Abstract and Applied Analysis

Volume 2015, Article ID 627260, 7 pages

http://dx.doi.org/10.1155/2015/627260

## A New Method for Proving Existence Theorems for Abstract Hammerstein Equations

^{1}African University of Science and Technology, Abuja 900241, Nigeria^{2}Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849-5168, USA^{3}Department of Mathematical Sciences, Bayero University, Kano 700241, Nigeria

Received 11 December 2014; Accepted 23 February 2015

Academic Editor: Simeon Reich

Copyright © 2015 C. E. Chidume et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An abstract Hammerstein equation is an equation of the form . A *new method* is introduced to prove the existence of a solution of this equation where and are nonlinear accretive (monotone) operators. The method does not involve the complicated
technique of factorizing a linear map via a Hilbert space and does not
involve the use of deep variational techniques.

#### 1. General Introduction

Let be a real normed space and let . The space is said to have* Gâteaux differentiable norm* if the limit exists for all ; in this case is said to be smooth. is said to have* uniformly Gâteaux differentiable norm* if, for each , the limit is attained uniformly for . Further, is said to be* uniformly smooth* if the limit is attained uniformly for . The* modulus of smoothness* of , , is defined by is equivalently said to be* smooth* if . Let ; is said to be *-uniformly smooth* (or to have a* modulus of smoothness of power type *) if there exists such that .

, , and the Sobolev space , , are all -uniformly smooth. In factFurthermore (see, e.g., [1]), Let denote the* generalized duality mapping* from to defined bywhere denotes the dual space of and denotes the generalized duality pairing. It is well known (see, e.g., Xu [2]) that where denotes (called the* normalized duality mapping*). It is well known that if is strictly convex, is single-valued. For more information and examples concerning (generalized) duality mappings, one may see the book of Cioranescu [3] and its review by Reich [4]. In the sequel, we will denote the single-valued duality map by .

A map is called* accretive* if, for all , there exists such that the following inequality holds:If is a real Hilbert space, the map is called* monotone*. In this case, satisfies the following condition:The map is called* strongly accretive* if there exists such that, for all , there exists , such that

A nonlinear integral equation of Hammerstein type (see, e.g., Hammerstein [5]) has the formwhere is a -finite measure on ; the kernel is defined on , is a real-valued function defined on and is, in general, nonlinear, and is a function on . Settingand on , then integral equation (9) can be put in abstract operator form as follows:where, without loss of generality, we have taken .

Interest in (9) 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 function, can, as a rule, be transformed into the form of (9).

Furthermore, equations of Hammerstein type play crucial role in the theory of optimal control systems, in automation, and in network theory (see, e.g., Dolezale [6]).

Several existence theorems for the solution of (9) have been proved by a host of distinguished mathematicians using various techniques (see, e.g., Browder and Gupta [7, 8], Chepanovich [9], and Petryshyn and Fitzpatrick [8]). In the remaining part of this section, we highlight the techniques used by Browder and Gupta [7] and Petryshyn and Fitzpatrick [8]. To do this, we first give definitions of some terms which are required in the theorems.

In the sequel, the symbol “” denotes strong convergence while “” denotes weak convergence.

*Definition 1 (see, e.g., [7]). *A mapping is said to be* hemicontinuous* if it is continuous from each line segment of to the weak topology of . That is, , , and such that and for sufficiently large and we have .

*Definition 2 (see, e.g., [7]). *Let be a bounded monotone linear mapping. is said to be* angle-bounded* with constant if, for all , in , . (This is well defined since and by the linearity and monotonicity of .)

In [7] Browder and Gupta proved the following theorem.

Theorem 3 (Browder-Gupta [7]). *Let be a real Banach space and its conjugate dual space. Let be a monotone angle-bounded continuous linear mapping of into with constant of angle-boundedness . Let be a hemicontinuous (possibly nonlinear) mapping of into such that, for a given constant ,**for all and in . Suppose finally that there exists a constant with such that for in **Then, there exists exactly one solution in of the nonlinear equation*

*The main tool used by the authors in proving Theorem 3 is that of splitting the linear operator via a Hilbert space and then applying a deep result of Minty [10]. Precisely, they proved that if is a real Banach space, is its dual space, and is a bounded linear mapping of into which is monotone and angle-bounded, then there exist a Hilbert space , a continuous linear mapping of into with adjoint injective, and a bounded skew-symmetric linear mapping of into such that(see Figure 1).*