Posts by Radu Precup

Abstract

We study the existence and uniqueness of equations

u′′(t)=f(t,u(t),u′(t),u(g₁(t)),…,u(g_{m}(t))) 1.1

u(t)=p(t),)t∈I╲intI,(1.2)

in a Banach space. This problem is regarded as a particular case of the Dirichlet problem

xxx

We study the existence and uniqueness of equations

\(u\prime\prime(t)=f(t,u(t),u\prime(t),u(g\U{2081} (t)),\ldots,u(g\_\{m\}(t))))\ 1.1

\(u(t)=p(t),)t\in I\diagdown intI)\,(1.2)

in a Banach space. This problem is regarded as a particular case of the
Dirichlet problem

??

for the differential equation

??

The main result regarding the existence of the solution of problem

??

is contained in Theorem 1, where the mapping in the right hand side must obey
a weaker condition than being compact.

This condition is expressed with the aid of Kuratowski noncompactness measure.

Authors

Radu Precup
”University Babeș-Bolyai” Cluj-Napoca, Romania

Keywords

?

PDF

Cite this paper as:

R. Precup, Measure of noncompactness and second order differential equations with deviating argument, Studia Univ. Babeş-Bolyai Math., 34 (1989) no. 2, pp. 25-35.

About this paper

Journal

Studia Universitatis Babes-Bolyai Mathematica

Publisher Name

Babes-Bolyai University

DOI

Not available yet.

Print ISSN

Not available yet.

Online ISSN

Not available yet.

MR: 91k:34094.

References

[1] Ahmerov, R.R., Kamenskii, M.I., Potapov, A.S., Rodkina, A.E. Sadovskii, B.N., Measure of noncompactness and condensing operators (Russian), Novosibirsk, Nauka, 1986.
[2] Ambrosetti A., Un theorema di esistenza per le equazioni differenziali negli spazi di Banach, Rend. Sem. Mat. Univ. Padova, 39(1967), 349-361.
[3] Bernfeld, S.R., Lakshmikantham V., An introduction to nonlinear boundary value  problems, Academic Press, New York, London, 1974.
[4] Bernstein S.N., Sur les equations du calcul des variations, Ann. Sci. Ecole Norm. Sup. 29 (1912), 431-485.
[5] Chandra, J., Lakshmikantham, V., Mitchell, A., Existence of solutions of boundary value problems for nonlinear second order systems in a Banach space, J. Nonlinear Analysis 2 (1978), 157-168.
[6] Dugundju, J., Granas, A., Fixed point theory, vol.1., Warszawa, 1982.
[7] Granas, A., Sur la methode de continuite de Poincare, C.R. Acad. Sci., Paris 282 (1976), 983-985.
[8] Granas, A., Guenther, R., Lee, J., Nonlinear boundary value problems for ordinary differential equations, Dissertationes Mathematicae, CCXIV, Warszawa, 1985.
[9] Hartman, P., On boundary value problem for systems of ordinary nonlinear second order differential equations. Trans. Amer. Math. Soc. 96 (1960), 493-509.
[10] Hartman, P., Ordinary differential equations, John Wiley & Sons. Inc., New York, 1964.
[11] Lakshmikantham, V., Abstract boundary value problems, Nonlinear Equations in Abstract Spaces  (Ed. V. Lakshmikantham), Academic Press, New Yprk, San Francisco, London, 1978, 117-123.
[12] Lasota, A., Yorke, J.A., Existence of solutions of two-point boundary value problems for nonlinear systems, J. Differential Equations 11 (1972), 509-518.
[13] Mawhin, J., Nonlinear boundary value problems for ordinary differential equations from Schauder theorem to stable homotopy, Universite Catholique de Louvain, Rapportr  No. 86 (1976).
[14] Precup, R., Nonlinear boundary value problems for infinite systems of second-order functional differential equations, Univ. Babes-Bolyai, Cluj-Napoca, Preprint nr.8, 1988, 17-30.
[15] Rus, A.I., Maximum principles for some nonlinear differential equations with deviating arguments, Studia Univ. Babes-Bolyai, 32, nr.2 (1987), 53-57.
[16] Scorza-Dragoni, G., Sul problema dei valori ai limiti per i sistemi di equazioni diferentiali del secondo ordine. Bull. U.M.I. 14 (1935), 225-230.
[17] Schmitt, K., Thompson, R., Boundary value problems for infinite systems of second-order differential equations, J. Differential Equations 18 (1975), 277-295.
[18] Thompson, R.C., Differential inequalities for infinite second order systems and an application to the method of lines, J. Differential Equations 17 (1975), 421-434.

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