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In this manuscript, we investigate some appropriate conditions which ensure the existence of at least one solution to a class of fractional order differential equations (FDEs) provided by{−CDqz(t)=θ(t,z(t));t∈J=[0,1],q∈(1,2],z(t)|t=0=ϕ(z),z(1)=δCDpz(η),p,η∈(0,1).$$textstylebegin{cases} -{}^{C}mathbf{D}^{q} z(t)=theta (t,z(t)); quad tin mathfrak{J}=[0,1], qin (1, 2],  z(t)vert _{t=0}=phi (z),qquad z(1)=delta{}^{C}mathbf{D}^{p} z(eta ),quad p, eta in (0,1). end{cases} $$The nonlinear functionθ:J×R→R$theta :mathfrak{J}times mathbf{R} rightarrow mathbf{R}$is continuous. Further,δ∈(0,1)$delta in (0, 1)$andϕ∈C(J,R)$phi in C(mathfrak{J},mathbf{R})$is a non-local function. We establish some adequate conditions for the existence of at least one solution to the considered problem by using Gronwall’s inequality and a priori estimate tools called the topological degree method. We provide two examples to verify the obtained results.

Study of a class of arbitrary order differential equations by a coincidence degree method