## On the application of measure of noncompactness to existence theorems.(English)Zbl 0589.45007

The measure of noncompactness $$\alpha$$ is used to prove existence theorems for Hammerstein integral equations, boundary value problems for nonlinear ordinary differential equations of second order and quasilinear evolution equations. In the case of Hammerstein integral equation $$(*)\quad x(t)=p(t)+\lambda \int_{I}K(t,s)f(s,x(s))ds$$ assume: 1) p is a continuous function from $$I:=[0,a]$$ into the Banach space E; 2) let f be a continuous function from $$I\times E$$ into the Banach space F satisfying: (i) f is continuous in x and strongly measurable in s; (ii) for any $$r>0$$ there exists an integrable function $$m_ r:I\to {\mathbb{R}}_+$$ such that $$\| f(s,x)\| \leq m_ r(s)$$ for all $$s\in I$$ and $$\| x\| \leq r$$; 3) K is a continuous function from $$I^ 2$$ into the space of bounded linear mappings $$F\to E$$. In addition assume that there exists an integrable function $$h:I\to {\mathbb{R}}_+$$ such that for any bounded subset X of E there exists a closed subset $$I_{\epsilon}$$ of I such that $$\mu (I\setminus I_{\epsilon})<\epsilon$$ and $$\alpha (f(T\times X))\leq \sup_{s\in T}h(s)\alpha (X)$$ for each closed subset T of $$I_{\epsilon}$$. Then there exists $$\rho >0$$ such that for any $$\lambda\in {\mathbb{R}}$$ with $$| \lambda | <\rho$$ the equation (*) has at least one continuous solution.
Reviewer: W.Petry

### MSC:

 45N05 Abstract integral equations, integral equations in abstract spaces 45G10 Other nonlinear integral equations 47H10 Fixed-point theorems
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### References:

  A. Ambrosetti , Un teorema di esistenza per le equazioni differenziali negli spazi di Banach , Rend. Semin. Mat. Univ. Padova , 39 ( 1967 ), pp. 349 - 360 . Numdam | MR 222426 | Zbl 0174.46001 · Zbl 0174.46001  F. Browder , Nonlinear equations of evolution , Ann. of Math. , 80 ( 1964 ), pp. 485 - 523 . MR 173960 | Zbl 0127.33602 · Zbl 0127.33602  F. Browder , Nonlinear operators and nonlinear equations of evolution , Proc. Symp. Nonlin. Funct. Anal. Chicago , AMS , 20 II, Providence, R. I ., 1972 . · Zbl 0244.57007  A. Cellina , On the existence of solutions of ordinary differential equations in Banach spaces , Funkcial. Ekvac. , 14 ( 1971 ), pp. 129 - 136 . Article | MR 304805 | Zbl 0271.34071 · Zbl 0271.34071  J. Chandra - V. LAKSHMIKANTHAM - A. MITCHELL, Existence of solutions of boundary value problems for nonlinear second order systems in a Banach space, J . Nonlinear Anal. , 2 ( 1978 ), pp. 157 - 168 . MR 512279 | Zbl 0385.34035 · Zbl 0385.34035  J. Daneš , On densifying and related mappings and their applications in nonlinear functional analysis, Theory of nonlinear operators , Akademie-Verlag , Berlin ( 1974 ), pp. 15 - 56 . MR 361946 | Zbl 0295.47058 · Zbl 0295.47058  G. Darbo , Punti uniti in trasformazioni a condominio non compatto , Rend. Sem. Mat. Univ. Padova , 24 ( 1955 ), pp. 84 - 92 . Numdam | MR 70164 | Zbl 0064.35704 · Zbl 0064.35704  K. Deimling , Ordinary differential equations in Banach spaces , Lecture Notes Math. no. 596 , Berlin , Heidelberg , New York , 1977 . MR 463601 | Zbl 0361.34050 · Zbl 0361.34050  K. Goebel - W. Rzymowski , An existence theorem for the equation x’ = f (t, x) in Banach spaces , Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. , 18 ( 1970 ), pp. 367 - 370 . MR 269957 | Zbl 0202.10003 · Zbl 0202.10003  P. Hartman , Ordinary differential equations , New York , London , Sydney , 1964 . MR 171038 | Zbl 0125.32102 · Zbl 0125.32102  A. Koschelev - M. Krasnoselskii - S. Michlin - L. Rakovschik - V. Stecenko - P. Zabreiko , Integral equations , SMB , Moskva , 1968 .  K. Kuratowski , Topologie , Warszawa , 1958 . Zbl 0078.14603 · Zbl 0078.14603  H. Mönch , Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, J . Nonlinear Anal. , 4 ( 1980 ), pp. 985 - 999 . MR 586861 | Zbl 0462.34041 · Zbl 0462.34041  G. Pianigiani , Existence of solutions of ordinary differential equations in Banach spaces , Bull. Acad. Pelon. Sci. Sér. Sci. Math. Astronom. Phys. , 23 ( 1975 ), pp. 853 - 857 . MR 393710 | Zbl 0317.34050 · Zbl 0317.34050  J Prüß , On semilinear evolution equations in Banach spaces , J. Reine Angew Math. , 303 - 304 ( 1978 ), pp. 144 - 158 . MR 514677 | Zbl 0398.34057 · Zbl 0398.34057  B. Sadovskii , On a fixed point principle , Frunkcjcnalnyj Analiz. , 1 ( 1976 ), pp. 74 - 76 . MR 211302  G. Scorza-Dragoni , Sul problema dei valori ai limiti per i sistemi di equazioni differenziali del secondo ordine , Bcll. U.M.I. , 14 ( 1935 ), pp. 225 - 230 . JFM 61.1238.08 · Zbl 0012.25702  S. Szufla , Some remarks on ordinary differential equations in Banach spaces , Bull. Acad. Pclon. Sci. S\theta r. Sci. Math. Astronom. Phys. , 16 ( 1968 ), pp. 795 - 800 . Zbl 0177.18902 · Zbl 0177.18902  S. Szufla , Równania różniczkowe w przestrzeniach Banacha , Ph. D. Thesis, Poznań , 1972 .  S. Szufla , On the existence of solutions of differential equations in Banach spaces , Bull. Acad. Polon. Sci. Math. , 30 ( 1982 ), pp. 507 - 515 . MR 718727 | Zbl 0532.34045 · Zbl 0532.34045  S. Szufla , On Volterra integral equations in Banach spaces , Funkcial. Ekvac. , 20 ( 1977 ), pp. 247 - 258 . Article | MR 511230 | Zbl 0379.45025 · Zbl 0379.45025
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