### Existence and uniqueness of solutions to a first-order differential equation via fixed point theorem in orthogonal metric space

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#### Abstract

In this paper, among the other things, we show that the solution of the first-order

differential equation is a fixed point of an integral operator from an orthogonal metric space into itself. This approach provides a new proof of the classical existence and uniqueness theorems of solutions to a first-order differential equation.

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R.P. Agarwal, M. Benchohra and S. Hamani, Boundary value problems for fractional differential equations, Georgian Math. J., 16 (2009) No.3, 401-411.

R.P. Agarwal, D. O'Regan and S. Stanek, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, J. Math. Anal. Appl. 371 (2010) 57-68.

R.P. Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109 (2010) 973-1033.

R.P. Agarwal, D. Franco and D. O'Regan, Singular boundary value problems for first and second order impulsive differential equations, Aequat. Math., 69 (2005) 83-96.

B. Ahmad and J.J. Nieto, Boundary Value Problems for a Class of Sequential Integro differential Equations of Fractional Order, J. Func. Space. Appl. (2013) Article ID 149659.

A. Amini-Harandi and H. Emami, A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal. 72 (2010) 2238-2242.

V.I. Arnold, Ordinary Differential Equations, Translated and Edited by Richard A. Silverman, The M.I.T.Pess(1998).

bibitem {Arn1} V.I. Arnold, textit{Ordinary Differential Equations, Translated from the Russian by Roger Cooke, Springer-verlog(1992).

H.Baghani and M.Ramezani, Contractive gauge functions in strongly orthogonal metric spaces, Int. J. Nonlinear Anal. Appl. Article in press ISSN: 2008-6822 (electronic).

H.Baghani, M.Eshaghi Gordji and M.Ramezani, Orthogonal sets:their relation to the axiom of choice and a generalized fixed point theorem, Journal of Fixed Point Theory and Applications,Volume 18, Issue 3, pp 465–477, (2016).

S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund. Math. 3 (1922) 133-181.

M. Belmekki, J.J. Nieto and Rosana Rodrيguez-Lَpez, Existence of solution to a periodic boundary value problem for a nonlinear impulsive fractional differential equation, E. J. Qualitative Theory of Diff. Equ. (2014) No. 16, 1-27.

M. Eshaghi Gordji, M. Ramezani, M. De La Sen and Y.J. Cho, On orthogonal sets and Banach fixed point theorem, accepted in Fixed point theorey.

A.A. Ivanov, Fixed point theory, Journal of Soviet Mathematics, 12 (1979) 1-64.

E. Karapinar and R.P. Agarwal, A note on 'Coupled fixed point theorems for $alpha-psi$-contractive-type mappings in partially ordered metric spaces, Fixed Point Theory and Applications (2013) 2013:216, 16 pp.1 .

J.J. Nieto, R.L. Pouso and R. Rodrguez-Lopez, Fixed point theorems in ordered abstract sets, Proc. Amer. Math. Soc. 135 (2007) 2505-2517.

J.J. Nieto, R.L. Pouso and R. Rodrguez-Lopez, Contractive mapping theorems in partially ordere sets and applications to ordinary differential equations, Order 22 (2005) 223-239.

J.J. Nieto, R.L. Pouso and R. Rodrguez-Lopez, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. 23(2007) 2205-2212.

M.Ramezani, Orthogonal metric space and convex contractions, Int. J. Nonlinear Anal. Appl. 6 (2015) No. 2, 127-132 ISSN: 2008-6822 (electronic).

DOI: https://doi.org/10.22190/FUMI1901123G

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