Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces

Abstract

The concept of coupled 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: ζ(v)=ð(v)+∫0MΞ(v,β)Ω(β,ζ(β),ξ(β))dβ,v∈[0,H],ξ(v)=ð(v)+∫0MΞ(v,β)Ω(β,ξ(β),ζ(β))dβ,v∈[0,H],\left\{\begin{array}{ll}\zeta \left({\mathfrak{v}})=ð\left({\mathfrak{v}})+\underset{0}{\overset{{\mathfrak{M}}}{\displaystyle \int }}\Xi \left({\mathfrak{v}},\beta )\Omega \left(\beta ,\zeta \left(\beta ),\xi \left(\beta )){\rm{d}}\beta ,& {\mathfrak{v}}\in \left[0,{\mathscr{H}}],\\ \xi \left({\mathfrak{v}})=ð\left({\mathfrak{v}})+\underset{0}{\overset{{\mathfrak{M}}}{\displaystyle \int }}\Xi \left({\mathfrak{v}},\beta )\Omega \left(\beta ,\xi \left(\beta ),\zeta \left(\beta )){\rm{d}}\beta ,& {\mathfrak{v}}\in \left[0,{\mathscr{H}}],\end{array}\right. where (a)ð:M→Rð:{\mathfrak{M}}\to {\mathbb{R}} and Ω:M×R×R→R\Omega :{\mathfrak{M}}\times {\mathbb{R}}\times {\mathbb{R}}\to {\mathbb{R}} are continuous;(b)Ξ:M×M\Xi :{\mathfrak{M}}\times {\mathfrak{M}} is continuous and measurable at β∈M,∀\beta \in {\mathfrak{M}},\hspace{0.33em}\forall v∈M{\mathfrak{v}}\in {\mathfrak{M}};(c)Ξ(v,β)≥0,∀v,β∈M\Xi \left({\mathfrak{v}},\beta )\ge 0,\hspace{0.33em}\forall {\mathfrak{v}},\beta \in {\mathfrak{M}} and ∫0HΞ(v,β)dβ≤1,∀v∈M{\int }_{0}^{{\mathscr{H}}}\Xi \left({\mathfrak{v}},\beta ){\rm{d}}\beta \le 1,\hspace{0.33em}\forall {\mathfrak{v}}\in {\mathfrak{M}}.

Keywords

• orthogonal set
• orthogonal metric space
• orthogonal continuous
• orthogonal preserving
• orthogonal 𝔉-contraction
• coupled fixed point
• 47h10
• 54h25