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A note on weakly isotone maps and common solutions for differential systems. (English) Zbl 1112.47049

The authors offer remarks on the existence of common solutions of the first order differential Cauchy problems studied by B.C.Dhage [Bull.Korean Math.Soc.36, No.3, 565–578 (1999; Zbl 0940.47043)], B.C.Dhage, S.Heikkilä and M. Kumpulainen [Indian J.Pure Appl.Math. 26, No.5, 411–416 (1995; Zbl 0839.47035)], H.K.Pathak [Riv.Mat.Univ.Parma, V.Ser.3, 193–202 (1994; Zbl 0876.47032)] and the reviewer [Riv.Mat.Univ.Parma. IV.Ser.16, 205–212 (1990; Zbl 0725.45013)].
They conclude that some of the fundamental hypotheses used by them are too strong to obtain interesting results for common solutions of differential systems in the space \(\mathbb R^n\).

MSC:

47H10 Fixed-point theorems
47G20 Integro-differential operators
34C11 Growth and boundedness of solutions to ordinary differential equations
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References:

[1] Singh, S. L.: Application of fixed point theorems to nonlinear integrodifferential equations. Riv. Mat. Univ. Parma, 16(4), 205–212 (1990) · Zbl 0725.45013
[2] Pathak, H. K.: Application of fixed point theorems to abstract Volterra integrodifferential equations. Riv. Mat. Univ. Parma, 3(5), 193–202 (1994) · Zbl 0876.47032
[3] Dhage, B. C., Heikkilä, S., Kumpulainen, M.: On common fixed points with applications to differential equations in ordered Banach space. Indian J. Pure Appl. Math., 26(5), 411–416 (1995) · Zbl 0839.47035
[4] Dhage, B. C.: Existence theorem for common solution of differential equations in locally convex spaces. An. Ştiinţ. Univ. Al. I. Cuza Iaşi Secţ. I a Mat., 39(3), 251–257 (1993)
[5] Dhage, B. C.: Existence theorems for common solutions of differential equations in Banach space. Jñānābha, 27, 47–52 (1997) · Zbl 1026.34068
[6] Dhage, B. C.: Condensing mappings and applications to existence theorems for common solution of differential equations. Bull. Korean Math. Soc., 36(3), 565–578 (1999) · Zbl 0940.47043
[7] Heikkilä, S., On the solvability of implicit functional equations with applications to discontinuous differential equations. Acta Mathematica Sinica, English Series, 22(1), 223–232 (2006) · Zbl 1103.39023
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