The viscosity technique for the implicit midpoint rule of nonexpansive mappings in

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1 Xu et al. Fixed Point Theory and Applications 05) 05:4 DOI 0.86/s R E S E A R C H Open Access The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces Hong-Kun Xu,*, Maryam A Alghamdi 3 and Naseer Shahzad * Correspondence: xuhk@hdu.edu.cn; xuhk@math.nsysu.edu.tw Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, 3008, China Department of Mathematics, King Abdulaziz University, P.O. Box 8003, Jeddah, 589, Saudi Arabia Full list of author information is available at the end of the article Abstract The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces is established. The strong convergence of this technique is proved under certain assumptions imposed on the sequence of parameters. Moreover, it is shown that the limit solves an additional variational inequality. Applications to variational inequalities, hierarchical minimization problems, and nonlinear evolution equations are included. MSC: 47J5; 47N0; 34G0; 65J5 Keywords: viscosity; implicit midpoint rule; nonexpansive mapping; projection; variational inequality; hierarchical minimization; nonlinear evolution equation Introduction The viscosity technique for nonexpansive mappings in Hilbert spaces was introduced by Moudafi [], following the ideas of Attouch []. Refinements in Hilbert spaces and extensions to Banach spaces were obtained by Xu [3]. This technique uses strict) contractions to regularize a nonexpansive mapping for the purpose of selecting a particular fixed point of the nonexpansive mapping, for instance, the fixed point of minimal norm or of a solution to another variational inequality. Let H be a Hilbert space, let T : H H be a nonexpansive mapping i.e., Tx Ty x y for all x, y H), and let f : H H be a contraction i.e., f x) f y) α x y for all x, y H and some α [0, )). The explicit viscosity method for nonexpansive mappings generates a sequence {x n } through the iteration process: x n+ = α n f x n )+ α n )Tx n, n 0,.) where I is the identity of H and {α n } is a sequence in 0, ). It is well known [, 3] that under certain conditions, the sequence {x n } converges in norm to a fixed point q of T which solves the variational inequality VI) I f )q, x q 0, x S,.) where S is the set of fixed points of T,namely,S = {x H : Tx = x}. 05 Xu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

2 Xu et al. Fixed Point Theory and Applications 05) 05:4 Page of The implicit midpoint rule IMR) is one of the powerful methods for solving ordinary differential equations; see [4 9] and the references therein. For instance, consider the initial value problem for the differential equation y t)=f yt)) with the initial condition y0) = y 0,wheref is a continuous function from R d to R d. The IMR is an implicit method that generates a sequence {y n } via the relation h y n+ y n )=f yn+ + y n ). In the case of nonlinear dissipative evolution equations in a Hilbert space H, the function f is of the form f = I T with I the identity and T a nonexpansive mapping of H. The equilibrium problem is reduced to the fixed point problem x = Tx. The IMR has therefore been extended [0] to nonexpansive mappings, which generates a sequence {x n } by the implicit procedure: xn + x n+ x n+ = t n )x n + t n T, n 0,.3) where the initial guess x 0 H is arbitrarily chosen, t n 0, ) for all n. In the present paper we will apply the viscosity technique to the implicit midpoint rule for nonexpansive mappings. More precisely, we consider the following semi-implicit algorithm which we call viscosity implicit midpoint rule VIMR, for short): xn + x n+ x n+ = α n f x n )+ α n )T, n 0..4) The idea is to use contractions to regularize the implicit midpoint rule for nonexpansive mappings. We will prove that the VIMR converges in norm to a fixed point of T which, in addition, also solves the VI.). The structure of the paper is set as follows. In Section, weintroducethenotionof nearest point projections, the demiclosedness principle of nonexpansive mappings, and a convergence lemma. The viscosity implicit midpoint rule for nonexpansive mappings is introduced in Section 3. The main result, that is, the strong convergence of this method, is proved also in this section. Applications to variational inequalities, hierarchical minimization problems and nonlinear evolution equations are presented in the final section, Section 4. Preliminaries Assume that H is a Hilbert space with inner product, and norm,respectively, and let C be a nonempty, closed, and convex subset of H.Wethenhavethenearestpoint projection from H onto C, P C,definedby P C x := arg min z C x z, x H..) Namely, P C x is the only point in C that minimizes the objective x z over z C. Note that P C x is characterized as follows: P C x C and x P C x, z P C x 0 for all z C..)

3 Xu et al. Fixed Point Theory and Applications 05) 05:4 Page 3 of Recall that a mapping T : C C is said to be nonexpansive if Tx Ty x y, x, y C. The set of fixed points of T is written FixT), that is, FixT)={x C : Tx = x}. Notethat FixT) is always closed and convex; further note that if, in addition, C is bounded, then FixT)isnonemptycf. []). The demiclosedness principle of nonexpansive mappings is quite helpful in verifying the weak convergence of an algorithm to a fixed point of a nonexpansive mapping. Lemma. [] The demiclosedness principle) Let H be a Hilbert space, Caclosedconvex subset of H, and T : C C a nonexpansive mapping with FixT). If {x n } is a sequence in C such that i) {x n } weakly converges to x and ii) {I T)x n } converges strongly to 0, then x = Tx. In proving the strong convergence of a sequence {x n } toapoint x, we always consider the real sequence { x n x } and then apply the following convergence lemma. Lemma. [] Assume {a n } is a sequence of nonnegative real numbers such that a n+ γ n )a n + δ n, n 0, where {γ n } is a sequence in 0, ) and {δ n } is a sequence in R such that i) n= γ n =, and ii) either lim sup n δ n /γ n 0 or n= δ n <. Then lim n a n =0. 3 The viscosity technique for implicit midpoint rule Let H be a Hilbert space, C a nonempty, closed, and convex subset of H, andt : C C a nonexpansive mapping such that FixT). Moreover,letf : C C be a contraction with coefficient α [0, ). The viscosity method for nonexpansive mappings is essentially a regularization method of nonexpansive mappings by contractions. In this section we consider the viscosity technique for the implicit midpoint rule of nonexpansive mappings which generates a sequence {x n } in the semi-implicit manner: xn + x n+ x n+ = α n f x n )+ α n )T, n 0, 3.) where α n 0, ) for all n. Note that the scheme 3.) is well defined for all n. We will employ the following conditions on {α n }: C) lim n α n =0, C) n=0 α n =, C3) either n=0 α α n+ α n < or lim n+ n α n =. The main result of this paper is the following result, the proof of which seems nontrivial. Theorem 3. Let H be a Hilbert space, C a closed convex subset of H, T : C C a nonexpansive mapping with S := FixT), and f : C C a contraction with coefficient

4 Xu et al. Fixed Point Theory and Applications 05) 05:4 Page 4 of α [0, ). Let {x n } be generated by the viscosity implicit midpoint rule 3.). Assume the conditions C)-C3). Then {x n } converges in norm to a fixed point q of T, which is also the unique solution of the variational inequality I f )q, x q 0, x S. 3.) In other words, q is the unique fixed point of the contraction P S f, that is, P S f q)=q. Proof We divide the proof into several steps. Step. We prove that {x n } is bounded. To see this we take p S to deduce that x n+ p α n ) T xn + x n+ p + α n f xn ) p α n ) x n + x n+ p + α n f xn ) fp) ) + f p) p It then follows that +α n and, moreover, α n xn p + x n+ p ) + α n α xn p + f p) p ). x n+ p +α )α n x n p + α n f p) p x n+ p +α )α n x n p + α n f p) p +α n +α n = α)α ) n x n p + α)α n f p) p ). +α n +α n α Consequently, we get { x n+ p max x n p, By induction we readily obtain { x n p max x 0 p, f p) p }. α } f p) p α for all n.itturnsoutthat{x n } is bounded. Step. lim n x n+ x n = 0. To see this we apply 3.)toget x n+ x n = α xn + x n+ nf x n )+ α n )T ) xn + x n α n f x n )+ α n )T ) = α xn + x n+ xn + x n n) T T

5 Xu et al. Fixed Point Theory and Applications 05) 05:4 Page 5 of ) xn + x n +α n α n ) T f x n ) + α n f xn ) fx n ) ) α n ) x n+ x n ) + M α n α n + αα n x n x n α n) x n+ x n + x n x n ) + M α n α n + αα n x n x n. Here M > 0 is a constant such that M sup T xn + x n+ f x n ). n 0 It turns out that +α n x n+ x n α n)+αα n x n x n + M α n α n. Consequently, we arrive at x n+ x n +α )α n x n x n + M α n α n +α n +α n = α)α ) n x n x n + M α n α n. 3.3) +α n +α n By virtue of the conditions C) and C3), we can apply Lemma. to 3.3)toobtain x n+ x n 0asn,asrequired. Step 3. lim n x n Tx n = 0. This follows from the argument below: x n Tx n x n x n+ + x xn + x n+ n+ T + T xn + x n+ Tx n xn + x n+ x n x n+ + α n f x n ) T + x n x n+ 3 x n x n+ + Mα n 0 asn ). Step 4. We prove that ω w x n ) FixT). Here ω w x n )= { x H :thereexistsasubsequenceof{x n } weakly converging to x } is the weak ω-limit set of {x n }. This is now a straightforward consequence of Step 3 and Lemma.. Step 5. We claim that lim sup q f q), q xn 0, 3.4) n where q S is the unique fixed point of the contraction P S f,thatis,q = P S f q)). Asamatteroffact,wecanfindasubsequence{x nj } of {x n } such that {x nj } converges weakly to a point p and moreover, lim sup q f q), q xn = lim q f q), q xnj. 3.5) n j

6 Xu et al. Fixed Point Theory and Applications 05) 05:4 Page 6 of Since p FixT)byStep4,wecancombine3.4)and3.5)anduse.)toconclude lim sup q f q), q xn = q f q), q p 0. n Step 6. We finally prove that x n q in norm. Here again q FixT) is the unique fixed point of the contraction P S f or in other words, q = P S f q). We present the details as follows: Let ) x n+ q = α xn + x n+ n) T q + α n f xn ) q ) = α n ) xn + x n+ T q + αn f x n ) q xn + x n+ +α n α n ) T q, f x n ) q α n ) x n + x n+ q + αn f x n ) q xn + x n+ +α n α n ) T q, f x n ) fq) xn + x n+ +α n α n ) T q, f q) q α n ) x n + x n+ q + αn f xn ) q +αα n α n ) x n + x n+ q x n q xn + x n+ +α n α n ) T q, f q) q. β n = αn f xn ) q +α n α n ) T It turns out that α n ) x n + x n+ xn + x n+ q +αα n α n ) x n q ) q, f q) q. 3.6) x n + x n+ q + β n x n+ q 0. Solving this quadratic inequality for x n+x n+ q yields x n + x n+ q { αα α n ) n α n ) x n q + 4α α n α n) x n q 4 α n ) β n x n+ q )} = αα n x n q + α α n x n q + x n+ q β n α n.

7 Xu et al. Fixed Point Theory and Applications 05) 05:4 Page 7 of This implies that x n+ q + x n q αα n x n q + α αn x n q + x n+ q β n. α n We therefore get αn ) x n+ q + +α )α n xn q ) α αn 4 x n q + x n+ q β n, which is reduced to the inequality 4 α n) x n+ q + xn +α )αn q 4 + α n) +α )α n ) xn q x n+ q α α n x n q + x n+ q β n, which is further reduced by using the elementary inequality x n q x n+ q x n q + x n+ q to the following inequality: 4 α n) 4 α n) ) +α )α n x n+ q +α )αn α n) ) +α )α n α αn x n q + β n. Solving for x n+ q yields x n+ q 4 + α )α n) + 4 α n) + α )α n ) α α n 4 α n) 4 α n) + α )α n ) x n q + γ n, 3.7) where γ n = Observing and β n 4 α n) 4 α n) + α )α n ). 3.8) 4 α n) 4 α n) +α )α n ) = α n) α)α n ) +α )αn α n) +α )α n α αn = +α )αn ) α)αn ) α α n,

8 Xu et al. Fixed Point Theory and Applications 05) 05:4 Page 8 of we can rewrite 3.7)as x n+ q + α )α n) α)α n ) α α n α n) α)α n ) x n q + γ n. 3.9) Consider the function ht):= t { + α )t) α)t) α t t) α)t) }, t >0. It is not hard after certain manipulations) to rewrite ht) as ht)= α) α) t + α t. t) α)t) It turns out that lim t 0 ht)=4 α)>0. Let δ 0 > 0 satisfy ht)>ε 0 := 3 α)>0, 0<t < δ 0. In other words, we have + α )t) α)t) α t t) α)t) < ε 0 t, 0<t < δ 0. As α n 0asn,wehaveanintegerN 0 big enough so that α n < δ for all n N 0.It then turns out from 3.9)that,foralln N 0, x n+ q ε 0 α n ) x n q + γ n. 3.0) Notice that by Steps and 3, we have T xn + x n+ x n 0 asn ). It then turns out from the definition 3.6)ofβ n and 3.4)that lim sup n β n α n 0, which in turn implies that lim sup n γ n α n 0. 3.) Finally, 3.) and the conditions C) and C) enable us to apply Lemma. to the inequality 3.0)toconcludethatlim n x n q =0,namely,x n q in norm. The proof is therefore complete.

9 Xu et al. Fixed Point Theory and Applications 05) 05:4 Page 9 of 4 Applications 4. Application to variational inequalities Consider the variational inequality VI) Ax, x x 0, x C, 4.) where A is a single-valued) monotone operator in H and C isaclosedconvexsubsetofh. We assume C doma).an exampleof 4.) is the constrained minimization problem min ϕx), 4.) x C where ϕ : H R is a lower-semicontinuous convex function. If ϕ is Fréchet) differentiable, then the minimization 4.) is equivalently reformulated as 4.) witha = ϕ. Notice that the VI 4.) is equivalent to the fixed point problem, for any λ >0, Tx = x, Tx := P C I λa)x. 4.3) If A is Lipschitzian and strongly monotone, then, for λ >0smallenough,T is a contraction and its unique fixed point is also the unique solution of the VI 4.). However, if A is not strongly monotone, T is no longer a contraction,in general.in thiscase we must deal with nonexpansive mappings for solving the VI 4.). More precisely, we assume A) A is L-Lipschitzian for some L >0,thatis, Ax Ay L x y, x, y H. A) A is μ-inverse strongly monotone μ-ism) for some μ >0,namely, Ax Ay, x y μ Ax Ay, x, y H. Note that if ϕ is L-Lipschtzian, then ϕ is L -ism. Under the conditions A) and A), it is well known [3] that the operator T = P C I λa) is nonexpansive provided 0 < λ <μ. It turns out that for this range of values of λ, fixed point algorithms can be applied to solve the VI 4.). Applying Theorem 3. we get the result below. Theorem 4. Assume the VI 4.) is solvable. Assume also A satisfies A) and A), and 0<λ <μ. Let f : C C be a contraction. Define a sequence {x n } by the viscosity implicit midpoint rule: xn + x n+ x n+ = α n f x n )+ α n )P C I λa), n 0. In addition, assume {α n } satisfies the conditions C)-C3). Then {x n } converges in norm to a solution x of the VI 4.) which is also a solution to the VI I f )x, x x 0, x A 0).

10 Xu et al. Fixed Point Theory and Applications 05) 05:4 Page 0 of 4. Application to hierarchical minimization We next consider a hierarchical minimization problem see [4] and references therein). Let ϕ 0, ϕ : H R be lower semicontinuous convex functions. Consider the hierarchical minimization min ϕ x), x S 0 S 0 := arg min x H ϕ 0x). 4.4) Here we always assume that S 0 is nonempty. Let S = arg min x S0 ϕ x) and assume S. Assume ϕ 0 and ϕ are differentiable and their gradients satisfy the Lipschitz continuity conditions: ϕ0 x) ϕ 0 y) L0 x y, ϕ x) ϕ y) L x y. 4.5) Note that the condition 4.5)impliesthat ϕ i is L i -ism i =0,).Nowlet T 0 = I γ 0 ϕ 0, T = I γ ϕ, where γ 0 >0andγ >0.NotethatT i is averaged) nonexpansive [3] if0<γ i </L i i = 0, ). Also, it is easily seen that S 0 = FixT 0 ). The optimality condition for x S 0 to be a solution of the hierarchical minimization 4.4)istheVI: x S 0, ϕ x ), x x 0, x S ) This is the VI 4.)withC = S 0 and A = ϕ. We therefore have the following result. Theorem 4. Assume the hierarchical minimization problem 4.4) is solvable. Assume 4.5) and 0<γ i </L i i =0,).Let f : C C be a contraction. Define a sequence {x n } by the viscosity implicit midpoint rule: xn + x n+ x n+ = α n f x n )+ α n )P S0 I λ ϕ ), n 0. In addition, assume {α n } satisfies the conditions C)-C3). Then {x n } converges in norm to a solution x of the VI 4.) which also solves the VI I f )x, x x 0, x S. 4.3 Application to nonlinear evolution equation Browder [5] proved the existence of a periodic solution of the time-dependent nonlinear evolution equation in a real) Hilbert space H, du + At)u = f t, u), dt t >0, 4.7) where At), a family of closed linear operators in H,andf : R H H satisfy the following conditions:

11 Xu et al. Fixed Point Theory and Applications 05) 05:4 Page of B) At) and f t, u) are periodic in t of period ξ >0. B) For each t and each pair u, v H, f t, u) f t, v), u v 0. B3) For each t andeach u DAt)), At)u, u 0. B4) There exists a mild solution u of 4.7)onR + for each initial value v H.Recallthat u is a mild solution of 4.7)withinitialvalueu0) = v if, for each t >0, ut)=ut,0)v + t 0 Ut, s)f s, us) ) ds, where {Ut, s)} t s 0 is the evolution system for the homogeneous linear system du + At)u =0 dt t > s). 4.8) B5) There exists some R >0such that f t, u), u <0 for u = R and all t [0, ξ]. Note that under the conditions B)-B5), the solution u has period ξ and u0) < R. We now apply our viscosity technique for IMR to 4.7). To this end, we define a mapping T : H H by Tv := uξ), v H, where u is the solution of 4.7) satisfying the initial condition u0) = v. It is easy to find that T is nonexpansive. Moreover, the assumption B5) implies that T is a self-mapping of the closed ball B := {v H : v R}. Consequently,T has a fixed point in B whichwedenotebyv, and the corresponding solution u of 4.7) is the periodic solution of 4.7) with period ξ with the initial condition u0) = v. In other words, finding a periodic solution of 4.7) is equivalent to finding a fixed point of T. Therefore, our viscosity technique for IMR is applicable to 4.7). It turns out that the sequence {v n } defined by the IMR vn + v n+ v n+ = α n f v n )+ α n )T 4.9) with {α n } satisfying the conditions C)-C3) of Theorem 3., converges weakly to a fixed point v of T, and then the corresponding mild solution u of 4.7)withinitialvalueu0) = ξ is a periodic solution of 4.7). Note that the iteration procedure 4.9) is essentially to find a mild solution of the nonlinear evolution system 4.7) with the initial value v n + v n+ )/. Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

12 Xu et al. Fixed Point Theory and Applications 05) 05:4 Page of Author details Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, 3008, China. Department of Mathematics, King Abdulaziz University, P.O. Box 8003, Jeddah, 589, Saudi Arabia. 3 Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University, P.O. Box 4087, Jeddah, 49, Saudi Arabia. Acknowledgements The authors are grateful to the anonymous referees for their helpful comments and suggestions, which improved the presentation of this manuscript. This project was funded by the Deanship of Scientific Research DSR), King Abdulaziz University, under grant No HiCi). The authors, therefore, acknowledge technical and financial support of KAU. Received: December 04 Accepted: 9 February 05 References. Moudafi, A: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 4, ). Attouch, H: Viscosity approximation methods for minimization problems. SIAM J. Optim. 63), ) 3. Xu, HK: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 98, ) 4. Auzinger, W, Frank, R: Asymptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case. Numer. Math. 56, ) 5. Bader, G, Deuflhard, P: A semi-implicit mid-point rule for stiff systems of ordinary differential equations. Numer. Math. 4, ) 6. Deuflhard, P: Recent progress in extrapolation methods for ordinary differential equations. SIAM Rev. 74), ) 7. Schneider, C: Analysis of the linearly implicit mid-point rule for differential-algebra equations. Electron. Trans. Numer. Anal.,-0 993) 8. Somalia, S: Implicit midpoint rule to the nonlinear degenerate boundary value problems. Int. J. Comput. Math. 793), ) 9. Van Veldhuxzen, M: Asymptotic expansions of the global error for the implicit midpoint rule stiff case). Computing 33, ) 0. Alghamdi, MA, Alghamdi, MA, Shahzad, N, Xu, HK: The implicit midpoint rule for nonexpansive mappings. Fixed Point Theory Appl. 04, 96 04). Goebel, K, Kirk, WA: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge 990). Xu, HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, ) 3. Xu, HK: Averaged mappings and the gradient-projection algorithm. J. Optim. Theory Appl. 50, ) 4. Cabot, A: Proximal point algorithm controlled by a slowly vanishing term: applications to hierarchical minimization. SIAM J. Optim. 5, ) 5. Browder, FE: Existence of periodic solutions for nonlinear equations of evolution. Proc. Natl. Acad. Sci. USA 53, )