**1. Introduction**

The subject of fractional order boundary value problems has been addressed by many researchers in recent years. The interest in the subject owes to its extensive applications in natural and social sciences. Examples include bio-engineering [1], ecology [2], financial economics [3], chaos and fractional dynamics [4], etc. One can find many interesting results using boundary value problems dealing with Caputo, Riemann–Liouville and Hadamard type fractional derivatives and equipped with a variety of boundary conditions in [5–17].

Integro-differential equations constitute an important area of investigation due to their occurrence in several applied fields, such as heat transfer phenomena [18,19], fractional power law [20], etc. Fractional integro-differential equations complemented with different kinds of boundary conditions have also been studied by many researchers, for example, [21–28]. In a recent paper [29], a nonlocal boundary value problem containing left Caputo and right Riemann–Liouville fractional derivatives, and both left and right Riemann–Liouville fractional integral operators was discussed.

Motivated by aforementioned work on integro-differential equations, we introduce and investigate a nonlinear Caputo–Riemann–Liouville type fractional integro-differential boundary value problem involving multi-point sub-strip boundary conditions given by

$$\begin{cases} \,^cD^q \mathbf{x}(t) + \sum\_{i=1}^k l^{p\_i} g\_i(t, \mathbf{x}(t)) = f(t, \mathbf{x}(t)), \; 0 < t < 1, \\\ \mathbf{x}(0) = a, \; \mathbf{x}'(0) = 0, \; \mathbf{x}''(0) = 0, \dots, \mathbf{x}^{(m-2)}(0) = 0, \\\ \mathbf{a} \mathbf{x}(1) + \beta \mathbf{x}'(1) = \gamma\_1 \int\_0^{\zeta} \mathbf{x}(s) ds + \sum\_{j=1}^p a\_j \mathbf{x}(\eta\_j) + \gamma\_2 \int\_{\zeta}^1 \mathbf{x}(s) ds, \end{cases} \tag{1}$$

where *cDq* represents the Caputo fractional derivative operator of order *q* ∈ (*m* − 1, *<sup>m</sup>*], *m* ∈ N, *m* ≥ 2, *pi* > 0, 0 < *ζ*, *η*1, *η*2, ... , *ηp*, *ξ* < 1, *f* , *gi* : [0, 1] × R → R, (*i* = 1, ... , *k*) are continuous functions *a*, *α*, *β*, *γ*1, *γ*2 ∈ R and *αj* ∈ R, *j* = 1, 2, ... , *p*. Notice that the fixed/nonlocal points involved in the problem (1) are non-singular.

We emphasize that the problem considered in this paper is novel in the sense that the fractional integro-differential equation involves many finitely Riemann–Liouville fractional integral type nonlinearities together with a non-integral nonlinearity. In the literature, one can find results on linear integro-differential equations [30], fractional integro-differential equations with nonlinearity depending on the linear integral terms [31,32], and initial value problems involving two nonlinear integral terms [33]. In contrast to the aforementioned work, our problem contains many finitely nonlinear integral terms of fractional order, which reduce to the nonlinear integral terms by fixing *pi* = 1, ∀*i* = 1, ... , *k*. For specific applications of integral-differential equations in the mathematical modeling of physical problems such as the spreading of disease by the dispersal of infectious individuals, and reaction-–diffusion models in ecology, see [1,2]. In particular, one can find more details on the topic in [34] and the references cited therein. For some recent work on fractional integro-differential equations, see [35,36]. In a more recent work [37], the authors studied the existence of solutions for a fractional integro-differential equation supplemented with dual anti-periodic boundary conditions. Concerning the boundary condition at the terminal position *t* = 1, the linear combination of the unknown function and its derivative is associated with the contribution due to two sub-strips (0, *ζ*) and (*ξ*, 1) and finitely many nonlocal positions between them within the domain [0, 1]. This boundary condition covers many interesting situations, for example, it corresponds to the two-strip aperture condition for all *αj* = 0, *j* = 1, ... , *p*. By taking *γ*1 = 0 = *γ*2, this condition takes the form of a multi-point nonlocal boundary condition. It is interesting to note that the role of integral boundary conditions in studying practical problems such as blood flow problems [38] and bacterial self-regularization [39], etc., is crucial. For the application of strip conditions in engineering and real world problems, see [40,41]. On the other hand, the concept of nonlocal boundary conditions plays a significant role when physical, chemical or other processes depend on the interior positions (non-fixed points or segments) of the domain, for instance, see [42–45] and the references therein.

The rest of the paper is arranged as follows. Section 2 contains some related concepts of fractional calculus and an auxiliary result concerning a linear version of the problem (1). We prove the existence and uniqueness of solutions for the problem (1) by applying Banach and Krasnosel'ski˘i's fixed point theorems in Section 3. Finally, examples illustrating the main results are demonstrated in Section 4.
