In this article, using the common (CLR) property, common fixed point results for two pairs of weakly compatible mappings satisfying contractive condition of integral type on metric spaces are established. Furthermore, the existence and uniqueness of common solution for system of functional equations arising in dynamic programming are discussed as an application of a common fixed point theorem presented in this paper. MSC: 47H10; 54H25

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This paper is devoted by developing sufficient condition required for the existence of solution to a nonlinear fractional order boundary value problem Dγfrakturufalse(ℓfalse)=ψfalse(ℓ,frakturufalse(λℓfalse)false),0.1emℓ∈frakturZ=false[0,1false], with fractional integral boundary conditions p1u(0)+q1u(1)=1Γ(γ)∫01(1−ρ)γ−1g1(ρ,u(ρ))dρ, and p2u′(0)+q2u′(1)=1Γ(γ)∫01(1−ρ)γ−1g2(ρ,u(ρ))dρ, where γ ∈ (1, 2], 0 < λ < 1, D denotes the Caputo fractional derivative (in short CFD), ψ,g1,g2:frakturZ×frakturR→frakturR are continuous functions and frakturpi,frakturqifalse(i=1,2false) are positive real numbers. Using topological degree theory sufficient results are constructed for the existence of at least one and unique solution to the concerned problem. For the validity of our result, a concrete example is presented in the end.

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In this paper, we study the existence of solution for the following non-linear matrix equations: