Summary
This paper deals with the existence of solutions for the implicit Cauchy problem F(t, x, x)=ϑB, x(t0)=x0 in a Banach space B. By using the Kuratowski and the Hausdorff measure of non compactness, we prove an existence theorem for the previous problem (Teorema 1.1) and its extension to non compact intervals (Teorema 2.1). These results generalize the previous ones by R.Conti [1] (in the case B=R), G.Pulvirenti [2] and T. Dominguez Benavides [3], [4] (in the general case). In particular, we relax a Lipschitz condition assumed by all of the abovementioned authors. Some applications of Teorema 2.1 are presented.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Bibliografia
R. Conti,Sulla risoluzione dell'equazione F(t, x, dx/dt)=0, Ann. Mat. Pura Appl., (4)48 (1959), pp. 97–102.
G. Pulvirenti,Equazioni differenziali in forma implicita in uno spazio di Banach, Ann. Mat. Pura Appl., (4)56 (1961), pp. 177–192.
T. Dominguez Benavides,An existence theorem for implicit differential equations in a Banach space, Ann. Mat. Pura Appl., (4)118 (1978), pp. 119–130.
T. Dominguez Benavides,Continuous dependence for implicit differential equations in Banach spaces, Collect. Math.,31 (1980), pp. 205–216.
B. Ricceri,Sull'esistenza delle soluzioni delle equazioni differenziali ordinarie negli spazi di Banach in ipotesi di Carathéodory, Boll. Un. Mat. Ital., (5)18-C (1981), pp. 1–19.
K. Goebel -W. Rzymowski,An existence theorem for the equations x′=f(t, x) in Banach space, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astronom. Phys.,18 (1970), pp. 367–370
Author information
Authors and Affiliations
Additional information
Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.
Rights and permissions
About this article
Cite this article
Emmanuele, G., Ricceri, B. Sull'esistenza delle soluzioni delle equazioni differenziali ordinarie in forma implicita negli spazi di Banach. Annali di Matematica pura ed applicata 129, 367–382 (1981). https://doi.org/10.1007/BF01762150
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01762150