Existence results for a generalized nonlinear Hammerstein equation on \(L_{1}\) spaces. (English) Zbl 1128.45006

Latrach, Khalid; Taoudi, M. Aziz

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 66, No. 10, 2325-2333 (2007).

The authors show the existence of at least one solution for the following nonlinear Hammerstein integral equation \[ \psi(t)=g(t,\psi(t))+\lambda\int_\Omega\zeta(t,s)f(s,\psi(s))\,ds \tag{1} \] in \(L^1(\Omega,X)\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) and \(X\) is a finite dimensional Banach space. Their assumptions are the following:
(i) \(g:\Omega\times X\rightarrow X\), is a contraction with respect to its second variable.
(ii) \(f:\Omega\times X\rightarrow Y\), is Caratheodory and its corresponding Nemitskii operator \(N_f\) maps \(L^1(\Omega,X)\) into \(L^1(\Omega,Y)\), where \(Y\) is another finite dimensional Banach space.
(iii) \(\zeta:\Omega\times\Omega\rightarrow L(Y,X)\), (where \(L(Y,X)\) is the space of bounded linear operators from \(Y\) into \(X\)), is strongly measurable and the linear operator \(B\) defined by \[ (B\psi)(t)=\int_\Omega\zeta(t,s)\psi(s)\,ds \] maps \(L^1(\Omega,X)\) into \(L^1(\Omega,Y)\).
(iv) \(\rho(t):\Omega\rightarrow L(Y,X)\), defined by \(\rho(t)(s)=\zeta(t,s)\) belongs to \(L^\infty(\Omega, L(Y,X))\).
(v) \(\alpha+\eta| \lambda| \,\|B\|<1\), where \(\alpha\) is the contraction constant of \(g\) and \(\eta\) is a constant related to the function \(f\).
The authors solve problem (1) using a variant of the Krasnosel’skii fixed-point theorem, concerning the sum of two operators \(A\) and \(B\). In particular, they replace the compactness of the operator \(A\) (assumed in the original version of the theorem) with a weak type of contractiveness for \(A+B\), involving the De Blasi measure of weak noncompactness.

Reviewer: Nikolaos G. Yannakakis (Athens)

MSC:

45G10	Other nonlinear integral equations
47H10	Fixed-point theorems
47H30	Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Keywords:

measure of weak noncompactness; nonlinear Hammerstein integral equation; Banach space; Krasnosel’skii fixed-point theorem

References:

[1]	Appell, J.; De Pascale, E., Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili, Boll. unione mat. ital. B, 3, 6, 497-515, (1984) · Zbl 0507.46025
[2]	Banas, J., Integrable solutions of Hammerstein and uryshon integral equations, J. aust. math. soc. A, 46, 61-68, (1989) · Zbl 0666.45008
[3]	Banas, J.; Knap, Z., Measure of weak noncompactness and nonlinear integral equations of convolution type, J. math. anal. appl., 146, 353-362, (1990) · Zbl 0699.45002
[4]	Barroso, C.S., Krasnoselskii’s fixed point theorem for weakly continuous maps, Nonlinear anal., 55, 25-31, (2003) · Zbl 1042.47035
[5]	C.S. Barroso, E.V. Teixeira, A fixed point theorem for sum of operators and applications (preprint)
[6]	Burton, T.A., A fixed point theorem of Krasnoselskii, Appl. math. lett., 11, 85-88, (1998) · Zbl 1127.47318
[7]	Burton, T.A.; Furumochi, T., Krasnoselskii’s fixed point theorem and stability, Nonlinear anal., 49, 445-454, (2002) · Zbl 1015.34046
[8]	De Blasi, F.S., On a property of the unit sphere in Banach spaces, Bull. math. soc. sci. math. roumanie, 21, 259-262, (1977) · Zbl 0365.46015
[9]	Dunford, N.; Schwartz, J.T., Linear operators, part I: general theory, (1958), Interscience Publishers New York
[10]	Emmanuele, G., Integrable solutions of a functional-integral equation, J. integral equations appl., 4, 89-94, (1992) · Zbl 0755.45005
[11]	Emmanuele, G., An existence theorem for Hammerstein integral equations, Port. math., 51, 607-611, (1994) · Zbl 0823.45004
[12]	Krasnosel’skii, M.A., On the continuity of the operator \(F u(x) = f(x, u(x))\), Dokl. akad. nauk. SSSR, 77, 185-188, (1951), (in Russian)
[13]	Krasnosel’skii, M.A., Topological methods in the theory of nonlinear integral equations, (1964), Pergamon Press New York
[14]	Latrach, K.; Taoudi, M.A.; Zeghal, A., Some fixed point theorems of the Schauder and krasnosel’skii type and application to nonlinear transport equations, J. differential equations, 221, 256-271, (2006) · Zbl 1091.47046
[15]	Lucchetti, R.; Patrone, F., On nemytskii’s operator and its application to the lower semicontinuity of integral functionals, Indiana univ. math. J., 29, 5, 703-735, (1980) · Zbl 0476.47049
[16]	O’regan, D., Fixed-point theory for the sum of two operators, Appl. math. lett., 9, 1-8, (1996) · Zbl 0858.34049
[17]	O’regan, D., Fixed-point theory for weakly sequentially continuous mappings, Math. comput. modelling, 27, 1-14, (1998) · Zbl 1185.34026
[18]	Smart, D.R., Fixed point theorems, (1980), Cambridge University Press Cambridge · Zbl 0427.47036