Abstract
We study two hybrid and non-hybrid fractional boundary value problems via the Caputo–Hadamard type derivatives. We seek the existence criteria for these two problems separately. By utilizing the generalized Dhage’s theorem, we derive desired results for an integral structure of solutions for the hybrid problems. Also by considering the special case as a non-hybrid boundary value problem (BVP), we establish other results based on the existing tools in the topological degree theory. In the end of the article, we examine our theoretical results by presenting some numerical examples to show the applicability of the analytical findings.
Highlights
The fractional calculus has always been one of the most widely used branches of mathematics in other applied and computational sciences
By reviewing other papers published in recent years, we find that some researchers have combined the existence theory with the topological degree theory and studied different models using the existing analytical notions in this theory
For this non-hybrid boundary value problem (BVP), we will apply a new approach based on the topological degree theory
Summary
The fractional calculus has always been one of the most widely used branches of mathematics in other applied and computational sciences. This novel aspect of fractional modeling initiated with a research manuscript proposed by Dhage and Lakshmikantham in 2010 (see [40]) They turned to a new family of differential equation entitled hybrid differential equation and established some useful existence criteria of extremal solutions by utilizing some basic inequalities [40]. BVP (1)–(2) reduces to the following Caputo–Hadamard fractional non-hybrid BVP: For this non-hybrid BVP, we will apply a new approach based on the topological degree theory. Note that both hybrid and non-hybrid BVPs (1)–(2) and (3) are novel in the sense that boundary conditions are written as mixed Hadamard integral and Caputo–Hadamard derivative simultaneously
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