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We consider the equation , where and and are metric spaces. This operator equation is compared with the "model" equation , where the function is continuous, nondecreasing in the first argument, and nonincreasing in the second argument. Conditions are obtained under which the existence of solutions of this operator equation follows from the solvability of the "model" equation. Conditions for the stability of the solutions under small variations in the mapping are established. The statements proved in the present paper extend the Kantorovich fixed-point theorem for differentiable mappings of Banach spaces, as well as its generalizations to coincidence points of mappings of metric spaces. [ABSTRACT FROM AUTHOR]

We study solvability of the inverse problems of recovering an unknown function on the nonlinear right-hand side of a first order operator-differential equation in some Banach space. The equation is furnished with the Cauchy data, and the overdetermination condition is the value of some operator at a solution. The existence and uniqueness theorems local in time are established. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we prove the existence of solutions of certain kinds of nonlinear fractional integrodifferential equations in Banach spaces. Further, Cauchy problems with nonlocal initial conditions are discussed for the aforementioned fractional integrodifferential equations. At the end, an example is presented. [Copyright &y& Elsevier]

In this paper, we study and investigate an interesting Caputo fractional derivative and Riemann–Liouville integral boundary value problem (BVP): c D 0 + q u (t) = f (t , u (t)) , t ∈ [ 0 , T ] , u (k) (0) = ξ k , u (T) = ∑ i = 1 m β i R L I 0 + p i u (η i) , where n − 1 < q < n , n ≥ 2 , m , n ∈ N , ξ k , β i ∈ R , k = 0 , 1 , ... , n − 2 , i = 1 , 2 , ... , m , and c D 0 + q is the Caputo fractional derivatives, f : [ 0 , T ] × C ([ 0 , T ] , E) → E , where E is the Banach space. The space E is chosen as an arbitrary Banach space; it can also be R (with the absolute value) or C ([ 0 , T ] , R) with the supremum-norm. RL I 0 + p i is the Riemann–Liouville fractional integral of order p i > 0 , η i ∈ (0 , T) , and ∑ i = 1 m β i η i p i + n − 1 Γ (n) Γ (n + p i) ≠ T n − 1 . Via the fixed point theorems of Krasnoselskii and Darbo, the authors study the existence of solutions to this problem. An example is included to illustrate the applicability of their results. [ABSTRACT FROM AUTHOR]