In this paper, we study ϕ-Laplacian problems for differential inclusions with Dirichlet boundary conditions. We prove the existence of solutions under both convexity and nonconvexity conditions on the multi-valued right-hand side. The nonlinearity satisfies either a Nagumo-type growth condition or an integrably boundedness one. The proofs rely on the Bonhnenblust-Karlin fixed point theorem and the Bressan-Colombo selection theorem respectively. Two applications to a problem from control theory are...

We develop the existence theory for sequential fractional differential equations involving Liouville-Caputo fractional derivative equipped with anti-periodic type (non-separated) and nonlocal integral boundary conditions. Several existence criteria depending on the nonlinearity involved in the problems are presented by means of a variety of tools of the fixed point theory. The applicability of the results is shown with the aid of examples. Our results are not only new in the given configuration...

Cauchy problem, boundary value problems with a boundary value condition and Sturm-Liouville problems related to the operator differential equation $X^{\left(2\right)}-AX=0$ are studied for the general case, even when the algebraic equation $X^2-A=0$ is unsolvable. Explicit expressions for the solutions in terms of data problem are given and computable expressions of the solutions for the finite-dimensional case are made available.

The paper deals with the quasi-linear ordinary differential equation ${\left(r\left(t\right)\varphi \left(u^{\text{'}}\right)\right)}^{\text{'}}+g(t,u)=0$ with $t\in [0,\infty )$. We treat the case when $g$ is not necessarily monotone in its second argument and assume usual conditions on $r\left(t\right)$ and $\varphi \left(u\right)$. We find necessary and sufficient conditions for the existence of unbounded non-oscillatory solutions. By means of a fixed point technique we investigate their growth, proving the coexistence of solutions with different asymptotic behaviors. The results generalize previous ones due to Elbert–Kusano, [Acta...

It is well known that a power of a right invertible operator is again right invertible, as well as a polynomial in a right invertible operator under appropriate assumptions. However, a linear combination of right invertible operators (in particular, their sum and/or difference) in general is not right invertible. It will be shown how to solve equations with linear combinations of right invertible operators in commutative algebras using properties of logarithmic and antilogarithmic mappings. The...

In the first part, we investigate the singular BVP $${\frac{d}{dt}}^c{D}^{\alpha }u+{(a/t)}^c{D}^{\alpha }u=\mathscr{H}u$$ , u(0) = A, u(1) = B, c D α u(t)|t=0 = 0, where $$\mathscr{H}$$ is a continuous operator, α ∈ (0, 1) and a < 0. Here, c D denotes the Caputo fractional derivative. The existence result is proved by the Leray-Schauder nonlinear alternative. The second part establishes the relations between solutions of the sequence of problems $${\frac{d}{dt}}^c{D}^{{\alpha }_n}u+{(a/t)}^c{D}^{{\alpha }_n}u=f(t,u{,}^c{D}^{{\beta }_n}u)$$ , u(0) = A, u(1) = B, $${\left.{}^c{D}^{{\alpha }_n}u\left(t\right)\right|}_{t=0}=0$$ where a < 0, 0 < β n ≤ α n < 1, limn→∞ β n = 1, and solutions of u″+(a/t)u′ = f(t, u, u′) satisfying...

In this paper, we establish sufficient conditions for the existence of mild solutions for fractional integro-differential inclusions with state-dependent delay. The techniques rely on fractional calculus, multivalued mapping on a bounded set and Bohnenblust-Karlin's fixed point theorem. Finally, we present an example to illustrate the theory.