In this paper, an extension of Darbo fixed point theorem is introduced. By applying our extension, we obtain a coupled fixed point theorem and a solution for an integral equation. The proofs of our results are based on the technique of measure of noncompactness. Definition 1.1. [6] A mapping µ : M E −→ [0, ∞) is called a measure of noncompactness if it satisfies the following conditions: (1) The family Kerµ = {X ∈ M E : µ(X) = 0} is nonempty and Kerµ ⊆ N E. (2) X ⊆ Y =⇒ µ(X) ≤ µ(Y). (3) µ(X) = µ(X). (4) µ(Conv(X)) = µ(X).

Here, some extensions of Darbo fixed point theorem associated with measures of noncompactness are proved. Then, as an application, our attention is focused on the existence of solutions of the integral equationx(t)=F(t,f(t,x(α1(t)),  x(α2(t))),((Tx)(t)/Γ(α))×∫0t‍(u(t,s,max⁡[0,r(s)]⁡|x(γ1(τ))|,  max⁡[0,r(s)]⁡|x(γ2(τ))|)/(t-s)1-α)ds,  ∫0∞v(t,s,x(t))ds),    0<α≤1,t∈[0,1]in the space of real functions defined and continuous on the interval[0,1].

In the present article, we study a new class of sequential boundary value problems of fractional order differential equations and inclusions involving ψ-Hilfer fractional derivatives, supplemented with integral multi-point boundary conditions. The main results are obtained by employing tools from fixed point theory. Thus, in the single-valued case, the existence of a unique solution is proved by using the classical Banach fixed point theorem while an existence result is established via Krasnosel’skiĭ’s fixed point theorem. The Leray–Schauder nonlinear alternative for multi-valued maps is the basic tool to prove an existence result in the multi-valued case. Finally, our results are well illustrated by numerical examples.

In this paper, some existence theorems involving generalized contractive conditions with respect to a measure are proved. By applying our results, we study some coupled fixed point theorems, and discuss the existence of solutions for a class of the system of integral equations. Finally, an example is included to show the efficiency of our results.

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