Existence of a common solution for a system of nonlinear integral equations via fixed point methods in b-metric spaces

TLDR: In this paper, the authors introduce a property and use this property to prove some common fixed point theorems in b-metric space, which can be regarded as consequences of their main results.

Abstract: Abstract In this paper we introduce a property and use this property to prove some common fixed point theorems in b-metric space. We also give some fixed point results on b-metric spaces endowed with an arbitrary binary relation which can be regarded as consequences of our main results. As applications, we applying our result to prove the existence of a common solution for the following system of integral equations: x (t) =  ∫ a b K 1  (t,r,x(r)) dr, x (t) =  ∫ a b K 2  (t,r,x(r)) dr,       $$\\matrix {x (t) = \\int \\limits_a^b {{K_1}} (t, r, x(r))dr, & & x(t) = \\int \\limits_a^b {{K_2}}(t, r, x(r))dr,} $$ where a, b ∈ ℝ with a < b, x ∈ C[a, b] (the set of continuous real functions defined on [a, b] ⊆ ℝ) and K1, K2 : [a, b] × [a, b] × ℝ → ℝ are given mappings. Finally, an example is also given in order to illustrate the effectiveness of such result.