Measure of noncompactness and second order differential equations with deviating argument

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.

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

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https://ictp.acad.ro/wp-content/uploads/2022/08/1989a-Precup-Measure-of-noncompactness-and-second-order-differential-equations-with-deviating-argument.pdf

http://www.cs.ubbcluj.ro/~studia-m/old_issues/subbmath_1989_34_02.pdf

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.

MR: 91k:34094.

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