In this work the authors considered the existence of positive solutions to a Hadamard-type fractional integral boundary value problem using the fixed point index method and multiplicity of solutions using Leggit-Williams fixed point theorem.
The authors expressed the solution of the boundary value problem as a Hammerstein integral equation (Lemma 1, Lemma 2) while properties of the kernel of the integral equation also known as the green’s function relevant to the study were obtained (Lemma 3). The authors defined a cone \(P\) and operators \(T\) and \(L^*_{\mu, \nu}\), where \(\mu, \ \nu\) are two positive constants. The spectral radius of \(L^*_{\mu, \nu}\) is found to be greater than 0 (Lemma 4).
The authors stated six initial hypothesis for this work (H1)–(H6). The fixed point index theory is used in Theorem 1 and 3 to prove existence of at least one positive solution. hypothesis (H1)–(H4) are used in Theorem 1 while Theorem 2 uses (H1)–(H2) and (H5)–(H6). Both theorems shows the operators are self map and completely continuous.
The Leggett-Williams fixed point theorem is used to prove multiplicity of solutions in Theorem 3. The authors stated an additional three hypothesis (H7)–(H9) which were used in Theorem 3 to prove existence of at least three positive solution. The work concludes with an example to demonstrate the results obtained.