (1986) 5. Turinici, M: Ran and Reurings theorems in ordered metric spaces. J. Indian Math. Soc. 78, 7-14 (11) 6. Jachymski, J: The contraction principle for mappings on a metric space with a graph. Proc. m. Math. Soc. 136, 1359-1373 (8) 7. Browder, FE: Nonexpansive nonlinear operators in a Banach space. Proc. Natl. cad. Sci. US 54, 141-144 (1965) 8. Ishikawa, S: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proc. m. Math. Soc. 59, 65-71 (1976) 9. Krasnoselskii, M: Two observations about the method of successive approximations. Usp. Mat. Nauk 1, 13-17 (1955) 1. Goebel, K, Kirk, W: Iteration processes for nonexpansive mappings. Contemp. Math. 1, 115-13 (1983) 11. Goebel, K, Kirk, W: Topics in Metric Fixed Point Theory. Cambridge Stud. dv. Math., vol. 8. Cambridge University Press, Cambridge (199) 1. Opial, Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. m. Math. Soc. 73, 591-597 (1967) 13. Smith, JL: Monotone Dynamical Systems. Mathematical Surveys and Monographs, vol. 41. m. Math. Soc., Providence (1995)