General Viscosity Implicit Midpoint Rule For Nonexpansive Mapping
aa r X i v : . [ m a t h . O C ] S e p General Viscosity Implicit Midpoint Rule For Nonexpansive Mapping
Shuja Haider Rizvi ∗ Department of Mathematics, Babu Banarasi Das University, Lucknow 226028, India
Abstract:
In this work, we suggest a general viscosity implicit midpoint rule for nonexpansive mappingin the framework of Hilbert space. Further, under the certain conditions imposed on the sequence ofparameters, strong convergence theorem is proved by the sequence generated by the proposed iterativescheme, which, in addition, is the unique solution of the variational inequality problem. Furthermore,we provide some applications to variational inequalities, Fredholm integral equations, and nonlinear evo-lution equations. The results presented in this work may be treated as an improvement, extension andrefinement of some corresponding ones in the literature.
Keywords:
General viscosity implicit midpoint rule; Nonexpansive mapping; Fixed-point problem; Itera-tive scheme.
Primary 65K15; Secondary: 47J25 65J15 90C33.
Throughout the paper unless otherwise stated, H denotes a real Hilbert space, we denote the norm andinner product of H by k · k , and h ., . i respectively. Let K be a nonempty, closed and convex subset of H . Let { x n } be any sequence in H , then x n → x (respectively, x n ⇀ x ) will denote strong (respectively,weak) convergence of the sequence { x n } .A mapping S : H → H is said to be contraction mapping if there exists a constant α ∈ (0 ,
1) suchthat k Sx − Sy k ≤ α k x − y k , for all x, y ∈ H . If α = 1 then S : H → H is said to be nonexpansive mapping i.e., k Sx − Sy k ≤ k x − y k ,for all x, y ∈ H . We use Fix( S ) to denote the set of fixed points of S . An operator B : H → H is said tobe strongly positive bounded linear operator , if there exists a constant ¯ γ > h Bx, x i ≥ ¯ γ k x k , ∀ x ∈ H. The viscosity approximation method of selecting a particular fixed point of a given nonexpansivemapping was proposed by Moudafi [1] in the framework of a Hilbert space, which generates the sequence ∗ Corresponding author; E-mail address:[email protected] (S.H. Rizvi) x n } by the following iterative scheme: x n +1 = α n Q ( x n ) + (1 − α n ) Sx n , n ≥ , (1.1)where { α n } ⊂ [0 ,
1] and Q is a contraction mapping on H . Note that the iterative scheme (1.1) generalizethe results of Browder [2] and Halpern [3] in another direction. The convergence of the explicit iterativescheme (1.1) has been the subject of many authors because under suitable conditions these iterationconverge strongly to the unique solution q ∈ Fix( S ) of the variational inequality h ( I − Q ) q, x − q i ≥ , ∀ x ∈ Fix( S ) . (1.2)This fact allows us to apply this method to convex optimization, linear programming and monotoneinclusions. In 2004, Xu [4] extended the result of Moudafi [1] to uniformly smooth Banach spaces andobtained strong convergence theorem. For related work, see [5–7].In 2006, Marino and Xu [8] introduced the following iterative scheme based on viscosity approximationmethod, for fixed point problem for a nonexpansive mapping S on H : x n +1 = α n γQ ( x n ) + ( I − α n B ) Sx n , n ≥ , (1.3)where Q is a contraction mapping on H with constant α > B is a strongly positive self-adjoint boundedlinear operator on H with constant ¯ γ > γ ∈ (0 , ¯ γα ). They proved that the sequence { x n } generatedby (1.3) converge strongly to the unique solution of the variational inequality h ( B − γQ ) z, x − z i ≥ , ∀ x ∈ Fix( S ) , (1.4)which is the optimality condition for the minimization problemmin x ∈ Fix( S ) h Bx, x i − h ( x ) , where h is the potential function for γQ .The implicit midpoint rule is one of the powerful numerical methods for solving ordinary differentialequations and differential algebraic equations. For related works, we refer to [9–16] and the referencescited therein. For instance, consider the initial value problem for the differential equation y ′ ( t ) = f ( y ( t ))with the initial condition y (0) = y , where f is a continuous function from R d to R d . The implicitmidpoint rule in which generates a sequence { y n } by the following the recurrence relation1 h ( y n +1 − y n ) = f (cid:18) y n +1 − y n (cid:19) . In 2014, implicit midpoint rule has been extended by Alghamdi et al. [17] to nonexpansive mappings,2hich generates a sequence { x n } by the following implicit iterative scheme: x n +1 = α n x n + (1 − α n ) S (cid:18) x n + x n +1 (cid:19) , n ≥ , (1.5)Recently, Xu et al. [18] extended and generalized the results of Alghamdi et al. [17] and presented thefollowing viscosity implicit midpoint rule for nonexpansive mapping, which generates a sequence { x n } bythe following implicit iterative scheme: x n +1 = α n Q ( x n ) + (1 − α n ) S (cid:18) x n + x n +1 (cid:19) , n ≥ , (1.6)where { α n } ⊂ [0 ,
1] and S is a nonexpansive mapping. They proved that under some mild conditions,the sequence generated by (1.6) converge in norm to fixed point of nonexpansive mapping, which, inaddition, solves the variational inequality (1.2). Further related work, see [19, 20].Motivated by the work of Moudafi [1], Xu [4], Marino and Xu [8], Alghamdi et al. [17] and Xu etal. [18], and by the ongoing research in this direction, we suggest and analyze general viscosity implicitmidpoint iterative scheme for fixed point of nonexpansive mapping in real Hilbert space. Further, basedon these general viscosity implicit midpoint iterative scheme, we prove the strong convergence theoremsfor a nonexpansive mapping. Furthermore, some consequences from these theorems are also derived. Theresults and methods presented here extend and generalize the corresponding results and methods givenin [1, 4, 8, 17, 18]. We recall some concepts and results which are needed in sequel.For every point x ∈ H , there exists a unique nearest point in K denoted by P K x such that k x − P K x k ≤ k x − y k , ∀ y ∈ K. (2.1) Remark 2.1. [21]
It is well known that P K is nonexpansive mapping and satisfies h x − y, P K x − P K y i ≥ k P K x − P K y k , ∀ x, y ∈ H. (2.2) Moreover, P K x is characterized by the fact P K x ∈ K and h x − P K x, y − P K x i ≤ . (2.3)The following Lemma is the well known demiclosedness principles for nonexpansive mappings. Lemma 2.1. [21, 22]
Assume that S be a nonexpansive self mapping of a closed and convex subset K ofa Hilbert space H . If S has a fixed point, then I − S is demiclosed, i.e., whenever { x n } is a sequence n K converging weakly to some x ∈ K and the sequence { ( I − S ) x n } converges strongly to some y , itfollows that ( I − S ) x = y . Lemma 2.2. [21, 22]
In real Hilbert space H , the following hold:(i) k x + y k ≤ k x k + 2 h y, x + y i , ∀ x, y ∈ H ; (2.4) (ii) k λx + (1 − λ ) y k = λ k x k + (1 − λ ) k y k − λ (1 − λ ) k x − y k , (2.5) for all x, y ∈ H and λ ∈ (0 , . Lemma 2.3. [8]
Assume that B is a strongly positive self-adjoint bounded linear operator on a Hilbertspace H with constant ¯ γ > and < ρ ≤ k B k − . Then k I − ρB k ≤ − ρ ¯ γ . Lemma 2.4. [4] . Let { a n } be a sequence of nonnegative real numbers such that a n +1 ≤ (1 − β n ) a n + δ n , n ≥ , where { β n } is a sequence in (0 , and { δ n } is a sequence in R such that(i) ∞ P n =1 β n = ∞ ; (ii) lim sup n →∞ δ n β n ≤ or ∞ P n =1 | δ n | < ∞ .Then lim n →∞ a n = 0 . In this section, we prove a strong convergence theorem based on the general viscosity implicit midpointrule for fixed point of nonexpansive mapping.
Theorem 3.1.
Let H be a real Hilbert space and B : H → H be a strongly positive bounded linearoperator with constant ¯ γ > such that < γ < ¯ γα < γ + α and Q : H → H be a contraction mapping withconstant α ∈ (0 , . Let S : H → H be a nonexpansive mapping such that Fix( S ) = ∅ . Let the iterativesequence { x n } be generated by the following general viscosity implicit midpoint iterative schemes: x n +1 = α n γQ ( x n ) + (1 − α n B ) S (cid:18) x n + x n +1 (cid:19) , n ≥ , (3.1) where { α n } is the sequence in (0 , and satisfying the following conditions(i) lim n →∞ α n = 0 ;(ii) ∞ P n =0 α n = ∞ ; iii) ∞ P n =1 | α n − α n − | < ∞ or lim n →∞ α n +1 α n = 1 .Then the sequence { x n } converge strongly to z ∈ Fix( S ) , where z = P Fix( S ) Q ( z ) . In other words, whichis also unique solution of variational inequality (1.4) .Proof. Note that from condition (i), we may assume without loss of generality that α n ≤ (1 − β n ) k B k − for all n . From Lemma 2.3, we know that if 0 < ρ ≤ k B k − , then k I − ρB k ≤ − ρ ¯ γ . We will assumethat k I − B k ≤ − ¯ γ .Since B is strongly positive self-adjoint bounded linear operator on H , then k B k = sup {|h Bu, u i| : u ∈ H, k u k = 1 } . Observe that h ( I − α n B ) u, u i = 1 − α n h Bu, u i≥ − α n k B k ≥ , which implies that (1 − α n B ) is positive. It follows that k ( I − α n B k = sup {h ((1 − α n B ) u, u i : u ∈ H, k u k = 1 } = sup { − α n h Bu, u i : u ∈ H, k u k = 1 }≤ − α n ¯ γ. Let q = P Fix( S ) . Since Q is a contraction mapping with constant α ∈ (0 , k q ( I − B + γQ )( x ) − q ( I − B + γQ )( y ) k ≤ k ( I − B + γQ )( x ) − ( I − B + γQ )( x ) k≤ k I − B kk x − y k + γ k Q ( x ) − Q ( y ) k≤ (1 − ¯ γ ) k x − y k + γα k x − y k≤ (1 − (¯ γ − γα )) k x − y k , for all x, y ∈ H . Therefore, the mapping q ( I − B + γQ ) is a contraction mapping from H into itself. Itfollows from Banach contraction principle that there exists an element z ∈ H such that z = q ( I − B + γQ ) z = P Fix( S ) ( I − B + γQ )( z ).Let p ∈ Fix( S ), we compute k x n +1 − p k = (cid:13)(cid:13)(cid:13) α n γQ ( x n ) + (1 − α n B ) S (cid:18) x n + x n +1 (cid:19) − p (cid:13)(cid:13)(cid:13) ≤ α n k γQ ( x n ) − Bp k + (1 − α n ¯ γ ) (cid:13)(cid:13)(cid:13)(cid:13) S (cid:18) x n + x n +1 (cid:19) − p (cid:13)(cid:13)(cid:13)(cid:13) ≤ α n [ γ k Q ( x n ) − Q ( p ) k + k γQ ( p ) − Bp k ] + (1 − α n ¯ γ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) x n + x n +1 (cid:19) − p (cid:13)(cid:13)(cid:13)(cid:13) α n γα k x n − p k + α n k γQ ( p ) − Bp k + (1 − α n ¯ γ )2 ( k x n − p k + k x n +1 − p k ) , which implies that(1 + α n ¯ γ )2 k x n +1 − p k ≤ (cid:20) α n γα + (1 − α ¯ γ )2 (cid:21) k x n − p k + α n k γQ ( p ) − Bp kk x n +1 − p k ≤ (cid:20) γα − ¯ γ ) α n α n ¯ γ (cid:21) k x n − p k + 2 α n α n ¯ γ k γQ ( p ) − Bp k≤ (cid:20) − γ − γα ) α n α n ¯ γ (cid:21) k x n − p k + 2 α n α n ¯ γ k γQ ( p ) − Bp k≤ (cid:20) − γ − γα ) α n α n ¯ γ (cid:21) k x n − p k + 2 α n (¯ γ − γα )1 + α n ¯ γ k γQ ( p ) − Bp k (¯ γ − γα ) . Consequently, we get k x n +1 − p k ≤ max n k x n − p k , k γQ ( p ) − Bp k ¯ γ − γα o . Therefore by using induction, we obtain k x n +1 − p k ≤ max n k x − p k , k γQ ( p ) − Bp k ¯ γ − γα o . (3.2)Hence the sequence { x n } is bounded.Next, we show that the sequence { x n } is asymptotically regular, i.e., lim n →∞ k x n +1 − x n k = 0. It followsfrom (3.1) that k x n +1 − x n k = (cid:13)(cid:13)(cid:13) α n γQ ( x n ) + (1 − α n B ) S (cid:18) x n + x n +1 (cid:19) − h α n − γQ ( x n − ) + (1 − α n − B ) (cid:18) x n − + x n (cid:19) i(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) (1 − α n B ) (cid:20) S (cid:18) x n + x n +1 (cid:19) − S (cid:18) x n − + x n (cid:19)(cid:21) +( α n − B − α n B ) (cid:20) S (cid:18) x n − + x n (cid:19) − γQ ( x n − ) (cid:21) + α n ( γQ ( x n ) − γQ ( x n − )) (cid:13)(cid:13)(cid:13) ≤ (1 − α n ¯ γ ) (cid:13)(cid:13)(cid:13) S (cid:18) x n + x n +1 (cid:19) − S (cid:18) x n − + x n (cid:19) (cid:13)(cid:13)(cid:13) + M | α n − − α n | + α n γ k Q ( x n ) − Q ( x n − ) k≤ (1 − α n ¯ γ )2 h k x n +1 − x n k + k x n − x n − k i + M | α n − − α n | + α n γα k x n − x n − k , where M := sup (cid:26) S (cid:18) x n + x n +1 (cid:19) + γ k Q ( x n ) k : n ∈ N (cid:27) . It follows that(1 + α n ¯ γ )2 k x n +1 − x n k ≤ (1 − α n ¯ γ )2 k x n − x n − k + M | α n − − α n | + α n γα k x n − x n − kk x n +1 − x n k ≤ γα − ¯ γ ) α n α n ¯ γ k x n − x n − k + 2 M α n ¯ γ | α n − − α n |≤ (cid:18) − γα − ¯ γ ) α n α n ¯ γ (cid:19) k x n − x n − k + 2 M α n ¯ γ | α n − − α n | .
6y using the conditions (i)-(iii) of Lemma 2.4, we obtainlim n →∞ k x n +1 − x n k = 0 . (3.3)Next, we show that lim n →∞ k x n − Sx n k = 0 . We can write k x n − Sx n k ≤ k x n − x n +1 k + (cid:13)(cid:13)(cid:13) x n +1 − S (cid:18) x n + x n +1 (cid:19) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) S (cid:18) x n + x n +1 (cid:19) − Sx n (cid:13)(cid:13)(cid:13) ≤ k x n − x n +1 k + α n (cid:13)(cid:13)(cid:13) γQ ( x n ) + (1 − γB ) S (cid:18) x n + x n +1 (cid:19) (cid:13)(cid:13)(cid:13) + 12 k x n +1 − x n k≤ k x n +1 − x n k + α n (cid:13)(cid:13)(cid:13) γQ ( x n ) − S (cid:18) x n + x n +1 (cid:19) (cid:13)(cid:13)(cid:13) ≤ k x n +1 − x n k + α n M. It follows from condition (i) and (3.3), we obtainlim n →∞ k x n − Sx n k = 0 . Since { x n } is bounded, there exists a subsequence { x n k } of { x n } such that x n k ⇀ ˆ x say. Next, weclaim that lim sup n →∞ h Q ( z ) − z, x n − z i ≤
0, where z = P Fix( S ) ( I − B + γQ ) z . To show this inequality, weconsider a subsequence { x n k } of { x n } such that x n k ⇀ ˆ x ,lim sup n →∞ h ( B − γQ ) z − z, x n − z i = lim sup n →∞ h ( B − γQ ) z − z, x n − z i = lim sup k →∞ h ( B − γQ ) z − z, x n k − z i = h ( B − γQ ) z − z, ˆ x − z i ≤ . (3.4)Finally, we show that x n → z . It follows from Lemma 2.2 that k x n +1 − z k = (cid:13)(cid:13)(cid:13) α n γQ ( x n ) + ( I − α n B ) S (cid:18) x n + x n +1 (cid:19) − z (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) α n ( γQ ( x n ) − Bz ) + ( I − α n B ) S (cid:18) x n + x n +1 (cid:19) − z (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) ( I − α n B ) S (cid:18) x n + x n +1 (cid:19) − z (cid:13)(cid:13)(cid:13) + 2 α n h γQ ( x n ) − Bz, x n +1 − z i≤ (1 − α n ¯ γ ) (cid:13)(cid:13)(cid:13) S (cid:18) x n + x n +1 (cid:19) − z (cid:13)(cid:13)(cid:13) + 2 α n γ k Q ( x n ) − Q ( z ) kk x n +1 − z k +2 α n h γQ ( z ) − Bz, x n +1 − z i≤ (1 − α n ¯ γ ) (cid:13)(cid:13)(cid:13) x n + x n +1 − z (cid:13)(cid:13)(cid:13) + 2 α n γα k x n − z kk x n +1 − z k α n h γQ ( z ) − Bz, x n +1 − z i≤ (1 − α n ¯ γ ) k x n − z k + 2 α n γα k x n − z kk x n +1 − z k +2 α n h γQ ( z ) − Bz, x n +1 − z i≤ (1 − α n ¯ γ ) h k x n − z k + 12 k x n +1 − z k − k x n +1 − x n k i +2 α n γα h k x n − z k + k x n +1 − z k i + 2 α n h γQ ( z ) − Bz, x n +1 − z i≤ h (1 − α n ¯ γ ) α n γα i ( k x n − z k + k x n +1 − z k )+2 α n h γQ ( z ) − Bz, x n +1 − z i≤ − α n ¯ γ + 2 α n γα k x n − z k + k x n +1 − z k ) + α n ¯ γ M +2 α n h γQ ( z ) − Bz, x n +1 − z i . This implies that k x n +1 − z k ≤ − γ − γα ) α n γ − γα ) α n k x n − z k + 2 α n ¯ γ γ − γα ) α n M + 4 α n γ − γα ) α n h γQ ( z ) − Bz, x n +1 − z i = h − γ − γα ) α n γ − γα ) α n i k x n − z k + 2 α n ¯ γ γ − γα ) α n M + 4 α n γ − γα ) α n h γQ ( z ) − Bz, x n +1 − z i = (1 − δ n ) k x n − z k + δ n σ n , (3.5)where M := sup {k x n − z k : n ≥ } , δ n = 4(¯ γ − γα ) α n γ − γα ) α n and σ n = ( α n ¯ γ ) M γ − γα ) α n ) + 4 α n γ − γα ) α n h γQ ( z ) − Bz, x n +1 − z i . Sincelim n →∞ α n = 0 and ∞ P n =0 α n = ∞ , it is easy to see that lim n →∞ δ n = 0, ∞ P n =0 δ n = ∞ and lim sup n →∞ σ n ≤
0. Hencefrom (3.4), (3.5) and Lemma 2.4, we deduce that x n → z . This completes the proof.As a direct consequences of Theorem 3.1, we obtain the following result due to Xu et al. [18] for fixedpoint of nonexpansive mapping. Take γ := 1 and B := I in Theorem 3.1 then the following Corollary isobtained. Corollary 3.1. [18]
Let H be a real Hilbert space and Q : H → H be a contraction mapping with constant α ∈ (0 , . Let S : H → H be a nonexpansive mapping such that Fix( S ) = ∅ . Let the iterative sequence { x n } be generated by the following general viscosity implicit midpoint iterative schemes: x n +1 = α n Q ( x n ) + (1 − α n ) S (cid:18) x n + x n +1 (cid:19) , n ≥ , (3.6) where { α n } is the sequence in (0 , and satisfying the conditions (i)-(iii) of Theorem . Then thesequence { x n } converge strongly to z ∈ Fix( S ) , which, in addition also solves variational inequality (1.2) . et al. [17] for fixed point problem of nonexpansive mapping.Take γ := 1 and Q, B := I in Theorem 3.1 then the following Corollary is obtained. Corollary 3.2. [17]
Let H be a real Hilbert space and Q : H → H be a contraction mapping with constant α ∈ (0 , . Let S : H → H be a nonexpansive mapping such that Fix( S ) = ∅ . Let the iterative sequence { x n } be generated by the following general viscosity implicit midpoint iterative schemes: x n +1 = α n x n + (1 − α n ) S (cid:18) x n + x n +1 (cid:19) , n ≥ , (3.7) where { α n } is the sequence in (0 , and satisfying the conditions (i)-(iii) of Theorem . Then thesequence { x n } converge strongly to z ∈ Fix( S ) . Remark 3.1.
Theorem extends and generalize the viscosity implicit midpoint rule of Xu et al. [4] and the implicit midpoint rule of Alghamdi et al. [17] to a general viscosity implicit midpoint rule for anonexpansive mappings, which also includes the results of [1, 8] as special cases.
We consider the following classical variational inequality problem (In short, VIP): Find x ∗ ∈ K such that h Ax ∗ , x − x ∗ i ≥ , ∀ x ∈ K, (4.1)where A is a single-valued monotone mapping on H and K is a closed and convex subset of H . We assume K ⊂ dom ( A ). An example of VIP (4.1) is the constrained minimization problem : Find x ∗ ∈ K suchthat min x ∈ K φ ( x ∗ ) (4.2)where φ : H → R is a lower-semicontinuous convex function. If φ is (Frechet) differentiable, then theminimization problem (4.2) is equivalently reformulated as VIP (4.1) with A = ∇ φ . Notice that the VIP(4.1) is equivalent to the following fixed point problem, for any λ > Sx ∗ = x ∗ , Sx := P K ( I − λA ) x. (4.3)If A is Lipschitz continuous and strongly monotone, then, for λ > S is a contractionmapping and its unique fixed point is also the unique solution of the VIP (4.1). However, if A is notstrongly monotone, S is no longer a contraction, in general. In this case we must deal with nonexpansivemappings for solving the VIP (4.1). More precisely, we assume(i) A is θ - Lipschitz continuous for some θ >
0, i.e., k Ax − Ay k ≤ θ k x − y k , ∀ x, y ∈ H. A is µ - inverse strongly monotone ( µ -ism) for some µ >
0, namely, h Ax − Ay, x − y ≥ µ k Ax − Ay k , ∀ x, y ∈ H. It is well known that by using the conditions (i) and (ii), the operator S = P K ( I − λA ) is nonexpansiveprovided that 0 < λ < µ . It turns out that for this range of values of λ , fixed point algorithms can beapplied to solve the VIP (4.1). Applying Theorem 3.1, we get the following result. Theorem 4.1.
Assume that
VIP (4.1) is solvable in which A satisfies the conditions (i) and (ii) with < λ < µ . Let B : H → H be a strongly positive bounded linear operator with constant ¯ γ > such that < γ < ¯ γα < γ + α and Q : H → H be a contraction mapping with constant α ∈ (0 , . Let the iterativesequence { x n } be generated by the following general viscosity implicit midpoint iterative schemes: x n +1 = α n γQ ( x n ) + (1 − α n B ) P K ( I − λA ) (cid:18) x n + x n +1 (cid:19) , n ≥ , (4.4) where { α n } is the sequence in (0 , and satisfying the conditions (i)-(iii) of Theorem . Then thesequence { x n } converge strongly to a solution z of VIP (4.1) , which is also unique solution of variationalinequality (1.4) . Consider a
Fredholm integral equation of the following form x ( t ) = g ( t ) + Z t F ( t, s, x ( s )) ds, t ∈ [0 , , (4.5)where g is a continuous function on [0 ,
1] and F : [0 , × [0 , × R → R is continuous. Note that if F satisfies the Lipschitz continuity condition, i.e., | F ( t, s, x ) − F ( t, s, y ) | ≤ | x − y | , ∀ t, s ∈ [0 , , x, y ∈ R , then equation (4.5) has at least one solution in L [0 ,
1] (see [23]). Define a mapping S : L [0 , → L [0 , Sx )( t ) = g ( t ) + Z t F ( t, s, x ( s )) ds, t ∈ [0 , . (4.6)It is easy to observe that S is nonexpansive. In fact, we have, for x, y ∈ L [0 , k Sx k = Z | ( Sx )( t ) − ( Sy )( t ) | dt = Z (cid:12)(cid:12)(cid:12)(cid:12)Z ( F ( t, s, x ( s )) − F ( t, s, x ( s ))) ds (cid:12)(cid:12)(cid:12)(cid:12) dt ≤ Z (cid:12)(cid:12)(cid:12)(cid:12)Z | x ( s ) − y ( s ) | ds (cid:12)(cid:12)(cid:12)(cid:12) dt Z | x ( s ) − y ( s ) | ds = k x − y k . This means that to find the solution of integral equation (4.5) is reduced to finding a fixed point ofthe nonexpansive mapping S in the Hilbert space L [0 , x ∈ L [0 , { x n } in L [0 ,
1] generated by the general viscosity implicit midpoint iterativescheme: x n +1 = α n γQ ( x n ) + (1 − α n B ) S (cid:18) x n + x n +1 (cid:19) , n ≥ , (4.7)where { α n } is the sequence in (0 ,
1) and satisfying the conditions (i)-(iii) of Theorem 3.1. Then thesequence { x n } converges strongly in L [0 ,
1] to the solution of integral equation (4.5).
Consider the following time-dependent nonlinear evolution equation in a Hilbert space H , dudt + A ( t ) u = f ( t, u ) , t > , (4.8)where A ( t ) is a family of closed linear operators in H and f : R × H → H . The following result is theexistence of periodic solutions of nonlinear evolution equation (4.8) due to Browder [24]. Theorem 4.2. [24]
Suppose that A ( t ) and f ( t, u ) are periodic in t of period ω > and satisfy thefollowing assumptions:(i) For each t and each pair u, v ∈ H , Re h f ( t, u ) − f ( t, v ) , u − v i ≤ . (ii) For each t and each u ∈ D ( A ( t )) , Re h A ( t ) u, u i ≥ .(iii) There exists a mild solution u of equation (4.8) on R + for each initial value v ∈ H . Recall that u is a mild solution of (4.8) with the initial value u (0) = v if, for each t > , u ( t ) = U ( t, v + Z U ( t, s ) f s, u ( s ) ds, where {U ( t, s ) } t ≥ s ≥ is the evolution system for the homogeneous linear system dudt + A ( t ) u = 0 , ( t > s ) . (4.9) (iv) There exists some R > such that Re h f ( t, u ) , u i < , for k u k = R and all t ∈ [0 , ω ] . hen there exists an element v of H with k v k < R such that the mild solution of equation (4.8) with theinitial condition u (0) = v is periodic of period ω . Next, we apply the general viscosity implicit midpoint rule for nonexpansive mappings to provide animplicit iterative scheme for finding a periodic solution of (4.8). As a matter of fact, define a mapping S : H → H by assigning to each v ∈ H the value u( ω ), where u is the solution of (4.8) satisfying theinitial condition u (0) = v . Namely, we define S by Sv = u ( ω ), where u solves (4.8) with u (0) = v .We then find that S is nonexpansive. Moreover, condition (iv) of Theorem 4.2 forces S to map theclosed ball B := { v ∈ H : k v k ≤ R } into itself. Consequently, S has a fixed point which we denote by v ,and the corresponding solution u of (4.8) with the initial condition u (0) = v is a desired periodic solutionof (4.8) with period ω . In other words, to find a periodic solution u of (4.8) is equivalent to finding afixed point of S . Therefore the general viscosity implicit midpoint rule is applicable to solve (4.8), inwhich { x n } is generated by the general viscosity implicit midpoint iterative scheme: x n +1 = α n γQ ( x n ) + (1 − α n B ) S (cid:18) x n + x n +1 (cid:19) , (4.10)where { α n } is the sequence in (0 ,
1) and satisfying the conditions (i)-(iii) of Theorem 3.1. Then thesequence { x n } converges weakly to a fixed point v of S , and the solution of (4.8) with the initial value u (0) = ω is a periodic solution of (4.8). Conclusion:
The present work has been aimed to study the general viscosity implicit midpoint rule fornonexpansive mapping and proved the strong convergence theorem for solving fixed point for a nonex-pansive mapping. Theorem 3.1 extends and generalize the viscosity implicit midpoint rule of Xu et al. [4]and the implicit midpoint rule of Alghamdi et al. [17] to a general viscosity implicit midpoint rule for anonexpansive mappings, which also includes the results of [1, 8] as special cases.
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