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Solution of integral equations via coupled fixed point theorems in \(\mathfrak{F}\)-complete metric spaces. (English) Zbl 1482.54068

Summary: The concept of coupled \(\mathfrak{F}\)-orthogonal contraction mapping is introduced in this paper, and some coupled fixed point theorems in orthogonal metric spaces are proved. The obtained results generalize and extend some of the well-known results in the literature. An example is presented to support our results. Furthermore, we apply our result to obtain the existence theorem for a common solution of the integral equations: \[ \begin{cases} \zeta (\mathfrak{v}) = \eth (\mathfrak{v})+\underset{0}{\overset{\mathfrak{M}}{\displaystyle\int}}\Xi (\mathfrak{v},\beta) \Omega (\beta,\zeta (\beta), \xi (\beta)) \mathrm{d}\beta, & \mathfrak{v}\in [0,\mathscr{H}], \\ \xi (\mathfrak{v}) =\eth (\mathfrak{v}) + \underset{0}{\overset{\mathfrak{M}}{\displaystyle\int}} \Xi (\mathfrak{v},\beta) \Omega (\beta, \xi (\beta), \zeta (\beta)) \mathrm{d}\beta, & \mathfrak{v}\in [0,\mathscr{H}], \end{cases} \] where
(a)
\(\eth :\mathfrak{M}\to\mathbb{R}\) and \(\Omega :\mathfrak{M} \times\mathbb{R}\times\mathbb{R}\to\mathbb{R}\) are continuous;
(b)
\(\Xi : \mathfrak{M}\times\mathfrak{M}\) is continuous and measurable at \(\beta \in\mathfrak{M},\forall\mathfrak{v}\in\mathfrak{M}\);
(c)
\(\Xi (\mathfrak{v},\beta)\geq 0,\; \forall\mathfrak{v}, \beta \in\mathfrak{M}\) and \(\int_0^{\mathscr{H}} \Xi (\mathfrak{v},\beta) \mathrm{d}\beta \leq 1,\; \forall\mathfrak{v}\in\mathfrak{M}\).

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
54E35 Metric spaces, metrizability
45G15 Systems of nonlinear integral equations
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