Elsevier

Applied Mathematics and Computation

Volume 231, 15 March 2014, Pages 140-147
Applied Mathematics and Computation

Strong convergence theorem for integral equations of Hammerstein type in Hilbert spaces

https://doi.org/10.1016/j.amc.2013.12.157Get rights and content

Abstract

Let H be a real Hilbert space. Let F:HH be a bounded, coercive and maximal monotone mapping. Let K:HH be a bounded and maximal monotone mapping. Let K and F satisfy the range condition. Suppose that uH is a solution to Hammerstein equation u+KFu=0. We construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the Hammerstein equation. Our iterative scheme in this paper seems far simpler than the iterative scheme used by Chidume and Ofoedu (2011) [13] and their strong assumption is dispensed with. We give some examples of our result so that it will find serious applications and be of much interest to our readers.

Introduction

Let E be a real normed space and let S{xE:x=1}. E is said to have a Gâteaux differentiable norm (and E is called smooth) if the limitlimt0x+ty-xtexists for each x,yS;E is said to have a uniformly Gâteaux differentiable norm if for each yS the limit is attained uniformly for xS. Further, E is said to be uniformly smooth if the limit exists uniformly for (x,y)S×S. The modulus of smoothness of E is defined byρE(τ)supx+y+x-y2-1:x=1,y=τ;τ>0.

E is equivalently said to be smooth if ρE(τ)>0,τ>0. Let q>1,E is said to be q-uniformly smooth (or to have a modulus of smoothness of power type q>1) if there exists c>0 such that ρE(τ)cτq. Hilbert spaces, Lp(orlp) spaces, 1<p<, and the Sobolev spaces, Wmp,1<p<, are q-uniformly smooth. Hilbert spaces are 2-uniformly smooth whileLp(orp)orWmpisp-uniformly smooth if1<p22-uniformly smooth ifp2.Let E be a real normed space and let Jq,(q>1) denote the generalized duality mapping from E into 2E given byJq(x)={fE:x,f=||x||qand||f||=||x||q-1},where E denotes the dual space of E and .,. denotes the generalized duality pairing. It is well known (see, for example, Xu [24]) that Jq(x)=||x||q-2J(x) if x0. For q=2, the mapping J=J2 from E to 2E is called normalized duality mapping. It is well known that if E is uniformly smooth, then J is single-valued (see, e.g., [24], [25]). In the sequel, we shall denote the single-valued normalized duality mapping by j.

A mapping A:D(A)EE is said to be accretive if x,yD(A), there exists j(x-y)J(x-y) such thatAx-Ay,j(x-y)0.For some real number η>0, A is called η-strongly accretive if x,yD(A), there exists jq(x-y)Jq(x-y) such thatGx-Gy,jq(x-y)η||x-y||q.

In Hilbert spaces, accretive operators are called monotone. The accretive operators were introduced independently in 1967 by Browder [6] and Kato [18]. Interest in such mappings stems from their firm connection with equations of evolution. For more on accretive/monotone mappings and connections with evolution equations, the reader may consult any of the following references [1], [2], [11], [14], [21].

A nonlinear integral equation of Hammerstein type (see, e.g., [17]) is one of the formu(x)+Ωk(x,y)f(y,u(y))dy=h(x)where dy is a σ-finite measure on the measure space Ω; the real kernel k is defined on Ω×Ω,f is a real-valued function defined on Ω×R and is, in general, nonlinear and h is a given function on Ω. If we now define an operator K byKv(x)=Ωk(x,y)v(y)dy;xΩ,and the so-called superposition or Nemyskii operator by Fu(y)f(y,u(y)) then, the integral equation (1.1) can be put in the operator theoretic form as follows:u+KFu=0,where, without loss of generality, we have taken h0.

Interest in Eq. (1.2) stems mainly from the fact that several problems that arise in differential equations, for instance, elliptic boundary value problems whose linear parts possess Green’s functions can, as a rule, be transformed into the form (1.2). Among these, we mention the problem of the forced oscillations of finite amplitude of a pendulum (see, e.g., [20, Chapter IV]).

Example 1.1

The amplitude of oscillation v(t) is a solution of the problemd2vdt2+a2sinv(t)=z(t),t[0,1]v(0)=v(1)=0,where the driving force z(t) is periodical and odd. The constant a0 depends on the length of the pendulum and on gravity. Since the Green’s function for the problemv(t)=0,v(0)=v(1)=0,is the triangular functionk(t,x)=t(1-x),0tx,x(1-t),xt1,problem (1.3) is equivalent to the nonlinear integral equationv(t)=-01k(t,x)[z(x)-a2sinv(x)]dx.If01k(t,x)z(x)dx=g(t)andv(t)+g(t)=u(t),then (1.4) can be written as the Hammerstein equationu(t)+01k(t,x)f(x,u(x))dx=0,where f(x,u(x))=a2sin[u(x)-g(x)].

Equations of Hammerstein type play a crucial role in the theory of optimal control systems and in automation and network theory (see, e.g., [16]). Several existence and uniqueness theorems have been proved for equations of the Hammerstein type (see, e.g., [3], [4], [5], [6], [7], [8], [10], [15]). The Mann iteration scheme (see, e.g., [19]) has successfully been employed (see, e.g., the recent monographs of Berinde [1] and Chidume [11]). The recurrence formulas used involved K-1 which is also assumed to be strongly monotone and this, apart from limiting the class of mappings to which such iterative schemes are applicable, is also not convenient in applications. Part of the difficulty is the fact that the composition of two monotone operators need not be monotone.

Recently, Chidume and Ofoedu [13] introduced a coupled explicit iterative scheme and proved the following strong convergence theorem for approximation of solution of a nonlinear integral equation of Hammerstein type in 2-uniformly smooth real Banach space. In particular, they proved the following theorem.

Theorem 1.2 Chidume and Ofoedu, [13]

Let E be a 2-uniformly smooth real Banach space. Let F,K:EE be bounded and accretive mappings. Let {un}n=1 and {vn}n=1 be sequences in E defined iteratively from arbitrary u1,v1E byun+1=un-λnαn(Fun-vn)-λnθn(un-u1),vn+1=vn-λnαn(Kvn+un)-λnθn(vn-v1),where {λn}n=1,{αn}n=1 and {θn}n=1 are real sequences in (0,1) such that λn=o(θn),αn=o(θn) and i=1λnθn=+. Suppose that u+KFu=0 has a solution in E. Then, there exist real constants ε0,ε1>0 and a set ΩW=E×E such that if αnε0θn and λnε1θn,nn0, for some n0N and w(u,v)Ω (where v=Fu), the sequence {un}n=1 converges strongly to u.

We make the following remarks about Theorem 1.2.

  • 1.

    The significance of Theorem 1.2 is not clear because of the requirement that u+KFu=0 has a solution u and that (u,Fu)Ω. In fact, the set Ω is referred to in the statement of Theorem 1.2, but without knowing what Ω is in the statement of theorem and the definition of Ω is given in the proof of Theorem 1.2.

  • 2.

    The requirement that (u,Fu)Ω seems to be a strong assumption.

  • 3.

    We suspect the result itself will never find serious applications or be of much interest (except to the authors) unless some examples are given.

It is our purpose in this paper to sharpen Theorem 1.2 with a new explicit iteration scheme in the settings of real Hilbert spaces without the requirement that u+KFu=0 has a solution u and that (u,Fu)Ω. Furthermore, we give some examples so that our result will find serious applications and be of much interest to the readers. Also, the conditions imposed on the iteration parameters in Theorem 1.2 are too strong compared to the conditions we shall impose on our iteration parameters in Section 3. Thus, our result improves significantly on the result of Chidume and Ofoedu [13] in the settings of real Hilbert spaces.

Section snippets

Preliminaries

We shall make use of the following lemmas in the sequel.

Lemma 2.1

Let H be a real Hilbert space. Then||x+y||2||x||2+2y,x+y,for all x,yH.

Lemma 2.2

see, e.g., [2], [23]

Let {an}n=1 be a sequence of nonnegative real numbers satisfying the following relation:an+1(1-αn)an+αnσn+γn,n1,where {αn}n=1,{σn}n=1 and {γn}n=1 satisfy the conditions:

  • (i)

    {αn}n=1[0,1],n=1αn=;

  • (ii)

    limsupnσn0,

  • (iii)

    γn0(n1),n=1γn<. Then, an0 as n.

Lemma 2.3

Shioji and Takahashi, [22]

Let (x0,x1,x2,)l be such that μnxn0 for all Banach limits μ. If limsupn(xn+1-xn)0, then limsupnxn0.

Lemma 2.4

Xu, [24]

Let H

Main results

Lemma 3.1

Let H be a real Hilbert space. Let F:HH be a bounded, coercive and maximal monotone mapping. Let K:HH be a bounded and maximal monotone mapping. Let {un}n=1 and {vn}n=1 be sequences in E defined iteratively from arbitrary u1,v1E byun+1=un-βn2(Fun-vn)-βn(un-u1),vn+1=vn-βn2(Kvn+un)-βn(vn-v1),n1where {βn}n=1 is a real sequence in (0,1) such that limnβn=0. Suppose that uH is a solution to u+KFu=0. Then, the sequences {un}n=1 and {vn}n=1 are bounded.

Proof

Let WH×H with the norm ||w||W(||u||2

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