## Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations.(English)Zbl 1220.54025

The purpose of this paper is to present, in the context of an ordered metric space, some fixed point theorems for continuous, nondecreasing mappings which satisfy Caristi type conditions with perturbed metrics. For example, the following theorem is proved.
Theorem 2.1. Let $$(X,\leq)$$ be a partially ordered set and suppose that there exists a metric $$d$$ in $$X$$ such that $$(X,d)$$ is a complete metric space. Let $$f:X\to X$$ be a continuous and nondecreasing mapping such that $\psi(d(f(x),f(y)))\leq \psi(d(x,y))-\phi(d(x,y))\text{ for all }x,y\in X,\;x\geq y,$ where $$\psi,\phi:\mathbb{R}_+\to \mathbb{R}_+$$ are some altering distance functions. If there exists $$x_0\in X$$ with $$x_0\leq f(x_0)$$, then $$f$$ has a fixed point in $$X$$.
As an application, an existence result for a first-order periodic problem for a differential equation is given.

### MSC:

 54H25 Fixed-point and coincidence theorems (topological aspects) 54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
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### References:

 [1] Khan, M. S.; Swaleh, M.; Sessa, S., Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc., 30, 1, 1-9 (1984) · Zbl 0553.54023 [2] Babu, G. V.R.; Lalitha, B.; Sandhya, M. L., Common fixed point theorems involving two generalized altering distance functions in four variables, Proc. Jangjeon Math. Soc., 10, 83-93 (2007) · Zbl 1139.54321 [3] Naidu, S. V.R., Some fixed point theorems in metric spaces by altering distances, Czechoslovak Math. J., 53, 205-212 (2003) · Zbl 1013.54011 [4] Sastry, K. P.R.; Babu, G. V.R., Some fixed point theorems by altering distances between the points, Indian J. Pure Appl. Math., 30, 641-647 (1999) · Zbl 0938.47044 [5] Alber, Ya. I.; Guerre-Delabriere, S., Principles of weakly contractive maps in Hilbert spaces, (Gohberg; Lyubich, Y., New Results in Operator Theory and its Applications. New Results in Operator Theory and its Applications, Operator Theory: Advances and Applications, vol. 98 (1997), Birkhäuser: Birkhäuser Basel), 7-22 · Zbl 0897.47044 [6] Rhoades, B. E., Some theorems on weakly contractive maps, Nonlinear Anal., 47, 2683-2693 (2001) · Zbl 1042.47521 [7] Dhutta, P. N.; Choudhury, B. S., A generalization of contraction principle in metric spaces, Fixed Point Theory Appl. (2008), Article ID 406368 · Zbl 1177.54024 [8] Agarwal, R. P.; El-Gebeily, M. A.; O’Regan, D., Generalized contractions in partially ordered metric spaces, Appl. Anal, 87, 109-116 (2008) · Zbl 1140.47042 [9] Burgic, Dz.; Kalabusic, S.; Kulenovic, M. R.S., Global attractivity results for mixed monotone mappings in partially ordered complete metric spaces, Fixed Point Theory Appl. (2009), Article ID 762478 · Zbl 1168.54327 [10] Ciric, L.; Cakid, N.; Rajovic, M.; Uma, J. S., Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed Point Theory Appl. (2008), Article ID 131294 · Zbl 1158.54019 [11] Gnana Bhaskar, T.; Lakshmikantham, V., Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65, 1379-1393 (2006) · Zbl 1106.47047 [12] Harjani, J.; Sadarangani, K., Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal., 71, 3403-3410 (2009) · Zbl 1221.54058 [13] Lakshmikantham, V.; Ciric, L., Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., 70, 4341-4349 (2009) · Zbl 1176.54032 [14] Nieto, J. J.; Rodríguez-López, R., Existence of extremal solutions for quadratic fuzzy equations, Fixed Point Theory Appl., 321-342 (2005) · Zbl 1102.54004 [15] Nieto, J. J.; Rodríguez-López, R., Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22, 223-239 (2005) · Zbl 1095.47013 [16] Nieto, J. J.; Rodríguez-López, R., Applications of contractive — like mapping principles to fuzzy equations, Rev. Math. Complut., 19, 361-383 (2006) · Zbl 1113.26030 [17] Nieto, J. J.; Pouso, R. L.; Rodríguez-López, R., Fixed point theorems in ordered abstract spaces, Proc. Amer. Math. Soc.., 135, 2505-2517 (2007) · Zbl 1126.47045 [18] Nieto, J. J.; Rodríguez-López, R., Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sinica, 23, 2205-2212 (2007) · Zbl 1140.47045 [19] O’Regan, D.; Petrusel, A., Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl., 341, 1241-1252 (2008) · Zbl 1142.47033 [20] Petrusel, A.; Rus, I. A., Fixed point theorems in ordered L-spaces, Proc. Amer. Math. Soc.., 134, 411-418 (2006) · Zbl 1086.47026 [21] Ran, A. C.M.; Reurings, M. C.B., A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc.., 132, 1435-1443 (2004) · Zbl 1060.47056 [22] Wu, Y., New fixed point theorems and applications of mixed monotone operator, J. Math. Anal. Appl., 341, 883-893 (2008) · Zbl 1137.47044 [23] Cabada, A.; Nieto, J. J., Fixed points and approximate solutions for nonlinear operator equations, J. Comput. Appl. Math., 113, 17-25 (2000) · Zbl 0954.47038
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