Abstract
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.
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1 Introduction
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 [7].
For the ordinary differential equation
IMR generates a sequence by the recursion procedure
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 .
If we write the function f in the form , then differential equation (1.1) becomes
and the process (1.2) is rewritten as
The equilibrium problem associated with differential equation (1.3) is the fixed point problem
This motivates us to transplant IMR (1.4) to the solving of the fixed point equation
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.
Algorithm I Initialize arbitrarily and iterate
where for all n.
Our second IMR is an algorithm that generates a sequence as follows.
Algorithm II Initialize arbitrarily and iterate
where for all n.
We observe that Algorithm I is equivalent to Algorithm II since it is easy to rewrite (1.7), by partially solving for , as
where
Consequently, we may concentrate on Algorithm II.
The purpose of this paper is to study the convergence of two IMR (1.7) and (1.8) in the case where the mapping T is a nonexpansive mapping in a general Hilbert space H, that is,
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 [17].
2 Convergence analysis
Throughout this section we always assume that H is a Hilbert space with the inner product and the norm and that is a nonexpansive mapping with a fixed point. We use to denote the set of fixed points of T. Namely, . It is not hard to find that both IMR (1.7) and (1.8) are well defined. As a matter of fact, for each fixed and , the mapping
is a contraction with coefficient . That is,
This is immediately clear due to the nonexpansivity of T.
It is also easily seen that the mapping
is a contraction with coefficient .
2.1 Properties of Algorithm II
We first discuss the properties of Algorithm II.
Lemma 2.1 Let be the sequence generated by Algorithm II. Then
-
(i)
for all and .
-
(ii)
.
-
(iii)
.
Proof Let . We deduce that
It turns out that
and
It is then immediately evident that
Moreover, since , (2.4) also implies that
and
The proof of the lemma is complete. □
Lemma 2.2 Let be the sequence generated by Algorithm II. Suppose that for all and some . Then
Proof By definition (1.8) of Algorithm II, we derive that
Hence
Using the assumption that , we further derive that
Now (2.6) and (2.7) imply that
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 ,
(C2) .
These two conditions are not restrictive. As a matter of fact, it is not hard to find that, for each , the sequence
satisfies (C1) and (C2).
Lemma 2.3 Assume (C1) and (C2). Then the sequence generated by Algorithm II satisfies the property
Proof From (1.8) it follows that
Now condition (C2) implies that for all large enough n. Hence from Lemma 2.2, we immediately get
Conclusion (2.9) now follows from the following inference:
□
To prove the convergence of Algorithm II, we need the following so-called demiclosedness principle for nonexpansive mappings.
Lemma 2.4 ([18])
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 [19]).
Lemma 2.5 Let K be a nonempty closed convex subset of a Hilbert space H, and let be a bounded sequence in H. Assume that
-
(i)
exists for all ,
-
(ii)
.
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.
Proof Since is also generated by algorithm (1.9), it suffices to verify that the sequence defined in (1.10) satisfies conditions (C1) and (C2). As and satisfies (C2), it is evident that satisfies (C2) as well. To see that also fulfils (C1), we argue as follows, using the fact that satisfies (C1):
□
3 Applications
3.1 Periodic solution of a nonlinear evolution equation
Consider the time-dependent nonlinear evolution equation in a (possibly complex) Hilbert space H,
where is a family of closed linear operators in H and .
Browder [20] proved the following existence of periodic solutions of equation (3.1).
Theorem 3.1 ([20])
Suppose that and are periodic in t of period and satisfy the following assumptions:
-
(i)
For each t and each pair ,
-
(ii)
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 ,
where is the evolution system for the homogeneous linear system
-
(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).
As a matter of fact, define a mapping by assigning to each the value , where u is the solution of (3.1) satisfying the initial condition . Namely, we define T by
We then find that T is nonexpansive. Moreover, assumption (iv) forces T to map the closed ball into itself. Consequently, T has a fixed point which we denote by v, and the corresponding solution u of (3.1) with the initial condition is a desired periodic solution of (3.1) with period ξ. In other words, to find a periodic solution u of (3.1) is equivalent to finding a fixed point of T. Our IMR is thus applicable to (3.1). It turns out that the sequence defined by the IMR
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
Consider a Fredholm integral equation of the form
where g is a continuous function on and is continuous. The existence of solutions has been investigated in the literature (see [21] and the references therein). In particular, if F satisfies the Lipschitz continuity condition
then equation (3.6) has at least one solution in ([[21], Theorem 3.3]). Define a mapping by
It is easily seen that T is nonexpansive. As a matter of fact, we have, for ,
This means that to find the solution of integral equation (3.6) is reduced to finding a fixed point of the nonexpansive mapping T in the Hilbert space . Hence our IMR is again applicable. Initiating with any function , we define a sequence of functions in by
Then the sequence converges weakly in to the solution of integral equation (3.6).
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Acknowledgements
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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Alghamdi, M.A., Alghamdi, M.A., Shahzad, N. et al. The implicit midpoint rule for nonexpansive mappings. Fixed Point Theory Appl 2014, 96 (2014). https://doi.org/10.1186/1687-1812-2014-96
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DOI: https://doi.org/10.1186/1687-1812-2014-96