The implicit midpoint rule for nonexpansive mappings
© Alghamdi et al.; licensee Springer. 2014
Received: 21 January 2014
Accepted: 16 March 2014
Published: 9 April 2014
The implicit midpoint rule (IMR) for nonexpansive mappings is established. The IMR generates a sequence by an implicit algorithm. Weak convergence of this algorithm is proved in a Hilbert space. Applications to the periodic solution of a nonlinear time-dependent evolution equation and to a Fredholm integral equation are included.
MSC:47J25, 47N20, 34G20, 65J15.
The implicit midpoint rule (IMR) is one of the powerful numerical methods for solving ordinary differential equations (in particular, the stiff equations) [1–6] and differential-algebra equations .
where is a stepsize. It is known that if is Lipschitz continuous and sufficiently smooth, then the sequence converges to the exact solution of (1.1) as uniformly over for any fixed .
where T is, in general, a nonlinear operator in a Hilbert space. We below introduce our implicit midpoint rule (IMR) for the fixed point problem (1.6) in two iterative algorithms. The first algorithm generates a sequence in the following manner.
where for all n.
Our second IMR is an algorithm that generates a sequence as follows.
where for all n.
Consequently, we may concentrate on Algorithm II.
The iterative methods for finding fixed points of nonexpansive mappings have received much attention due to the fact that in many practical problems, the governing operators are nonexpansive (cf. [8, 9]). Two iterative methods are basic and they are Mann’s method [10, 11] and Halpern’s method [12–16]. An implicit method is also proposed in .
2 Convergence analysis
This is immediately clear due to the nonexpansivity of T.
is a contraction with coefficient .
2.1 Properties of Algorithm II
We first discuss the properties of Algorithm II.
for all and .
The proof of the lemma is complete. □
This in turn implies (2.8). □
2.2 Convergence of Algorithms I and II
As Algorithm I is a variant of Algorithm II, we focus on the convergence of Algorithm II. To this end, we need two conditions for the sequence of parameters as follows:
(C1) for all and some ,
satisfies (C1) and (C2).
To prove the convergence of Algorithm II, we need the following so-called demiclosedness principle for nonexpansive mappings.
Lemma 2.4 ()
Let C be a nonempty closed convex subset of a Hilbert space H, and let be a nonexpansive mapping with a fixed point. Assume that is a sequence in C such that weakly and strongly. Then (i.e., ).
We use the notation to denote the set of all weak cluster points of the sequence .
The following result is easily proved (see ).
exists for all ,
Then weakly converges to a point in K.
We are now in a position to state and prove the main convergence result of this paper.
Theorem 2.6 Let H be a Hilbert space and be a nonexpansive mapping with . Assume that is generated by IMR (1.8) where the sequence of parameters satisfies conditions (C1) and (C2). Then converges weakly to a fixed point of T.
Proof By Lemmas 2.3 and 2.4, we have . Furthermore, by Lemma 2.1, exists for all . Consequently, we can apply Lemma 2.5 with to assert the weak convergence of to a point in . □
We then have the following convergence result for IMR (1.7).
Theorem 2.7 Let H be a Hilbert space and be a nonexpansive mapping with . Assume that is generated by IMR (1.7) where the sequence of parameters satisfies conditions (C1) and (C2). Then converges weakly to a fixed point of T.
3.1 Periodic solution of a nonlinear evolution equation
where is a family of closed linear operators in H and .
Theorem 3.1 ()
- (i)For each t and each pair ,
For each t and each , .
- (iii)There exists a mild solution u of equation (3.1) on for each initial value . Recall that u is a mild solution of (3.1) with the initial value if, for each ,
- (iv)There exists some such that
for and all .
Then there exists an element v of H with such that the mild solution of equation (3.1) with the initial condition is periodic of period ξ.
We next apply our IMR for nonexpansive mappings to provide an iterative method for finding a periodic solution of (3.1).
converges weakly to a fixed point v of T, and the mild solution of (3.1) with the initial value is a periodic solution of (3.1). Note that the iteration method (3.3) is essentially to find a mild solution of (3.1) with the initial value of .
3.2 Fredholm integral equation
Then the sequence converges weakly in to the solution of integral equation (3.6).
This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under Grant No. 2-363-1433-HiCi. The authors, therefore, acknowledge technical and financial support of KAU.
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